Daniel Liberzon's research
interests

### Hybrid
control:

We are interested in developing control algorithms in which
continuous dynamics are coupled with discrete logic. The resulting
closed-loop systems are known as *hybrid* because they
are described by an interaction of differential equations and
discrete automata. The motivating applications are those in
which the controller is implemented on a digital computer or
communicates with the process remotely over a network. A particular
focus of our research is on developing hybrid control laws for
nonlinear systems subject to communication constraints.
Relevant Publications
Specific research directions within this domain include **finite-data-rate control** and **topological entropy**, for which I recommend the following sources.

**Analysis
of switched systems:**

The aim of this research is to provide a theoretical foundation
for the analysis of systems that result from applying switching control
methods described above. We are
developing stability criteria for switched systems which utilize common,
multiple, and weak Lyapunov functions. These are complemented by tools for
hybrid systems developed by computer scientists to verify stability.
We are also exploring
mathematical methods from the theory of Lie algebras. Going beyong
stability, we are investigating basic properties of switched systems
with inputs and outputs. Relevant Publications
One specific research direction within this domain is **commutation relations and stability under arbitrary switching**, for which I recommend the following sources

### Nonlinear control
theory:

We are working on several aspects of nonlinear systems and control theory.
A specific topic of interest is the use of control Lyapunov
functions for achieving various forms of disturbance attenuation,
such as input-to-state stability.
These problems are motivated in part by questions that arise
in the design of logic-based switching control algorithms for nonlinear
systems. Our recent work also re-examines the concepts of a
minimum-phase
nonlinear system and of nonlinear observability and controllability.
Relevant Publications
One specific research direction within this domain is **robust nonlinear observers and synchronization**, for which I recommend the following sources.

### Supervisory
control of uncertain systems:

This research area can be regarded as a subtopic of hybrid control,
which focuses on systems with large modeling uncertainties.
The basic paradigm is to design a high-level decision maker, called
a *supervisor*, which orchestrates logic-based switching
among a family of candidate controllers. Such switching control
techniques provide an alternative to more traditional continuously
tuned adaptive control laws, and are more suitable
for computer implementation. Our primary objective is to develop
a systematic supervisory control methodology for nonlinear uncertain systems.
Relevant Publications

**Stochastic differential equations:
**

This research is concerned with investigating steady-state
properties of systems described by stochastic differential equations.
The most recent direction is stability of stochastic switched
systems.
We are also exploring
how the steady-state properties of controlled stochastic systems
are affected by the choice of control.
Relevant Publications

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Liberzon