**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.

"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)
*

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) | 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
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.
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.

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

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

**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.

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:
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:
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:
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:
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.

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: 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: 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:
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:
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.

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: 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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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: 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: 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.

Lyapunov small-gain theorems for networks of
not necessarily ISS hybrid systems (with A. Mironchenko and G. Yang),
submitted to Automatica, Aug 2016.

Abstract:
We prove a novel Lyapunov-based small-gain theorem for networks of $n \ge 2$ 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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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: 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: 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:
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:
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:
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:
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: 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: 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: 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: 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: 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: 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: 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

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:
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:
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:
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.

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

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

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

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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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: 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:
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:
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:
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:
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:
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: 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: 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: 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.

Quantized adaptive stabilization of minimum-phase systems (with A. Selivanov and A. L. Fradkov),
submitted to IEEE Transactions on Automatic Control, Jun 2016

Abstract:
We consider a linear minimum-phase system of an arbitrary relative
degree with an unknown bounded disturbance and dynamically quantized
measurements. The shunting method (parallel feedforward compensator) is
applied to obtain hyper-minimum-phase augmented system that is further
stabilized by a passification-based adaptive controller. By
constructing a switching procedure for the adaptive controller
parameters, we ensure convergence of the system state from an arbitrary
set to an ellipsoid, whose size depends on the disturbance bound. The
results are demonstrated by an example of an aircraft flight control.

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*, to appear.

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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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: 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.

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:
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:
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:
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:
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.

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

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

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

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

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

Abstract: We propose and analyze passication-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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

See also the slides of the talk.

On quantized control and geometric optimization (with F. Bullo), in Proceedings of the

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

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

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

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.

Quantized adaptive stabilization of minimum-phase systems (with A. Selivanov and A. L. Fradkov),
submitted, Jun 2016

Abstract:
We consider a linear minimum-phase system of an arbitrary relative
degree with an unknown bounded disturbance and dynamically quantized
measurements. The shunting method (parallel feedforward compensator) is
applied to obtain hyper-minimum-phase augmented system that is further
stabilized by a passification-based adaptive controller. By
constructing a switching procedure for the adaptive controller
parameters, we ensure convergence of the system state from an arbitrary
set to an ellipsoid, whose size depends on the disturbance bound. The
results are demonstrated by an example of an aircraft flight control.

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:
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:
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:
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:
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:
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:
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: 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: 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.

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: 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:
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:
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: 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.

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:
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.

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: 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:
We propose and analyze passication-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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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: 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.

Stabilizing randomly switched systems
(with D. Chatterjee),
SIAM Journal on Control and Optimization, vol. 49, no. 5, pp. 2008-2031, 2011.

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:
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:
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: 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:
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.

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:
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:
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:
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: 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: 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.

**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:
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).

**Disclaimer:**

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searches, and timely dissemination of
scholarly and technical
work.
Copyright and all rights therein are retained by original copyright holders.

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