Daniel Liberzon's publications and preprints

First, some personal thoughts on publishing: I think we (as a research community) publish way too much. Everyone is always busy trying to meet a paper submission deadline or reviewing other people's papers. I wish we had more time to think about our ideas, discuss them with colleagues, and select and develop the best ones. If we produce fewer papers, their quality will be higher and it will be easier for others to keep up with them now and find them in the future, which will mean higher long-term value and impact of our work. I am certainly guilty of writing too many papers myself, even though I actually write fewer papers than many of my colleagues. The current academic system unfortunately puts pressure on all of us to publish more than we should, and so we need to (gradually) change the system. We should resist "bean counting" when it comes to hiring and promotions, and keep ourselves and our students focused on depth and quality. In fact, even in the current system - at least at top institutions - it is more beneficial for one's career to publish a few really good papers than a lot of mediocre ones, but not everyone seems to realize it. Read more on this

"An author who is willing to take credit for a paper must also bear responsibility for its contents." On Being a Scientist: Responsible Conduct in Research (2nd edition, 1995)

Books

CALCULUS OF VARIATIONS AND OPTIMAL CONTROL THEORY: A CONCISE INTRODUCTION, Princeton University Press, 2012. ISBN 978-0-691-15187-8. Online preview (HTML) | Free preliminary copy (PDF)| Errata | Sample quotes | Front and back cover | Courses that use the book This is a textbook for a first-year graduate course on calculus of variations and optimal control theory. Its aim is to provide a concise and rigorous introduction to the subject without requiring extensive background in control theory or advanced mathematics from the student. After presenting the main results in calculus of variations, the book makes a gradual transition to optimal control and discusses the maximum principle, the Hamilton-Jacobi-Bellman theory of dynamic programming, and linear-quadratic optimal control. Unique features of the book include: Logical and notational consistency in the exposition of classical and modern topics provides a unified view of the field A complete proof of the maximum principle is the centerpiece of the book, with special attention paid to preparing the ground for this proof beforehand Remarks tracing the historical evolution of ideas and results complement the technical developments All of the material in the book can be covered in one semester. Around sixty exercises are integrated into the text, and their solutions are available to instructors. Notes and references at the end of each chapter and the final chapter on advanced topics indicate avenues for further study. From reviewers' comments: The manuscript is a very scholarly - and indeed concise - introduction to optimal control theory. While many books on the topic have been published, the author argues convincingly in the preface for the utility of his contribution, which is marked by a nice balance of rigor and accessibility, an elegant historical progression, and a careful and complete proof of the maximum principle. Without doubt, the author has achieved the objectives stated in the preface. A course based on this manuscript should be a pleasure to take, an enriching experience guided by careful mathematics, fascinating historical perspectives, and thought-provoking exercises. The notes and references at the end of each chapter are excellent, as is the flow of the chapters. I can not think of another text which can serve as a brief introduction to the field, which explains the maximum principle well, and which is accessible to graduate students in engineering. The author put a lot of effort into explaining most of the technical material, clearly with the students in mind. The organization of the material is good, so is the choice of presented topics. The size of the manuscript is very manageable. The exercises appear challenging but well-chosen and should greatly help students understand the material.

Other reviews of this book: AMS Math Reviews, Zentralblatt Math, Amazon.com.

More info about this book on books.google.com

SWITCHING IN SYSTEMS AND CONTROL, Birkhauser, Boston, MA, Jun 2003. Volume in series Systems and Control: Foundations and Applications. ISBN 978-0-8176-4297-6. Available through SpringerLink here (subscription required). This book examines switched systems from a control-theoretic perspective, focusing on stability analysis and control synthesis of systems that combine continuous dynamics with switching events. The theory of such switched systems is related to the study of hybrid systems, which has recently attracted considerable attention among control theorists, computer scientists, and practicing engineers. Aimed at readers with a background in systems and control, this book bridges the gap between classical mathematical control theory and the interdisciplinary field of hybrid systems. The book is divided into three main parts: Part I introduces the classes of systems studied in the book Part II develops stability theory for switched systems; it covers single and multiple Lyapunov function analysis methods, Lie-algebraic stability criteria, stability under limited-rate switching, and switched systems with various types of useful special structures Part III is devoted to switching control design; it describes several wide classes of continuous-time control systems for which the logic-based switching paradigm emerges naturally as a control design tool. Switching control algorithms for several specific problems are discussed. The text adopts a progressive approach, presenting elementary concepts informally and more advanced topics with greater rigor. Results are first derived for linear systems and then extended to nonlinear systems. Full proofs for most results are provided. An extensive bibliography and a section of technical and historical notes complete the work. Requiring only familiarity with the basic theory of linear systems, the book is suitable as a text for a graduate course on switched systems and switching control. It may also serve as an introduction to this active area of research for control theorists and mathematicians, as well as a useful reference for experts in the field.

More info about this book on books.google.com

Read reviews of this book in IEEE Control Systems Magazine, IEEE Transactions on Automatic Control, AMS Math Reviews, Zentralblatt Math, and on Amazon.com.

Selected papers

For a chronological list of journal papers, please see my CV.

Go to this page for more information about my co-authors.

All papers by topic

Click on a topic to see its expanded (and more accurate) name. Sometimes the same paper is listed under several topics. Papers within each topic are arranged by category: journal articles, then book chapters (if any), then conference articles. Within each category, more recent papers are closer to the top.
Note: The topics below are quite broad. For recommended resources on more specific directions of my recent research, see this page.

Nonlinear systems | Switched systems | Quantized control | Adaptive control | Stochastic systems

Nonlinear systems and control:

On topological entropy of interconnected nonlinear systems, IEEE Control Systems Letters, vol. 5, no. 6, pp. 2210-2214, Dec 2021.
↓Abstract ↑Abstract: We study topological entropy of a nonlinear system represented as an interconnection of smaller subsystems. Under suitable assumptions on the Jacobian matrices characterizing the interconnection, we obtain an explicit upper bound on the entropy of the overall system, and show that it can be related to upper bounds on the entropies of the subsystems. We also analyze in detail the special case of a cascade connection of two subsystems, establishing an upper bound on the entropy which is more tightly linked to individual entropy bounds for the subsystems. See also the video of the CDC 2021 talk.

Robust synchronization of electric power generators (with O. Ajala and A. D. Dominguez-Garcia), Systems and Control Letters, vol. 152, article 104937, pp. 1-9, Jun 2021.
↓Abstract ↑Abstract: We consider the problem of synchronizing two electric power generators, one of which (the leader) is serving a time-varying electrical load, so that they can ultimately be connected to form a single power system. Each generator is described by a second-order reduced state-space model. We assume that the generator not serving an external load initially (the follower) has access to measurements of the leader's phase angle, corrupted by some additive disturbances. By using these measurements, and leveraging results on reduced-order observers with ISS-type robustness, we propose a procedure that drives (i) the angular velocity of the follower close enough to that of the leader, and (ii) the phase angle of the follower close enough to that of the point at which both systems will be electrically connected. An explicit bound on the synchronization error in terms of the measurement disturbance and the variations in the electrical load served by the leader is computed. We illustrate the procedure via numerical simulations.

A library of second-order models for synchronous machines (with O. Ajala, A. D. Dominguez-Garcia, and P. Sauer), IEEE Transactions on Power Systems, vol. 35, no. 6, pp. 4803-4814, Nov 2020.
↓Abstract ↑Abstract: This paper presents a library of second-order reduced models for synchronous machines that can be utilized in controller design and power system analysis tasks. The reduced models are developed through the systematic reduction of a nineteenth-order model, using singular perturbation techniques, and the dynamic states of the resulting models are the power angle and angular speed. Using two test cases, the reduced models are validated by comparing their voltage, frequency and phase profiles with that of the high-order model, and that of the so-called classical model.

Almost Lyapunov functions for nonlinear systems (with S. Liu and V. Zharnitsky), Automatica, vol. 113, article 108758, pp. 1-13, Mar 2020.
↓Abstract ↑Abstract: We study convergence of nonlinear systems in the presence of an "almost Lyapunov" function which, unlike the classical Lyapunov function, is allowed to be nondecreasing -- and even increasing -- on a nontrivial subset of the phase space. Under the assumption that the vector field is free of singular points (away from the origin) and that the subset where the Lyapunov function does not decrease is sufficiently small, we prove that solutions approach a small neighborhood of the origin. A nontrivial example where this theorem applies is constructed.

Nonlinear observers robust to measurement disturbances in an ISS sense (with H. Shim), IEEE Transactions on Automatic Control, vol. 61, no. 1, pp. 48-61, Jan 2016.
↓Abstract ↑Abstract: This paper formulates and studies the concept of quasi-Disturbance-to-Error Stability (qDES) which characterizes robustness of a nonlinear observer to an output measurement disturbance. In essence, an observer is qDES if its error dynamics are input-to-state stable (ISS) with respect to the disturbance as long as the plant's input and state remain bounded. We develop Lyapunov-based sufficient conditions for checking the qDES property for both full-order and reduced-order observers. We use these conditions to show that several well-known observer designs yield qDES observers, while some others do not. Our results also enable the design of novel qDES observers, as we demonstrate with examples. When combined with a state feedback law robust to state estimation errors in the ISS sense, a qDES observer can be used to achieve output feedback control design with robustness to measurement disturbances. As an application of this idea, we treat a problem of stabilization by quantized output feedback.

An asymptotic ratio characterization of input-to-state stability (with H. Shim), IEEE Transactions on Automatic Control, vol. 60, no. 12, pp. 3401-3404, Dec 2015.
↓Abstract ↑Abstract: For continuous-time nonlinear systems with inputs, we introduce the notion of an asymptotic ratio ISS Lyapunov function. The derivative of such a function along solutions is upper-bounded by the difference of two terms whose ratio is asymptotically smaller than 1 for large states. This asymptotic ratio condition is sometimes more convenient to check than standard ISS Lyapunov function conditions. We show that the existence of an asymptotic ratio ISS Lyapunov function is equivalent to ISS. A related notion of ISS with non-uniform convergence rate is also explored.

Norm-controllability of nonlinear systems (with M. Müller and F. Allgöwer), IEEE Transactions on Automatic Control, vol. 60, no. 7, pp. 1825-1840, Jul 2015.
↓Abstract ↑Abstract: In this paper, we introduce and study the notion of norm-controllability for nonlinear systems. This property captures the responsiveness of a system with respect to applied inputs, which is quantified via the norm of an output. As a main contribution, we obtain several Lyapunov-like sufficient conditions for norm-controllability, some of which are based on higher-order derivatives of a Lyapunov-like function. Various aspects of the proposed concept and the sufficient conditions are illustrated by several examples, including a chemical reactor application. Furthermore, for the special case of linear systems, we establish connections between norm-controllability and standard controllability. Addendum

Lyapunov conditions for input-to-state stability of impulsive systems (with J. P. Hespanha and A. R. Teel), Automatica, vol. 44, no. 11, pp. 2735-2744, Nov 2008.
↓Abstract ↑Abstract: This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for impulsive systems, i.e., dynamical systems that evolve according to ordinary differential equations most of the time, but occasionally exhibit discontinuities (or impulses). We provide a set of Lyapunov-based sufficient conditions for establishing these ISS properties. When the continuous dynamics are ISS but the discrete dynamics that govern the impulses are not, the impulses should not occur too frequently, which is formalized in terms of an average dwell-time (ADT) condition. Conversely, when the impulse dynamics are ISS but the continuous dynamics are not, there must not be overly long intervals between impulses, which is formalized in terms of a novel reverse ADT condition. We also investigate the cases where (i) both the continuous and discrete dynamics are ISS and (ii) one of these is ISS and the other only marginally stable for the zero input, while sharing a common Lyapunov function. In the former case we obtain a stronger notion of ISS, for which a necessary and sufficient Lyapunov characterization is available. The use of the tools developed herein is illustrated through examples from a Micro- Electro-Mechanical System (MEMS) oscillator and a problem of remote estimation over a communication network. See also a more complete version.

Nonlinear norm-observability notions and stability of switched systems (with J. P. Hespanha, D. Angeli, and E. D. Sontag), IEEE Transactions on Automatic Control, vol. 50, no. 2, pp. 154-168, Feb 2005.
↓Abstract ↑Abstract: This paper proposes several definitions of observability for nonlinear systems and explores relationships among them. These observability properties involve the existence of a bound on the norm of the state in terms of the norms of the output and the input on some time interval. A Lyapunov-like sufficient condition for observability is also obtained. As an application, we prove several variants of LaSalle's stability theorem for switched nonlinear systems. These results are demonstrated to be useful for control design in the presence of switching as well as for developing stability results of Popov type for switched feedback systems.

Output-input stability implies feedback stabilization, Systems and Control Letters, vol. 53, no. 3/4, pp. 237-248, Nov 2004.
↓Abstract ↑Abstract: We study the recently introduced notion of output-input stability, which is a robust variant of the minimum-phase property for general smooth nonlinear control systems. This paper develops the theory of output-input stability in the multi-input, multi-output setting. We show that output-input stability is a combination of two system properties, one related to detectability and the other to left-invertibility. For systems affine in controls, we derive a necessary and sufficient condition for output-input stability, which relies on a global version of the nonlinear structure algorithm. This condition leads naturally to a globally asymptotically stabilizing state feedback strategy for affine output-input stable systems.

Universal construction of feedback laws achieving ISS and integral-ISS disturbance attenuation (with E. D. Sontag and Y. Wang), Systems and Control Letters, vol. 4, no. 2, pp. 111-127, Jun 2002.
↓Abstract ↑Abstract: We study nonlinear systems with both control and disturbance inputs. The main problem addressed in the paper is design of state feedback control laws that render the closed-loop system integral-input-to-state stable (iISS) with respect to the disturbances. We introduce an appropriate concept of control Lyapunov function (iISS-CLF), whose existence leads to an explicit construction of such a control law. The same method applies to the problem of input-to-state stabilization. Converse results and techniques for generating iISS-CLFs are also discussed. Erratum

Output-input stability and minimum-phase nonlinear systems (with A. S. Morse and E. D. Sontag), IEEE Transactions on Automatic Control, vol. 47, no. 3, pp. 422-436, Mar 2002.
↓Abstract ↑Abstract: This paper introduces and studies the notion of output-input stability, which represents a variant of the minimum-phase property for general smooth nonlinear control systems. The definition of output-input stability does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. In the spirit of the "input-to-state stability'' philosophy, it requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of output-input stable systems thus defined includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts. As an application, we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control.

Conferences:

On topological entropy of interconnected nonlinear systems, in Proceedings of the 60th IEEE Conference on Decision and Control, Austin, TX (online), Dec 2021, pp. 3448-3453.
↓Abstract ↑Abstract: We study topological entropy of a nonlinear system represented as an interconnection of smaller subsystems. Under suitable assumptions on the Jacobian matrices characterizing the interconnection, we obtain an explicit upper bound on the entropy of the overall system, and show that it can be related to upper bounds on the entropies of the subsystems. We also analyze in detail the special case of a cascade connection of two subsystems, establishing an upper bound on the entropy which is more tightly linked to individual entropy bounds for the subsystems. See also the video of the talk.

On almost Lyapunov functions for systems with inputs (with S. Liu), in Proceedings of the 58th IEEE Conference on Decision and Control, Nice, France, Dec 2019, pp. 468-473.
↓Abstract ↑Abstract: In this work an almost Lyapunov function theorem from our recent work is generalized to systems with inputs. It is shown that if for any inputs and initial conditions, the time that solutions of the system can stay in a "bad" region where the Lyapunov function does not decrease fast enough has a sufficiently small upper bound, then the system is globally exponentially stable uniformly with respect to the inputs. In our analysis, the almost Lyapunov function is directly expressed as a function of time along arbitrary solution and the upper bound of the ratio of this function at the time the solution trajectory leaves and enters the "bad" region is found to be less than 1. Consequently all solutions are shown to converge to the origin asymptotically with some careful justification. It is also concluded that a system with inputs is exponentially inputto- state stable if its auxiliary system satisfies all the hypotheses in our main theorem. The result is then applied on an example adopted and modified from our previous work and it shows an improvement in the sense that stability can still be verified even when there is stronger perturbation to the example's stable dynamics.See also a more complete version.

Higher order derivatives of Lyapunov functions for stability of systems with inputs (with S. Liu), in Proceedings of the 58th IEEE Conference on Decision and Control, Nice, France, Dec 2019, pp. 6146-6151.
↓Abstract ↑Abstract: In this paper we study an alternative method for determining stability of dynamical systems by inspecting higher order derivatives of a Lyapunov function. The system can be time invariant or time varying; in both cases we define the higher order derivatives when there are inputs. We then claim and prove that if there exists a linear combination of those higher order derivatives with non-negative coefficients (except that the coefficient of the 0-th order term needs to be positive) which is negative semi-definite, then the system is globally uniformly asymptotically stable. The proof involves repeated applications of comparison principle for first order differential relations. We also show that a system with inputs whose auxiliary system admits a Lyapunov function satisfying the aforementioned conditions is input-to-state stable. See also a more complete version.

A second-order synchronous machine model for multi-swing stability analysis (with O. Ajala, A. D. Dominguez-Garcia, and P. Sauer), in Proceeding of the North American Power Symposium, Wichita, KS, Oct 2019.
↓Abstract ↑Abstract: This paper presents a second-order model for synchronous machines that can be utilized in power system dynamic performance analysis and control design tasks. The model has a similar structure to the classical model in that it comprises two dynamic states, the power angle and the angular speed. However, unlike the classical model, the model finds applications beyond first swing stability analysis; for example, they can also be utilized in transient stability studies. The model is developed through a systematic model-order reduction of a nineteenth-order state-space model by using singular perturbation techniques. It is validated by comparing its response with that of the nineteenth-order model and that of the classical model.

An approach to robust synchronization of electric power generators (with O. Ajala and A. D. Dominguez-Garcia), in Proceedings of the 57th IEEE Conference on Decision and Control, Miami Beach, FL, Dec 2018, pp. 1586-1591.
↓Abstract ↑Abstract: We consider the problem of synchronizing two electric power generators, one of which (the leader) is serving a time-varying load, so that they can ultimately be connected to form a single power system. Both generators are described by second-order reduced state-space models, and we assume that the generator not serving any load initially (the follower) has access to measurements of the leader generator phase angle corrupted by some additive disturbances. By using these measurements, and leveraging results on reduced-order observers with ISS-type robustness, we propose a procedure that drives (i) the angular velocity of the follower close enough to that of the leader, and (ii) the phase angle of the follower close enough to that of the point at which both systems will be electrically connected. An explicit bound on the synchronization error in terms of the measurement disturbance and the variations in the electrical load served by the leader is computed. We illustrate the procedure via numerical simulations.

On almost Lyapunov functions for non-vanishing vector fields (with S. Liu and V. Zharnitsky), in Proceedings of the 55th IEEE Conference on Decision and Control, Las Vegas, NV, Dec 2016, pp. 5557-5562.
↓Abstract ↑Abstract: We study convergence properties of nonlinear systems in the presence of Lyapunov functions which decrease along solutions in a given region not everywhere but rather on the complement of a set of small volume. The structure is quite general except that the system dynamics never vanishes in a region that is away from the equilibrium. It is shown that solutions starting inside the region will approach a small set around the origin as long as the volume where the Lyapunov function does not decrease fast enough is sufficiently small. The main theorem of this paper is established by tracking the change of Lyapunov function value when the solution passes through the above mentioned volume and finding an upper bound of the volume swept out by a neighborhood along the solution before it can achieve an overall gain in its Lyapunov function value. The result shows that the convergence rate is traded off against the size of such small volume that the system can have. In the end a non-trivial example where our theorem is applicable is demonstrated.

On almost Lyapunov functions (with C. Ying and V. Zharnitsky), in Proceedings of the 53rd IEEE Conference on Decision and Control, Los Angeles, CA, Dec 2014, pp. 3083-3088.
↓Abstract ↑Abstract: We study asymptotic stability properties of nonlinear systems in the presence of "almost Lyapunov" functions which decrease along solutions in a given region not everywhere but rather on the complement of a set of small volume. Nothing specific about the structure of this set is assumed besides an upper bound on its volume. We show that solutions starting inside the region approach a small set around the origin whose volume depends on the volume of the set where the Lyapunov function does not decrease, as well as on other system parameters. The result is established by a perturbation argument which compares a given system trajectory with nearby trajectories that lie entirely in the set where the Lyapunov function is known to decrease, and trades off convergence speed of these trajectories against the expansion rate of the distance to them from the given trajectory. See also the slides of the talk.

Norm-controllability, or how a nonlinear system responds to large inputs (with M. Müller and F. Allgöwer), in Proceedings of the 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2013) (semi-plenary paper), Toulouse, France, Sep 2013, pp. 104-109.
↓Abstract ↑Abstract: The purpose of this paper is to survey and discuss recent results on norm-controllability of nonlinear systems. This notion captures the responsiveness of a nonlinear system with respect to the applied inputs in terms of the norm of an output map, and can be regarded as a certain type of gain concept and/or a weaker notion of controllability. We state several Lyapunov-like sufficient conditions for this property in a simplified formulation, and illustrate the concept with several examples. See also the slides of the talk.

Relaxed conditions for norm-controllability of nonlinear systems (with M. Müller and F. Allgöwer), in Proceedings of the 51st IEEE Conference on Decision and Control, Maui, HI, Dec 2012, pp. 314-319.
↓Abstract ↑Abstract: In this paper, we further study the notion of norm-controllability, which captures the responsiveness of a nonlinear system with respect to applied inputs in terms of the norm of an output map. We give sufficient conditions for this property based on higher-order lower directional derivatives, which generalize the results obtained in our earlier work and help to establish norm-controllability for outputs with relative degree greater than one. Furthermore, we illustrate the obtained results by means of a chemical reaction example. See also a more complete version.

On norm-controllability of nonlinear systems (with M. Müller and F. Allgöwer), in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, Dec 2011, pp. 1741-1746.
↓Abstract ↑Abstract: In this paper, we introduce and study the notion of "norm-controllability'' for nonlinear systems. This property captures the responsiveness of a system with respect to the applied inputs, which is quantified via the norm of an output map. As a main contribution, we obtain a Lyapunov-like sufficient condition for norm-controllability. Several examples illustrate the various aspects of the proposed concept, and we also further elaborate norm-controllability for the special case of linear systems.

Quasi-ISS reduced-order observers and quantized output feedback (with H. Shim and J.-S. Kim), in Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China, Dec 2009, pp. 6680-6685.
↓Abstract ↑Abstract: We formulate and study the problem of designing nonlinear observers whose error dynamics are input-to-state stable (ISS) with respect to additive output disturbances as long as the plant's input and state remain bounded. We present a reduced-order observer design which achieves this quasi-ISS property when there exists a suitable state-independent error Lyapunov function. We show that our construction applies to several classes of nonlinear systems previously studied in the observer design literature. As an application of this robust observer concept, we prove that quantized output feedback stabilization is achievable when the system possesses a quasi-ISS reduced-order observer and a state feedback law that yields ISS with respect to measurement errors. A worked example is included. See also the slides of the talk.

On input-to-state stability of impulsive systems (with J. P. Hespanha and A. R. Teel), in Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, Dec 2005, pp. 3992-3997.
↓Abstract ↑Abstract: This paper introduces appropriate concepts of input-to-state stability (ISS) and integral-ISS for systems with impulsive effects. We provide a set of Lyapunov-based sufficient conditions to establish these properties. When the continuous dynamics are stabilizing but the impulsive effects are destabilizing, the impulses should not occur too frequently, which can be formalized in terms of an average dwell-time condition. Conversely, when the impulses are stabilizing and the continuous dynamics is destabilizing, there must not be overly long intervals between impulses, which is formalized in terms of a reverse average dwell-time condition. We also investigate limiting cases of systems that remain stable for arbitrarily small/large average dwell-times. See also a more complete version.

Output-input stability and feedback stabilization of multivariable nonlinear control systems, in Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, Dec 2003, pp. 1550-1555.
↓Abstract ↑Abstract: We study the recently introduced notion of output-input stability, which is a robust variant of the minimum-phase property for general smooth nonlinear control systems. This paper develops the theory of output-input stability in the multi-input, multi-output setting. We show that output-input stability is a combination of two system properties, one related to detectability and the other to left-invertibility. For systems affine in controls, we derive a necessary and sufficient condition for output-input stability, which relies on a global version of the nonlinear structure algorithm. This condition leads naturally to a globally asymptotically stabilizing state feedback strategy for affine output-input stable systems. See also the slides of the talk.

Nonlinear observability and an invariance principle for switched systems (with J. P. Hespanha and E. D. Sontag), in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, Dec 2002, pp. 4300-4305.
↓Abstract ↑Abstract: This paper proposes several definitions of observability for nonlinear systems and explores relationships between them. These observability properties involve the existence of a bound on the norm of the state in terms of the norm of the output on some time interval. As an application, we prove a LaSalle-like stability theorem for switched nonlinear systems. See also the slides of the talk.

Output-input stability of nonlinear systems and input/output operators, in Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems (MTNS), South Bend, IN, Aug 2002.
↓Abstract ↑Abstract: The notion of output-input stability, recently proposed in [2], represents a variant of the minimum-phase property for general smooth nonlinear control systems. In the spirit of the input-to-state stability (ISS) philosophy, the definition of output-input stability requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The present work extends this concept to the setting of input/output operators. We show that output-input stability of a system implies output-input stability of the associated input/output operator, and that under suitable reachability and observability assumptions, a converse result also holds. See also the slides of the talk.

Output-input stability: a new variant of the minimum-phase property for nonlinear systems (with A. S. Morse and E. D. Sontag), in Proceedings of the 5th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2001), St. Petersburg, Russia, Jul 2001, pp. 743-748.
↓Abstract ↑Abstract: This paper studies the notion of output-input stability, which is a variant of the minimum-phase property for general smooth nonlinear control systems. In the spirit of the "input-to-state stability'' philosophy, the definition of the new concept requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of output-input stable systems includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts. A characterization of output-input stability for SISO systems is given in terms of suitable relative degree and detectability concepts. See also the slides of the talk.

A new definition of the minimum-phase property for nonlinear systems, with an application to adaptive control (with A. S. Morse and E. D. Sontag), in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, Dec 2000, pp. 2106-2111.
↓Abstract ↑Abstract: We introduce a new definition of the minimum-phase property for general smooth nonlinear control systems. The definition does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. It requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of minimum-phase systems thus defined includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts. We explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control. See also the slides of the talk

ISS and integral-ISS disturbance attenuation with bounded controls, in Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, Dec 1999, pp. 2501-2506.
↓Abstract ↑Abstract: We consider the problem of achieving disturbance attenuation in the ISS and integral-ISS sense for nonlinear systems with bounded controls. For the ISS case we derive a "universal" formula which extends an earlier result of Lin and Sontag to systems with disturbances. For the integral-ISS case we give two constructions, one resulting in a smooth control law and the other in a switching control law. We also briefly discuss some issues related to input-to-state stability of switched and hybrid systems. See also the slides of the talk.

On integral-input-to-state stabilization (with E. D. Sontag and Y. Wang), in Proceedings of the 1999 American Control Conference, San Diego, CA, Jun 1999, pp. 1598-1602.
↓Abstract ↑Abstract: This paper continues the investigation of the recently introduced integral version of input-to-state stability (iISS). We study the problem of designing control laws that achieve iISS disturbance attenuation. The main contribution is a concept of control Lyapunov function (iISS-CLF) whose existence leads to an explicit construction of such a control law. The results are compared with the ones available for the ISS case. See also the slides of the talk.

Switched and hybrid systems:

Journals: (jump down to Conferences)

Topological entropy of switched nonlinear and interconnected systems (with G. Yang and J. P. Hespanha), Mathematics of Control, Signals, and Systems (special issue dedicated to Eduardo Sontag's 70th birthday), vol. 35, no. 3, pages 641-683, Sep 2023.
↓Abstract ↑Abstract: This paper studies topological entropy of switched nonlinear and interconnected systems. We construct a general upper bound for the topological entropy of a switched nonlinear system, in terms of an average of the asymptotic suprema of measures of Jacobian matrices of system functions of individual modes, weighted by their active rates. We also construct a general lower bound in a similar form, in terms of the asymptotic infima of traces of these Jacobian matrices. The general upper bound is readily used to construct an upper bound for entropy in the interconnected case. We also construct less conservative upper bounds for entropy in diagonal and block-diagonal cases, as well as more conservative upper bounds that require less information on the switching in all cases. The upper bounds for entropy and their relations are illustrated by numerical examples of a switched Lotka-Volterra ecosystem model See also a more complete version.

Estimation entropy, Lyapunov exponents, and quantizer design for switched linear systems (with G. S. Vicinansa), SIAM Journal on Control and Optimization, vol. 61, no. 1, pp. 198-224, 2023.
↓Abstract ↑Abstract: In this paper, we study connections between the estimation entropy of a switched linear system and its Lyapunov exponents. We prove lower and upper bounds for the estimation entropy in terms of the Lyapunov exponents and show that, under the so-called regularity assumption, those bounds coincide. Also, using a geometric object called Oseledets' filtration of the system, we prove that the estimation entropy of the system is given in terms of the Lyapunov exponents. Further, we show how to use the exponents and the Oseledets' filtration to design a quantization scheme for state estimation of switched linear systems. Then, we prove that we can make this algorithm work at an average data-rate arbitrarily close to the upper bound we provided for the estimation entropy of the given system. Furthermore, we can choose the average data-rate to be arbitrarily close to the estimation entropy whenever the switched linear system is regular. We show that, under the regularity assumption, the quantization scheme is completely causal in the sense that it only depends on information that is available up to the current time instant. We show that regularity is a natural property of many practical systems, such as Markov Jump Linear Systems, and give sufficient conditions for it.

Integral-input-to-state stability of switched nonlinear systems under slow switching (with S. Liu, A. Russo, and A. Cavallo), IEEE Transactions on Automatic Control, vol. 67, no. 11, pp. 5841-5855, Nov 2022.
↓Abstract ↑Abstract: In this paper we study integral-input-to-state stability (iISS) of nonlinear switched systems with jumps. We demonstrate by examples that, unlike ISS, iISS is not always preserved under slow enough dwell-time switching. We then present sufficient conditions for iISS to be preserved under slow switching. These conditions involve, besides a sufficiently large dwell time, some additional properties of comparison functions characterizing iISS of the individual modes. When these properties are only partially satisfied, we are able to conclude weaker variants of iISS, also introduced in this work. As an illustration, we show that switched systems with bilinear zero-input-stable modes are always iISS under sufficiently large dwell time.

ISS and integral-ISS of switched systems with nonlinear supply functions (with S. Liu and A. Tanwani), Mathematics of Control, Signals, and Systems, vol. 34, no. 2, pp. 297-327, Jun 2022.
↓Abstract ↑Abstract: The problem of input-to-state stability (ISS), and its integral version (iISS), are considered for switched nonlinear systems with inputs, resets and possibly unstable subsystems. For the dissipation inequalities associated with the Lyapunov function of each subsystem, it is assumed that the supply functions, which characterize the decay rate and ISS/iISS gains of the subsystems, are nonlinear. The change in the value of Lyapunov functions at switching instants is described by a sum of growth and gain functions, which are also nonlinear. Using the notion of average dwell-time (ADT) to limit the number of switching instants on an interval, and the notion of average activation time (AAT) to limit the activation time for unstable systems, a formula relating ADT and AAT is derived to guarantee ISS/iISS of the switched system. Case studies of switched systems with saturating dynamics and switched bilinear systems are included for illustration of the results.

Topological entropy of switched linear systems: general matrices and matrices with commutation relations (with G. Yang, A. J. Schmidt, and J. P. Hespanha), Mathematics of Control, Signals, and Systems, vol. 32, no. 3, pp. 411-453, Sep 2020.
↓Abstract ↑Abstract: This paper studies a notion of topological entropy for switched systems, formulated in terms of the minimal number of trajectories needed to approximate all trajectories with a finite precision. For general switched linear systems, we prove that the topological entropy is independent of the set of initial states, and construct an upper bound in terms of an average of the measures of system matrices of individual modes, weighted by their corresponding active times. We also construct a lower bound in terms of an active-time-weighted average of their traces. For switched linear systems with scalar-valued state and those with pairwise commuting matrices, we establish formulae for the topological entropy in terms of active-time-weighted averages of the eigenvalues of system matrices of individual modes. For the more general case with simultaneously triangularizable matrices, we construct an upper bound for the topological entropy that only depends on the eigenvalues, their order in a simultaneous triangularization, and the active times. In each case above, we also establish upper bounds that are more conservative but require less information on the system matrices or on the switching, with their relations illustrated by numerical examples. Stability conditions inspired by the upper bounds for the topological entropy are presented as well.

Unified stability criteria for slowly time-varying and switched linear systems (with X. Gao, J. Liu, and T. Basar), Automatica, vol. 96, pp. 110-120, Oct 2018.
↓Abstract ↑Abstract: This paper presents a unified approach to formulating stability conditions for slowly time-varying linear systems and switched linear systems. The concept of total variation of a matrix-valued function is introduced to characterize the variation of the system matrix. Using this concept, a result generalizing existing stability conditions for slowly time-varying linear systems is derived. As special cases of this result, two sets of stability conditions are derived for switched linear systems, which match known results in the literature.

Feedback stabilization of a switched linear system with unknown disturbances under data-rate constraints (with G. Yang), IEEE Transactions on Automatic Control, vol. 63, no. 7, pp. 2107-2122, Jul 2018.
↓Abstract ↑Abstract: We study the problem of stabilizing a switched linear system with a completely unknown disturbance using sampled and quantized state feedback. The switching is assumed to be slow enough in the sense of combined dwell-time and average dwell-time, each individual mode is assumed to be stabilizable, and the data-rate is assumed to be large enough but finite. Extending the approach of reachable-set approximation and propagation from an earlier result on the disturbance-free case, we develop a communication and control strategy that achieves a variant of input-to-state stability with exponential decay. An estimate of the disturbance bound is introduced to counteract the unknown disturbance, and a novel algorithm is designed to properly adjust the estimate and recover the state when it escapes the range of quantization.

Lyapunov small-gain theorems for networks of not necessarily ISS hybrid systems (with A. Mironchenko and G. Yang), Automatica, vol. 88, pp. 10-20, Feb 2018.
↓Abstract ↑Abstract: We prove a novel Lyapunov-based small-gain theorem for networks composed of n >= 2 hybrid systems which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems proposed in the last few years. We also show how average dwell-time (ADT) clocks and reverse ADT clocks can be used to modify the Lyapunov functions for subsystems and to enlarge the applicability of the derived small-gain theorems.

Generalized switching signals for input-to-state stability of switched systems (with A. Kundu and D. Chatterjee), Automatica, vol. 64, pp. 270-277, Feb 2016.
↓Abstract ↑Abstract: This article deals with input-to-state stability (ISS) of continuous-time switched nonlinear systems. Given a family of systems with exogenous inputs such that not all systems in the family are ISS, we characterize a new and general class of switching signals under which the resulting switched system is ISS. Our stabilizing switching signals allow the number of switches to grow faster than an affine function of the length of a time interval, unlike in the case of average dwell time switching. We also recast a subclass of average dwell time switching signals in our setting and establish analogs of two representative prior results.

A Lyapunov-based small-gain theorem for interconnected switched systems (with G. Yang), Systems and Control Letters, vol. 78, pp. 47-54, Apr 2015.
↓Abstract ↑Abstract: Stability of an interconnected system consisting of two switched systems is investigated in the scenario where in both switched systems there may exist some subsystems that are not input-to-state stable (non-ISS). We show that, providing the switching signals neither switch too frequently nor activate non-ISS subsystems for too long, a small-gain theorem can be used to conclude global asymptotic stability (GAS) of the interconnected system. For each switched system, with the constraints on the switching signal being modeled by an auxiliary timer, a correspondent hybrid system is defined to enable the construction of a hybrid ISS Lyapunov function. Apart from justifying the ISS property of their corresponding switched systems, these hybrid ISS Lyapunov functions are then combined to establish a Lyapunov-type small-gain condition which guarantees that the interconnected system is globally asymptotically stable.

Lyapunov-based small-gain theorems for hybrid systems (with D. Nesic and A. R. Teel), IEEE Transactions on Automatic Control, vol. 59, no. 6, pp. 1395-1410, Jun 2014.
↓Abstract ↑Abstract: Constructions of strong and weak Lyapunov functions are presented for a feedback connection of two hybrid systems satisfying certain Lyapunov stability assumptions and a small-gain condition. The constructed strong Lyapunov functions can be used to conclude input-to-state stability (ISS) of hybrid systems with inputs and global asymptotic stability (GAS) of hybrid systems without inputs. In the absence of inputs, we also construct weak Lyapunov functions nondecreasing along solutions and develop a LaSalle-type theorem providing a set of sufficient conditions under which such functions can be used to conclude GAS. In some situations, we show how average dwell time (ADT) and reverse average dwell time (RADT) "clocks" can be used to construct Lyapunov functions that satisfy the assumptions of our main results. The utility of these results is demonstrated for the "natural" decomposition of a hybrid system as a feedback connection of its continuous and discrete dynamics, and in several design-oriented contexts: networked control systems, event-triggered control, and quantized feedback control.

Finite data-rate feedback stabilization of switched and hybrid linear systems, Automatica, vol. 50, no. 2, pp. 409-420, Feb 2014.
↓Abstract ↑Abstract: We study the problem of asymptotically stabilizing a switched linear control system using sampled and quantized measurements of its state. The switching is assumed to be slow enough in the sense of combined dwell time and average dwell time, each individual mode is assumed to be stabilizable, and the available data rate is assumed to be large enough. Our encoding and control strategy is rooted in the one proposed in our earlier work on non-switched systems, and in particular the data-rate bound used here is the data-rate bound from that earlier work maximized over the individual modes. The main technical step that enables the extension to switched systems concerns propagating over-approximations of reachable sets through sampling intervals, during which the switching signal is not known. Our primary focus is on systems with time-dependent switching (switched systems) but the setting of state-dependent switching (hybrid systems) is also discussed.

On observability for switched linear systems: characterization and observer design (with A. Tanwani and H. Shim), IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 891-904, Apr 2013.
↓Abstract ↑Abstract: This paper presents a characterization of observability and an observer design method for switched linear systems with state jumps. A necessary and sufficient condition is presented for observability, globally in time, when the system evolves under predetermined mode transitions. Because this characterization depends upon the switching signal under consideration, the existence of singular switching signals is studied alongside developing a sufficient condition that guarantees uniform observability with respect to switching times. Furthermore, while taking state jumps into account, a relatively weaker characterization is given for determinability, the property that concerns with recovery of the original state at some time rather than at all times. Assuming determinability of the system, a hybrid observer is designed for the most general case to estimate the state of the system and it is shown that the estimation error decays exponentially. Since the individual modes of the switched system may not be observable, the proposed strategy for designing the observer is based upon a novel idea of accumulating the information from individual subsystems. Erratum

Input/output-to-state stability and state-norm estimators for switched nonlinear systems (with M. Müller), Automatica, vol. 48, no. 9, pp. 2029-2039, Sep 2012.
↓Abstract ↑Abstract: In this paper, the concepts of input-/output-to-state stability (IOSS) and state-norm estimators are considered for switched nonlinear systems under average dwell-time switching signals. We show that if the average dwell-time is large enough, a switched system is IOSS if all of its constituent subsystems are IOSS. Moreover, under the same conditions, a non-switched state-norm estimator exists for the switched system. Furthermore, if some of the constituent subsystems are not IOSS, we show that still IOSS can be established for the switched system, if the activation time of the non-IOSS subsystems is not too big. Again, under the same conditions, a state-norm estimator exists for the switched system. However, in this case, the state-norm estimator is a switched system itself, consisting of two subsystems. We show that this state-norm estimator can be constructed such that its switching times are independent of the switching times of the switched system it is designed for.

Switched nonlinear differential algebraic equations: solution theory, Lyapunov functions, and stability (with S. Trenn), Automatica, vol. 48, no. 5, pp. 954-963, May 2012.
↓Abstract ↑Abstract: We study switched nonlinear differential algebraic equations (DAEs) with respect to existence and nature of solutions as well as stability. We utilize piecewise-smooth distributions introduced in earlier work for switched linear DAE to establish a solution framework for switched nonlinear DAEs. In particular, we allow induced jumps in the solutions. To study stability, we first generalize Lyapunov's direct method to non-switched DAEs and afterwards obtain Lyapunov criteria for asymptotic stability of switched DAEs. Developing appropriate generalizations of the concepts of a common Lyapunov function and multiple Lyapunov functions for DAEs, we derive sufficient conditions for asymptotic stability under arbitrary switching and under sufficiently slow average dwell-time switching, respectively.

On robust Lie-algebraic stability conditions for switched linear systems (with A. A. Agrachev and Y. Baryshnikov), Systems and Control Letters, vol. 61, no. 2, pp. 347-353, Feb 2012.
↓Abstract ↑Abstract: This paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching, which involve the commutators (Lie brackets) among the given matrices generating the switched system. The main novel feature of these stability criteria is that, unlike their earlier counterparts, they are robust with respect to small perturbations of the system parameters. Two distinct approaches are investigated. For discrete-time switched linear systems, we formulate a stability condition in terms of an explicit upper bound on the norms of the Lie brackets. For continuous-time switched linear systems, we develop two stability criteria which capture proximity of the associated matrix Lie algebra to a solvable or a "solvable plus compact" Lie algebra, respectively.

Stabilizing randomly switched systems (with D. Chatterjee), SIAM Journal on Control and Optimization, vol. 49, no. 5, pp. 2008-2031, 2011.
↓Abstract ↑Abstract: This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunov-like functions, and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control design.

An inversion-based approach to fault detection and isolation in switching electrical networks (with A. Tanwani and A. D. Dominguez-Garcia), IEEE Transactions on Control Systems Technology, vol. 19, no. 5, pp. 1059-1074, Sep 2011.
↓Abstract ↑Abstract: This paper proposes a framework for fault detection and isolation (FDI) in electrical energy systems based on techniques developed in the context of invertibility of switched systems. In the absence of faults -- the nominal mode of operation -- the system behavior is described by one set of linear differential equations or more in the case of systems with natural switching behavior, e.g., power electronics systems. Faults are categorized as hard and soft. A hard fault causes abrupt changes in the system structure, which results in an uncontrolled transition from the nominal mode of operation to a faulty mode governed by a different set of differential equations. A soft fault causes a continuous change over time of certain system structure parameters, which results in unknown additive disturbances to the set(s) of differential equations governing the system dynamics. In this setup, the dynamic behavior of an electrical energy system (with possible natural switching) can be described by a switched state-space model where each mode is driven by possibly known and unknown inputs. The problem of detection and isolation of hard faults is equivalent to uniquely recovering the switching signal associated with uncontrolled transition caused by hard faults. The problem of detection and isolation of soft faults is equivalent to recovering the unknown additive disturbance caused by the fault. Uniquely recovering both switching signal and unknown inputs is the concern of the (left) invertibility problem in switched systems, and we are able to adopt theoretical results on that problem, developed earlier, to the present FDI setting. The application of the proposed framework to fault detection and isolation in switching electrical networks is illustrated with several examples.

Invertibility of nonlinear switched systems (with A. Tanwani), Automatica, vol. 46, no. 12, pp. 1962-1973, Dec 2010.
↓Abstract ↑Abstract: This article addresses the invertibility problem for switched nonlinear systems affine in controls. The problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state. We extend the concept of switch-singular pairs, introduced recently, to nonlinear systems and develop a formula for checking if the given state and output form a switch-singular pair. A necessary and sufficient condition for the invertibility of switched nonlinear systems is given, which requires the invertibility of individual subsystems and the nonexistence of switch-singular pairs. When all the subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included to illustrate these newly developed concepts.

Verifying average dwell time of hybrid systems (with S. Mitra and N. Lynch), ACM Transactions on Embedded Computing Systems, vol. 8, article 3, pp. 1-37, Dec 2008.
↓Abstract ↑Abstract: Average dwell time (ADT) properties characterize the rate at which a hybrid system performs mode switches. In this article, we present a set of techniques for verifying ADT properties. The stability of a hybrid system A can be verified by combining these techniques with standard methods for checking stability of the individual modes of A. We introduce a new type of simulation relation for hybrid automata - switching simulation - for establishing that a given automaton A switches more rapidly than another automaton B. We show that the question of whether a given hybrid automaton has ADT can be answered either by checking an invariant or by solving an optimization problem. For classes of hybrid automata for which invariants can be checked automatically, the invariant-based method yields an automatic method for verifying ADT; for automata that are outside this class, the invariant has to be checked using inductive techniques. The optimization-based method is automatic and is applicable to a restricted class of initialized hybrid automata. A solution of the optimization problem either gives a counterexample execution that violates the ADT property, or it confirms that the automaton indeed satisfies the property. The optimization and the invariant-based methods can be used in combination to find the unknown ADT of a given hybrid automaton.

Invertibility of switched linear systems (with L. Vu), Automatica, vol. 44, no. 4, pp. 949-958, Apr 2008.
↓Abstract ↑Abstract: We address the invertibility problem for switched systems, which is the problem of recovering the switching signal and the input uniquely given an output and an initial state. In the context of hybrid systems, this corresponds to recovering the discrete state and the input from partial measurements of the continuous state. In solving the invertibility problem, we introduce the concept of singular pairs for two systems. We give a necessary and sufficient condition for a switched system to be invertible, which says that the individual subsystems should be invertible and there should be no singular pairs. When the individual subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included.

On stability of randomly switched nonlinear systems (with D. Chatterjee), IEEE Transactions on Automatic Control, vol. 52, no. 12, pp. 2390-2394, Dec 2007.
↓Abstract ↑Abstract: This article is concerned with stability analysis and stabilization of randomly switched systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switchings are triggered by a stochastic process which is independent of the state of the system, and between two consecutive switching instants the dynamics are deterministic. Our results provide sufficient conditions for almost sure stability and stability in the mean using Lyapunov-based methods, when individual subsystems are stable and a certain "slow switching" cndition holds. This slow switching condition takes the form of an asymptotic upper bound on the probability mass function of the number of switches that occur between the initial and current time instants. This condition is shown to hold for switching signals coming from the states of finite-dimensional continuous-time Markov chains; our results therefore hold for Markovian jump systems in particular. For systems with control inputs we provide explicit control schemes for feedback stabilization using the universal formula for stabilization of nonlinear systems.

Input-to-state stability of switched systems and switching adaptive control (with L. Vu and D. Chatterjee), Automatica, vol. 43, no. 4, pp. 639-646, Apr 2007.
↓Abstract ↑Abstract: In this paper we prove that a switched nonlinear system has several useful ISS-type properties under average dwell-time switching signals if each constituent dynamical system is ISS. This extends available results for switched linear systems. We apply our result to stabilization of uncertain nonlinear systems via switching supervisory control, and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide ISS properties with respect to the estimation errors. Detailed illustrative examples are included.

Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions (with D. Chatterjee), SIAM Journal on Control and Optimization, vol. 45, no. 1, pp. 174-206, 2006.
↓Abstract ↑Abstract: This paper presents a general framework for analyzing stability of nonlinear switched systems, by combining the method of multiple Lyapunov functions with a suitably adapted comparison principle in the context of stability in terms of two measures. For deterministic switched systems, this leads to a unification of representative existing results and an improvement upon the current scope of the method of multiple Lyapunov functions. For switched systems perturbed by white noise, we develop new results which may be viewed as natural stochastic counterparts of the deterministic ones. In particular, we study stability of deterministic and stochastic switched systems under average dwell-time switching.

Lie-algebraic stability conditions for nonlinear switched systems and differential inclusions (with M. Margaliot), Systems and Control Letters, vol. 55, no. 1, pp. 8-16, Jan 2006.
↓Abstract ↑Abstract: We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, Unsolved Problems in Mathematical Systems Theory and Control, 2004]. To prove the result, we consider an optimal control problem which consists in finding the "most unstable" trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. This property is obtained as a special case of a reachability result by bang-bang controls which is of independent interest. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion.

Common Lyapunov functions for families of commuting nonlinear systems (with L. Vu), Systems and Control Letters, vol. 54, no. 5, pp. 405-416, May 2005.
↓Abstract ↑Abstract: We present constructions of a local and global common Lyapunov function for a finite family of pairwise commuting globally asymptotically stable nonlinear systems. The constructions are based on an iterative procedure, which at each step invokes a converse Lyapunov theorem for one of the individual systems. Our results extend a previously available one which relies on exponential stability of the vector fields.

Nonlinear norm-observability notions and stability of switched systems (with J. P. Hespanha, D. Angeli, and E. D. Sontag), IEEE Transactions on Automatic Control, vol. 50, no. 2, pp. 154-168, Feb 2005.
↓Abstract ↑Abstract: This paper proposes several definitions of observability for nonlinear systems and explores relationships among them. These observability properties involve the existence of a bound on the norm of the state in terms of the norms of the output and the input on some time interval. A Lyapunov-like sufficient condition for observability is also obtained. As an application, we prove several variants of LaSalle's stability theorem for switched nonlinear systems. These results are demonstrated to be useful for control design in the presence of switching as well as for developing stability results of Popov type for switched feedback systems.

Common Lyapunov functions and gradient algorithms (with R. Tempo), IEEE Transactions on Automatic Control, vol. 49, no. 6, pp. 990-994, Jun 2004.
↓Abstract ↑Abstract: This paper is concerned with the problem of finding a quadratic common Lyapunov function for a large family of stable linear systems. We present gradient iteration algorithms which give deterministic convergence for finite system families and probabilistic convergence for infinite families.

Lie-algebraic stability criteria for switched systems (with A. A. Agrachev), SIAM Journal on Control and Optimization, vol. 40, no. 1, pp. 253-269, Jun 2001.
↓Abstract ↑Abstract: It was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that the corresponding switched linear system is exponentially stable for arbitrary switching. In this paper we prove that the same properties hold under the weaker condition that the Lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact Lie group. The corresponding local stability result for nonlinear switched systems is also established. Moreover, we demonstrate that if a Lie algebra fails to satisfy the above condition, then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable. Relevant facts from the theory of Lie algebras are collected at the end of the paper for easy reference. Note: I subsequently learned that a result essentially equivalent to Theorem 2 of this paper was proved earlier by S. A. Kutepov (Absolute stability of bilinear systems with a compact Levi factor, in Russian, Kibernet. i Vychisl. Tekhn., vol. 62, pp. 28-33, 1984). Details

Basic problems in stability and design of switched systems (with A. S. Morse), IEEE Control Systems Magazine, vol. 19, no. 5, pp. 59-70, Oct. 1999.
↓Abstract ↑Abstract: By a switched system, we mean a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that orchestrates the switching between them. This article surveys recent developments in three basic problems regarding stability and design of switched systems. These problems are: stability for arbitrary switching sequences, stability for certain useful classes of switching sequences, and construction of stabilizing switching sequences. We also provide motivation for studying these problems by discussing how they arise in connection with various questions of interest in control theory and applications.

Stability of switched systems: a Lie-algebraic condition (with J. P. Hespanha and A. S. Morse), Systems and Control Letters, vol. 37, no. 3, pp. 117-122, Jul 1999.
↓Abstract ↑Abstract: We present a sufficient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function. We also discuss the implications of this result for switched nonlinear systems. See also the slides of the talk given at the Brockettfest, Cambridge, MA, Oct 1998.
Addendum

Book chapters:

Observer design for switched linear systems with state jumps (with A. Tanwani and H. Shim), in Hybrid Dynamical Systems: Observation and Control, Lecture Notes in Control and Information Sciences, vol. 457 (M. Djemai and M. Defoort, Eds.), Springer International Publishing, Switzerland, 2015, pp. 179-203.
↓Abstract ↑Abstract: An observer design for switched linear systems with state resets is proposed based on the geometric conditions for large-time observability from our recent work. Without assuming the observability of individual subsystems, the basic idea is to combine the maximal information available from each mode to obtain a good estimate of the state after a certain time interval (overwhich the switched system is observable) has passed. We first study systems where state reset maps at switching instants are invertible, in which case it is possible to collect all the observable and unobservable information separately at one time instant. One can then annihilate the unobservable component of all the modes and obtain an estimate of the state by introducing an error correction map at that time instant. However, for the systems with non-invertible jump maps, this approach needs to be modified and a recursion-based error correction scheme is proposed. In both approaches, the criterion for choosing the output injection matrices is given, which leads to the asymptotic recovery of the system state.

Stability analysis of hybrid systems via small-gain theorems (with D. Nesic), in Proceedings of the Ninth International Workshop on Hybrid Systems: Computation and Control, Santa Barbara, CA, Mar 2006, Lecture Notes in Computer Science, vol. 3927 (J. Hespanha and A. Tiwari, Eds.), Springer, Berlin, pp. 421-435.
↓Abstract ↑Abstract: We present a general approach to analyzing stability of hybrid systems, based on input-to-state stability (ISS) and small-gain theorems. We demonstrate that the ISS small-gain analysis framework is very naturally applicable in the context of hybrid systems. Novel Lyapunov-based and LaSalle-based small-gain theorems for hybrid systems are presented. The reader does not need to be familiar with ISS or small-gain theorems to be able to follow the paper.

Verifying average dwell time by solving optimization problems (with S. Mitra and N. Lynch), in Proceedings of the Ninth International Workshop on Hybrid Systems: Computation and Control, Santa Barbara, CA, Mar 2006, Lecture Notes in Computer Science, vol. 3927 (J. Hespanha and A. Tiwari, Eds.), Springer, Berlin, pp. 476-490.
↓Abstract ↑Abstract: In the switched system model, discrete mechanisms of a hybrid system are abstracted away in terms of an exogenous switching signal which brings about the mode switches. The Average Dwell time (ADT) property defines restricted classes of switching signals which provide sufficient conditions for proving stability of switched systems. In this paper, we use a specialization of the Hybrid I/O Automaton model to capture both the discrete and the continuous mechanisms of hybrid systems. Based on this model, we develop methods for automatically verifying ADT properties and present simulation relations for establishing equivalence of hybrid systems with respect to ADT. Given a candidate ADT for a hybrid system, we formulate an optimization problem; a solution of this problem either establishes the ADT property or gives an execution fragment of the system that violates it. For two special classes of hybrid systems, we show that the corresponding optimization problems can be solved using standard mathematical programming techniques.We formally define equivalence of two hybrid systems with respect to ADT and present a simulation relation-based method for proving this equivalence. The proposed methods are applied to verify ADT properties of a linear hysteresis switch and a nondeterministic thermostat. See also a more complete version.

Switched systems, Handbook of Networked and Embedded Control Systems (D. Hristu-Varsakelis and W. S. Levine, Eds.), Birkhauser, Boston, 2005, pp. 559-574.

Lie algebras and stability of switched nonlinear systems, Unsolved Problems in Mathematical Systems Theory and Control (V. D. Blondel and A. Megretski, Eds.), Princeton University Press, 2004, pp. 203-207. See also Open Problems Book of the 15th International Symposium on Mathematical Theory of Networks and Systems (MTNS), South Bend, IN, Aug 2002, pp. 90-92.
See also a partial solution.

Conferences:

Topological entropy of switched nonlinear systems (with G. Yang and J. P. Hespanha), in Proceedings of the 24th ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2021), Nashville, TN (online), May 2021.
↓Abstract ↑Abstract: This paper studies topological entropy of switched nonlinear systems. We construct a general upper bound for the topological entropy in terms of an average of the asymptotic suprema of the measures of Jacobian matrices of individual modes, weighted by the corresponding active rates. A general lower bound is also established in terms of an active-rate-weighted average of the asymptotic infima of the traces of these Jacobian matrices. For switched systems with diagonal modes, we establish upper and lower bounds that only depend on the eigenvalues of Jacobian matrices, their relative order among individual modes, and the active rates. For both cases, we also establish upper bounds that are more conservative but require less information on the switching, with their relations illustrated by numerical examples of a switched Lotka--Volterra ecosystem model.

Quantizer design for linear switched systems with minimal data-rate (with G. S. Vicinansa), in Proceedings of the 24th ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2021), Nashville, TN (online), May 2021.
↓Abstract ↑Abstract: In this paper, we present a quantization scheme that reconstructs the state of linear switched systems with a prescribed exponential decaying rate for the state estimation error. We show how to use the Lyapunov exponents and a geometric object called Oseledets' filtration to design such a quantization scheme. Then, we prove that this algorithm works at an average data-rate close to the estimation entropy of the given system. Furthermore, we can choose the average data-rate to be arbitrarily close to the estimation entropy whenever the linear switched system has the so-called regularity property. We show that, under the regularity assumption, the quantization scheme is completely causal in the sense that it only depends on information that is available at the current time instant. Finally, we present simulation results for a Markov Jump Linear System, a class of systems for which the realizations are known to be regular with probability 1.

Average dwell-time bounds for ISS and integral ISS of switched systems using Lyapunov functions (with S. Liu and A. Tanwani), in Proceedings of the 59th IEEE Conference on Decision and Control, Jeju Island, Korea (online), Dec 2020, pp. 6291-6296.
↓Abstract ↑Abstract: The problem of input-to-state stability (ISS), and its integral version (iISS), is considered for switched systems with inputs and resets. The individual subsystems are assumed to be ISS (resp. iISS) with nonlinear decay rates in dissipation inequalities associated with the Lyapunov function of each subsystem. The change in the value of Lyapunov functions at switching instants is described by a nonlinear growth function. A generalized lower bound is computed for average dwell-time (ADT) to guarantee ISS/iISS of the switched system. In particular, an explicit formula of ADT lower bound is given for switched bilinear systems with zero-input-stable subsystems.

Quasi-integral-input-to-state stability for switched nonlinear systems (with A. Russo, S. Liu, and A. Cavallo), in Proceedings of the 21st IFAC World Congress, Berlin, Germany (online), Jul 2020.
↓Abstract ↑Abstract: In this paper we introduce and give sufficient conditions for the quasi-iISS property for switched nonlinear system under dwell-time switching signals. Unlike previous works, our dwell-time bound does not rely on the knowledge of the state but it relies only on the system initial condition and the bound on the input energy. We prove, through a counterexample, that knowledge of the system initial state and bound on input energy is necessary for the estimation of a dwell-time that guarantees quasi-iISS for the switched system. An illustrative example is also included.

Estimation entropy for regular linear switched systems (with G. S. Vicinansa), in Proceedings of the 58th IEEE Conference on Decision and Control, Nice, France, Dec 2019, pp. 5754-5759.
↓Abstract ↑Abstract: In this paper, we introduce the concept of regular switched systems and we present some consequences of this property on the estimation entropy's calculation. For systems of such a class, one can derive a formula for the estimation entropy in terms of the system's Lyapunov exponents. We also present some sufficient conditions on the system that guarantee regularity. Some of those conditions include the cases of periodic switching, simultaneously triangularizable systems, and a large class of randomly switched systems that contains Markov Jump Linear Systems (MJLS) as a special case. For that last part, we use tools from ergodic theory to draw conclusions that hold almost surely. See also a more complete version.

On topological entropy and stability of switched linear systems (with G. Yang and J. P. Hespanha), in Proceedings of the 22nd ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2019), Montreal, Canada, April 2019, pp. 119-127 (Best Paper Award winner).
↓Abstract ↑Abstract: This paper studies topological entropy and stability properties of switched linear systems. First, we show that the exponential growth rates of solutions of a switched linear system are essentially upper bounded by its topological entropy. Second, we estimate the topological entropy of a switched linear system by decomposing it into a part that is generated by scalar multiples of the identity matrix and a part that has zero entropy, and proving that the overall topological entropy is upper bounded by that of the former. Third, we prove that a switched linear system is globally exponentially stable if its topological entropy remains zero under a destabilizing perturbation. Finally, the entropy estimation via decomposition and the entropy-based stability condition are applied to three classes of switched linear systems to construct novel upper bounds for topological entropy and novel sufficient conditions for global exponential stability.

On stability of nonlinear slowly time-varying and switched systems (with X. Gao and T. Basar), in Proceedings of the 57th IEEE Conference on Decision and Control, Miami Beach, FL, Dec 2018, pp. 6458-6463.
↓Abstract ↑Abstract: In this paper, we apply a total variation approach to bridge the stability criteria for nonlinear time-varying systems and nonlinear switched systems. In particular, we derive a set of stability conditions applying to both nonlinear time-varying systems and nonlinear switched systems. We show that the derived stability conditions, when applied to nonlinear time-varying systems and nonlinear switched systems, recover the existing stability results in the literature. We also show that the derived stability conditions can be applied to qualitatively recover a unified stability criterion for slowly time-varying linear systems and switched linear systems proposed in our recent work. See also the slides of the talk.

On topological entropy of switched linear systems with diagonal, triangular, and general matrices (with G. Yang and A. J. Schmidt), in Proceedings of the 57th IEEE Conference on Decision and Control, Miami Beach, FL, Dec 2018, pp. 5682-5687.
↓Abstract ↑Abstract: This paper introduces a notion of topological entropy for switched systems, formulated using the minimal number of initial states needed to approximate all initial states within a finite precision. We show that it can be equivalently defined using the maximal number of initial states separable within a finite precision, and introduce switching-related quantities such as the active time of each mode, which prove to be useful in calculating the topological entropy of switched linear systems. For general switched linear systems, we show that the topological entropy is independent of the set of initial states, and establish upper and lower bounds using the active-time-weighted averages of the norms and traces of system matrices in individual modes, respectively. For switched linear systems with scalar-valued state or simultaneously diagonalizable matrices, we derive formulae for the topological entropy using active-time-weighted averages of eigenvalues, which can be extended to the case with simultaneously triangularizable matrices to obtain an upper bound. In these three cases with special matrix structure, we also provide more general but more conservative upper bounds for the topological entropy.

Global stability and asymptotic gain imply input-to-state stability for state-dependent switched systems (with S. Liu), in Proceedings of the 57th IEEE Conference on Decision and Control, Miami Beach, FL, Dec 2018, pp. 2360-2365.
↓Abstract ↑Abstract: In this paper we study several stability properties for state-dependent switched systems. We examine the gap between global asymptotic stability and uniform global asymptotic stability, and illustrate it with an example. Several regularity assumptions are proposed in order to obtain the equivalence between these two stability properties. Based on this equivalence, we are able to show that global stability and asymptotic gain imply input-to-state stability for state-dependent switched systems, which is the main result of the paper. The proof consists of a bypass via an auxiliary system which takes in a bounded disturbance, and showing that this system is uniformly globally asymptotically stable.See also a more complete version.

Stabilization of interconnected switched control-affine systems via a Lyapunov-based small-gain approach (with G. Yang and J.-P. Jiang), in Proceedings of the American Control Conference, Seattle, WA, May 2017, pp. 5182-5187.
↓Abstract ↑Abstract: We study the feedback stabilization of interconnected switched control-affine systems with both input-to-state stable (ISS) and non-ISS modes. Provided that the switching is slow in the sense of average dwell-time and the active time of non-ISS modes is short in proportion, suitable feedback controls are designed to achieve input-to-state practical stability (ISpS) with an arbitrarily small constant. We devise such feedback controls by extending a previous small-gain theorem on stability of interconnected switched systems to the ISpS context, and proposing a novel Lyapunov-based gain-assignment scheme.

Analysis of different Lyapunov function constructions for interconnected hybrid systems (with G. Yang and A. Mironchenko), in Proceedings of the 55th IEEE Conference on Decision and Control, Las Vegas, NV, Dec 2016, pp. 465-470.
↓Abstract ↑Abstract: This paper studies stability of interconnections of hybrid dynamical systems, in the general scenario that the continuous or discrete dynamics of subsystems may have destabilizing effects. We analyze two existing methods of constructing Lyapunov functions for the interconnection based on candidate ISS Lyapunov functions for subsystems, small-gain conditions, and auxiliary timers modeling restrictions on jump frequency in terms of average dwell-time and reverse average dwell-time. We compare their feasibility and limitations for different types of subsystem dynamics, and examine a case that the combination of them is needed to establish global asymptotic stability.

Finite data-rate stabilization of a switched linear system with unknown disturbance (with G. Yang), in Proceedings of the 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2016), Monterey, CA, Aug 2016, pp. 1103-1108.
↓Abstract ↑Abstract: We study the stabilization of a switched linear system with unknown disturbance using sampled and quantized state feedback. The switching is slow in the sense of combined dwell-time and average dwell-time, while the active mode is unknown except at sampling times. Each mode of the switched system is stabilizable, and the disturbance admits an unknown bound. A communication and control strategy is designed to achieve practical stability and exponential convergence w.r.t. the initial state with a nonlinear gain on the disturbance, provided the data-rate meets given lower bounds. Compared with previous results, a more involved algorithm is developed to handle effects of the unknown disturbance based on employing an iteratively updated estimate of the disturbance bound and expanding the over-approximations of reachable sets over sampling intervals from the case without disturbance.

Connections between stability conditions for slowly time-varying and switched linear systems (with X. Gao, J. Liu, and T. Basar), in Proceedings of the 54th IEEE Conference on Decision and Control, Osaka, Japan, Dec 2015, pp. 2339-2334.
↓Abstract ↑Abstract: This paper establishes an explicit relationship between stability conditions for slowly time-varying linear systems and switched linear systems. The concept of total variation of a matrix-valued function is introduced to characterize the variation of the system matrix. Using this concept, a result generalizing existing stability conditions for slowly time-varying linear systems is derived. As a special case of this result, it is shown that a switched linear system is globally exponentially stable if the average dwell time of the switching signal is large enough, which qualitatively matches known results in the literature.

Stabilizing a switched linear system with disturbance by sampled-data quantized feedback (with G. Yang), in Proceedings of the American Control Conference, Chicago, IL, Jul 2015, pp. 2193-2198.
↓Abstract ↑Abstract: We study the problem of stabilizing a switched linear system with disturbance using sampled and quantized measurements of its state. The switching is assumed to be slow in the sense of combined dwell-time and average dwell-time, while the active mode is unknown except at sampling times. Each mode of the switched linear system is assumed to be stabilizable, and the magnitude of the disturbance is constrained by a known bound. A communication and control strategy is designed to guarantee bounded-input-bounded-state (BIBS) stability of the switched linear system and an exponential convergence rate with respect to the initial state, providing the data rate satisfies certain lower bounds. Such lower bounds are established by expanding the over-approximation bounds of reachable sets over sampling intervals derived in a previous paper to accommodate effects of the disturbance.

Input-to-state stability for switched systems with unstable subsystems: a hybrid Lyapunov construction (with G. Yang), in Proceedings of the 53rd IEEE Conference on Decision and Control, Los Angeles, CA, Dec 2014, pp. 6240-6245.
↓Abstract ↑Abstract: The input-to-state stability (ISS) of a nonlinear switched system is investigated in the scenario where there may exist some subsystems that are not input-to-state stable (non-ISS). We show that, providing the switching signal neither switches too frequently nor activates non-ISS subsystems for too long, a hybrid ISS Lyapunov function can be constructed to guarantee ISS of the switched system. With the constraints on the switching signal being modeled by a novel auxiliary timer, a hybrid system is defined so that the solutions to the two systems are correspondent. After the construction and verification of an ISS Lyapunov function, ISS of all complete solutions to the hybrid system, and therefore all solutions to the switched system, is conveniently proved.

Lyapunov small-gain theorems for not necessarily ISS hybrid systems (with A. Mironchenko and G. Yang), in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS), Groningen, the Netherlands, Jul 2014, pp. 1001-1008.
↓Abstract ↑Abstract: We prove a novel Lyapunov-based small-gain theorem for interconnections of n hybrid systems, which are not necessarily input-to-state stable. This result unifies and extends several small-gain theorems for hybrid and impulsive systems, proposed in the last few years. Also we show how the average dwell-time (ADT) clocks and reverse ADT clocks can be used to modify the Lyapunov functions for subsystems and to enlarge the applicability of derived small-gain theorems.

Robust stability conditions for switched linear systems: commutator bounds and the Lojasiewicz inequality (with Y. Baryshnikov), in Proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, Dec 2013, pp. 722-726.
↓Abstract ↑Abstract: This paper discusses conditions for stability of switched linear systems under arbitrary switching, formulated in terms of smallness of appropriate commutators of the matrices generating the switched system. Such conditions provide robust variants of well-known stability conditions requiring these commutators to vanish and leading to the existence of a common quadratic Lyapunov function. The main contribution of the paper is to apply the Lojasiewicz inequality to characterize the persistence of a common quadratic Lyapunov function as the matrices are perturbed so that their commutators no longer vanish but instead are sufficiently small. It is shown how known constructions of common quadratic Lyapunov functions for commuting matrices and for matrices generating nilpotent or solvable Lie algebras can be used, in conjunction with the Lojasiewicz inequality, to estimate allowable deviations of the commutators from zero.

Limited-information control of hybrid systems via reachable set propagation, in Proceedings of the 16th ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2013), Philadelphia, PA, Apr 2013, pp. 11-19.
↓Abstract ↑Abstract: This paper deals with control of hybrid systems based on limited information about their state. Specifically, measurements being passed from the system to the controller are sampled and quantized, resulting in finite data-rate communication. The main ingredient of our solution to this control problem is a novel method for propagating over-approximations of reachable sets for hybrid systems through sampling intervals, during which the discrete mode is unknown. In addition, slow-switching conditions of the (average) dwell-time type and multiple Lyapunov functions play a central role in the analysis. See also the slides of the talk.

Small-gain theorems of LaSalle type for hybrid systems (with D. Nesic and A. R. Teel), in Proceedings of the 51st IEEE Conference on Decision and Control, Maui, HI, Dec 2012, pp. 6825-6830.
↓Abstract ↑Abstract: We study stability of hybrid systems described as feedback interconnections of smaller subsystems, within a Lyapunov-based ISS small-gain analysis framework. We focus on constructing a weak (nonstrictly decreasing) Lyapunov function for the overall hybrid system from weak ISS-Lyapunov functions for the subsystems in the interconnection. Asymptotic stability of the hybrid system is then concluded by applying results of LaSalle type. The utility of this approach is illustrated on feedback systems arising in event-triggered control and quantized control. See also the slides of the talk.

Stabilizing a switched linear system by sampled-data quantized feedback, in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, Dec 2011, pp. 8321-8326.
↓Abstract ↑Abstract: We study the problem of asymptotically stabilizing a switched linear control system using sampled and quantized measurements of its state. The switching is assumed to be slow enough in the sense of combined dwell time and average dwell time, each individual mode is assumed to be stabilizable, and the available data rate is assumed to be large enough. Our encoding and control strategy is rooted in the one proposed in our earlier work on non-switched systems, and in particular the data-rate bound used here is the data-rate bound from that earlier work maximized over the individual modes. The main technical step that enables the extension to switched systems concerns propagating over-approximations of reachable sets through sampling intervals, during which the switching signal is unknown. See also the slides of the talk.

Commutativity and asymptotic stability for linear switched DAEs (with S. Trenn and F. Wirth), in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, Dec 2011, pp. 417-422.
↓Abstract ↑Abstract: For linear switched ordinary differential equations with asymptotically stable constituent systems, it is well known that commutativity of the coefficient matrices implies asymptotic stability of the switched system under arbitrary switching. This result is generalized to linear switched differential algebraic equations (DAEs). Although the solutions of a switched DAE can exhibit jumps it turns out that it suffices to check commutativity of the "flow" matrices. As in the ODE case we are also able to construct a common quadratic Lyapunov function.

Robust invertibility of switched linear systems (with A. Tanwani), in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, Dec 2011, pp. 441-446.
↓Abstract ↑Abstract: In this paper, we address the effects of uncertainties in output measurements and initial conditions on invertibility of switched systems -- the problem concerned with the recovery of the input and the switching signal using the output and the initial state. By computing the reachable sets and maximal error in the propagation of state trajectories, we derive conditions under which it is possible to recover the exact switching signal over a certain time interval, provided the uncertainties are bounded in some sense. In addition, we discuss separately the case where each subsystem is minimum phase and it is possible to recover the exact switching signal globally in time. The input, though, is recoverable only up to a neighborhood of the original input.

Observability implies observer design for switched linear systems (with A. Tanwani and H. Shim), in Proceedings of the 14th ACM International Conference on Hybrid Systems: Computation and Control, Chicago, IL, Apr 2011, pp. 3-12.
↓Abstract ↑Abstract: This paper presents a unified framework for observability and observer design for a class of hybrid systems. A necessary and sufficient condition is presented for observability, globally in time, when the system evolves under predetermined mode transitions. A relatively weaker characterization is given for determinability, the property that concerns with unique recovery of the state at some time rather than at all times. These conditions are then utilized in the construction of a hybrid observer that is feasible for implementation in practice. The observer, without using the derivatives of the output, generates the state estimate that converges to the actual state under persistent switching. Erratum

Towards robust Lie-algebraic stability conditions for switched linear systems (with A. A. Agrachev and Y. Baryshnikov), in Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, GA, Dec 2010, pp. 408-413.
↓Abstract ↑Abstract: This paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching, which involve the commutators (Lie brackets) among the given matrices generating the switched system. The main novel feature of these stability criteria is that, unlike their earlier counterparts, they are robust with respect to small perturbations of the system parameters. Two distinct approaches are investigated. For discrete-time switched linear systems, we formulate a stability condition in terms of an explicit upper bound on the norms of the Lie brackets. For continuous-time switched linear systems, we develop two stability criteria which capture proximity of the associated matrix Lie algebra to a solvable or a "solvable plus compact" Lie algebra, respectively. See also the slides of the talk.

State-norm estimators for switched nonlinear systems under average dwell-time (with M. Müller), in Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, GA, Dec 2010, pp. 1275-1280.
↓Abstract ↑Abstract: In this paper, we consider the concept of state-norm estimators for switched nonlinear systems under average dwell-time switching signals. State-norm estimators are closely related to the concept of input/output-to-state stability (IOSS). We show that if the average dwell-time is large enough, a non-switched state-norm estimator for a switched system exists in the case where each of its constituent subsystems is IOSS. Furthermore, we show that a switched state-norm estimator, consisting of two subsystems, exists for a switched system in the case where only some of its constituent subsystems are IOSS and others are not, provided that the average dwell-time is large enough and the activation time of the non-IOSS subsystems is not too large. In both cases, the stated sufficient conditions are also sufficient for the switched system to be IOSS. For the case where some subsystems are not IOSS, we also show that the switched state-norm estimator can be constructed in such a way that its switching times are independent of the switching times of the switched system it is designed for. See also a more complete version.

Input/output-to-state stability of switched nonlinear systems (with M. Müller), in Proceedings of the American Control Conference, Baltimore, MD, Jul 2010, pp. 1708-1712.
↓Abstract↑Abstract: In this paper, we study the property of input/output-to-state stability (IOSS) for switched nonlinear systems under average dwell-time switching signals, both when each of the constituent systems is IOSS as well as when only some of the constituent systems are IOSS and others are not. This extends available results on input-to-state stability for switched nonlinear systems whose constituent systems are all ISS. We show that if the average dwell-time is big enough and if the fraction of time where one of the non-IOSS systems is active is not too big, then IOSS of the switched system can be established.

On stability of linear switched differential algebraic equations (with S. Trenn), in Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China, Dec 2009, pp. 2156-2161.
↓Abstract↑Abstract: This paper studies linear switched differential algebraic equations (DAEs), i.e., systems defined by a finite family of linear DAE subsystems and a switching signal that governs the switching between them. We show by examples that switching between stable subsystems may lead to instability, and that the presence of algebraic constraints leads to a larger variety of possible instability mechanisms compared to those observed in switched systems described by ordinary differential equations (ODEs). We prove two sufficient conditions for stability of switched DAEs based on the existence of suitable Lyapunov functions. The first result states that a common Lyapunov function guarantees stability under arbitrary switching when an additional condition involving consistency projectors holds (this extra condition is not needed when there are no jumps, as in the case of switched ODEs). The second result shows that stability is preserved under switching with sufficiently large dwell time.

On new sufficient conditions for stability of switched linear systems, in Proceedings of the 2009 European Control Conference, Budapest, Hungary, Aug 2009, pp. 3257-3262.
↓Abstract↑Abstract: This work aims to connect two existing approaches to stability analysis of switched linear systems: stability conditions based on commutation relations between the subsystems and stability conditions of the slow-switching type. The proposed sufficient conditions for stability have an interpretation in terms of commutation relations; at the same time, they involve only elementary computations of matrix products and induced norms, and possess robustness to small perturbations of the subsystem matrices. These conditions are also related to slow switching, in the sense that they rely on the knowledge of how slow the switching should be to guarantee stability; however, they cover situations where the switching is actually not slow enough, by accounting for relations between the subsystems. Numerical examples are included for illustration. See also the slides of the talk.

Invertibility of nonlinear switched systems (with A. Tanwani), in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec 2008, pp. 286-291.
↓Abstract↑Abstract: This article addresses the invertibility problem for switched nonlinear systems affine in controls. The problem is concerned with finding the input and switching signal uniquely from given output and initial state. We extend the concept of switch-singular pairs, introduced in [1], to nonlinear systems and develop a formula for checking if given state and output form a switch-singular pair. We give a necessary and sufficient condition for a switched system to be invertible, which says that the subsystems should be invertible and there should be no switch-singular pairs. When all the subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs. Detailed examples are included to illustrate these newly developed concepts. See also a more complete version.

Towards ISS disturbance attenuation for randomly switched systems (with D. Chatterjee), in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, Dec 2007, pp. 5612-5617.
↓Abstract↑Abstract: We are concerned with input-to-state stability (ISS) of randomly switched systems. We provide preliminary results dealing with sufficient conditions for stochastic versions of ISS for randomly switched systems without control inputs, and with the aid of universal formulae we design controllers for ISS-disturbance attenuation when control inputs are present. Two types of switching signals are considered: the first is characterized by a statistically slow-switching condition, and the second by a class of semi-Markov processes.

Stability of interconnected switched systems and adaptive control of time-varying plants (with L. Vu), in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, Dec 2007, pp. 4021-4026.
↓Abstract↑Abstract: We discuss stability of a loop consisting of two asynchronous switched systems, in which the first switched system influences the input and the switching signal of the second switched system and the second switched system affects the first switched system's jump map. We show that when the first switched system has a small dwell-time and is switching slowly in the spirit of average dwell-time switching, all the states of the closed loop are bounded. We show how this result relates to supervisory adaptive control of time-varying plants. When the uncertain plant takes the form of a switched system with an unknown switching signal, we show that all the states of the closed-loop control system are guaranteed to be bounded provided that the plant's switching signal varies slowly enough.

On invertibility of switched linear systems (with L. Vu), in Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, Dec 2006, pp. 4081-4086.
↓Abstract↑Abstract: We address a new problem - the invertibility problem for continuous-time switched linear systems, which is the problem of recovering the switching signal and the input uniquely given an output and an initial state. In the context of hybrid systems, this corresponds to recovering the discrete state and the input from partial measurements of the continuous state. In solving the invertibility problem, we introduce the concept of singular pairs for two systems. We give a necessary and sufficient condition for a switched system to be invertible, which says that the subsystems should be invertible and there should be no singular pairs. When all the subsystems are invertible, we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs.

Stability and stabilization of randomly switched systems (with D. Chatterjee), in Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, Dec 2006, pp. 2643-2648.
↓Abstract↑Abstract: This article is concerned with stability analysis and stabilization of randomly switched systems with control inputs. The switching signal is modeled as a jump stochastic process independent of the system state; it selects, at each instant of time, the active subsystem from a family of deterministic systems. Three different types of switching signals are considered: the first is a jump stochastic process that satisfies a statistically slow switching condition; the second and the third are jump stochastic processes with independent identically distributed values at jump times together with exponential and uniform holding times, respectively. For each of the three cases we first establish sufficient conditions for stochastic stability of the switched system when the subsystems do not possess control inputs; not every subsystem is required to be stable in the latter two cases. Thereafter we design feedback controllers when the subsystems are affine in control and are not all zeroinput stable, with the control space being general subsets of $R^m$. Our analysis results and universal formulae for feedback stabilization of nonlinear systems for the corresponding control spaces constitute the primary tools for control design. See also a more complete version.

ISS of switched systems and applications to switching adaptive control (with L. Vu and D. Chatterjee), in Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, Dec 2005, pp. 120-125.
↓Abstract↑Abstract: In this paper we prove that a switched nonlinear system has several useful ISS-type properties under average dwell-time switching signals if each constituent dynamical system is ISS. This extends available results for switched linear systems. We apply our result to stabilization of uncertain nonlinear systems via switching supervisory control, and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide ISS properties with respect to the estimation errors. Detailed illustrative examples are included. See also the slides of the talk.

A small-gain approach to stability analysis of hybrid systems (with D. Nesic), in Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, Dec 2005, pp. 5409-5414.
↓Abstract↑Abstract: We propose to use ISS small-gain theorems to analyze stability of hybrid systems. We demonstrate that the small-gain analysis framework is very naturally and generally applicable in the context of hybrid systems, and thus has a potential to be useful in many applications. The main idea is illustrated on specific problems in the context of control with limited information, where it is shown to provide novel interpretations, powerful extensions, and a more unified treatment of several previously available results. The reader does not need to be familiar with ISS or small-gain theorems to be able to follow the paper. See also the slides of the talk.

A Lie-algebraic condition for stability of switched nonlinear systems (with M. Margaliot), in Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, Dec 2004, pp. 4619-4624.
↓Abstract↑Abstract: We present a stability criterion for switched nonlinear systems which involves Lie brackets of the individual vector fields but does not require that these vector fields commute. A special case of the main result says that a switched system generated by a pair of globally asymptotically stable nonlinear vector fields whose third-order Lie brackets vanish is globally uniformly asymptotically stable under arbitrary switching. This generalizes a known fact for switched linear systems and provides a partial solution to the open problem posed in [D. Liberzon, Lie algebras and stability of switched nonlinear systems, Unsolved Problems in Mathematical Systems Theory and Control, 2004]. To prove the result, we consider an optimal control problem which consists in finding the "most unstable" trajectory for an associated control system, and show that there exists an optimal solution which is bang-bang with a bound on the total number of switches. By construction, our criterion also automatically applies to the corresponding relaxed differential inclusion. See also the slides of the talk.

On stability of stochastic switched systems (with D. Chatterjee), in Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, Dec 2004, pp. 4125-4127.
↓Abstract↑Abstract: In this paper we propose a method for stability analysis of switched systems perturbed by a Wiener process. It utilizes multiple Lyapunov-like functions and is analogous to an existing result for deterministic switched systems. See also the slides of the talk.

Stability of hybrid automata with average dwell time: an invariant approach (with S. Mitra), in Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, Dec 2004, pp. 1394-1399.
↓Abstract↑Abstract: A formal method based technique is presented for proving the average dwell time property of a hybrid system, which is useful for establishing stability under slow switching. The Hybrid Input/Output Automaton (HIOA) is used as the model for hybrid systems, and it is shown that some known stability theorems from system theory can be adapted to be applied in this framework. The average dwell time property of a given automaton, is formalized as an invariant of a corresponding transformed automaton, such that the former has average dwell time if and only if the latter satisfies the invariant. Formal verification techniques can be used to check this invariance property. In particular, the HIOA framework facilitates inductive invariant proofs by systematically breaking them down into cases for the discrete actions and continuous trajectories of the automaton. The invariant approach to proving the average dwell time property is illustrated by analyzing the hysteresis switching logic unit of a supervisory control system. See also the slides of the talk.

Gradient algorithms for finding common Lyapunov functions (with R. Tempo), in Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, Dec 2003, pp. 4782-4787.
↓Abstract↑Abstract: This paper is concerned with the problem of finding a quadratic common Lyapunov function for a large family of stable linear systems. We present gradient iteration algorithms which give deterministic convergence for finite system families and probabilistic convergence for infinite families. See also the slides of the talk.

Nonlinear observability and an invariance principle for switched systems (with J. P. Hespanha and E. D. Sontag), in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, Dec 2002, pp. 4300-4305.
↓Abstract↑Abstract: This paper proposes several definitions of observability for nonlinear systems and explores relationships between them. These observability properties involve the existence of a bound on the norm of the state in terms of the norm of the output on some time interval. As an application, we prove a LaSalle-like stability theorem for switched nonlinear systems. See also the slides of the talk.

A note on uniform global asymptotic stability of nonlinear switched systems in triangular form (with D. Angeli), in Proceedings of the 14th International Symposium on Mathematical Theory of Networks and Systems (MTNS), Perpignan, France, Jun 2000.
↓Abstract↑Abstract: This note examines stability properties of systems that result from switching between globally asymptotically stable nonlinear systems in triangular form. We show by means of a counterexample that, unlike in the linear case, such a switched system might not be uniformly globally asymptotically stable for arbitrary switching signals. We then proceed to formulate conditions that guarantee uniform global asymptotic stability.

Lie-algebraic conditions for exponential stability of switched systems (with A. A. Agrachev), in Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, Dec 1999, pp. 2679-2684.
↓Abstract↑Abstract: It has recently been shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that the corresponding switched linear system is exponentially stable for arbitrary switching. In this paper we prove that the same properties hold under the weaker condition that the Lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact Lie group. The corresponding local stability result for nonlinear switched systems is also established. Moreover, we demonstrate that if a Lie algebra fails to satisfy the above condition, then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable. Relevant facts from the theory of Lie algebras are collected at the end of the paper for easy reference. See also the slides of the talk.

ISS and integral-ISS disturbance attenuation with bounded controls, in Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, Dec 1999, pp. 2501-2506.
↓Abstract↑Abstract: We consider the problem of achieving disturbance attenuation in the ISS and integral-ISS sense for nonlinear systems with bounded controls. For the ISS case we derive a "universal" formula which extends an earlier result of Lin and Sontag to systems with disturbances. For the integral-ISS case we give two constructions, one resulting in a smooth control law and the other in a switching control law. We also briefly discuss some issues related to input-to-state stability of switched and hybrid systems. See also the slides of the talk.

Stabilizing a linear system with finite-state hybrid output feedback, in Proceedings of the 7th IEEE Mediterranean Conference on Control and Automation, Haifa, Israel, Jun 1999, pp. 176-183.
↓Abstract↑Abstract: The purpose of this short note is to establish and explore a link between the problem of stabilizing a linear system using finite-state hybrid output feedback and the problem of finding a stabilizing switching sequence for a switched linear system with unstable individual matrices, each of which separately has recently received attention in the literature. See also the slides of the talk.

Control under communication constraints:

Estimation entropy, Lyapunov exponents, and quantizer design for switched linear systems (with G. S. Vicinansa), SIAM Journal on Control and Optimization, vol. 61, no. 1, pp. 198-224, 2023.
↓Abstract ↑Abstract: In this paper, we study connections between the estimation entropy of a switched linear system and its Lyapunov exponents. We prove lower and upper bounds for the estimation entropy in terms of the Lyapunov exponents and show that, under the so-called regularity assumption, those bounds coincide. Also, using a geometric object called Oseledets' filtration of the system, we prove that the estimation entropy of the system is given in terms of the Lyapunov exponents. Further, we show how to use the exponents and the Oseledets' filtration to design a quantization scheme for state estimation of switched linear systems. Then, we prove that we can make this algorithm work at an average data-rate arbitrarily close to the upper bound we provided for the estimation entropy of the given system. Furthermore, we can choose the average data-rate to be arbitrarily close to the estimation entropy whenever the switched linear system is regular. We show that, under the regularity assumption, the quantization scheme is completely causal in the sense that it only depends on information that is available up to the current time instant. We show that regularity is a natural property of many practical systems, such as Markov Jump Linear Systems, and give sufficient conditions for it.

Entropy and minimal bit rates for state estimation and model detection (with S. Mitra), IEEE Transactions on Automatic Control, vol. 63, no. 10, pp. 3330-3344, Oct 2018.
↓Abstract ↑Abstract: We study a notion of estimation entropy for continuous-time nonlinear systems, formulated in terms of the number of system trajectories that approximate all other trajectories up to an exponentially decaying error. We also consider an alternative definition of estimation entropy which uses approximating functions that are not necessarily trajectories of the system, and show that the two entropy notions are equivalent. We establish an upper bound on the estimation entropy in terms of the sum of the desired convergence rate and an upper bound on the matrix measure of the Jacobian, multiplied by the system dimension. A lower bound on the estimation entropy is developed as well. We describe an iterative procedure that uses quantized and sampled state measurements to generate state estimates that converge to the true state at the desired exponential rate. The average bit rate utilized by this procedure matches the derived upper bound on the estimation entropy, and no other algorithm of this type can perform the same estimation task with bit rates lower than the estimation entropy. Finally, we discuss an application of the estimation procedure in determining, from the quantized state measurements, which of two competing models of a dynamical system is the true model. We show that under a mild assumption of "exponential separation" of the candidate models, detection always happens in finite time.

Feedback stabilization of a switched linear system with unknown disturbances under data-rate constraints (with G. Yang), IEEE Transactions on Automatic Control, vol. 63, no. 7, pp. 2107-2122, Jul 2018.
↓Abstract ↑Abstract: We study the problem of stabilizing a switched linear system with a completely unknown disturbance using sampled and quantized state feedback. The switching is assumed to be slow enough in the sense of combined dwell-time and average dwell-time, each individual mode is assumed to be stabilizable, and the data-rate is assumed to be large enough but finite. Extending the approach of reachable-set approximation and propagation from an earlier result on the disturbance-free case, we develop a communication and control strategy that achieves a variant of input-to-state stability with exponential decay. An estimate of the disturbance bound is introduced to counteract the unknown disturbance, and a novel algorithm is designed to properly adjust the estimate and recover the state when it escapes the range of quantization.

Robustness of Pecora-Carroll synchronization under communication constraints (with B. Andrievsky and A. L. Fradkov), Systems and Control Letters, vol. 111, pp. 27-33, Jan 2018.
↓Abstract ↑Abstract: We study synchronization of nonlinear systems with robustness to disturbances that arise when measurements sent from the master system to the slave system are affected by quantization and time sampling. Viewing the synchronization problem as an observer design problem, we invoke a recently developed theory of nonlinear observers robust to output measurement disturbances and formulate a sufficient condition for robust synchronization. The approach is illustrated by a detailed analysis of the Pecora-Carroll synchronization scheme for the Lorenz system, for which an explicit bound on the synchronization error depending on the quantizer range and sampling period is derived.

Control with minimal cost-per-symbol encoding and quasi-optimality of event-based encoders (with J. Pearson and J. P. Hespanha), IEEE Transactions on Automatic Control, vol. 62, no. 5, pp. 2286-2301, May 2017.
↓Abstract↑Abstract: We consider the problem of stabilizing a continuous-time linear time-invariant system subject to communication constraints. A noiseless finite-capacity communication channel connects the process sensors to the controller/actuator. The sensor's state measurements are encoded into symbols from a finite alphabet, transmitted through the channel, and decoded at the controller/actuator. We suppose that the transmission of each symbol costs one unit of communication resources, except for one special symbol in the alphabet that is "free" and effectively signals the absence of transmission. We present a necessary and sufficient condition for the existence of a stabilizing controller and encoder/decoder pair, which depends on the encoder's average bit-rate, its average resource consumption, and the unstable eigenvalues of the process. The paper concludes with an analysis of a simple emulation-based controller and event-based encoder/decoder pair that are easy to implement, stabilize the process, and have average bit-rate and resource consumption within a constant factor of the optimal bound. See also a more complete technical report.

Energy control of a pendulum with quantized feedback (with R. Seifullaev and A. L. Fradkov), Automatica, vol. 67, pp. 171-177, May 2016.
↓Abstract↑Abstract: The problem of controlling a nonlinear system to an invariant manifold using quantized state feedback is considered by the example of controlling the pendulum's energy. A feedback control law based on the speed gradient algorithm is chosen. The main result consisting in precisely characterizing allowed quantization error bounds and resulting energy deviation bounds is presented.

Adaptive control of passifiable linear systems with quantized measurements and bounded disturbances (with A. Selivanov and A. L. Fradkov), Systems and Control Letters, vol. 88, pp. 62-67, Feb 2016.
↓Abstract↑Abstract: We consider a linear uncertain system with an unknown bounded disturbance under a passification-based adaptive controller with quantized measurements. First, we derive conditions ensuring ultimate boundedness of the system. Then we develop a switching procedure for an adaptive controller with a dynamic quantizer that ensures convergence to a smaller set. The size of the limit set is defined by the disturbance bound. Finally, we demonstrate applicability of the proposed controller to polytopic-type uncertain systems and its efficiency by the example of a yaw angle control of a flying vehicle.

Nonlinear observers robust to measurement disturbances in an ISS sense (with H. Shim), IEEE Transactions on Automatic Control, vol. 61, no. 1, pp. 48-61, Jan 2016.
↓Abstract↑Abstract: This paper formulates and studies the concept of quasi-Disturbance-to-Error Stability (qDES) which characterizes robustness of a nonlinear observer to an output measurement disturbance. In essence, an observer is qDES if its error dynamics are input-to-state stable (ISS) with respect to the disturbance as long as the plant's input and state remain bounded. We develop Lyapunov-based sufficient conditions for checking the qDES property for both full-order and reduced-order observers. We use these conditions to show that several well-known observer designs yield qDES observers, while some others do not. Our results also enable the design of novel qDES observers, as we demonstrate with examples. When combined with a state feedback law robust to state estimation errors in the ISS sense, a qDES observer can be used to achieve output feedback control design with robustness to measurement disturbances. As an application of this idea, we treat a problem of stabilization by quantized output feedback.

Compensation of disturbances for MIMO systems with quantized output (with I. Furtat and A. L. Fradkov), Automatica, vol. 60, pp. 239-244, Oct 2015.
↓Abstract↑Abstract: The paper deals with the robust output feedback discrete control of continuous-time multi input multi output (MIMO) linear plants with arbitrary relative degree under parametric uncertainties and external bounded disturbances with quantized output signal. The parallel reference model (auxiliary loop) to the plant is used for extracting information about the uncertainties acting on the plant. The proposed algorithm guarantees that the output of the plant tracks the reference output with the required accuracy.

Lyapunov-based small-gain theorems for hybrid systems (with D. Nesic and A. R. Teel), IEEE Transactions on Automatic Control, vol. 59, no. 6, pp. 1395-1410, Jun 2014.
↓Abstract↑Abstract: Constructions of strong and weak Lyapunov functions are presented for a feedback connection of two hybrid systems satisfying certain Lyapunov stability assumptions and a small-gain condition. The constructed strong Lyapunov functions can be used to conclude input-to-state stability (ISS) of hybrid systems with inputs and global asymptotic stability (GAS) of hybrid systems without inputs. In the absence of inputs, we also construct weak Lyapunov functions nondecreasing along solutions and develop a LaSalle-type theorem providing a set of sufficient conditions under which such functions can be used to conclude GAS. In some situations, we show how average dwell time (ADT) and reverse average dwell time (RADT) "clocks" can be used to construct Lyapunov functions that satisfy the assumptions of our main results. The utility of these results is demonstrated for the "natural" decomposition of a hybrid system as a feedback connection of its continuous and discrete dynamics, and in several design-oriented contexts: networked control systems, event-triggered control, and quantized feedback control.

Finite data-rate feedback stabilization of switched and hybrid linear systems, Automatica, vol. 50, no. 2, pp. 409-420, Feb 2014.
↓Abstract↑Abstract: We study the problem of asymptotically stabilizing a switched linear control system using sampled and quantized measurements of its state. The switching is assumed to be slow enough in the sense of combined dwell time and average dwell time, each individual mode is assumed to be stabilizable, and the available data rate is assumed to be large enough. Our encoding and control strategy is rooted in the one proposed in our earlier work on non-switched systems, and in particular the data-rate bound used here is the data-rate bound from that earlier work maximized over the individual modes. The main technical step that enables the extension to switched systems concerns propagating over-approximations of reachable sets through sampling intervals, during which the switching signal is not known. Our primary focus is on systems with time-dependent switching (switched systems) but the setting of state-dependent switching (hybrid systems) is also discussed.

Supervisory control of uncertain systems with quantized information (with L. Vu), International Journal of Adaptive Control and Signal Processing (special issue on recent trends on the use of switching and mixing in adaptive control), vol. 26, no. 8, pp. 739-756, Aug 2012.
↓Abstract↑Abstract: This work addresses the problem of stabilizing uncertain systems with quantized outputs using the supervisory control framework, in which a finite family of candidate controllers are employed together with an estimator-based switching logic to select the active controller at every time. For static quantizers, we provide a relationship between the quantization range and the quantization error bound that guarantees closed-loop stability. Such a condition also implies a lower bound on the number of information bits needed to guarantee stability of a supervisory control scheme with quantized information. For dynamic quantizers that can vary the quantization parameters in real time, we show that the closed loop can be asymptotically stabilized, provided that additional conditions on the quantization range and the quantization error bound are satisfied

Input-to-state stabilizing controller for systems with coarse quantization (with Y. Sharon), IEEE Transactions on Automatic Control, vol. 57, no. 4, pp. 830-844, Apr 2012.
↓Abstract↑Abstract: We consider the problem of achieving input-to-state stability (ISS) with respect to external disturbances for control systems with quantized measurements. Quantizers considered in this paper take finitely many values and have adjustable "center" and "zoom" parameters. Both the full state feedback and the output feedback cases are considered. Similarly to previous techniques from the literature, our proposed controller switches repeatedly between "zooming out" and "zooming in." However, here we use two modes to implement the "zooming in" phases, which gives us the important benefit of using the minimal number of quantization regions. Our analysis is trajectory-based and utilizes a cascade structure of the closed-loop hybrid system. We further show that our method is robust to modeling errors in the plant dynamics using a specially adapted small-gain theorem. The main results are developed for linear systems, but we also discuss their extension to nonlinear systems under appropriate assumptions.

Rendezvous without coordinates (with J. Yu and S. LaValle), IEEE Transactions on Automatic Control, vol. 57, no. 2, pp. 421-434, Feb 2012.
↓Abstract↑Abstract: We study minimalism in sensing and control by considering a multi-agent system in which each agent moves like a Dubins car and has a limited sensor that reports only the presence of another agent within some sector of its windshield. Using a simple quantized control law with three values, each agent tracks another agent assigned to it by maintaining that agent within this windshield sector. We use Lyapunov analysis to show that by acting autonomously in this way, the agents will achieve rendezvous given a connected initial assignment graph and a merging assumption. We then proceed to show that, by making the quantized control law slightly stronger, a connected initial assignment graph is not required and the sensing model can be weakened further. A distinguishing feature of our approach is that it does not involve any estimation procedure aimed at reconstructing coordinate information. Our scenario thus appears to be the first example in which an interesting task is performed with extremely coarse sensing and control, and without state estimation. The system was implemented in computer simulation, accessible through the Web, of which the results are presented in the paper.

A unified framework for design and analysis of networked and quantized control systems (with D. Nesic), IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 732-747, Apr 2009.
↓Abstract↑Abstract: We generalize and unify a range of recent results in quantized control systems (QCS) and networked control systems (NCS) literature and provide a unified framework for controller design for control systems with quantization and time scheduling via an emulation-like approach. A crucial step in our proofs is finding an appropriate Lyapunov function for the quantization/time-scheduling protocol which verifies its uniform global exponential stability (UGES). We construct Lyapunov functions for several representative protocols that are commonly found in the literature, as well as some new protocols not considered previously. Our approach is flexible and amenable to further extensions which are briefly discussed.

Nonlinear control with limited information, Communications in Information and Systems (Roger Brockett Legacy special issue), vol. 9, pp. 41-58, 2009.
↓Abstract↑Abstract: This paper discusses several recent results by the author and collaborators, which are united by the common goal of making nonlinear control theory more robust to imperfect information. These results are also united by common technical tools, centering around input-to-state stability (ISS), small-gain theorems, Lyapunov functions, and hybrid systems. The goal of this paper is to present an overview of these results which highlights their unifying features and which is more accessible to a general audience than the original technical articles.

Input-to-state stabilization of linear systems with quantized state measurements (with D. Nesic), IEEE Transactions on Automatic Control, vol. 52, no. 5, pp. 767-781, May 2007.
↓Abstract↑Abstract: We consider the problem of achieving input-to-state stability (ISS) with respect to external disturbances for control systems with linear dynamics and quantized state measurements. Quantizers considered in this paper take finitely many values and have an adjustable "zoom" parameter. Building on an approach applied previously to systems with no disturbances, we develop a control methodology that counteracts an unknown disturbance by switching repeatedly between "zooming out" and "zooming in". Two specific control strategies that yield ISS are presented. The first one is implemented in continuous time and analyzed with the help of a Lyapunov function, similarly to earlier work. The second strategy incorporates time sampling, and its analysis is novel in that it is completely trajectory-based and utilizes a cascade structure of the closed-loop hybrid system. We discover that in the presence of disturbances, time-sampling implementation requires an additional modification which has not been considered in previous work.

Quantization, time delays, and nonlinear stabilization, IEEE Transactions on Automatic Control, vol. 51, no. 7, pp. 1190-1195, Jul 2006.
↓Abstract↑Abstract: The purpose of this note is to demonstrate that a unified study of quantization and delay effects in nonlinear control systems is possible by merging the quantized feedback control methodology recently developed by the author and the small-gain approach to the analysis of functional differential equations with disturbances proposed earlier by Teel. We prove that under the action of a robustly stabilizing feedback controller in the presence of quantization and time delays satisfying suitable conditions, solutions of the closed-loop system starting in a given region remain bounded and eventually enter a smaller region. We present several versions of this result and show how it enables global asymptotic stabilization via a dynamic quantization strategy.

Quantized control via locational optimization (with F. Bullo), IEEE Transactions on Automatic Control, vol. 51, no. 1, pp. 2-13, Jan 2006.
↓Abstract↑Abstract: This paper studies state quantization schemes for feedback stabilization of control systems with limited information. The focus is on designing the least destabilizing quantizer subject to a given information constraint. We explore several ways of measuring the destabilizing effect of a quantizer on the closed-loop system, including (but not limited to) the worst-case quantization error. In each case, we show how quantizer design can be naturally reduced to a version of the so-called multicenter problem from locational optimization. Algorithms for obtaining solutions to such problems, all in terms of suitable Voronoi quantizers, are discussed. In particular, an iterative solver is developed for a novel weighted multicenter problem which most accurately represents the least destabilizing quantizer design. A simulation study is also presented.

Stabilization of nonlinear systems with limited information feedback (with J. P. Hespanha), IEEE Transactions on Automatic Control, vol. 50, no. 6, pp. 910-915, Jun 2005.
↓Abstract↑Abstract: This paper is concerned with the problem of stabilizing a nonlinear continuous-time system by using sampled encoded measurements of the state. We demonstrate that global asymptotic stabilization is possible if a suitable relationship holds between the number of values taken by the encoder, the sampling period, and a system parameter, provided that a feedback law achieving input-to-state stability with respect to measurement errors can be found. The issue of relaxing the latter condition is also discussed.

Hybrid feedback stabilization of systems with quantized signals, Automatica, vol. 39, no. 9, pp. 1543-1554, Sep 2003.
↓Abstract↑Abstract: This paper is concerned with global asymptotic stabilization of continuous-time systems subject to quantization. A hybrid control strategy originating in earlier work (R. W. Brockett and D. Liberzon, Quantized feedback stabilization of linear systems, IEEE Trans. Automat. Control, 45:1279-1289, 2000) relies on the possibility of making discrete on-line adjustments of quantizer parameters. We explore this method here for general nonlinear systems with general types of quantizers affecting the state of the system, the measured output, or the control input. The analysis involves merging tools from Lyapunov stability, hybrid systems, and input-to-state stability.

On stabilization of linear systems with limited information, IEEE Transactions on Automatic Control, vol. 48, no. 2, pp. 304-307, Feb 2003.
↓Abstract↑Abstract: We consider the problem of stabilizing a linear time-invariant system using sampled encoded measurements of its state or output. We derive a relationship between the number of values taken by the encoder and the norm of the transition matrix of the open-loop system over one sampling period, which guarantees that global asymptotic stabilization can be achieved. A coding scheme and a stabilizing control strategy are described explicitly.

Quantized feedback stabilization of linear systems (with R. W. Brockett), IEEE Transactions on Automatic Control, vol. 45, no. 7, pp. 1279-1289, Jul 2000.
↓Abstract↑Abstract: This paper addresses feedback stabilization problems for linear time-invariant control systems with saturating quantized measurements. We propose a new control design methodology, which relies on the possibility of changing the sensitivity of the quantizer while the system evolves. The equation that describes the evolution of the sensitivity with time (discrete rather than continuous in most cases) is interconnected with the given system (either continuous or discrete). When applied to systems that are stabilizable by linear time-invariant feedback, this approach yields global asymptotic stability. See also the slides of the talk given at the 4th SIAM Conference on Control and its Applications, Jacksonville, FL, May 1998.

Book chapters:

Input-to-state stabilization with quantized output feedback (with Y. Sharon), in Proceedings of the 11th International Workshop on Hybrid Systems: Computation and Control, St. Louis, MO, Apr 2008, Lecture Notes in Computer Science, vol. 4981 (M. Egerstedt and B. Mishra, Eds.), Springer, Berlin, pp. 500-513.
↓Abstract↑Abstract: We study control systems where the output subspace is covered by a finite set of quantization regions, and the only information available to a controller is which of the quantization regions currently contains the system's output. We assume the dimension of the output subspace is strictly less than the dimension of the state space. The number of quantization regions can be as small as 3 per dimension of the output subspace. We show how to design a controller that stabilizes such a system, and makes the system robust to an external unknown disturbance in the sense that the closed-loop system has the Input-to-State Stability property. No information about the disturbance is required to design the controller. Achieving the ISS property for continuous- time systems with quantized measurements requires a hybrid approach, and indeed our controller consists of a dynamic, discrete-time observer, a continuous-time state-feedback stabilizer, and a switching logic that switches between several modes of operation. Except for some properties that the observer and the stabilizer must possess, our approach is general and not restricted to a specic observer or stabilizer. Examples of specic observers that possess these properties are included.

Stability analysis of hybrid systems via small-gain theorems (with D. Nesic), in Proceedings of the Ninth International Workshop on Hybrid Systems: Computation and Control, Santa Barbara, CA, Mar 2006, Lecture Notes in Computer Science, vol. 3927 (J. Hespanha and A. Tiwari, Eds.), Springer, Berlin, pp. 421-435.
↓Abstract↑Abstract: We present a general approach to analyzing stability of hybrid systems, based on input-to-state stability (ISS) and small-gain theorems. We demonstrate that the ISS small-gain analysis framework is very naturally applicable in the context of hybrid systems. Novel Lyapunov-based and LaSalle-based small-gain theorems for hybrid systems are presented. The reader does not need to be familiar with ISS or small-gain theorems to be able to follow the paper.

On quantization and delay effects in nonlinear control systems, in Proceedings of the Workshop on Networked Embedded Sensing and Control, University of Notre Dame, South Bend, IN, Oct 2005, Lecture Notes in Control and Information Sciences, vol. 331 (P. J. Antsaklis and P. Tabuada, Eds.), Springer, Berlin, pp. 219-229.
↓Abstract↑Abstract: The purpose of this paper is to demonstrate that a unified study of quantization and delay effects in nonlinear control systems is possible by merging the quantized feedback control methodology recently developed by the author and the small-gain approach to the analysis of functional differential equations with disturbances proposed earlier by Teel. We prove that under the action of a robustly stabilizing feedback controller in the presence of quantization and sufficiently small delays, solutions of the closed-loop system starting in a given region remain bounded and eventually enter a smaller region. We present several versions of this result and show how it enables global asymptotic stabilization via a dynamic quantization strategy. See also the slides of the talk.

Nonlinear stabilization by hybrid quantized feedback, in Proceedings of the Third International Workshop on Hybrid Systems: Computation and Control, Pittsburgh, PA, Mar 2000, Lecture Notes in Computer Science, vol. 1790 (N. Lynch and B. H. Krogh, Eds.), Springer, Berlin, pp. 243-257.
↓Abstract↑Abstract: This paper is concerned with asymptotic stabilization of continuous-time control systems by means of quantized feedback. For linear systems, a hybrid control strategy for dealing with this problem was recently proposed by Roger Brockett and the author. The solution is based on making discrete on-line adjustments to the sensitivity of the quantizer. In the present paper we extend this method to a class of nonlinear systems. See also the slides of the talk.

Conferences:

Quantizer design for linear switched systems with minimal data-rate (with G. S. Vicinansa), in Proceedings of the 24th ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2021), Nashville, TN (online), May 2021.
↓Abstract ↑Abstract: In this paper, we present a quantization scheme that reconstructs the state of linear switched systems with a prescribed exponential decaying rate for the state estimation error. We show how to use the Lyapunov exponents and a geometric object called Oseledets' filtration to design such a quantization scheme. Then, we prove that this algorithm works at an average data-rate close to the estimation entropy of the given system. Furthermore, we can choose the average data-rate to be arbitrarily close to the estimation entropy whenever the linear switched system has the so-called regularity property. We show that, under the regularity assumption, the quantization scheme is completely causal in the sense that it only depends on information that is available at the current time instant. Finally, we present simulation results for a Markov Jump Linear System, a class of systems for which the realizations are known to be regular with probability 1.

How to park a car blindfolded (with G. S. Vicinansa), in Proceedings of the 8th IFAC Workshop on Distributed Estimation and Control in Networked Systems (NecSys), Chicago, IL, Sep 2019, pp. 211-216.
↓Abstract ↑Abstract: In this paper, we present an algorithm that achieves global asymptotic stability for a broad class of driftless nonholonomic systems using limited data rate. Moreover, we prove in a constructive way that the minimum data-rate for this class of systems is zero, hence providing a data-rate theorem, which was an open problem in the literature [Li and Baillieul (2004)]. Finally, we present the Dubin's car as an example of system in such a class. See also a more complete version and the poster.

Robust observers and Pecora-Carroll synchronization with limited information (with B. Andrievsky and A. L. Fradkov), in Proceedings of the 56th IEEE Conference on Decision and Control, Melbourne, Australia, Dec 2017, pp. 2416-2421.
↓Abstract ↑Abstract: We study synchronization of nonlinear systems with robustness to disturbances that arise when measurements sent from the master system to the slave system are affected by quantization and time sampling. Viewing the synchronization problem as an observer design problem, we invoke a recently developed theory of nonlinear observers robust to output measurement disturbances and formulate a sufficient condition for robust synchronization. The approach is illustrated by a detailed analysis of the Pecora-Carroll synchronization scheme for the Lorenz system, for which an explicit bound on the synchronization error depending on the quantizer range and sampling period is derived. See also the slides of the talk.

Entropy notions for state estimation and model detection with finite-data-rate measurements (with S. Mitra), in Proceedings of the 55th IEEE Conference on Decision and Control, Las Vegas, NV, Dec 2016, pp. 7335-7340.
↓Abstract ↑Abstract: We study a notion of estimation entropy for continuous-time nonlinear systems, formulated in terms of the number of system trajectories that approximate all other trajectories up to an exponentially decaying error. We also consider an alternative definition of estimation entropy which uses approximating functions that are not necessarily trajectories of the system, and show that the two entropy notions are equivalent. We establish an upper bound on the estimation entropy in terms of the sum of the desired convergence rate and an upper bound on the matrix measure of the Jacobian, multiplied by the system dimension. We describe an iterative procedure that uses quantized and sampled state measurements to generate state estimates that converge to the true state at the desired exponential rate. The average bit rate utilized by this procedure matches the derived upper bound on the estimation entropy. We also show that no other algorithm of this type can perform the same estimation task with bit rates lower than the estimation entropy. Finally, we discuss an application of the estimation procedure in determining, from the quantized state measurements, which of two competing models of a dynamical system is the true model. We show that under a mild assumption of exponential separation of the candidate models, detection always happens in finite time. See also the slides of the talk.

Finite data-rate stabilization of a switched linear system with unknown disturbance (with G. Yang), in Proceedings of the 10th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2016), Monterey, CA, Aug 2016, pp. 1103-1108.
↓Abstract ↑Abstract: We study the stabilization of a switched linear system with unknown disturbance using sampled and quantized state feedback. The switching is slow in the sense of combined dwell-time and average dwell-time, while the active mode is unknown except at sampling times. Each mode of the switched system is stabilizable, and the disturbance admits an unknown bound. A communication and control strategy is designed to achieve practical stability and exponential convergence w.r.t. the initial state with a nonlinear gain on the disturbance, provided the data-rate meets given lower bounds. Compared with previous results, a more involved algorithm is developed to handle effects of the unknown disturbance based on employing an iteratively updated estimate of the disturbance bound and expanding the over-approximations of reachable sets over sampling intervals from the case without disturbance.

Entropy and minimal data rates for state estimation and model detection (with S. Mitra), in Proceedings of the 19th ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2016), Vienna, Austria, Apr 2016, pp. 247-256.
↓Abstract ↑Abstract: We investigate the problem of constructing exponentially converging estimates of the state of a continuous-time system from state measurements transmitted via a limited- data-rate communication channel, so that only quantized and sampled measurements of continuous signals are available to the estimator. Following prior work on topological entropy of dynamical systems, we introduce a notion of estimation entropy which captures this data rate in terms of the number of system trajectories that approximate all other trajectories with desired accuracy. We also propose a novel alternative definition of estimation entropy which uses approximating functions that are not necessarily trajectories of the system. We show that the two entropy notions are actually equivalent. We establish an upper bound for the estimation entropy in terms of the sum of the system's Lipschitz constant and the desired convergence rate, multiplied by the system dimension. We propose an iterative procedure that uses quantized and sampled state measurements to generate state estimates that converge to the true state at the desired exponential rate. The average bit rate utilized by this procedure matches the derived upper bound on the estimation entropy. We also show that no other estimator (based on iterative quantized measurements) can perform the same estimation task with bit rates lower than the estimation entropy. Finally, we develop an application of the estimation procedure in determining, from the quantized state measurements, which of two competing models of a dynamical system is the true model. We show that under a mild assumption of exponential separation of the candidate models, detection is always possible in finite time. Our numerical experiments with randomly generated affine dynamical systems suggest that in practice the algorithm always works.

Quasi-optimality of event-based encoders (with J. Pearson and J. P. Hespanha), in Proceedings of the 54th IEEE Conference on Decision and Control, Osaka, Japan, Dec 2015, pp. 4800-4805.
↓Abstract ↑Abstract: We present an event-triggered controller for stabilizing a continuous-time linear time-invariant system subject to communication constraints. We model the communication constraints as a noiseless finite-capacity communication channel between the process sensors and the controller/actuator. An encoder converts the process state into symbols to send across the channel to the controller/actuator, which converts the symbols into a state estimate to be used in a simple emulation-based state-feedback control law. We derive a sufficient condition for this scheme to stabilize the process. The condition depends on the encoder's average bit-rate, its average consumption of communication resources, and the eigenvalues of the process. The proposed encoding scheme is order-optimal in the sense that its stability condition is within a constant factor of the optimal bound from previous work.

Stabilizing a switched linear system with disturbance by sampled-data quantized feedback (with G. Yang), in Proceedings of the American Control Conference, Chicago, IL, Jul 2015, pp. 2193-2198.
↓Abstract ↑Abstract: We study the problem of stabilizing a switched linear system with disturbance using sampled and quantized measurements of its state. The switching is assumed to be slow in the sense of combined dwell-time and average dwell-time, while the active mode is unknown except at sampling times. Each mode of the switched linear system is assumed to be stabilizable, and the magnitude of the disturbance is constrained by a known bound. A communication and control strategy is designed to guarantee bounded-input-bounded-state (BIBS) stability of the switched linear system and an exponential convergence rate with respect to the initial state, providing the data rate satisfies certain lower bounds. Such lower bounds are established by expanding the over-approximation bounds of reachable sets over sampling intervals derived in a previous paper to accommodate effects of the disturbance.

Control with minimum communication cost per symbol (with J. Pearson and J. P. Hespanha), in Proceedings of the 53rd IEEE Conference on Decision and Control, Los Angeles, CA, Dec 2014, pp. 6050-6055.
↓Abstract ↑Abstract: We address the problem of stabilizing a continuous-time linear time-invariant process under communication constraints. We assume that the sensor that measures the state is connected to the actuator through a finite capacity communication channel over which an encoder at the sensor sends symbols from a finite alphabet to a decoder at the actuator. We consider a situation where one symbol from the alphabet consumes no communication resources, whereas each of the others consumes one unit of communication resources to transmit. This paper explores how the imposition of limits on an encoder's bit-rate and average resource consumption affect the encoder/decoder/controller's ability to keep the process bounded. The main result is a necessary and sufficient condition for a bounding encoder/decoder/controller which depends on the encoder's bit-rate, its average resource consumption, and the unstable eigenvalues of the process.

Cyclic pursuit without coordinates: convergence to regular polygon formations (with M. Arnold and Y. Baryshnikov), in Proceedings of the 53rd IEEE Conference on Decision and Control, Los Angeles, CA, Dec 2014, pp. 6191-6196.
↓Abstract ↑Abstract: We study a multi-agent cyclic pursuit model where each of the identical agents moves like a Dubins car and maintains a fixed heading angle with respect to the next agent. We establish that stationary shapes for this system are regular polygons. We derive a sufficient condition for local convergence to such regular polygon formations, which takes the form of an inequality connecting the angles of the regular polygon with the heading angle of the agents. A block-circulant structure of the system's linearization matrix in suitable coordinates facilitates and elucidates our analysis. Our results are complementary to the conditions for rendezvous obtained in earlier work [Yu et al., IEEE Trans. Autom. Contr., Feb. 2012]. See also the slides of the talk.

Passification-based adaptive control with quantized measurements (with A. Selivanov and A. L. Fradkov), in Proceedings of the 19th IFAC World Congress, Cape Town, South Africa, Aug 2014.
↓Abstract ↑Abstract: We propose and analyze passification-based adaptive controller for linear uncertain systems with quantized measurements. Since the effect of the quantization error is similar to the effect of a disturbance, the adaptation law with sigma-modification is used. To ensure convergence to a smaller set, the parameters of the adaptation law are being switched during the evolution of the system and a dynamic quantizer is used. It is proved that if the quantization error is small enough then the proposed controller ensures convergence of the state of a hyper-minimum-phase system to an arbitrarily small vicinity of the origin. Applicability of the proposed controller to polytopic-type uncertain systems and its efficiency is demonstrated by the example of yaw angle control of a flying vehicle.

Robust control with compensation of disturbances for systems with quantized output (with I. Furtat and A. L. Fradkov), in Proceedings of the 19th IFAC World Congress, Cape Town, South Africa, Aug 2014.
↓Abstract ↑Abstract: The paper deals with the robust output feedback discrete control of continuous-time linear plants with arbitrary relative degree under parametric uncertainties and external bounded disturbances with quantized output signal. The parallel reference model (auxiliary loop) to the plant is used for obtaining the uncertainties acting on the plant. The proposed algorithm guarantees that the output of the plant tracks the reference output with the required accuracy.

Limited-information control of hybrid systems via reachable set propagation, in Proceedings of the 16th ACM International Conference on Hybrid Systems: Computation and Control (HSCC 2013), Philadelphia, PA, Apr 2013, pp. 11-19.
↓Abstract↑Abstract: This paper deals with control of hybrid systems based on limited information about their state. Specifically, measurements being passed from the system to the controller are sampled and quantized, resulting in finite data-rate communication. The main ingredient of our solution to this control problem is a novel method for propagating over-approximations of reachable sets for hybrid systems through sampling intervals, during which the discrete mode is unknown. In addition, slow-switching conditions of the (average) dwell-time type and multiple Lyapunov functions play a central role in the analysis. See also the slides of the talk.

Small-gain theorems of LaSalle type for hybrid systems (with D. Nesic and A. R. Teel), in Proceedings of the 51st IEEE Conference on Decision and Control, Maui, HI, Dec 2012, pp. 6825-6830.
↓Abstract↑Abstract: We study stability of hybrid systems described as feedback interconnections of smaller subsystems, within a Lyapunov-based ISS small-gain analysis framework. We focus on constructing a weak (nonstrictly decreasing) Lyapunov function for the overall hybrid system from weak ISS-Lyapunov functions for the subsystems in the interconnection. Asymptotic stability of the hybrid system is then concluded by applying results of LaSalle type. The utility of this approach is illustrated on feedback systems arising in event-triggered control and quantized control. See also the slides of the talk.

Stabilizing a switched linear system by sampled-data quantized feedback, in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, Orlando, FL, Dec 2011, pp. 8321-8326.
↓Abstract↑Abstract: We study the problem of asymptotically stabilizing a switched linear control system using sampled and quantized measurements of its state. The switching is assumed to be slow enough in the sense of combined dwell time and average dwell time, each individual mode is assumed to be stabilizable, and the available data rate is assumed to be large enough. Our encoding and control strategy is rooted in the one proposed in our earlier work on non-switched systems, and in particular the data-rate bound used here is the data-rate bound from that earlier work maximized over the individual modes. The main technical step that enables the extension to switched systems concerns propagating over-approximations of reachable sets through sampling intervals, during which the switching signal is unknown. See also the slides of the talk.

Adaptive control using quantized measurements with application to vision-only landing control (with Y. Sharon and Y. Ma), in Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, GA, Dec 2010, pp. 2511-2516.
↓Abstract↑Abstract: We consider a class of control systems where the plant model is unknown and the observed states are only measured through a quantizer. In order to estimate the plant model for the controller, we follow up on an approach where an optimization is taking place over both the plant model parameters and the state of the plant. We propose a computationally efficient algorithm for solving the optimization problem, and prove that the algorithm converges using tools from convex and non-smooth analysis. We demonstrate both the importance of this class of control systems and our method of solution using the following application: having a fixed wing airplane follow a desired glide slope on approach to landing. The only available feedback is from a camera mounted at the front of the airplane and looking at a runway of unknown dimensions. The quantization is due to the finite resolution of the camera. Using this application we also compare our method to the basic method which is prevalent in the literature, where the optimization is only taking place over the plant model parameters.

Stabilization of linear systems under coarse quantization and time delays (with Y. Sharon), in Proceedings of the 2nd IFAC Workshop on Distributed Estimation and Control in Networked Systems (NecSys), Annecy, France, Sep 2010, pp. 31-36.
↓Abstract↑Abstract: We consider the problem of stabilizing a control system using a coarse state quantizer in the presence of time delays. We assume the quantizer has an adjustable "center'' and "zoom'' parameters, and employ an alternating "zoom out''/"zoom in'' mechanism in order to achieve a large region of attraction while having the system converge to a small region around the origin. This mechanism is adopted from our previous work where delays were not considered. Here we show that the control system, using the same mechanism and without making any changes in order to accommodate delays explicitly, remains stable under small delays. The main tool we use to prove the result is the nonlinear small-gain theorem. See also the poster.

Quasi-ISS reduced-order observers and quantized output feedback (with H. Shim and J.-S. Kim), in Proceedings of the 48th IEEE Conference on Decision and Control, Shanghai, China, Dec 2009, pp. 6680-6685.
↓Abstract↑Abstract: We formulate and study the problem of designing nonlinear observers whose error dynamics are input-to-state stable (ISS) with respect to additive output disturbances as long as the plant's input and state remain bounded. We present a reduced-order observer design which achieves this quasi-ISS property when there exists a suitable state-independent error Lyapunov function. We show that our construction applies to several classes of nonlinear systems previously studied in the observer design literature. As an application of this robust observer concept, we prove that quantized output feedback stabilization is achievable when the system possesses a quasi-ISS reduced-order observer and a state feedback law that yields ISS with respect to measurement errors. A worked example is included. See also the slides of the talk.

Rendezvous without coordinates (with J. Yu and S. LaValle), in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec 2008, pp. 1803-1808.
↓Abstract↑Abstract: We study minimalism in sensing and control by considering a multi-agent system in which each agent moves like a Dubins car and has a limited sensor that reports only the presence of another agent within some sector of its windshield. Using a very simple quantized control law with three values, each agent tracks another agent assigned to it by maintaining that agent within this windshield sector. We use Lyapunov analysis to show that by acting autonomously in this way, the agents will achieve rendezvous if the initial assignment graph is connected. A distinguishing feature of our approach is that it does not involve any estimation procedure aimed at reconstructing coordinate information. Our scenario thus appears to be the first example in which an interesting task is performed with extremely coarse sensing and control, and without state estimation. The system was implemented in computer simulation, accessible through the Web, of which the results are presented in the paper. See also a more complete version.

Stabilizing uncertain systems with dynamic quantization (with L. Vu), in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec 2008, pp. 4681-4686.
↓Abstract↑Abstract: We consider the problem of stabilizing uncertain linear systems with quantization. The plant uncertainty is dealt with by the supervisory adaptive control framework, which employs switching among a finite family of candidate controllers. For a static quantizer, we quantify a relationship between the quantization range and the quantization error bound that guarantees closed loop stability. Using a dynamic quantizer which can vary the quantization parameters in real time, we show that the closed loop is asymptotically stabilized provided a certain condition on the quantization range and the quantization error bound is satisfied. Our results extend previous results on stabilization of known systems with quantization to the case of uncertain systems. See also a more complete version.

Observer-based quantized output feedback control of nonlinear systems, in Proceedings of the 17th IFAC World Congress, Seoul, Korea, Jul 2008, pp. 8039-8043.
↓Abstract↑Abstract: This paper addresses the problem of stabilizing a nonlinear system by means of quantized output feedback. A conceptual framework is presented in which the control input is generated by an observer-based feedback controller acting on quantized output measurements. A stabilization result is established under the assumption that this observer-based controller possesses robustness with respect to output measurement errors in an input-to-state stability (ISS) sense. Designing such observers and controllers is a largely open problem, some partial results on which are discussed. The main goal of the paper is to encourage further work on this important topic. See also the slides of the talk.

Input-to-state stabilization with minimum number of quantization regions (with Y. Sharon), in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, Dec 2007, pp. 20-25.
↓Abstract↑Abstract: We study control systems where the state measurements are quantized and time-sampled, and an unknown disturbance is being applied. We present a dynamic quantization scheme that switches between three modes of operation. We show that by using this scheme with a continuous static feedback controller we achieve a closed-loop system which has the Input-to-State Stability property (ISS). Our design does not use any characterization of the disturbance; as long as the disturbance is bounded the system will remain stable. We show that three quantization regions per dimension is sufficient to achieve the ISS property, and furthermore we show that the ISS property is achievable using a data rate that is arbitrarily close to the minimum required data rate when no disturbance is applied.

A unified approach to controller design for systems with quantization and time scheduling (with D. Nesic), in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, Dec 2007, pp. 3939-3944.
↓Abstract↑Abstract: We generalize and unify a range of recent results in quantized control systems (QCS) and networked control systems (NCS) literature and provide a unified framework for controller design for control systems with quantization and time scheduling via an emulation-like approach. A crucial step in our approach is finding an appropriate Lyapunov function for the quantization/time-scheduling protocol which verifies its uniform global exponential stability (UGES). We construct Lyapunov functions for several representative protocols that are commonly found in the literature, as well as some new protocols not considered previously.

Observer-based quantized output feedback control of nonlinear systems, in Proceedings of the 15th Mediterranean Conference on Control and Automation, Athens, Greece, Jun 2007.
↓Abstract↑Abstract: This paper addresses the problem of stabilizing a nonlinear system by means of quantized output feedback. A framework is presented in which the control input is generated by an observer-based feedback controller acting on quantized output measurements. A stabilization result is established under the assumption that this observer-based controller possesses robustness with respect to output measurement errors in an input-to-state stability (ISS) sense. Designing such observers and controllers is a largely open problem, some partial results on which are discussed. The main goal of the paper is to encourage further work on this important topic. See also the slides of the talk.

Input-to-state stabilization of linear systems with quantized feedback (with D. Nesic), in Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, Dec 2005, pp. 8197-8202.
↓Abstract↑Abstract: We consider the problem of achieving input-to-state stability (ISS) with respect to external disturbances for control systems with linear dynamics and quantized state measurements. Quantizers considered in this paper take finitely many values and have an adjustable "zoom" parameter. Extending an approach developed previously for systems with no disturbances, we present a control methodology that counteracts an unknown disturbance by switching repeatedly between "zooming in" and "zooming out". Two specific control strategies that yield ISS are described. The first one is implemented in continuous time and analyzed with the help of a Lyapunov function, similarly to earlier work. The second strategy incorporates time sampling, and its analysis is novel in that it is completely trajectory-based and utilizes a cascade structure of the closed-loop hybrid system. We discover that in the presence of disturbances, time-sampling implementation requires an additional modification which has not been considered in previous work. See also a more complete version.

A small-gain approach to stability analysis of hybrid systems (with D. Nesic), in Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, Dec 2005, pp. 5409-5414.
↓Abstract↑Abstract: We propose to use ISS small-gain theorems to analyze stability of hybrid systems. We demonstrate that the small-gain analysis framework is very naturally and generally applicable in the context of hybrid systems, and thus has a potential to be useful in many applications. The main idea is illustrated on specific problems in the context of control with limited information, where it is shown to provide novel interpretations, powerful extensions, and a more unified treatment of several previously available results. The reader does not need to be familiar with ISS or small-gain theorems to be able to follow the paper. See also the slides of the talk.

Stabilizing a nonlinear system with limited information feedback, in Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, Dec 2003, pp. 182-186.
↓Abstract↑Abstract: This paper is concerned with the problem of stabilizing a nonlinear continuous-time system by using sampled encoded measurements of the state. We demonstrate that global asymptotic stabilization is possible if a suitable relationship holds between the number of values taken by the encoder, the sampling period, and a system parameter, provided that a feedback law achieving input-to-state stability with respect to measurement errors can be found. Also presented at the 1st International Symposium on Control, Communications and Signal Processing, Hammamet, Tunisia, Mar 2004 (click here for extended abstract).
See also the slides of the talk.

On quantized control and geometric optimization (with F. Bullo), in Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HI, Dec 2003, pp. 2567-2572.
↓Abstract↑Abstract: This paper studies state quantization schemes for feedback stabilization of linear control systems with limited information. The focus is on designing the least destabilizing quantizer subject to a given information constraint. We explore several ways of measuring the destabilizing effect of a quantizer on the closed-loop system, including (but not limited to) the worst-case quantization error. In each case, we show how quantizer design can be naturally reduced to a version of the so-called multicenter problem from locational optimization. Algorithms for obtaining solutions to such problems, all in terms of suitable Voronoi quantizers, are discussed. In particular, an iterative solver is developed for a novel weighted multicenter problem which most accurately represents the least destabilizing quantizer design.See also the slides of the talk.

A note on stabilization of linear systems using coding and limited communication, in Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, Dec 2002, pp. 836-841.
↓Abstract↑Abstract: We consider the problem of stabilizing a linear time-invariant system using sampled encoded measurements of its state or output. We derive a relationship between the number of values taken by the encoder and the norm of the transition matrix of the open-loop system over one sampling period, which guarantees that global asymptotic stabilization can be achieved. A coding scheme and a stabilizing control strategy are described explicitly. See also the slides of the talk.

Stabilization by quantized state or output feedback: a hybrid control approach, in Proceedings of the 15th IFAC World Congress, Barcelona, Spain, Jul 2002 (IFAC Young Author Prize paper).
↓Abstract↑Abstract: This paper deals with global asymptotic stabilization of continuous-time systems with quantized signals. A hybrid control strategy originating in earlier work relies on the possibility of making discrete on-line adjustments of quantizer parameters. We explore this method here for general nonlinear systems with general types of quantizers affecting the state of the system or the measured output. See also the slides of the talk.

A hybrid control framework for systems with quantization, in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, Dec 2001, pp. 1217-1222.
↓Abstract↑Abstract: This paper is concerned with global asymptotic stabilization of systems subject to quantization. A hybrid control strategy originating in earlier work relies on the possibility of making discrete on-line adjustments of quantizer parameters. We explore this method here for general nonlinear systems with general types of quantizers affecting the state of the system or the control input. The analysis involves merging tools from Lyapunov stability, hybrid systems, and input-to-state stability. See also the slides of the talk.

Logic-based switching control of uncertain systems:

Adaptive control of passifiable linear systems with quantized measurements and bounded disturbances (with A. Selivanov and A. L. Fradkov), Systems and Control Letters, vol. 88, pp. 62-67, Feb 2016.
↓Abstract↑Abstract: We consider a linear uncertain system with an unknown bounded disturbance under a passification-based adaptive controller with quantized measurements. First, we derive conditions ensuring ultimate boundedness of the system. Then we develop a switching procedure for an adaptive controller with a dynamic quantizer that ensures convergence to a smaller set. The size of the limit set is defined by the disturbance bound. Finally, we demonstrate applicability of the proposed controller to polytopic-type uncertain systems and its efficiency by the example of a yaw angle control of a flying vehicle.

The bang-bang funnel controller for uncertain nonlinear systems with arbitrary relative degree (with S. Trenn), IEEE Transactions on Automatic Control, vol. 58, no. 12, pp. 3126-3141, Dec 2013.
↓Abstract↑Abstract: The paper considers output tracking control of uncertain nonlinear systems with arbitrary known relative degree and known sign of the high frequency gain. The tracking objective is formulated in terms of a time-varying bound---a funnel---around a given reference signal. The proposed controller is bang-bang with two control values. The controller switching logic handles arbitrarily high relative degree in an inductive manner with the help of auxiliary derivative funnels. We formulate a set of feasibility assumptions under which the controller maintains the tracking error within the funnel. Furthermore, we prove that under mild additional assumptions the considered system class satisfies these feasibility assumptions if the selected control values are sufficiently large in magnitude. Finally, we study the effect of time delays in the feedback loop and we are able to show that also in this case the proposed bang-bang funnel controller works provided the slightly adjusted feasibility assumptions are satisfied.

Supervisory control of uncertain systems with quantized information (with L. Vu), International Journal of Adaptive Control and Signal Processing (special issue on recent trends on the use of switching and mixing in adaptive control), vol. 26, no. 8, pp. 739-756, Aug 2012.
↓Abstract↑Abstract: This work addresses the problem of stabilizing uncertain systems with quantized outputs using the supervisory control framework, in which a finite family of candidate controllers are employed together with an estimator-based switching logic to select the active controller at every time. For static quantizers, we provide a relationship between the quantization range and the quantization error bound that guarantees closed-loop stability. Such a condition also implies a lower bound on the number of information bits needed to guarantee stability of a supervisory control scheme with quantized information. For dynamic quantizers that can vary the quantization parameters in real time, we show that the closed loop can be asymptotically stabilized, provided that additional conditions on the quantization range and the quantization error bound are satisfied

Supervisory control of uncertain linear time-varying systems (with L. Vu), IEEE Transactions on Automatic Control, vol. 56, no. 1, pp. 27-42, Jan 2011.
↓Abstract↑Abstract: We consider the problem of adaptively stabilizing linear plants with unknown time-varying parameters in the presence of noise, disturbances, and unmodeled dynamics using the supervisory control framework, which employs multiple candidate controllers and an estimator based switching logic to select the active controller at every time. Time-varying uncertain linear plants can be stabilized by supervisory control, provided that the plant's parameter varies slowly enough in terms of mixed dwell-time switching and average dwell-time switching, the noise and disturbances are bounded and small enough in terms of the L-infinity norm, and the unmodeled dynamics are small enough in the input-to-state stability sense. This work extends previously reported works on supervisory control of linear time-invariant systems with constant unknown parameters to the case of linear time-varying uncertain systems. A numerical example is included, and limitations of the approach are discussed.

Input-to-state stability of switched systems and switching adaptive control (with L. Vu and D. Chatterjee), Automatica, vol. 43, no. 4, pp. 639-646, Apr 2007.
↓Abstract↑Abstract: In this paper we prove that a switched nonlinear system has several useful ISS-type properties under average dwell-time switching signals if each constituent dynamical system is ISS. This extends available results for switched linear systems. We apply our result to stabilization of uncertain nonlinear systems via switching supervisory control, and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide ISS properties with respect to the estimation errors. Detailed illustrative examples are included.

Overcoming the limitations of adaptive control by means of logic-based switching (with J. P. Hespanha and A. S. Morse), Systems and Control Letters, vol. 49, no. 1, pp. 49-65, May 2003.
↓Abstract↑Abstract: In this paper we describe a framework for adaptive control which involves logic-based switching among a family of candidate controllers. We compare it with more conventional adaptive control techniques that rely on continuous tuning, emphasizing how switching and logic can be used to overcome some of the limitations of traditional adaptive control. The issues are discussed in a tutorial, non-technical manner and illustrated with specific examples.

Hysteresis-based switching algorithms for supervisory control of uncertain systems (with J. P. Hespanha and A. S. Morse), Automatica, vol. 39, no. 2, pp. 263-272, Feb 2003.
↓Abstract↑Abstract: We address the problem of controlling a linear system with unknown parameters ranging over a continuum by means of switching among a finite family of candidate controllers. We present a new hysteresis-based switching logic, designed specifically for this purpose, and derive a bound on the number of switches produced by this logic on an arbitrary time interval. The resulting switching control algorithm is shown to provide stability and robustness to arbitrary bounded noise and disturbances and sufficiently small unmodeled dynamics.

Supervision of integral-input-to-state stabilizing controllers (with J. P. Hespanha and A. S. Morse), Automatica, vol. 38, no. 8, pp. 1327-1335, Aug 2002.
↓Abstract↑Abstract: The subject of this paper is hybrid control of nonlinear systems with large-scale uncertainty. We describe a high-level controller, called a "supervisor'', which orchestrates logic-based switching among a family of candidate controllers. We show that in this framework, the problem of controller design at the lower level can be reduced to finding an integral-input-to-state stabilizing control law for an appropriate system with disturbance inputs. Employing the recently introduced "scale-independent hysteresis'' switching logic, we prove that in the case of purely parametric uncertainty with unknown parameters taking values in a finite set the switching terminates in finite time and state regulation is achieved.

Output-input stability and minimum-phase nonlinear systems (with A. S. Morse and E. D. Sontag), IEEE Transactions on Automatic Control, vol. 47, no. 3, pp. 422-436, Mar 2002.
↓Abstract↑Abstract: This paper introduces and studies the notion of output-input stability, which represents a variant of the minimum-phase property for general smooth nonlinear control systems. The definition of output-input stability does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. In the spirit of the "input-to-state stability'' philosophy, it requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of output-input stable systems thus defined includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts. As an application, we explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control.

Multiple model adaptive control with safe switching (with B. D. O. Anderson, T. S. Brinsmead, and A. S. Morse), International Journal of Adaptive Control and Signal Processing (invited paper), vol. 15, pp. 445-470, 2001.
↓Abstract↑Abstract: The purpose of this paper is to marry the two concepts of Multiple Model Adaptive Control and Safe Adaptive Control. In its simplest form, Multiple Model Adaptive Control involves a supervisor switching among one of a finite number of controllers as more is learnt about the plant, until one of the controllers is finally selected and remains unchanged. Safe Adaptive Control is concerned with ensuring that when the controller is changed in an adaptive control algorithm, the frozen plant-controller combination is never (closed loop) unstable. This is a nontrivial task since by definition of an adaptive control problem, the plant is not fully known. The proposed solution method involves a frequency-dependent performance measure and employs the Vinnicombe metric. The resulting safe switching guarantees depend on the extent to which a closed-loop transfer function can be accurately identified.

Multiple model adaptive control, part 2: Switching (with B. D. O. Anderson, T. S. Brinsmead, F. De Bruyne, J. P. Hespanha, and A. S. Morse), International Journal on Robust and Nonlinear Control (invited paper), vol. 11, pp. 479-496, 2001.
↓Abstract↑Abstract: This paper addresses the problem of controlling a continuous-time linear system with large modeling errors. We employ an adaptive control algorithm consisting of a family of linear candidate controllers supervised by a high-level switching logic. Methods for constructing such controller families have been discussed in the recent paper by the authors. The present paper concentrates on the switching task in a multiple model context. We describe and compare two different switching logics, and in each case study the behavior of the resulting closed-loop hybrid system.

Multiple model adaptive control, part 1: Finite controller coverings (with B. D. O. Anderson, T. S. Brinsmead, F. De Bruyne, J. P. Hespanha, and A. S. Morse), International Journal on Robust and Nonlinear Control (invited paper), vol. 10, pp. 909-929, 2000.
↓Abstract↑Abstract: We consider the problem of determining an appropriate model set on which to design a set of controllers for a multiple model switching adaptive control scheme. We show that, given mild assumptions on the uncertainty set of linear time-invariant plant models, it is possible to determine a finite set of controllers such that for each plant in the uncertainty set, satisfactory performance will be obtained for some controller in the finite set. We also demonstrate how such a controller set may be found. The analysis exploits the Vinnicombe metric and the fact that the set of approximately band- and time-limited transfer functions is approximately finite-dimensional.

Logic-based switching control of a nonholonomic system with parametric modeling uncertainty (with J. P. Hespanha and A. S. Morse), Systems and Control Letters (special issue on hybrid systems), vol. 38, no. 3, pp. 167-177, Nov 1999.
↓Abstract↑Abstract: This paper is concerned with control of nonholonomic systems in the presence of parametric modeling uncertainty. The specific problem considered is that of parking a wheeled mobile robot of unicycle type with unknown parameters, whose kinematics can be described by Brockett's nonholonomic integrator after an appropriate state and control coordinate transformation. We employ the techniques of supervisory control to design a hybrid feedback control law that solves this problem. See also a parking movie.

Conferences:

Distributed linear supervisory control (with A. Khanafer and T. Basar), in Proceedings of the 53rd IEEE Conference on Decision and Control, Los Angeles, CA, Dec 2014, pp. 1458-1463.
↓Abstract↑Abstract: In this work, we propose a distributed version of the logic-based supervisory adaptive control scheme. Given a network of agents whose dynamics contain unknown parameters, the distributed supervisory control scheme is used to assist the agents to converge to a certain set-point without requiring them to have explicit knowledge of that set-point. Unlike the classical supervisory control scheme where a centralized supervisor makes switching decisions among the candidate controllers, in our scheme, each agent is equipped with a local supervisor that switches among the available controllers. The switching decisions made at a certain agent depend only on the information from its neighboring agents. We apply our framework to the distributed averaging problem in the presence of large modeling uncertainty and support our findings by simulations.

Passification-based adaptive control with quantized measurements (with A. Selivanov and A. L. Fradkov), in Proceedings of the 19th IFAC World Congress, Cape Town, South Africa, Aug 2014.
↓Abstract↑Abstract: We propose and analyze passification-based adaptive controller for linear uncertain systems with quantized measurements. Since the effect of the quantization error is similar to the effect of a disturbance, the adaptation law with sigma-modification is used. To ensure convergence to a smaller set, the parameters of the adaptation law are being switched during the evolution of the system and a dynamic quantizer is used. It is proved that if the quantization error is small enough then the proposed controller ensures convergence of the state of a hyper-minimum-phase system to an arbitrarily small vicinity of the origin. Applicability of the proposed controller to polytopic-type uncertain systems and its efficiency is demonstrated by the example of yaw angle control of a flying vehicle.

Robust control with compensation of disturbances for systems with quantized output (with I. Furtat and A. L. Fradkov), in Proceedings of the 19th IFAC World Congress, Cape Town, South Africa, Aug 2014.
↓Abstract↑Abstract: The paper deals with the robust output feedback discrete control of continuous-time linear plants with arbitrary relative degree under parametric uncertainties and external bounded disturbances with quantized output signal. The parallel reference model (auxiliary loop) to the plant is used for obtaining the uncertainties acting on the plant. The proposed algorithm guarantees that the output of the plant tracks the reference output with the required accuracy.

The bang-bang funnel controller: time delays and case study (with S. Trenn), in Proceedings of the 2013 European Control Conference, Zurich, Switzerland, Jul 2013, pp. 1669-1674.
↓Abstract↑Abstract: We investigate the recently introduced bang-bang funnel controller with respect to its robustness to time delays. We present slightly modified feasibility conditions and prove that the bang-bang funnel controller applied to a relative-degree-two nonlinear system can tolerate sufficiently small time delays. A second contribution of this paper is an extensive case study, based on a model of a real experimental setup, where implementation issues such as the necessary sampling time and the conservativeness of the feasibility assumptions are explicitly considered.

The bang-bang funnel controller (with S. Trenn), in Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, GA, Dec 2010, pp. 690-695.
↓Abstract↑Abstract: A bang-bang controller is proposed which is able to ensure reference signal tracking with prespecified time-varying error bounds (the funnel) for nonlinear systems with relative degree one or two. For the design of the controller only the knowledge of the relative degree is needed. The controller is guaranteed to work when certain feasibility assumptions are fulfilled, which are explicitly given in the main results. Linear systems with relative degree one or two are feasible if the system is minimum phase and the control values are large enough. See also a more complete version.

Adaptive control using quantized measurements with application to vision-only landing control (with Y. Sharon and Y. Ma), in Proceedings of the 49th IEEE Conference on Decision and Control, Atlanta, GA, Dec 2010, pp. 2511-2516.
↓Abstract↑Abstract: We consider a class of control systems where the plant model is unknown and the observed states are only measured through a quantizer. In order to estimate the plant model for the controller, we follow up on an approach where an optimization is taking place over both the plant model parameters and the state of the plant. We propose a computationally efficient algorithm for solving the optimization problem, and prove that the algorithm converges using tools from convex and non-smooth analysis. We demonstrate both the importance of this class of control systems and our method of solution using the following application: having a fixed wing airplane follow a desired glide slope on approach to landing. The only available feedback is from a camera mounted at the front of the airplane and looking at a runway of unknown dimensions. The quantization is due to the finite resolution of the camera. Using this application we also compare our method to the basic method which is prevalent in the literature, where the optimization is only taking place over the plant model parameters.

Stabilizing uncertain systems with dynamic quantization (with L. Vu), in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec 2008, pp. 4681-4686.
↓Abstract↑Abstract: We consider the problem of stabilizing uncertain linear systems with quantization. The plant uncertainty is dealt with by the supervisory adaptive control framework, which employs switching among a finite family of candidate controllers. For a static quantizer, we quantify a relationship between the quantization range and the quantization error bound that guarantees closed loop stability. Using a dynamic quantizer which can vary the quantization parameters in real time, we show that the closed loop is asymptotically stabilized provided a certain condition on the quantization range and the quantization error bound is satisfied. The results in this work extend previous results on stabilization of known systems with quantization to the case of uncertain systems. See also a more complete version.

Stability of interconnected switched systems and adaptive control of time-varying plants (with L. Vu), in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, Dec 2007, pp. 4021-4026.
↓Abstract↑Abstract: We discuss stability of a loop consisting of two asynchronous switched systems, in which the first switched system influences the input and the switching signal of the second switched system and the second switched system affects the first switched system's jump map. We show that when the first switched system has a small dwell-time and is switching slowly in the spirit of average dwell-time switching, all the states of the closed loop are bounded. We show how this result relates to supervisory adaptive control of time-varying plants. When the uncertain plant takes the form of a switched system with an unknown switching signal, we show that all the states of the closed-loop control system are guaranteed to be bounded provided that the plant's switching signal varies slowly enough.

ISS of switched systems and applications to switching adaptive control (with L. Vu and D. Chatterjee), in Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain, Dec 2005, pp. 120-125.
↓Abstract↑Abstract: In this paper we prove that a switched nonlinear system has several useful ISS-type properties under average dwell-time switching signals if each constituent dynamical system is ISS. This extends available results for switched linear systems. We apply our result to stabilization of uncertain nonlinear systems via switching supervisory control, and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide ISS properties with respect to the estimation errors. Detailed illustrative examples are included. See also the slides of the talk.

Hierarchical hysteresis switching (with J. P. Hespanha and A. S. Morse), in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, Dec 2000, pp. 484-489.
↓Abstract↑Abstract: We describe a new switching logic, called "hierarchical hysteresis switching'', and establish a bound on the number of switchings produced by this logic on a given interval. The motivating application is the problem of controlling a linear system with large modeling uncertainty. We consider a control algorithm consisting of a finite family of linear controllers supervised by the hierarchical hysteresis switching logic. In this context, the bound on the number of switchings enables us to prove stability of the closed-loop system in the presence of noise, disturbances, and unmodeled dynamics. See also the slides of the talk.

Bounds on the number of switchings with scale-independent hysteresis: applications to supervisory control (with J. P. Hespanha and A. S. Morse), in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, Dec 2000, pp. 3622-3627.
↓Abstract↑Abstract: In this paper we analyze the Scale-Independent Hysteresis Switching Logic introduced in recent work. We show that, under suitable "open-loop" assumptions, one can establish an upper bound on the number of switchings produced by the logic on any given interval. This bound comes as a function of the variation of the inputs to the logic on that interval. In this paper it is also shown that, in a supervisory control context, this leads to switching that is slow-on-the-average, allowing us to study the stability of hysteresis-based adaptive control systems in the presence of measurement noise.

A new definition of the minimum-phase property for nonlinear systems, with an application to adaptive control (with A. S. Morse and E. D. Sontag), in Proceedings of the 39th IEEE Conference on Decision and Control, Sydney, Australia, Dec 2000, pp. 2106-2111.
↓Abstract↑Abstract: We introduce a new definition of the minimum-phase property for general smooth nonlinear control systems. The definition does not rely on a particular choice of coordinates in which the system takes a normal form or on the computation of zero dynamics. It requires the state and the input of the system to be bounded by a suitable function of the output and derivatives of the output, modulo a decaying term depending on initial conditions. The class of minimum-phase systems thus defined includes all affine systems in global normal form whose internal dynamics are input-to-state stable and also all left-invertible linear systems whose transmission zeros have negative real parts. We explain how the new concept enables one to develop a natural extension to nonlinear systems of a basic result from linear adaptive control. See also the slides of the talk

Towards the supervisory control of uncertain nonholonomic systems (with J. P. Hespanha and A. S. Morse), in Proceedings of the 1999 American Control Conference, San Diego, CA, Jun 1999, pp. 3520-3524.
↓Abstract↑Abstract: This paper is concerned with control of nonholonomic systems in the presence of parametric modeling uncertainties. The specific problem considered is that of parking a wheeled mobile robot of unicycle type with unknown parameters, whose kinematics can be described by the nonholonomic integrator after an appropriate state and control coordinate transformation. We employ the techniques of supervisory control to design a hybrid feedback control law that solves this problem. See also the slides of the talk.

Stochastic systems:

Stabilizing randomly switched systems (with D. Chatterjee), SIAM Journal on Control and Optimization, vol. 49, no. 5, pp. 2008-2031, 2011.
↓Abstract↑Abstract: This article is concerned with stability analysis and stabilization of randomly switched systems under a class of switching signals. The switching signal is modeled as a jump stochastic (not necessarily Markovian) process independent of the system state; it selects, at each instant of time, the active subsystem from a family of systems. Sufficient conditions for stochastic stability (almost sure, in the mean, and in probability) of the switched system are established when the subsystems do not possess control inputs, and not every subsystem is required to be stable. These conditions are employed to design stabilizing feedback controllers when the subsystems are affine in control. The analysis is carried out with the aid of multiple Lyapunov-like functions, and the analysis results together with universal formulae for feedback stabilization of nonlinear systems constitute our primary tools for control design.

On stability of randomly switched nonlinear systems (with D. Chatterjee), IEEE Transactions on Automatic Control, vol. 52, no. 12, pp. 2390-2394, Dec 2007.
↓Abstract↑Abstract: This article is concerned with stability analysis and stabilization of randomly switched systems. These systems may be regarded as piecewise deterministic stochastic systems: the discrete switchings are triggered by a stochastic process which is independent of the state of the system, and between two consecutive switching instants the dynamics are deterministic. Our results provide sufficient conditions for almost sure stability and stability in the mean using Lyapunov-based methods, when individual subsystems are stable and a certain "slow switching" condition holds. This slow switching condition takes the form of an asymptotic upper bound on the probability mass function of the number of switches that occur between the initial and current time instants. This condition is shown to hold for switching signals coming from the states of finite-dimensional continuous-time Markov chains; our results therefore hold for Markovian jump systems in particular. For systems with control inputs we provide explicit control schemes for feedback stabilization using the universal formula for stabilization of nonlinear systems.

Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions (with D. Chatterjee), SIAM Journal on Control and Optimization, vol. 45, no. 1, pp. 174-206, 2006.
↓Abstract↑Abstract: This paper presents a general framework for analyzing stability of nonlinear switched systems, by combining the method of multiple Lyapunov functions with a suitably adapted comparison principle in the context of stability in terms of two measures. For deterministic switched systems, this leads to a unification of representative existing results and an improvement upon the current scope of the method of multiple Lyapunov functions. For switched systems perturbed by white noise, we develop new results which may be viewed as natural stochastic counterparts of the deterministic ones. In particular, we study stability of deterministic and stochastic switched systems under average dwell-time switching.

Nonlinear feedback systems perturbed by noise: steady-state probability distributions and optimal control (with R. W. Brockett), IEEE Transactions on Automatic Control, vol. 45, no. 6, pp. 1116-1130, Jun 2000.
↓Abstract↑Abstract: We describe a class of nonlinear feedback systems perturbed by white noise for which explicit formulas for steady-state probability densities can be found. We show that this class includes what has been called monotemperaturic systems in earlier work, and establish relationships with Lyapunov functions for the corresponding deterministic systems. We also treat a number of stochastic optimal control problems in the case of quantized feedback, with performance criteria formulated in terms of the steady-state probability density.

Spectral analysis of Fokker-Planck and related operators arising from linear stochastic differential equations (with R. W. Brockett), SIAM Journal on Control and Optimization, vol. 38, no. 5, pp. 1453-1467, May 2000.
↓Abstract↑Abstract: We study spectral properties of certain families of linear second-order differential operators arising from linear stochastic differential equations. We construct a basis in the Hilbert space of square-integrable functions using modified Hermite polynomials, and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed. In particular, we completely describe the spectrum of the Fokker-Planck operator on an appropriate invariant subspace of rapidly decaying functions. The eigenvalues of the Fokker-Planck operator provide information about the speed of convergence of the underlying stochastic process to steady state, which is important for stochastic estimation and control applications. We show that the operator families under consideration can be realized as solutions of differential equations in the double bracket form on an operator Lie algebra, which leads to a simple expression for the flow of their eigenfunctions.

Book chapters:

Stabilization of deterministic control systems under random sampling: overview and recent developments (with A. Tanwani and D. Chatterjee), in Uncertainty in Complex Networked Systems - In Honor of Roberto Tempo (T. Basar, Ed.), Systems & Control: Foundations & Applications, Birkhauser, 2018, pp. 209-246.
↓Abstract ↑Abstract: This chapter addresses the problem of stabilizing continuous-time deterministic control systems via a sample-and-hold scheme under random sampling. The sampling process is assumed to be a Poisson counter, and the open-loop system is assumed to be stabilizable in an appropriate sense. Starting from as early as mid-1950s, when this problem was studied in the Ph.D. dissertation of R.E. Kalman, we provide a historical account of several works that have been published thereafter on this topic. In contrast to the approaches adopted in these works, we use the framework of piecewise deterministic Markov processes to model the closed-loop system, and carry out the stability analysis by computing the extended generator. We demonstrate that for any continuous-time robust feedback stabilizing control law employed in the sample-and-hold scheme, the closed-loop system is asymptotically stable for all large enough intensities of the Poisson process. In the linear case, for increasingly large values of the mean sampling rate, the decay rate of the sampled process increases monotonically and converges to the decay rate of the unsampled system in the limit. In the second part of this article, we fix the sampling rate and address the question of whether there exists a feedback gain which asymptotically stabilizes the system in mean square under the sample-and-hold scheme. For the scalar linear case, the answer is in the affirmative and a constructive formula is provided here. For systems with dimension greater than 1 we provide an answer for a restricted class of linear systems, and we leave the solution corresponding to the general case as an open problem.

Conferences:

Towards ISS disturbance attenuation for randomly switched systems (with D. Chatterjee), in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, Dec 2007, pp. 5612-5617.
↓Abstract↑Abstract: We are concerned with input-to-state stability (ISS) of randomly switched systems. We provide preliminary results dealing with sufficient conditions for stochastic versions of ISS for randomly switched systems without control inputs, and with the aid of universal formulae we design controllers for ISS-disturbance attenuation when control inputs are present. Two types of switching signals are considered: the first is characterized by a statistically slow-switching condition, and the second by a class of semi-Markov processes.

Stability and stabilization of randomly switched systems (with D. Chatterjee), in Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, Dec 2006, pp. 2643-2648.
↓Abstract↑Abstract: This article is concerned with stability analysis and stabilization of randomly switched systems with control inputs. The switching signal is modeled as a jump stochastic process independent of the system state; it selects, at each instant of time, the active subsystem from a family of deterministic systems. Three different types of switching signals are considered: the first is a jump stochastic process that satisfies a statistically slow switching condition; the second and the third are jump stochastic processes with independent identically distributed values at jump times together with exponential and uniform holding times, respectively. For each of the three cases we first establish sufficient conditions for stochastic stability of the switched system when the subsystems do not possess control inputs; not every subsystem is required to be stable in the latter two cases. Thereafter we design feedback controllers when the subsystems are affine in control and are not all zeroinput stable, with the control space being general subsets of $R^m$. Our analysis results and universal formulae for feedback stabilization of nonlinear systems for the corresponding control spaces constitute the primary tools for control design. See also a more complete version.

On stability of stochastic switched systems (with D. Chatterjee), in Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, Dec 2004, pp. 4125-4127.
↓Abstract↑Abstract: In this paper we propose a method for stability analysis of switched systems perturbed by a Wiener process. It utilizes multiple Lyapunov-like functions and is analogous to an existing result for deterministic switched systems. See also the slides of the talk.

Quantized feedback systems perturbed by white noise (with R. W. Brockett), in Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, Dec 1998, pp. 1327-1328.
↓Abstract↑Abstract: This paper treats a class of nonlinear feedback systems perturbed by white noise, the nonlinearity being given by a piecewise constant function of a certain type. We obtain explicit formulae for steady-state probability densities associated with such systems. This result is used to address a stochastic optimal control problem that can be interpreted as minimization of the cost of implementing a feedback control law. See also the slides of the talk.

On explicit steady-state solutions of Fokker-Planck equations for a class of nonlinear feedback systems (with R. W. Brockett), in Proceedings of the 1998 American Control Conference, Philadelphia, PA, Jun 1998, pp. 264-268.
↓Abstract↑Abstract: We study the question of existence of steady-state probability distributions for systems perturbed by white noise. We describe a class of nonlinear feedback systems for which an explicit formula for the steady-state probability density can be found. These systems include what has been called monotemperaturic systems in earlier work. We also establish relationships between the steady-state probability densities and Liapunov functions for the corresponding deterministic systems. See also the slides of the talk.

Other publications

Letter: Leaving a lasting mark, IEEE Control Systems Magazine, vol. 35, no. 2, pp. 19-20, Apr 2015.

Interview: People in Control, IEEE Control Systems Magazine, vol. 27, no. 6, pp. 40-42, Dec 2007.

Book review: Liapunov Functions and Stability in Control Theory, 2nd edition by A. Bacciotti and L. Rosier, Automatica, vol. 41, no. 12, pp. 2183-2184, Dec 2005.

Book review: Hybrid Dynamical Systems: Controller and Sensor Switching Problems by A. V. Savkin and R. J. Evans, International Journal of Hybrid Systems, vol. 4, pp. 161-164, Mar/Jun 2004.

Book review: Qualitative Theory of Hybrid Dynamical Systems by A. S. Matveev and A. V. Savkin, Automatica, vol. 39, no. 2, pp. 368-369, Feb 2003.

Editorial: Switching and Logic in Adaptive Control, special issue of the International Journal of Adaptive Control and Signal Processing (edited by J. P. Hespanha and D. Liberzon), vol. 15, no. 3, 2001. Editorial.

Thesis: Asymptotic Properties of Nonlinear Feedback Control Systems, Ph.D. Thesis, Department of Mathematics, Brandeis University, Waltham, MA, Feb 1998.
↓Abstract↑Abstract: We study asymptotic behaviour of nonlinear feedback control systems, both deterministic and stochastic. Of particular interest is the case of quantized feedback, i.e., when the nonlinearity takes the form of a specific piecewise constant function. In the context of deterministic linear control systems with quantized measurements, we show how quantized feedback can be used to asymptotically stabilize the system (Chapter II).
For systems perturbed by white noise, we address the question of existence of steady-state probability distributions. In the linear case, the solution to the Fokker-Planck equation which describes the evolution of the probability density is well known. In particular, one has an expression for the steady-state probability density, which is an eigenfunction of the Fokker-Planck operator with eigenvalue zero. We show that other eigenvalues and eigenfunctions of the Fokker-Planck operator associated with a linear system can also be directly computed (Chapter III). In the nonlinear case, the situation is more complicated. We describe a class of nonlinear feedback systems for which explicit formulae for the steady-state probability densities can be found, and give two interpretations of this result, one related to certain concepts from statistical thermodynamics, and the other related to Lur'e problem of absolute stability (Chapter IV). We demonstrate how the solutions obtained here can be used to treat a number of stochastic optimal control problems (Chapter V).

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