Daniel Liberzon's publications
and preprints
Book
Switching
in Systems and
Control, Birkhauser, Boston, MA, Jun 2003.
Volume in series Systems and Control: Foundations and Applications.
ISBN 0-8176-4297-8.
Brief description:
This book presents theoretical developments in the field of
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. This
book is written with the purpose of bridging the gap between the
classical mathematical control theory and the interdisciplinary
field of hybrid systems. It is aimed primarily at the readers with
background in systems and control theory.
The first part introduces the classes of systems studied in the
book. The second part develops stability theory for switched
systems. Analysis methods based on single and multiple Lyapunov
functions are used to obtain stability results for systems with
suitable commutation relations, systems with constrained
switching, and systems with special structure. The third part of
the book is devoted to switching control design for
continuous-time systems. Logic-based switching control algorithms
are described for systems which cannot be stabilized by continuous
feedback, systems with sensor or actuator constraints, and systems
with large modeling uncertainty. The results are typically derived
for linear systems and then extended to nonlinear systems.
The book can be used as a text for a second-level graduate course
on switched systems and switching control. It can also serve as an
introduction to this active area of current research for control
theorists and mathematicians, as well as a comprehensive reference
source 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.
Papers by topic
The most recent submissions are marked with 
Click on a topic to see its
expanded (and more accurate) name.
Papers within each topic are arranged by category: journal articles, then book
chapters (if any), then conference articles. Within each category, the papers
are listed in reverse chronological order.
Nonlinear systems and control:
Journals:
Lyapunov conditions for
input-to-state stability for impulsive systems (with J. P. Hespanha
and
A. R. Teel),
Automatica, to appear.
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.
Conferences:
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, 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 '02), 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.
Switched and hybrid systems:
Journals:
Stabilizing randomly switched systems (with D. Chatterjee), submitted.
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.
Verifying average dwell time of hybrid systems
(with S. Mitra
and N. Lynch),
ACM Transactions in Embedded Computing Systems,
to appear.
Abstract:
The switched system model abstracts away the discrete mechanisms
of a hybrid system in terms of
an exogenous switching signal. Dwell Time and Average Dwell Time
(ADT) criteria, introduced
by Morse and Hespanha, define restricted classes of switching
signals that guarantee stability of
the whole system, provided the individual modes of the switched
system are stable. In this paper,
we present a set of techniques for establishing stability
through verification of ADT properties.
We introduce a new type of simulation relation for hybrid
automata---switching simulation---that
allows us to show that the ADT of one automaton is no less than
that of another. We show that
the question of whether a given hybrid automaton has ADT can
be answered by checking a
carefully designed invariant or by solving an optimization
problem. The invariant-based method
is applicable to any hybrid automaton. For suitable classes of
automata the invariant in question
can be checked automatically. The optimization-based method is
applicable to a restricted class
of initialized hybrid automata. For this class, 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-based approach
is automatic and complements the
invariant-based method in the sense that they 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.
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.
Book chapters:
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 '02), South Bend, IN, Aug 2002, pp. 90-92.
See also a partial solution.
Conferences:
Invertibility of nonlinear switched systems
(with A. Tanwani),
in Proceedings of the 47th IEEE Conference on Decision and
Control, Cancun,
Mexico,
Dec 2008, to appear.
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, 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, 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, June 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.
Control under communication
constraints:
Journals:
Nonlinear control with
limited information, submitted to Communications in Information
Systems, Roger Brockett Legacy special issue, Jun 2008.
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.
A unified framework for design and analysis of networked and quantized
control systems (with D. Nesic),
IEEE Transactions on Automatic Control, to appear.
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.
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.
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.
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.
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.
Book chapters:
Input-to-state stabilization with quantized output feedback
(with Y. Sharon),
in Proceedings of the 11th International Workshop
on Hybrid Systems: Computation and Control, St. Louis, MO, Apr 2008,
Lecture Notes
in
Computer Science, vol. 4981 (M.
Egerstedt and B. Mishra, Eds.),
Springer, Berlin, pp. 500-513.
Abstract:
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.
Conferences:
Rendezvous without coordinates
(with J. Yu and S. LaValle),
in Proceedings of the 47th IEEE Conference on Decision and
Control, Cancun,
Mexico,
Dec 2008, to appear.
Abstract:
We study minimalism in sensing and control by
considering a multi-agent system in which each agent moves
like a Dubins car and has a limited sensor that reports only the
presence of another agent within some sector of its windshield.
Using a very simple quantized control law with three values,
each agent tracks another agent assigned to it by maintaining
that agent within this windshield sector. We use Lyapunov
analysis to show that by acting autonomously in this way,
the agents will achieve rendezvous if the initial assignment
graph is connected. A distinguishing feature of our approach
is that it does not involve any estimation procedure aimed
at reconstructing coordinate information. Our scenario thus
appears to be the first example in which an interesting task
is performed with extremely coarse sensing and control, and
without state estimation. The system was implemented in
computer simulation, accessible
through the Web, of which
the results are presented in the paper.
See also a more
complete version.
Stabilizing uncertain systems with dynamic quantization (with L. Vu), in
Proceedings of the 47th IEEE Conference on Decision and Control, Cancun,
Mexico, Dec 2008, to appear.
Abstract:
We consider the problem of stabilizing uncertain
linear systems with quantization. The plant uncertainty
is dealt with by the supervisory adaptive control
framework, which employs switching among a finite family
of candidate controllers. For a static quantizer, we quantify
a relationship between the quantization range and the
quantization error bound that guarantees closed loop
stability. Using a dynamic quantizer which can vary the
quantization parameters in real time, we show that the
closed loop is asymptotically stabilized provided a certain
condition on the quantization range and the quantization
error bound is satisfied. Our results extend
previous results on stabilization of known systems with
quantization to the case of uncertain systems.
See also a more
complete version.
Observer-based
quantized output feedback control of nonlinear systems, in
Proceedings
of the 17th IFAC World Congress, Seoul, Korea, Jul 2008, pp.
8039-8043.
Abstract: This paper
addresses the problem of stabilizing a nonlinear system by means of quantized output
feedback. A conceptual framework is presented in which the control input
is generated by an
observer-based feedback controller acting on quantized output measurements. A
stabilization result is established under the assumption that this observer-based
controller possesses robustness with respect to output measurement errors in an
input-to-state stability (ISS) sense. Designing such observers and controllers is a
largely open problem, some partial results on which are discussed. The main goal of
the paper is to encourage further work on this important topic.
See also
the slides
of the
talk.
Input-to-state stabilization with minimum number of quantization regions
(with Y. Sharon),
in Proceedings of the 46th IEEE Conference on Decision and
Control, New Orleans,
LA,
Dec 2007, pp. 20-25.
Abstract:
We study control systems where the state measurements are
quantized and time-sampled, and an unknown disturbance is
being applied. We present a dynamic quantization scheme
that switches between three modes of operation. We show
that by using this scheme with a continuous static feedback
controller we achieve a closed-loop system which has the
Input-to-State Stability property (ISS). Our design does
not use any characterization of the disturbance; as long as
the disturbance is bounded the system will remain stable.
We show that three quantization regions per dimension is
sufficient to achieve the ISS property, and furthermore we
show that the ISS property is achievable using a data rate
that is arbitrarily close to the minimum required data rate
when no disturbance is applied.
A unified approach to controller design for systems with
quantization and time scheduling (with D. Nesic),
in Proceedings of the 46th IEEE Conference on Decision and
Control, New Orleans,
LA,
Dec 2007, pp. 3939-3944.
Abstract:
We generalize and unify a range of recent results in quantized control systems
(QCS) and networked control systems (NCS) literature and provide a unified
framework for controller design for control systems with quantization and time
scheduling via an emulation-like approach. A crucial step in our approach is
finding an appropriate Lyapunov function for the quantization/time-scheduling
protocol which verifies its uniform global exponential stability (UGES). We
construct Lyapunov functions for several representative protocols that are
commonly found in the literature, as well as some new protocols not considered
previously.
Observer-based
quantized output feedback control of nonlinear systems, in Proceedings
of the 15th Mediterranean Conference on Control and Automation, Athens,
Greece, Jun 2007.
Abstract: This paper
addresses the problem of stabilizing a nonlinear system by means of quantized output
feedback. A framework is presented in which the control input is generated by an
observer-based feedback controller acting on quantized output measurements. A
stabilization result is established under the assumption that this observer-based
controller possesses robustness with respect to output measurement errors in an
input-to-state stability (ISS) sense. Designing such observers and controllers is a
largely open problem, some partial results on which are discussed. The main goal of
the paper is to encourage further work on this important topic.
See also
the slides
of the
talk.
Input-to-state stabilization of linear systems
with quantized feedback (with D. Nesic),
in Proceedings of the 44th IEEE Conference on Decision and
Control, Seville, Spain, Dec 2005, pp. 8197-8202.
Abstract:
We consider the problem of achieving input-to-state stability
(ISS) with respect to external disturbances for control systems
with linear dynamics and quantized state measurements. Quantizers
considered in this paper take finitely many values and have an
adjustable ``zoom" parameter. Extending an approach developed
previously for systems with no disturbances, we present a control
methodology that counteracts an unknown disturbance by switching
repeatedly between ``zooming in" and ``zooming out". Two specific
control strategies that yield ISS are described. The first one is
implemented in continuous time and analyzed with the help of a
Lyapunov function, similarly to earlier work. The second strategy
incorporates time sampling, and its analysis is novel in that it
is completely trajectory-based and utilizes a cascade structure of
the closed-loop hybrid system. We discover that in the presence of
disturbances, time-sampling implementation requires an additional
modification which has not been considered in previous work.
See also a more
complete version.
A small-gain approach to stability analysis of hybrid systems (with D. Nesic),
in Proceedings of the 44th IEEE Conference on Decision and
Control, 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.
Stabilizing a nonlinear system
with limited information feedback,
in Proceedings of the 42nd IEEE Conference on Decision and
Control, Maui, HI, Dec 2003, pp. 182-186.
Abstract:
This paper is concerned with the problem of stabilizing a
nonlinear continuous-time system by using sampled encoded
measurements of the state. We demonstrate that global asymptotic
stabilization is possible if a suitable relationship holds between
the number of values taken by the encoder, the sampling period,
and a system parameter, provided that a feedback law achieving
input-to-state stability with respect to measurement errors can be
found.
Also presented at the 1st International Symposium on
Control, Communications and Signal Processing,
Hammamet, Tunisia, Mar 2004 (click here for extended abstract).
See also
the slides of the
talk.
On quantized control and geometric optimization (with
F. Bullo),
in Proceedings of the 42nd IEEE Conference on Decision and
Control, Maui, HI, Dec 2003, pp. 2567-2572.
Abstract:
This paper studies state quantization schemes for feedback
stabilization of linear control systems with limited information. The
focus is on designing the least destabilizing quantizer subject
to a given information constraint. We explore several ways of
measuring the destabilizing effect of a quantizer on the
closed-loop system, including (but not limited to) the
worst-case quantization error. In each case, we show how
quantizer design can be naturally reduced to a version of the
so-called multicenter problem from locational optimization.
Algorithms for obtaining solutions to such problems, all in terms of suitable
Voronoi quantizers, are discussed.
In
particular, an iterative solver is developed for a novel
weighted multicenter problem which most accurately represents
the least destabilizing quantizer design.
See also
the slides of the
talk.
A note on stabilization of linear systems using coding and limited communication,
in Proceedings of the 41st IEEE Conference on Decision and
Control, Las Vegas, NV, Dec 2002, pp. 836-841.
Abstract:
We consider the problem of stabilizing a linear time-invariant
system using sampled encoded measurements of its state or
output. We derive a relationship between the number of values
taken by the encoder and the norm of the transition matrix of the
open-loop system over one sampling period, which guarantees that
global asymptotic stabilization can be achieved. A coding scheme
and a stabilizing control strategy are described explicitly.
See also
the slides of the
talk.
Stabilization by quantized state or output feedback:
a hybrid control approach, in
Proceedings of the 15th IFAC World Congress, Barcelona, Spain,
Jul 2002 (IFAC Young Author
Prize paper).
Abstract:
This paper deals with global
asymptotic stabilization of continuous-time
systems with quantized signals. A hybrid control
strategy originating in earlier work relies on the possibility
of making discrete
on-line adjustments of quantizer parameters.
We explore this method here
for general nonlinear systems with general types
of quantizers affecting the state of the system or the measured output.
See also
the slides of the
talk.
A hybrid control framework for systems with quantization,
in Proceedings of the 40th IEEE Conference on Decision and
Control, Orlando, FL,
Dec 2001, pp. 1217-1222.
Abstract:
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.
Supervisory
control of uncertain systems:
Journals:
Switching adaptive control of time-varying plants (with L. Vu), submitted to IEEE
Transactions
on Automatic Control,
Nov 2007.
Abstract:
We study the problem of adaptively stabilizing time-varying plants using
supervisory control, which
employs multiple controllers and a supervisory unit that orchestrates switching
among the controllers. We
show that for bounded disturbances and noise, all the closed-loop signals remain
bounded provided that
the plant varies slowly enough and the unmodeled dynamics are small enough. Slow
variation is quantified
by a new class of switching signals which are characterized by dwell-time as well
as average dwell-time.
In proving stability of the closed loop, we also derive a new result on stability
of interconnected switched
systems in which the states and the switching signals of two switched systems are
mutually constrained.
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: In this paper we study the scale-independent hysteresis
switching logic introduced in earlier work, as well
as its new variant
called ``hierarchical hysteresis switching''. We derive
bounds on the number of switchings produced by these logics on
an arbitrary finite
interval. The motivating problem is that 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 arbitrary bounded disturbances and noise and
sufficiently small unmodeled dynamics.
The main advance over previous work is that we are able to deal
with parametric uncertainty ranging over a continuum,
without relying on a fixed
interval between switching times.
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.
Conferences:
Stabilizing uncertain systems with dynamic quantization (with L. Vu), in
Proceedings of the 47th IEEE Conference on Decision and Control, Cancun,
Mexico, Dec 2008, to appear.
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, 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.
Stochastic systems:
Journals:
Stabilizing randomly switched systems
(with D. Chatterjee),
submitted.
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.
Conferences:
Towards ISS disturbance attenuation for randomly switched systems (with D. Chatterjee), in
Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans,
LA,
Dec 2007, pp. 5612-5617.
Abstract:
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.
Other publications
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:
The above material is presented to ensure ease of reference and timely dissemination of
scholarly and technical
work.
Copyright and all rights therein are retained by original copyright holders.
Back to the
homepage of Daniel
Liberzon