Nonlinear Control Abstracts

         Rodolphe Sepulchre                   Eduardo D. Sontag
     Universite de Liege, Belgium          Rutgers University, USA      

We expect NCA to become a routine method for publicizing and 
communicating recent advances in the field, promoting a faster 
diffusion of new results and allowing for comments and exchanges 
prior to the publication (or even formal journal submission) of papers.
Submissions will be numbered and dated; thus, a "time-stamp" will be 
available for preprint announcements. The papers themselves will be 
available through Web links, or anonymous FTP, from authors' sites 
(we will not, at least for now, provide an archive site for preprints.) 

The focus of NCA is on theoretical contributions, but its scope also 
includes illustrative applications of nontrivial theoretical work, in 
the broad field of nonlinear control theory.

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Issue No. 1, March, 1998

TITLE: A passivation approach to power systems stabilization
AUTHORS: R. Ortega, A. Stankovi\'c and P. Stefanov 
         Lab des Signaux et Systemes, Sup\'{e}lec, 
         Northeastern University

ABSTRACT: In this paper we address the problem of supression of low
frequency oscillations in power systems. These oscillations
appear in strongly interconnected networks because of load
and topology changes, and they may cause loss of synchronism
and generator tripping. We propose the utilisation of passivation 
techniques to design power system stabilizers for the synchronous generators.
The generator to be controlled is described by a standard lagrange model, with 
three forcing terms: the mechanical
torque coming from the turbine, the terminal voltage of the network and 
the field voltage, which is our control variable. In view of the significant
differences between the mechanical and the electrical
time scales, the first signal can be treated as a constant disturbance.
The  terminal voltage may be viewed as the output of an operator, --defined 
by the remaining part of the network--, which is in feedback interconnection 
with the generator. Our basic assumption is that the network is always 
absorbing energy from the generator, whence the interconnection subsystem
(as viewed from the generator) is passsive. The control objective is then 
to close a loop around the  field voltage so as to passivize the generator
system. We characterize, in terms of a simple linear matrix inequality, a 
class of linear state--feedback controllers which achieve this objective.

NCA 1-2-980120

TITLE: On Hybrid Control of Nonlinear Systems under Slow Sampling: 
       Application to  Induction Machines
AUTHORS: H. Ludvigsen, R. Ortega, P. Albertos and O. Egeland
         Lab des Signaux et Systemes, Sup\'{e}lec

ABSTRACT: We study here the problem of controlling continuous--time
nonlinear dynamical systems whose inputs live on a finite set. Our main
concern is the situation when, due to technological or information 
transmition considerations, fast switching is {\em not possible}.  To 
compensate for the delay  we   incorporate into our scheme a {\em predictive}  
and    minimize the {\em average} behaviour of a Lyapunov function. This is 
of particular interest when the zero dynamics are {\em periodic}. Also, if 
during the transient we   cannot decrease the original Lyapunov function,
we propose  to switch to a partial one.   Instrumental for our study is a 
novel characterization  in {\em input space} of switching (and in particular 
sliding mode) control.  The new scheme is applied to the practically 
important problem of direct torque and flux control of induction motors, for
which a complete stability analysis is carried out.

NCA 1-3-980120

TITLE: 	Adaptive Controller Design and Disturbance Attenuation for
	SISO Linear Systems with Noisy Output Measurements 
AUTHORS: ZIGANG PAN, Polytechnic University   
	 TAMER BASAR, University of Illinois
KEYWORDS: 	Robust adaptive control; linear uncertain systems; 
		nonlinear H-infinity control; worst-case identification.
PS file retrieval: contact author by email for file

We address the worst-case adaptive controller design problem for
uncertain single-input single-output linear systems with noisy output
measurements, under the assumption that the (parametrically) unknown
system is minimum phase with a known relative degree and unknown
high-frequency gain of known sign.  Adopting a game-theoretic
framework, we first formulate this adaptive control problem as a
nonlinear H-infinity control problem with imperfect state measurements
-- a formulation that accommodates transient and asymptotic
performance, as well as robustness.  By employing  the cost-to-come
function analysis and integrator backstepping methodology, we derive
explicit expressions for the worst-case adaptive controllers.  By
utilizing the a priori knowledge of the bounded convex set where the
true parameters belong, we introduce soft projection in the parameter
estimates, which then guarantees the bounded-input bounded-output
property of the closed-loop system with respect to exogenous
disturbance inputs  without any assumption of persistency of excitation
of the reference signal.  We present a systematic optimality-guided
robust adaptive control design process that encompasses
parameter-identification, state-estimation, and nonlinear controller
design in a unified framework.


Title:  Comments on integral variants of ISS
Author(s):  Eduardo Sontag (Rutgers)
Email contact:
Keywords: input to state stability, Lyapunov functions, input/output stability
This note discusses two integral variants of the input-to-state stability
(ISS) property, which represent nonlinear generalizations of L2 stability, in
much the same way that ISS generalizes L-infinity stability. Both variants are
equivalent to ISS for linear systems. For general nonlinear systems, it is
shown that one of the new properties is strictly weaker than ISS, while the
other one is equivalent to it. For bilinear systems, a complete
characterization is provided of the weaker property. An interesting fact about
functions of type KL is proved as well.

Title:  A Lyapunov characterization of robust stabilization
Author(s):  Yu.S. Ledyaev (Western Michigan), Eduardo Sontag (Rutgers)
Email contact:
Keywords: Lyapunov functions, stabilization, actuator noise
One of the fundamental facts in control theory (Artstein's theorem) is the
equivalence, for systems affine in controls, between continuous feedback
stabilizability to an equilibrium and the existence of smooth control Lyapunov
functions. This equivalence breaks down for general nonlinear systems, not
affine in controls. One of the main results in this paper establishes that the
existence of smooth Lyapunov functions implies the existence of (in general,
discontinuous) feedback stabilizers which are insensitive to small errors in
state measurements.  Conversely, it is shown that the existence of such
stabilizers in turn implies the existence of smooth control Lyapunov
functions. Moreover, it is established that, for general nonlinear control
systems under persistently acting disturbances, the existence of smooth
Lyapunov functions is equivalent to the existence of (possibly) discontinuous)
feedback stabilizers which are robust with respect to small measurement errors
and small additive external disturbances.

Title:  Meagre functions and asymptotic behaviour of dynamical systems
Author(s):  W.Desch (Graz), H.Logemann and E.P.Ryan (Bath), E. Sontag (Rutgers)
Email contact:
Keywords: stability, integral invariance
A measurable function x from a subset J of R into a metric space X is said to
be C-meagre if C is non-empty subset of X and, for every closed subset K of X
disjoint from C, the preimage of K under x has finite Lebesgue measure. This
concept of meagreness, applied to trajectories, is shown to provide a unifying
framework which facilitates a variety of characterizations, extensions or
generalizations of diverse facts pertaining to asymptotic behaviour of
dynamical systems.

Title:  Remarks on universal nonsingular controls for discrete-time systems
Author(s):  F. Wirth (Bremen), Eduardo Sontag (Rutgers)
Email contact:
Analytic discrete-time systems, forward accessibility, universal controls
For analytic discrete-time systems, it is shown that uniform forward
accessibility implies the generic existence of universal nonsingular control
sequences. A particular application is given by considering forward accessible
systems on compact manifolds. For general systems, it is proved that the
complement of the set of universal sequences of infinite length is of the
first category. For classes of systems satisfying a descending chain
condition, and in particular for systems defined by polynomial dynamics,
forward accessibility implies uniform forward accessibility.

TITLE: Semiglobal Stabilization in the Presence of Minimum-Phase Dynamic Input
AUTHORS: L. Praly, CAS Ecole des Mines and Z.-P. Jiang, Department of
Electrical Engineering, University of Sydney
KEYWORDS: robust control, dynamic input uncertainties, output feedback

ABSTRACT: This paper presents a dynamic state feedback approach to the
semiglobal stabilization of nonlinear systems with minimum-phase dynamic
input uncertainties.  The assumption needed to get this new result is
weaker in one direction than the assumption of input feedback passivity or
that of nonlinear small gain.

TITLE: Further Results on Robust Semiglobal Stabilization with Dynamic
Input Uncertainties
AUTHORS: L. Praly, CAS Ecole des Mines and Z.-P. Jiang, Department of
Electrical Engineering, University of Sydney
KEYWORDS: robust control, dynamic input uncertainties, output feedback
ABSTRACT: This paper presents a dynamic state feedback approach to the
semiglobal stabilization of nonlinear systems with minimum-phase dynamic
input uncertainties. The assumption needed to get this new result is weaker
than the assumption of input feedback passivity or that of nonlinear small
gain considered up to now. Here we show how the result proposed in the
previous paper
can be extended to the general relative degree case. For ease of
presentation, we restrict ourselves the single input single output case.

TITLE: Slow peaking and low-gain designs for global stabilization
of nonlinear systems
AUTHORS: R. Sepulchre, University of Liege, Belgium
KEYWORDS: global stabilization, low-gain designs, homogeneous approximations
ABSTRACT:  This paper addresses  the  global  stabilization of a chain of
integrators perturbed by a vector field u p(x,u) which satisfies p(x,0)=0.
The paper characterizes  the growth conditions to be imposed on the perturbation
to guarantee global statilizability.  They  are expressed as a  higher-order
condition with respect to a  particular  weighted dilation related to the
peaking exponents of the nominal system. An explicit control law is given
which achieves global asymptotic statbility when the growth restrictions are 
The basic result is then extended to more general cascade systems. A tight
 illustration of the result is given on the  popular frictionless ball-and-beam 

TITLE: A new asymptotic stability criterion for non-linear time-variant
differential equations
AUTHORS: Dirk Aeyels, Joan Peuteman, SYSTeMS, Universiteit Gent 
KEYWORDS:Nonlinear dynamics, Time-varying systems, Liapunov, Linear pendulum.
ABSTRACT: A new sufficient condition for asymptotic stability of ordinary
differential equations is proposed. Unlike classical Liapunov theory, the
time derivative along solutions of the Liapunov function may take positive
and negative values. The classical Liapunov approach may be regarded as an
infinitesimal version of the present theorem. Verification in practical
problems is harder than in the classical case; an example is included in
order to indicate how the present theorem may be applied.

Issue No. 2, April, 1998

Title: Impulse-Free Output Regulation of Singular Nonlinear Systems
Authors: Jie Huang, The Chinese University of Hong Kong 
Ji-Feng Zhang, The Chinese Academy of Science  
Keywords: Singular nonlinear systems, output regulation, center manifold 
Abstract: This paper addresses the output regulation problem for a 
class of singular nonlinear systems, that is, designing of 
feedback controls for a class of singular nonlinear systems such that 
the output of the closed-loop system can track a class of reference inputs 
and reject a class of disturbances asymptotically. Here both the reference 
inputs and the disturbances are generated by an autonomous differential 
equation. A generalized version of the center manifold theorem that 
applies to singular nonlinear systems is established 
first. Then necessary and sufficient conditions are given for the solvability 
of the output regulation problem by singular feedback controls. 
Finally,  the output regulation problem by 
a normal feedback control is addressed. 
PS file: available upon request.

NCA 2-2-980309

TITLE: Sufficient Conditions for a Dynamical System to Possess an
       Unbounded Solution
AUTHORS: R. ORSI, University of Melbourne
         L. PRALY, CAS Ecole des Mines
         I. MAREELS, University of Melbourne
KEYWORDS: unbounded solutions, global stabilizability.
ABSTRACT: In this paper we present readily verifiable conditions under 
which a dynamical system of the form $\dot{x}=f(x)$ possesses an 
unbounded solution. This result is illustrated by showing it can be 
used to infer results about lack of global stabilizability for nonlinear 
control systems. The key observation in the paper is that behaviour at 
infinity can be studied using local methods applied to an auxiliary 
NCA 2-3-980316

TITLE: Input-to-State Stabilization of Linear Systems with Positive Outputs
AUTHORS: Dragan Nesic and Eduardo D. Sontag
         University of California, Santa Barbara; 
         Rutgers University, New Jersey.
ABSTRACT: This paper considers the problem of stabilization of linear 
systems for which only the magnitudes of outputs are measured. It is shown
that, if a system is controllable and observable, then one can find a 
stabilizing controller, which is robust with respect to observation
noise (in the ISS sense). The controller is sampled-data, periodic and in 
the absence of disturbances it yields a dead-beat behavior.


TITLE: A note on exponential stability of partially slowly time-varying
nonlinear systems
AUTHORS: Joan Peuteman, Dirk Aeyels, SYSTeMS, Universiteit Gent,
Technologiepark-Zwijnaarde 9, 9052 GENT (Zwijnaarde), BELGIUM
KEYWORDS: Nonlinear dynamics, Exponential stability, Circle criterion,
Linear pendulum
ABSTRACT: Consider a system $\dot{x} = f(x,t,\frac{t}{\beta})$ with a
time-varying vectorfield which contains a regular and a slow time scale
($\beta$ large). Assume there exists $\alpha(\tau)$ such that $\| x_{\tau}
(t,t_0,x_0) \| \leq K(\tau) \|x_0\| e^{\alpha(\tau) (t-t_0)}$ where
$x_{\tau} (t,t_0,x_0)$ is the solution of the system $\dot{x} = f(x,t,\tau)$
with initial state $x_0$ at $t_0$. We show that for $\beta$ sufficiently
large, $\dot{x} = f(x,t,\frac{t}{\beta})$ is exponentially stable when the
average of $\alpha(\tau)$ is negative. This result can be used to extend the
circle criterion i.e. to obtain a sufficient condition for exponential
stability of a feedback interconnection of a slowly time-varying linear
system and a sector nonlinearity. An example is included which shows that
the technique can be used to obtain an exponential stability result for a
pendulum with a nonlinear partially slowly time-varying friction attaining
positive and negative values.

Issue No. 3, May, 1998

TITLE:  Symmetries of flat rank two distributions and
        sub-Riemannian structures
AUTHOR: YU. L. SACHKOV, Program Systems Institute
KEYWORDS: sub-Riemannian geometry, symmetries, distributions,
sub-Riemannian structures.
ABSTRACT: Flat sub-Riemannian structures are local approximations ---
nilpotentizations --- of sub-Riemannian structures at regular points. Lie
algebras of symmetries of flat maximal growth distributions
and sub-Riemannian structures of rank two are computed in dimensions
3, 4, and 5.
NCA 3-2-980408 

TITLE:  Survey on Controllability of Invariant Systems on Solvable Lie Groups
AUTHOR: YU. L. SACHKOV, Program Systems Institute
KEYWORDS: Controllability, right-invariant systems, Lie groups,
solvable, Lie semigroups, bilinear systems
ABSTRACT: Known and new results on controllability of right-invariant
systems  on solvable Lie groups are presented and discussed. The main
ideas and technique used are outlined, illustrating examples are given.
Some open questions are suggested.
NCA 3-3-980502

TTITLE: Global stabilization of nonlinear systems using bounded control: A
        parametric optimization approach.
AUTHORS: J. Solis-Daun, R. Suarez and J. Alvarez-Ramirez.
        Division de Ciencias Basicas e Ingenieria,
        Universidad Autonoma Metropolitana-Iztapalapa,,,
KEYWORDS: bounded control, nonlinear systems, global stabilization,
 parametric programming.
ABSTRACT: A bounded feedback control is designed for the global asymptotic
stabilization of affine systems whose free dynamics are Lyapunov stable. In
general, the resulting procedure implies that gains, as state-functions, are
obtained from the solution of a c-parameterized nonlinear programming
problem. Special interest is focused on an important class of homogeneous
systems and on bilinear systems. In many cases (e.g.,
non-homogeneous bilinear systems), the resulting closed-loop system is 
implicitly defined, i.e., it consists of a system of
differential equations plus a nonlinear algebraic equation (required to
compute the control). Concerning the mentioned class of homogeneous systems
(that includes a class of globally asymptotically stabilizable systems by
linear feedback), the problem of inputs subject to global bounded rate is
also addressed.

NCA 3-4-98502

TITLE:   CLF Based Designs with Robustness to Dynamic Input Uncertainties 
AUTHORS: Mrdjan Jankovic (Ford Scientific Research Laboratories)
         Rodolphe Sepulchre (University of  Liege)
         Petar V.  Kokotovic(University of California, Santa Barbara)
KEYWORDS: stability margins, inverse optimality, Sontag's formula
ABSTRACT: The problem of robust stabilization of nonlinear systems in the 
presence of input uncertainties is of great importance in practical 
implementation.  Stabilizing control laws may not be robust to this type of 
uncertainty, especially if cancellation of nonlinearities is used in the 
design. By exploiting a connection between robustness and optimality, 
``domination redesign'' of the control Lyapunov function (CLF) based Sontag's 
formula has been shown  to possess robustness to static and dynamic  
input uncertainties. In this paper we provide a sufficient condition for
the domination redesign to apply. This condition relies on  properties
of  local approximations of the system and of the CLF. We show that a 
domination redesign may not exist when these conditions are
violated and illustrate how these conditions may guide  the choice of a CLF
which is suitable for domination redesign.


Issue No. 4, June, 1998


TITLE:  Input-to-state stability of exponentially stabilized semilinear control
        systems with inhomogeneous perturbations
        Universita di Roma "La Sapienza", Italy and
        J.-W.-Goethe Universitaet Frankfurt a.M., Germany
KEYWORDS: input-to-state stability, stabilizing Feedback control, robustness
ABSTRACT: In this paper we investigate the robustness of state feedback
stabilized semilinear systems subject to inhomogeneous perturbations in terms
of input-to-state stability. We consider a general class of exponentially
stabilizing feedback controls which covers sampled discrete feedbacks and
discontinuous mappings as well as classical feedbacks and derive a necessary
and sufficient condition for the corresponding closed loop systems to be 
input-to-state stable with exponential decay and linear dependence on the
perturbation. This condition is easy to check and admits a precise estimate for
the constants involved in the input-to-state stability formulation. Applying
this result to an optimal control based discrete feedback yields an equivalence
between (open loop) asymptotic null controllability and robust input-to-state
(state feedback) stabilizability.
NCA 4-2-980530

TITLE: Robust Control of a Class of Uncertain Nonlinear Systems in
Strict-Feedback Form.
AUTHORS: J. Alvarez-Ramirez, R. Suarez and R. Femat.
        Division de Ciencias Basicas e Ingenieria,
        Universidad Autonoma Metropolitana-Iztapalapa, M=E9xico.
KEYWORDS: Nonlinear control  systems;  Adaptive  control;  Robust  control;
Strict-feedback form.
ABSTRACT: A new design procedure for robust  control  of  nonlinear  systems
transformable into the strict-feedback  form  is  proposed.  The  controller
comprises  a  robust  globally  bounded  state  feedback   and  a  high-gain
observer. A characteristic of the controller is that it provides an estimate
of the matched uncertainties. The proposed control scheme and the stability
results  are  also  applicable  to single-input,  nonlinear  systems  that 
are  transformable  into   feedback linearizable systems. Examples  are  
presented to demonstrate the performance of the proposed control algorithm.

Issue No. 5, July, 1998

TITLE:   Energy--based Lyapunov functions for forced Hamiltonian systems
with dissipation 

AUTHORS: B. Maschke, R. Ortega  and A. J. van der Schaft 

ABSTRACT: It is well known that  the total energy is a suitable Lyapunov
function to study the stability of the trivial equilibrium of an isolated
standard Hamiltonian system. In many practical instances, however, the
system is in interaction with its environment through some forcing terms.
This gives rise to what we call forced Hamiltonian systems, for which the 
equilibria of interest are  now different from zero. When the system is 
linear a Lyapunov function can be immediately obtained by simply shifting 
the  coordinates in the total energy. However, for nonlinear systems there 
is no guarantee that this incremental energy  is, not even locally, a 
Lyapunov function. In  this paper we propose a constructive procedure to 
modify the total energy function of forced Hamiltonian systems with 
dissipation in order to generate Lyapunov functions for non--zero equilibria.
 A key step in the procedure, which is motivated from energy--balance 
considerations standard in network modeling of physical systems, is to 
embed the system into a larger Hamiltonian system for which a series of 
Casimir functions can be easily constructed. Interestingly  enough, for 
linear systems the resulting Lyapunov function is the incremental energy, 
thus our derivations provide a physical explanation to it. An easily 
verifiable necessary and sufficient condition for the applicability of 
the technique in the general nonlinear case is given.  Some examples that
illustrate the method are given.
NCA 5-2-980604 

TITLE:  Necessary and sufficient conditions for passivity of the LuGre
friction model  

AUTHORS:N. Barabanov and R. Ortega  

ABSTRACT: Friction is a nonlinear phenomenon difficult to describe 
analytically. To capture its effect  in  mechanical systems a
bristle--based dynamical model, known as the LuGre model, was  recently 
proposed in the literature. It is difficult to assess whether this 
(or any other) mathematical model constitutes a  bona fide friction 
model. It should however reflect the dissipative nature of friction, which 
mathematically translates into the requirement of defining a passive 
operator from velocity to friction force. In this paper we give necessary and
sufficient conditions for this property to hold for the LuGre model.
The conditions are expressed in terms of a simple  algebraic inequality 
involving the parameters of the model. If this inequality does not hold 
we construct an input signal that generates a periodic orbit along which the
passivity inequality is violated.
NCA 5-3-980605

Title: Analysis of the Local Robustness of Stability for Flows
Author(s):  A.D.B.~Paice, (ABB Corporate Research, Baden)
            F. Wirth (University of Bremen)
Email contact:
Keywords: Robust stability, genericity, semi-algebraic sets

Abstract: In this paper the problem of measuring the robustness of
stability for a perturbed continuous time nonlinear system at a
singular fixed point is studied.  Various stability radii are
introduced and their values for the nonlinear system and its
linearization are compared. It is shown that they generically
coincide.  This result may also be used to show generic continuity of
linear real stability radii.
Some examples are presented showing that it is sometimes necessary to
consider the nonlinear system directly, and not simply to rely on the
information provided by the linearization.
NCA 5-4-980617

TITLE: Towards the supervisory control of uncertain nonholonomic systems
KEYWORDS: nonholonomic system; modeling uncertainty; hybrid feedback 
control; estimator-based supervisor.
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. The proposed
procedure seems to have several advantages over conventional adaptive

Issue No. 6, August, 1998

TITLE:  Overview of complexity and decidability results for three classes of
elementary nonlinear systems  

AUTHORS: Vincent D. Blondel and John N. Tsitsiklis

ABSTRACT: It has become increasingly apparent this last decade that many 
problems in systems and control are NP-hard and, in some cases, undecidable. 
The inherent complexity of some of the most elementary problems in systems and 
control points to the necessity of using alternative approximate techniques to 
deal with  problems that are unsolvable or intractable when exact solutions 

We survey some of the decidability and complexity results available for three 
classes of discrete time nonlinear systems.  In each case, we draw the line
between the problems that are unsolvable, those that are NP-hard, and those 
which polynomial time algorithms are known.
NCA 6-2-980722 

TITLE:  A characterization of integral input to state stability

AUTHORS: D. Angeli, E.D. Sontag, Y. Wang

Just as input to state stability (ISS) generalizes the idea of finite gains
with respect to supremum norms, the new notion of integral input to state
stability (IISS) generalizes the concept of finite gain when using an
integral norm on inputs.
In this paper, we obtain a necessary and sufficient characterization of the
IISS property, expressed in terms of dissipation inequalities.
NCA 6-3-980722

TITLE:  VC Dimension of Neural Networks

AUTHORS: E.D. Sontag

ABSTRACT: The Vapnik-Chervonenkis (VC) dimension is an integer which helps to
characterize distribution-independent learning of binary concepts from
positive and negative samples.
This paper, based on lectures delivered at the Isaac Newton Institute in
August of 1997, presents a brief introduction, establishes various elementary
results, and discusses how to estimate the VC dimension in several examples of
interest in neural network theory.
(It does not address the learning and estimation-theoretic applications of
VC dimension, and the applications to uniform convergence theorems for
empirical probabilities, for which many suitable references are available.)
NCA 6-4-980722

TITLE:  Further results on controllability of recurrent neural networks

AUTHORS: Y. Qiao, E.D. Sontag

ABSTRACT: This paper studies controllability properties of recurrent neural
networks.  The new contributions are: (1) an extension of the result in a
previous paper ("Complete controllability of continuous-time recurrent neural
networks", with Sussmann) to a slightly different model, where inputs appear
in an affine form, (2) a formulation and proof of a necessary and sufficient
condition, in terms of local-local controllability, and (3) a complete
analysis of the 2-dimensional case for which the hypotheses made in previous
work do not apply.
NCA 6-5-980722

TITLE:  A learning result for continuous-time recurrent neural networks

AUTHOR: E.D. Sontag

The following learning problem is considered, for continuous-time recurrent
neural networks having sigmoidal activation functions.  Given a ``black box''
representing an unknown system, measurements of output derivatives are
collected, for a set of randomly generated inputs, and a network is used to
approximate the observed behavior.  It is shown that the number of inputs
needed for reliable generalization (the sample complexity of the learning
problem) is upper bounded by an expression that grows polynomially with the
dimension of the network and logarithmically with the number of output
derivatives being matched.
NCA 6-6-980722

TITLE:  Recurrent neural networks: Some systems-theoretic aspects

AUTHORS: E.D. Sontag

ABSTRACT: This paper provides an exposition of some recent results regarding
system-theoretic aspects of continuous-time recurrent (dynamic)
neural networks with sigmoidal activation functions.
The class of systems is introduced and discussed, and a result is cited
regarding their universal approximation properties.
Known characterizations of controllability, observability, and parameter
identifiability are reviewed, as well as a result on minimality.
Facts regarding the computational power of recurrent nets are also
NCA 6-7-980817

TITLE:	  Finite gain stabilization of discrete-time linear systems
	  subject to actuator saturation

AUTHORS:  Xiangyu Bao, University of Virginia
	  Zongli Lin, University of Virginia
	  Eduardo D. Sontag, Rutgers University


KEYWORD:  Input saturation, discrete-time linear systems,
          finite gain stability, Lyapunov functions

ABSTRACT: It is shown that, for neutrally stable discrete-time linear 
systems subject to actuator saturation, finite gain $l_p$ stabilization 
can be achieved by linear output feedback, for all $p\in(1,\infty]$.
An explicit construction of the corresponding feedback laws is given. 
The feedback laws constructed also result in a closed-loop system 
that is globally asymptotically stable, and in an input-to-state estimate.

Issue No. 7, October, 1998

NCA 7-1-980908

TITLE: Regularity of Solutions of Burgers' Equation with Globally
 Stabilizing Nonlinear Boundary Feedback

AUTHORS: A. Balogh and M. Krstic


ABSTRACT: We consider the viscous Burgers equation under the
second author's recently proposed nonlinear boundary conditions
which guarantee global asymptotic stabilization and semiglobal
exponential stabilization in the H^1 sense. We show global existence
and uniqueness of classical solutions with initial data which are
assumed to be only in L^2. To do this, we establish a priori estimates
of up to four spatial and two temporal derivatives, and then employ
the Banach fixed point theorem to the integral representation with
a heat kernel. Our result is global in time and allows arbitrary size
of initial data. It strengthens recent results by Byrnes, Gilliam,
and Shubov, Ly, Mease, and Titi, and Ito and Yan. We include a
numerical result which illustrates the performance of the boundary
NCA 7-2-980908

TITLE: Global Boundary Stabilization of the Korteweg-de Vries-Burgers

AUTHORS: W.-J. Liu and M. Krstic


ABSTRACT: The problem of global exponential stabilization by  boundary
feedback for the Korteweg-de Vries-Burgers equation on the domain
[0,1] is considered. We derive a control law of the form
            u(0)=u_x(1)= u_{xx}(1)-k[u(1)^3+u(1)]=0,
where k is a sufficiently large positive constant, and prove that it
guarantees L^2-global exponential stability, H^1-global asymptotic
stability, and H^1-semiglobal exponential stability. The closed-loop
system is shown to be well posed.
NCA 7-3-980908

TITLE: Stability Enhancement by Boundary Control in the
Kuramoto-Sivashinsky Equation

AUTHORS: W.-J. Liu and M. Krstic


ABSTRACT: We address the problem of Dirichlet and Neumann
boundary control of the Kuramoto-Sivashinsky equation on the
domain [0,1]. First we note that, while the uncontrolled Dirichlet
problem is asymptotically stable when an ``anti-diffusion"
parameter is small, and unstable when it is large (we determine
the critical value of the parameter), the uncontrolled Neumann
problem is never asymptotically stable. We develop a Neumann
feedback law that guarantees L^2-global exponential stability and
H^2-global asymptotic stability for small values of the
anti-diffusion parameter. The more interesting problem of boundary
stabilization when the anti-diffusion parameter is large remains
open. Our proof of global existence and uniqueness of solutions
of the closed-loop system involves construction of a Green
function and application of  the Banach contraction mapping principle.
NCA 7-4-980909

TITLE: On integral-input-to-state stabilization

AUTHORS: Daniel Liberzon (Yale University), Eduardo D. Sontag (Rutgers
University), Yuan Wang (Florida Atlantic University)

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 an appropriate concept of control
Lyapunov function (iISS-CLF), whose existence leads to an explicit
construction of such a control law.  The results are compared and
contrasted with the ones available for the ISS case. 


Issue No. 8, November, 1998

NCA 8-1-981015

TITLE: Ignored Input Dynamics and a new Characterization of Control
Lyapunov Functions

AUTHORS: B. Hamzi and L. Praly, CAS Ecole des Mines


KEYWORDS: robust control, dynamic input uncertainties, output feedback
stabilization, control Lyapunov functions, dissipativity

ABSTRACT: Our objective in this paper is to extend as much as possible the
dissipativity approach for the study of robustness of stability in
the presence of known/unknown but ignored input dynamics. This leads
us to:

- give a new characterization of control Lyapunov functions where
  ``$L_fV$'' is upper-bounded by a function of ``$L_gV$'',

- define the dissipativity approach as:
  - assuming the ignored dynamics are dissipative with storage
    function $W$ and (known) supply rate $w$,
  - analyzing closed-loop stability with the sum of the storage
    function $W$ and a CLF for the nominal part.

Stability margin is given in terms of an inequality the supply rate
should satisfy. However, in spite of this extension, the dissipativity
approach cannot cope with ignored dynamics which would have non zero
relative degree or would be non minimum phase.
NCA 8-2-981026

TITLE: Stability and stabilization:
       Discontinuities and the effect of disturbances 

AUTHOR: Eduardo D. Sontag


ABSTRACT: In this expository paper, we deal with several questions related to
stability and stabilization of nonlinear finite-dimensional continuous-time
systems. We review the basic problem of feedback stabilization, placing an
emphasis upon relatively new areas of research which concern stability with
respect to "noise" (such as errors introduced by actuators or sensors).
The table of contents is as follows:
Review of Stability and Asymptotic Controllability
The Problem of Stabilization
Obstructions to Continuous Stabilization
Control-Lyapunov Functions and Artstein's Theorem
Discontinuous Feedback
Nonsmooth CLF's
Insensitivity to Small Measurement and Actuator Errors
Effect of Large Disturbances: Input-to-State Stability
Comments on Notions Related to ISS
NCA 8-3-981026

TITLE: A polynomial-time algorithm for state equivalence in hybrid PL systems

AUTHORS: Bhaskar DasGupta and Eduardo D. Sontag


ABSTRACT: The area of hybrid systems concerns issues of modeling, computation,
and control for systems which combine discrete and continuous components.
The subclass of piecewise linear (PL) systems provides one systematic approach
to discrete-time hybrid systems, naturally blending switching mechanisms with
classical linear components.
PL systems model arbitrary interconnections of finite automata and linear
systems.  Tools from automata theory, logic, and related areas of computer
science and finite mathematics are used in the study of PL systems, in
conjunction with linear algebra techniques, all in the context of a "PL
algebra" formalism.
PL systems are of interest as controllers as well as identification models.
Basic questions for any class of systems are those of equivalence, and, in
particular, if state spaces are equivalent under a change of variables.
This paper studies this state-space equivalence problem for PL systems. The
problem was known to be decidable, but its computational complexity was
potentially exponential; here it is shown to be solvable in polynomial-time.
NCA 8-4-981113

TITLE:  A Survey of Computational Complexity Results in Systems and Control  

AUTHORS: Vincent D. Blondel and John N. Tsitsiklis

ABSTRACT: The purpose of the paper is twofold: 
(a) to provide a tutorial introduction to some key concepts from the theory
of computational complexity, highlighting their relevance to systems and
control theory, and
(b) to survey the relatively recent research activity lying at the interface
between these fields.
We begin with a brief introduction to models of computation, the concepts of
undecidability, polynomial time algorithms, NP-completeness, and the
implications of intractability results.
We then survey a number of problems that arise in systems and control
theory, some of them classical, some of them related to current research. 
We discuss them from the point of view of computational complexity and also
point out many open problems. In particular, we consider problems related to
stability or stabilizability of linear systems with parametric uncertainty,
robust control, time-varying linear systems, nonlinear and hybrid systems,
and stochastic optimal control.

Issue No. 9, December, 1998


NCA 9-1-981126

TITLE:  On the rate of convergence of infinite horizon discounted
        value functions

        Universita di Roma "La Sapienza", Italy and
        J.W. Goethe-Universitaet Frankfurt a.M., Germany,
        Universitaet Bremen, Germany


KEYWORDS: Nonlinear optimal control, optimal value functions, 
          rate of convergence

ABSTRACT: In this paper we investigate the rate of convergence for the optimal
value function of an infinite horizon discounted optimal control problem as
the discount rate tends to zero. Using the Integration Theorem for Laplace
transformations we provide conditions on averaged functionals along suitable
trajectories yielding at most quadratic pointwise convergence.  Under
appropriate controllability assumptions from this we derive criteria for at
most linear uniform convergence on control sets. Applications of these results
are given and an example is discussed in which both linear and slower rates of
convergence occur. or:

NCA 9-2-981203


        Universita di Roma "La Sapienza", Italy and
        J.-W. Goethe-Universitaet Frankfurt a.M., Germany          


KEYWORDS: Homogeneous system, state feedback stabilization, 
          control Lyapunov functions, Lyapunov exponents

We show that for any asymptotically controllable homogeneous system in
euclidian space (not necessarily Lipschitz at the origin) there exists a
homogeneous control Lyapunov function and a homogeneous, possibly
discontinuous state feedback law stabilizing the corresponding sampled closed
loop system.  If the system satisfies the usual local Lipschitz condition on
the whole space we obtain semi-global stability of the sampled closed loop
system for each sufficiently small fixed sampling rate, if the system
satisfies a global Lipschitz condition we obtain global exponential stability
for each sufficiently small fixed sampling rate.  The control Lyapunov
function and the feedback are based on the Lyapunov exponents of a suitable
auxiliary system and admit a numerical approximation.

NCA 9-3-981203

TITLE: Asymptotic stability equals exponential stability, 
       and ISS equals finite energy gain - if you twist your eyes  

AUTHORS: L. Gruene, E.D. Sontag and F.R. Wirth


KEYWORDS: asymptotic stability, exponential stability, input-to-state 
          stability, nonlinear H-infinity

ABSTRACT: In this paper we show that uniformly global asymptotic stability for
a family of ordinary differential equations is equivalent to uniformly global
exponential stability under a suitable nonlinear change of variables.  The
same is shown for input-to-state stability and input-to-state exponential
stability, and for input-to-state exponential stability and a nonlinear
H-infinity estimate.

Issue No. 10, January, 1999

NCA 10-1-990105

TITLE: Benchmark problems in stability and design of switched systems
AUTHORS: Daniel Liberzon and A. Stephen Morse (Yale University)
ABSTRACT: A switched system is a hybrid dynamical system consisting of a
family of continuous-time subsystems and a rule that governs the
between them. This paper 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.
NCA 10-2-99-0110

TITLE: Notions of Input to Output Stability
AUTHORS: Eduardo Sontag and Yuan Wang
KEYWORDS: input/output stability, ISS, nonlinear control, robust
stability, partial stability
ABSTRACT:  This paper deals with several related notions of output
stability with respect to inputs (which may be thought of as disturbances).
The main such notion is called input to output stability (IOS), and it
reduces to input to state stability (IDD) when the output equals the complete
For systems with no inputs, IOS provides a generalization of the classical
concept of partial stability.
everal variants, which formalize in different manners the transient
behavior, are introduced.
The main results provide a comparison among these notions.

NCA 10-3-990110

TITLE: Lyapunov Characterizations of Input to Output Stability
AUTHORS: Eduardo Sontag and Yuan Wang
KEYWORDS: Lyapunov functions, output stability, ISS, robust control
ABSTRACT: This paper presents necessary and sufficient characterizations of
several notions of input to output stability.
Similar Lyapunov characterizations have been found to play a key role in the
analysis of the input to state stability property, and the results given here
extend their validity to the case when the output, but not necessarily the
entire internal state, is being regulated.

NCA 10-4-990121

TITLE:  Controllability of Invariant Systems on Lie Groups and their
Homogeneous Spaces
AUTHOR: YU. L. SACHKOV, Program Systems Institute, Pereslavl-Zalessky,
KEYWORDS: Controllability, right-invariant systems, Lie groups,
Lie semigroups, homogeneous spaces, affine systems, bilinear systems
ABSTRACT:  The aim of this work is to give a comprehensive survey of
results on controllability of right-invariant control systems on Lie
groups and their homogeneous spaces.
This subject is an area of active research in the mathematical control
theory and the Lie semigroup theory during the last 25 years. The
motivations for this study are diverse: applications in mechanics and geometry,
connections with other important classes of nonlinear control systems
(bilinear and affine), the work on generalization of S. Lie's theory from the
group case to the semigroup case.

The contents of the survey is as follows:
1. Introduction
2. Definitions and general properties of right-invariant systems
3. Control systems subordinated to a group action
4. Lie saturate
5. Homogeneous systems
6. Compact Lie groups
7. Semidirect products of Lie groups
8. Semisimple Lie groups
9. Nilpotent Lie groups
10. Products of Lie groups
11. Lie groups with cocompact radical
12. Hypersurface systems
13. Completely solvable Lie groups
14. Lie groups differing from their derived subgroups
15. Metabelian Lie groups
16. Small-dimensional simply connected solvable Lie groups
17. Final remarks
Index and
NCA 10-5-990129

TITLE: Supervisory Control of Integral-Input-to-State Stabilizing

AUTHORS: Joao P. Hespanha and A. Stephen Morse


ABSTRACT: A high-level supervisor, employing switching and logic, is
proposed to orchestrate the switching between a family of candidate
controllers into feedback with an imprecisely modeled process so as to
stabilize it.  Each of the candidate controllers is required to
integral-input-to-state stabilize one particular admissible process
model, with respect to a suitably defined disturbance input. The
controller selection is made by (i) continuously comparing in real time
suitably defined ``normed'' output estimation errors or ``performance
signals'' and (ii) placing in the feedback-loop, from time to time, that
candidate controller whose corresponding performance signal is the
smallest. The use of integral-input-to-state stability in the context of
supervisory control of nonlinear systems, allowed us to weaken the
requirements on the candidate controllers being used. It also seems
quite natural when the performance signals are defined as ``integral norms''
of the output estimation errors.

Issue No. 11, March, 1999

NCA 11-1-990201

TITLE: Stabilization of Port--Controlled Hamiltonian Systems:
Passivation and Energy--Balancing

AUTHORS: Romeo Ortega, Arjan J. van der Schaft, Bernhard Maschke and
Gerardo Escobar


ABSTRACT: Passivity--based control (PBC) is a well--established technique that
has shown to be very powerful to design robust controllers for physical
systems described by Euler--Lagrange (EL) equations of motion. The application
of PBC in regulation problems of mechanical systems yields controllers that
have a clear physical interpretation in terms of interconnection of the system
with its environment. In particular, the total energy of the closed--loop is
the difference between the energy of the system and the energy supplied by the
controller. Furthermore, since the EL structure is preserved in closed--loop,
PBC is robust {\em vis \'a vis} unmodeled dissipative effects.  These features
can hardly be overestimated in practical implementations. Unfortunately, these
nice properties are lost when PBC is used in other applications, for instance,
in electrical and electromechanical systems. Our main objective in this paper
is to develop a new PBC theory encompassing a broader class of systems, and
preserving the aforementioned energy--balancing stabilization mechanism and
the structure invariance.  Towards this end, we depart from the EL description
of the systems and consider instead port--controlled Hamiltonian (PCH) models,
which result >from the network modeling of energy-conserving lumped-parameter
physical systems with independent storage elements, and strictly contain the
class of EL models. There are two key advantages of working with PCH models
for PBC, firstly, they capture the physical constraints of the system more
directly, and secondly the structural obstacles for energy shaping and damping
injection are better revealed.  We identify a class of PCH models for which
PBC ensures the Hamiltonian structure is preserved, with storage function the
energy balance. One final advantage of the method is that it is rather
systematic and the controller can be easily derived using symbolic


NCA 11-2-990226

TITLE: Forward completeness, unboundedness observability,
       and their Lyapunov characterizations 
AUTHORS: David Angeli and Eduardo Sontag

KEYWORDS: stability properties, Lyapunov methods,
          global existence of solutions, observability

ABSTRACT: A finite-dimensional continuous-time system is forward complete if
solutions exist globally, for positive time.
  This paper shows that forward completeness can be characterized in a
necessary and sufficient manner by means of smooth scalar growth
  Moreover, a version of this fact is also proved for systems with
inputs, and a generalization is also provided for systems with outputs and a
notion (unboundedness observability) of relative completeness.
  We apply these results to obtain a bound on reachable states in terms
of energy-like estimates of inputs.
NCA 11-3-990304

TITLE: Disturbance attenuation with bounded controls

AUTHORS: Daniel Liberzon (Yale University)


ABSTRACT:  We consider the problem of achieving disturbance attenuation
the ISS and integral-ISS sense for nonlinear systems using bounded
controls. For the ISS case we derive a universal formula which extends
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.

Issue No. 12, April, 1999

NCA 12-1-990324

TITLE: Cheap Control Performance of a Class of Non-Right-Invertible
       Nonlinear Systems

AUTHORS: J.H. Braslavsky*, R. Middleton* & J.S. Freudenberg+
	 *Department of E&CE, The University of Newcastle, Australia
	 +Department of EECS, University of Michigan, Ann Arbor, USA

KEYWORDS: nonlinear optimal regulation, performance limitations,
	  near-optimal cheap control, strict-feedback systems

ABSTRACT: For strict-feedback nonlinear systems, this paper shows that
it is impossible to reduce to zero the optimal cost in the regulation
of more states than the number of control inputs in the system, even
using unrestricted control effort. By constructing a near optimal
cheap control law, we characterise the infimum value of the optimal
regulation cost as the optimal value of a reduced-order regulator
problem where the states with lower relative degree drive those with
higher relative degree. We illustrate our results with two examples of
practical interest: the optimal regulation of the rotational motion of
a free rigid body, and the optimal control of a magnetic suspension
NCA 12-2-990330

TITLE: Asymptotic Properties of Adaptive Controllers Revisited - 
`Do Convergent Parameter Estimation Schemes Lead, Generically,
to Stabilizing Parameter Estimates?'

AUTHOR: Stuart Townley


ABSTRACT: We consider the following fundamental question in adaptive 
control: Does an adaptive controller converge to a non-adaptive
stabilizing `limit controller'? 
This result is true for adaptive controllers which invoke a
persistency of excitation (PE) condition. However, it is not obvious
that the same would be true for adaptive controllers in the absence
of PE.  First we recall some positive results valid for classes of 
so-called universal adaptive controllers. For such classes, the limit 
controller is stabilizing for a generic set of initial conditions. 

However, this generic result is not true in general.
Specifically we show, for a class of back-stepping controllers with
adaptive tuning functions, that the set of initial conditions for which 
the limit controller is destabilizing can have non-zero Lebesgue

Krstic has claimed [in IEEE Trans. Aut. Control, 41, pp. 817-829, 
(1996), Theroem 5.1] that the set of initial conditions leading to 
destabilizing limit controllers has zero Lebesgue measure. Our result 
shows that this claim is not true in general. Note that this claim is 
partially true in the sense that the set of initial conditions leading 
to a limit system with an unstable linearization does have zero 
Lebesgue measure. 

For more details see the full paper ``An Example of a Globally 
Stabilizing Adaptive Controller with a Generically Destabilizing 
Parameter Estimate'' due for publication in IEEE Trans. Aut. Control., 
May 2000.>
NCA 12-3-992204

TITLE: Remarks on continuous feedback
AUTHORS: E. Sontag and H.J. Sussmann

This is a scan (postscript and tiff formats both available) of the paper 
in the Proc. IEEE Conf. Decision and Control, Albuquerque, Dec.1980,
 pp.916-921, which dealt with obstructions to continuous stabilization 
and the use of time-varying/dynamic feedback to avoid such obstructions 
(for one-dimensional systems).
It was recently placed on the web, by request, since the 20-yr old 
proceedings are not easily available to everyone, and since no journal 
paper with this material was ever submitted.

Issue No. 13, July, 1999

NCA 13-1-990618

TITLE: A smooth Lyapunov function from a class-KL estimate
       involving two positive semidefinite functions

AUTHORS: A.R. Teel, UC Santa Barbara and L. Praly, CAS Ecole des Mines.


KEYWORDS: Stability in two measures, differential inclusions, converse
Lyapunov functions, partial stability.

ABSTRACT: We consider differential inclusions where a positive semidefinite
function of the solutions satisfies a class-KL estimate in terms of time and a
second positive semidefinite function of the initial condition.  We show that
a smooth converse Lyapunov function, i.e., one whose derivative along
solutions can be used to establish the class-KL estimate, exists if and only
if the class-KL estimate is robust, i.e., it holds for a larger, perturbed
differential inclusion. It remains an open question whether all class-KL
estimates are robust. One sufficient condition for robustness is that the
original differential inclusion is locally Lipschitz.

Another sufficient condition is that the two positive semidefinite functions
agree and a backward completability condition holds.  These special cases
unify and generalize many results on converse Lyapunov theorems for
differential equations and differential inclusions that have appeared in the
literature.  For instance, we obtain the counterpart for differential
inclusions of the celebrated result of Kurzweil for continuous differential
equations. More generally our result applies for the case of compact and also
noncompact attractors, and partial asymptotic stability.



NCA 13-2-990701

TITLE:    A note on stability of arbitrarily switched homogeneous 
AUTHOR:   David Angeli (DSI, Firenze)

ABSTRACT: A family of arbitrarily switched systems is exponentially stable if
there exists positive constants M and c such that the solution at time t
satisfies an estimate of the following kind:

                  |x(t)| <= M exp(-ct) |x(0)|,

for all possible switching sequences. Clearly exponential stability implies
attractivity of the origin; we shot that for homogeneous systems, (and as a
special case for linear ones) the converse implication is also true.


NCA 13-3-990701

TITLE:    Intrinsic robustness of global asymptotic stability
AUTHOR:   David Angeli (DSI, Firenze)

ABSTRACT: Equivalence is shown for discrete time systems between global
asymptotic stability and the so called integral Input-to-State Stability.  The
latter is a notion of robust stability with respect to exogenous disturbances
which informally translates into the statement ''no matter what is the initial
condition, if the energy of the inputs is small, then the state must
eventually be small''.


NCA 13-5-990708

TITLE: A Separation Principle for the Control of a Class of Nonlinear

AUTHORS: A.N. Atassi*, H.K. Khalil+
*Department of ECE, University of California, Santa Barbara, USA
+Department of ECE, Michigan State University, East Lansing, USA


KEYWORDS: Output feedback, Separation principle, High-gain observers,
Convergence to a set.

ABSTRACT: In this paper we give a general formulation of a number of control
problems. This formulation considers a wide class of systems and any globally
bounded state feedback controller that renders a certain compact set
positively invariant and asymptotically attractive. We prove that, by
implementing the control law using a high-gain observer, we can recover
asymptotic stability of the attractive set, an estimate of its region of
attraction (the whole region in the global case), and trajectories.

WWW access:

Issue No. 14, February, 2000


NCA 14-1-990813

TITLE: Stability Enhancement by Boundary Control in 2D Channel

AUTHORS: Andras Balogh, Weijiu Liu and Miroslav Krstic


ABSTRACT: In this paper we stabilize the parabolic equilibrium
profile in a 2D channel flow using actuators and sensors only at
the wall. The control of channel flow was previously considered
by Speyer and coworkers, and Bewley and coworkers, who derived
feedback laws based on linear optimal control, and implemented by
wall-normal actuation.  With an objective to achieve global
Lyapunov stabilization, we arrive at a feedback law using
tangential actuation (using teamed pairs of synthetic jets or
rotating disks) and only local measurements of wall shear stress,
allowing to embed the feedback in MEMS hardware, without need for
wiring. This feedback is shown to guarantee global stability in
at least $H^2$ norm, which by Sobolev's embedding theorem implies
continuity in space and time of both the flow field and the
control (as well as their convergence to the desired steady
state). The theoretical results are limited to low values of
Reynolds number, however, we present simulations that demonstrate
the effectiveness of the proposed feedback for values five order
of magnitude higher.

WWW access:

NCA 14-2-990813

TITLE: Adaptive   Control  of  Burgers'  Equation with Unknown

AUTHORS: Weijiu Liu and Miroslav Krstic


ABSTRACT: In this paper, we propose  a fortified boundary control
law and an  adaptation law for  Burgers' equation  with unknown
viscosity, where no a priori knowledge of a lower bound on
viscosity is needed. This control law is decentralized, i.e.,
implementable without the need for central computer and wiring.
Using the Lyapunov method, we prove that the  closed-loop system,
including  the parameter estimator as a dynamic component, is
globally $H^1$ stable and well posed. Furthermore, we show that
the state of the system  is regulated to zero by developing an
alternative to Barbalat's Lemma which can not be used in the
present situation.

WWW access:

NCA 14-3-990813

TITLE: Backstepping Boundary  Control   of Burgers' Equation with
Actuator Dynamics

AUTHORS: Weijiu Liu and Miroslav Krstic


ABSTRACT: In this paper we propose  a backstepping boundary
control law for  Burgers' equation  with actuator dynamics. While
the control law without actuator dynamics depends only on the
signals $u(0,t)$ and  $u(1,t)$, the backstepping control also
depends on  $u_{x}(0,t)$, $u_{x}(1,t)$, $u_{xx}(0,t)$ and
$u_{xx}(1,t)$, making the regularity of the control inputs the
key technical issue of the paper. With elaborate  Lyapunov
analysis, we prove that all these signals are sufficiently
regular and  the  closed-loop system, including  the boundary
dynamics, is globally $H^3$ stable and well posed.

WWW access:


NCA 14-4-990922

TITLE: The Explicit Linear Quadratic Regulator for Constrained
Systems AUTHORS: A. Bemporad (ETH Zurich), M. Morari (ETH Zurich),
          V. Dua (Imperial College, London), and E. N. Pistikopoulos
          (Imperial College, London)
E-MAIL: bemporad,,

ABSTRACT: For discrete time linear time invariant systems with
constraints on inputs and states, we develop an algorithm to
determine explicitly the state feedback control law which
minimizes a quadratic performance criterion. We show that the
control law is piecewise linear and continuous for both the
finite horizon problem (model predictive control) and the usual
infinite time measure (constrained linear quadratic regulation).
Thus, the on-line computation is reduced to a simple linear
function evaluation, instead of the expensive quadratic program
required up to now. Control based on on-line optimization has
long been recognized as a superior alternative for constrained
systems. The technique proposed in this paper is attractive for a
wide range of practical problems where the computational
complexity of on-line optimization is prohibitive.

NCA 14-5-992511

TITLE: On the Prevalence of Linear Parameter-Varying Systems
AUTHORS: D.J.Leith, W.E.Leithead
         Dept. of Electronic & Electrical Engineering,
         University of Strathclyde, U.K.

ABSTRACT: A number of recent gain-scheduling approaches assume
that the plant to be controlled is in so-called linear
parameter-varying form. However, present theory does not support
the reformulation of nonlinear systems into linear
parameter-varying form without, in general, considerable
restrictions either on the class of nonlinear systems considered
or on the allowable operating region.. By employing
velocity-based linearisation analysis, it is shown that a very
general class of nonlinear systems can, indeed, be transformed
into linear parameter-varying form.


NCA 14-6-992511

TITLE: Uniting Local and Global Controllers with Robustness to
Vanishing Noise

Christophe Prieur, Laboratoire d'analyse numerique, Universite Paris Sud
Laurent Praly, CAS, Ecole des Mines


nonlinear stabilization, hybrid control, time-varying control,
disturbance,measurement errors, generalized trajectories, Krasovskii's

We consider control systems for which we know two stabilizing
controllers. The former is ``optimal'' but local, the latter is
global. We look for a uniting control law providing
a globally stabilizing locally optimal controller. We study several
solutions based on continuous, discontinuous, hybrid, time varying
controllers. One criterion of selection of a controller is the
robustness of the stability to vanishing noise.
This leads us in particular to consider a kind of
generalization of Krasovskii trajectories for hybrid systems.



NCA 14-7-991217

TITLE: A note on periodic output feedback stabilization

AUTHORS: Luc Moreau and Dirk Aeyels, Universiteit Gent


KEYWORDS: stabilization, time-varying, output feedback

ABSTRACT: In (Brockett, 1998, A stabilization problem) the
following open problem was posed: given a linear time-invariant
continuous time system, find ---if any--- an exponentially
stabilizing static time-varying output feedback. In the present
note, we give necessary and sufficient conditions for this
problem to be solvable for the particular case of scalar-input
scalar-output second-order systems. The proof of the sufficiency
part is constructive and supplies the required feedback gain.



NCA 14-8-000217

TITLE: Input-output-to-state stability

AUTHORS: Mikhail Krichman, Eduardo D. Sontag, Yuan Wang


KEYWORDS: detectability - norm observers - Lyapunov functions -

ABSTRACT: This work explores Lyapunov characterizations of the
input-output-to-state stability (IOSS) property for nonlinear
systems.  The notion of IOSS is a natural generalization of the
standard zero-detectability property used in the linear case.
The main contribution of this work is to establish a complete
equivalence between the input-output-to-state stability property
and the existence of a certain type of smooth Lyapunov function.
As corollaries, one shows the existence of "norm-estimators", and
obtains characterizations of nonlinear detectability in terms of
relative stability and of finite-energy estimates.



NCA 14-9-000217

TITLE: Structure and stability of certain chemical networks and
applications to the kinetic proofreading model of T-cell receptor
signal transduction

AUTHORS: Eduardo D. Sontag


KEYWORDS: kinetic proofreading - chemical reactors - stability -
deficiency-zero networks

ABSTRACT: This paper deals with the theory of structure,
stability, robustness, and stabilization for an appealing class
of nonlinear systems which arises in the analysis of chemical
networks.  The results given here extend, but are also heavily
based upon, certain previous work by Feinberg, Horn, and Jackson,
of which a self-contained and streamlined exposition is
included.  The theoretical conclusions are illustrated through an
application to the kinetic proofreading model proposed by
McKeithan for T-cell receptor signal transduction.



NCA 14-10-000217

TITLE: Further equivalences and semiglobal versions of integral
input to state stability

AUTHORS: David Angeli, Eduardo D. Sontag, Yuan Wang


KEYWORDS: input to state stability - Lyapunov methods - system

ABSTRACT: This paper continues the study of the integral
input-to-state stability (iISS) property.  It is shown that the
iISS property is equivalent to one which arises from the
consideration of mixed norms on states and inputs, as well as to
the superposition of a ``bounded energy bounded state''
requirement and the global asymptotic stability of the unforced
system.  A semiglobal version of iISS is shown to imply the
global version, though a counterexample shows that the analogous
fact fails for input to state stability (ISS).  The results in
this note complete the basic theoretical picture regarding iISS
and ISS.



NCA 14-11-000217

TITLE: Remarks regarding the gap between continuous, Lipschitz,
and differentiable storage functions for dissipation inequalities
appearing in H-infinity control

AUTHORS: Lionel Rosier and Eduardo D. Sontag


KEYWORDS: storage functions - dissipation inequalities - Lyapunov
functions - stability - viscosity solutions - stability

ABSTRACT: This paper deals with the regularity of solutions of the
Hamilton-Jacobi Inequality which arises in H-infinity control. It
shows by explicit counterexamples that there are gaps between
existence of continuous and locally Lipschitz (positive definite
and proper) solutions, and between Lipschitz and continuously
differentiable ones. On the other hand, it is shown that it is
always possible to smooth-out solutions, provided that an
infinitesimal increase in gain is allowed.



NCA 14-12-000217

TITLE: Neural systems as nonlinear filters

AUTHORS: Wolfgang Maass and Eduardo D. Sontag


KEYWORDS: neural networks - spiking nets - Volterra series -

ABSTRACT: We analyze computations on temporal patterns and
spatio-temporal patterns in formal network models whose temporal
dynamics arises from empirically established quantitative models
for short term dynamics at biological synapses. We give a
complete characterization of all linear and nonlinear filters
that can be approximated by such dynamic network models: it is
the class of all filters that can be approximated by Volterra
series. This characterization is shown to be rather stable with
regard to changes in the model. For example it is shown that
synaptic facilitation and one layer of neurons suffices for
approximating arbitrary filters from this class. Our results
provide a new complexity hierarchy for all filters that are
approximable by Volterra series, which appears to be closer
related to the actual cost of implementing such filters in neural
hardware than preceding complexity measures. Our results also
provide a new parameterization for approximations to such filters
in terms of parameters that are arguable related to those that
are tunable in biological neural systems.



NCA 14-13-000217

TITLE: Universal formulas for feedback stabilization with respect
to Minkowski balls

AUTHORS: Michael Malisoff and Eduardo D. Sontag


KEYWORDS: constrained controls - control-Lyapunov functions

ABSTRACT: This note provides explicit algebraic stabilizing
formulas for clf's when controls are restricted to certain
Minkowski balls in Euclidean space. Feedbacks of this kind are
known to exist by a theorem of Artstein, but the proof of
Artstein's theorem is nonconstructive. The formulas are obtained
from a general feedback stabilization technique and are used to
construct approximation solutions to some stabilization problems.


Issue No. 15, July 2000


NCA 15-1-002303

TITLE: A generalization of Zubov's method to perturbed systems

AUTHORS: Fabio Camilli, Universita de l'Aquila, Italy
         Lars Gruene, J.-W. Goethe-Universitaet, Frankfurt a.M., Germany
         Fabian Wirth, Universitaet Bremen, Germany


KEYWORDS: robust domain of attraction, robust Lyapunov function,
          Zubov's method

ABSTRACT: A generalization of Zubov's theorem on representing the
domain of attraction via the solution of a suitable partial
differential equation is presented for the case of perturbed
systems with a singular fixed point. For the construction it is
necessary to consider solutions in the viscosity sense. As a
consequence maximal robust Lyapunov functions can be
characterized as viscosity solutions.


NCA 15-2-002303

 TITLE: A regularization of Zubov's equation for
robust domains of attraction

AUTHORS: Fabio Camilli, Universita de l'Aquila, Italy
         Lars Gruene, J.-W. Goethe-Universitaet, Frankfurt a.M., Germany
         Fabian Wirth, Universitaet Bremen, Germany


KEYWORDS: robust domain of attraction, robust Lyapunov function,
          Zubov's method, computational approach

ABSTRACT: We derive a method for the computation of robust
domains of attraction based on a recent generalization of Zubov's
theorem on representing robust domains of attraction for
perturbed systems via the viscosity solution of a suitable
partial differential equation. While a direct discretization of
the equation leads to numerical difficulties due to a singularity
at the stable equilibrium, a suitable regularization enables us
to apply a standard discretization technique for
Hamilton-Jacobi-Bellman equations. We present the resulting fully
discrete scheme and show a numerical example.


NCA 15-3-002303

 TITLE: Zubov's method for perturbed differential

AUTHORS: Fabio Camilli, Universita de l'Aquila, Italy
         Lars Gruene, J.-W. Goethe-Universitaet, Frankfurt a.M., Germany
         Fabian Wirth, Universitaet Bremen, Germany


KEYWORDS: asymptotically stable compact set, robust domain of
          robust Lyapunov function, Zubov's method

ABSTRACT: We present a further generalization of Zubov's method
to perturbed differential equations. The goal is to characterize
the domain of attraction of a compact set which is uniformly
locally asymptotically stable under all admissible time varying
perturbations. We show that also in this general setting the
straightforward generalization of the classical Zubov's equations
has a unique viscosity solution which characterizes the robust
domain of attraction as a suitable sublevel set.


NCA 15-4-002303

 TITLE: Computing control Lyapunov functions via a
Zubov type algorithm

AUTHORS: Lars Gruene, J.-W. Goethe-Universitaet, Frankfurt a.M.,
         Fabian Wirth, Universitaet Bremen, Germany


KEYWORDS: asymptotic controllability, domain of asymptotic
          control Lyapunov function, Zubov's method, computational

ABSTRACT: Under the assumption of local asymptotic
nullcontrollability we define the domain of asymptotic
nullcontrollability. On this set we define a control Lyapunov
function via an optimal control problem. It is shown that this
function can be characterized as the unique viscosity solution of
a partial differential equation which can be interpreted as a
generalization of Zubov's equation.


NCA 15-5-002303

TITLE: Higher order numerical schemes for affinely controlled
       nonlinear systems

AUTHORS: Lars Gruene, Peter Kloeden
         J.-W. Goethe-Universitaet, Frankfurt a.M., Germany


KEYWORDS: affine control systems, Taylor expansion, Taylor
          Runge-Kutta type schemes, multiple control integrals

ABSTRACT: A systematic method for the derivation of high order
schemes for affinely controlled nonlinear systems is developed.
Using an adaptation of the stochastic Taylor expansion for
control systems we construct Taylor schemes of arbitrary high
order and indicate how derivative free Runge-Kutta type schemes
can be obtained. Furthermore an approximation technique for the
multiple control integrals appearing in the schemes is proposed.


NCA 15-6-002303

 TITLE: Attractors under perturbation and

AUTHORS: Lars Gruene, J.-W. Goethe-Universitaet, Frankfurt a.M.,


KEYWORDS: attractors, perturbation, numerical approximation,
          robustness, input-to-state-stability

ABSTRACT: Using control theoretic techniques we give a necessary
and sufficient condition for the convergence of attractors in one
step discretizations of ordinary differential equations.
Furthermore, our approach allows estimates for the resulting
discretization error. The condition used for these results is
similar but slightly weaker than the input-to-state-stability
property well known in control theory.


NCA 15-7-000327

TITLE: On controllability of the real shifted inverse power

AUTHORS: Uwe Helmke, Universitaet Wuerzburg, Germany
         Fabian Wirth, Universitaet Bremen, Germany


KEYWORDS: inverse iteration, forward accessibility,
          controllability, polynomial matrix equations

ABSTRACT: We study the inverse power method well-known in
numerical linear algebra from a control point of view. In
particular, controllability properties of the inverse power
method on projective space are investigated. It is known that for
complex eigenvalue shifts a simple characterization of the
reachable sets in terms of invariant subspaces can be obtained.
In contrast, the real case under consideration in this paper is
more complicated. Using properties of universally regular
controls, necessary and sufficient conditions for complete
controllability are obtained in terms of the solvability of a
matrix equation. Partial results on conditions for the
solvability of this matrix equation are given.


NCA 15-8-000503

TITLE: Stabilization of Rotational Motion with Application to
Spacecraft Attitude Control

AUTHOR: Rafal Wisniewski


ABSTRACT: The objective of this paper is to develop a control
scheme for stabilization of a hamiltonian system. The method
generalizes the results available in the literature on motion
control in the Euclidean space to an arbitrary differential
manifold equipped with a metric. This modification is essencial
for global stabilization of a rotary motion.

Along with a model of the system formulated in the Hamilton's
canonical form the algorithm uses information about a required
potential energy  and a dissipation term. The control action is
the sum of the gradient of the potential energy and the
dissipation force. It is shown that this control law makes the
system uniformly asymptotically stable to the desired reference
point. The concept is very straightforward in the Euclidean space,
however a global rotation control can not be tackled. An
additional  modification is made to address a system which  flow
lies  on a Riemannian manifold. The Lyapunov stability theory is
adapted and reformulated to fit to the new framework of Riemannian
manifolds. To illustrate the results  a spacecraft attitude
control problem is considered. Firstly, a global canonical
representation for the spacecraft motion is found, then three
spacecraft control problems are addressed: stabilization in the
inertial frame, magnetic libration damping for the gravity
gradient stabilization and a slew maneuver with obstacle

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NCA 15-9-000516

TITLE: Adaptive grid generation for evolutive

AUTHORS: Lars Gruene, J.-W. Goethe-Universitaet, Frankfurt a.M.,


KEYWORDS: optimal control, Hamilton-Jacobi-Bellman equation,
          numerical solution, adaptive space discretization

ABSTRACT: We present an adaptive grid generation for a class of
evolutive Hamilton-Jacobi-Bellman equations. Using a two step
(semi-Lagrangian) discretization of the underlying optimal
control problem we define a-posteriori local error estimates for
the discretization error in space. Based on these estimates we
present an iterative procedure for the generation of adaptive
grids and discuss implementational details for a suitable
hierarchical data structure.


NCA 15-10-000615

TITLE: Minimum-phase nonlinear systems: a new definition

AUTHORS: D. Liberzon, A. S. Morse, E. D. Sontag


ABSTRACT: This paper introduces and studies 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. 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 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.  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.

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