Editors: Rodolphe Sepulchre Eduardo D. Sontag Universite de Liege, Belgium Rutgers University, USA r.sepulchre@ulg.ac.be sontag@control.rutgers.edu 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. -1- Contributions have to be sent to: "nca-submit@montefiore.ulg.ac.be" Please use the same format as the contributions below. -2- You can subscribe to the nca-letter by sending an e-mail message to "nca@montefiore.ulg.ac.be" carrying the subject 'add' or 'subscribe' in the body of the mail. You will be automatically subscribed and included in our mailing list. To unsubscribe from this list, send an e-mail message to "nca@montefiore.ulg.ac.be" with the subject 'remove', 'delete' or 'unsubscribe' in the body of the mail. -3- If your address changed first unsubscribe with the instruction `unsubscribe nca' and then subscribe again (using your new E-mail address).

******************************************************************************* NCA-1-1-980120 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 E-mail: rortega@lss.supelec.fr,astankov@cdsp.neu.eduhttp://www.supelec.fr/invi/lss/fr/personnels/ortega/Bienvenue.html

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 E-mail: rortega@lss.supelec.frhttp://www.supelec.fr/invi/lss/fr/personnels/ortega/Bienvenue.html

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} feature 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 EMAIL: tbasar@decision.csl.uiuc.edu KEYWORDS: Robust adaptive control; linear uncertain systems; nonlinear H-infinity control; worst-case identification. PS file retrieval: contact author by email for file ABSTRACT: 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. ******************************************************************************* NCA-1-4-980123 Title: Comments on integral variants of ISS Author(s): Eduardo Sontag (Rutgers) Email contact: sontag@control.rutgers.edu Keywords: input to state stability, Lyapunov functions, input/output stability Abstract: 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/iiss.ps.gz

****************************************************************************** NCA-1-5-980123 Title: A Lyapunov characterization of robust stabilization Author(s): Yu.S. Ledyaev (Western Michigan), Eduardo Sontag (Rutgers) Email contact: sontag@control.rutgers.edu Keywords: Lyapunov functions, stabilization, actuator noise Abstract: 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/yuri-clf.ps.gz

***************************************************************************** NCA-1-6-980123 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: sontag@control.rutgers.edu Keywords: stability, integral invariance Abstract: 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/ryan-logeman.ps.gz

***************************************************************************** NCA-1-7-980123 Title: Remarks on universal nonsingular controls for discrete-time systems Author(s): F. Wirth (Bremen), Eduardo Sontag (Rutgers) Email contact: sontag@control.rutgers.edu Analytic discrete-time systems, forward accessibility, universal controls Abstract: 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/wirth-scl.ps.gz

***************************************************************************** NCA-1-8-980223 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 EMAIL: praly@cas.ensmp.fr KEYWORDS: robust control, dynamic input uncertainties, output feedback stabilization 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.http://cas.ensmp.fr/~praly/Papers/In-Dist-Min-Phi-NOLCOS98.ps.gz

***************************************************************************** NCA-1-9-980223 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 EMAIL: praly@cas.ensmp.fr KEYWORDS: robust control, dynamic input uncertainties, output feedback stabilization 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 paperhttp://cas.ensmp.fr/~praly/Papers/In-Dist-Min-Phi-NOLCOS98.ps.gz

can be extended to the general relative degree case. For ease of presentation, we restrict ourselves the single input single output case.http://cas.ensmp.fr/~praly/Papers/In-Dist-Min-Phi-CDC98.ps.gz

***************************************************************************** NCA-1-10-980223 TITLE: Slow peaking and low-gain designs for global stabilization of nonlinear systems AUTHORS: R. Sepulchre, University of Liege, Belgium EMAIL: sepulchre@montefiore.ulg.ac.be 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 met. 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 model. FTP ACCESS: tournesol.auto.ucl.ac.be/pub/sepulchre/peaking.ps ***************************************************************************** NCA-1-11-980226 TITLE: A new asymptotic stability criterion for non-linear time-variant differential equations AUTHORS: Dirk Aeyels, Joan Peuteman, SYSTeMS, Universiteit Gent EMAIL: Dirk.Aeyels@rug.ac.be, Joan.Peuteman@rug.ac.be 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.http://ensmain.rug.ac.be/staff/pub_da_90.html

******************************************************************************* NCA-2-1-980309 Title: Impulse-Free Output Regulation of Singular Nonlinear Systems Authors: Jie Huang, The Chinese University of Hong Kong Email: jhuang@mae.cuhk.edu.hk Ji-Feng Zhang, The Chinese Academy of Science Email: jif@iss03.iss.ac.cn Keywords: Singular nonlinear systems, output regulation, center manifold theory. 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 EMAIL: r.orsi@ee.mu.oz.au, praly@cas.ensmp.fr, i.mareels@ee.mu.oz.au 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 system.http://www.ee.mu.oz.au/pgrad/orsi/unbounded.ps.gz

******************************************************************************* 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. E-mail: dragan@lagrange.ece.ucsb.eduhttp://www.math.rutgers.edu/~sontag/papers.html

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. ******************************************************************************** NCA-2-4-980319 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 EMAIL: Joan.Peuteman@rug.ac.be, Dirk.Aeyels@rug.ac.be 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.http://ensmain.rug.ac.be/staff/pub_da_90.html

******************************************************************************* NCA-3-1-980408 TITLE: Symmetries of flat rank two distributions and sub-Riemannian structures AUTHOR: YU. L. SACHKOV, Program Systems Institute EMAIL: sachkov@sys.botik.ru 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.http://www.botik.ru/PSI/CPRC/sachkov/sym_ps.zip

******************************************************************************* NCA 3-2-980408 TITLE: Survey on Controllability of Invariant Systems on Solvable Lie Groups AUTHOR: YU. L. SACHKOV, Program Systems Institute EMAIL: sachkov@sys.botik.ru 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.http://www.botik.ru/PSI/CPRC/sachkov/boulder2.ps

******************************************************************************* 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, E-MAIL:jesd@xanum.uam.mx, rsua@xanum.uam.mx, jjar@xanum.uam.mx 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. FTP ACCESS: ftp.mat.uam.mx/pub/ecdiff/rsuarez/Nonlinear.ps ****************************************************************************** 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) E-MAIL: mjankov1@ford.com 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. FTP ACCESS:http://www.montefiore.ulg.ac.be/~sepulch/web/scl98.ps

******************************************************************************* NCA-4-1-980528 TITLE: Input-to-state stability of exponentially stabilized semilinear control systems with inhomogeneous perturbations AUTHOR: L. GRUENE Universita di Roma "La Sapienza", Italy and J.-W.-Goethe Universitaet Frankfurt a.M., Germany EMAIL: gruene@math.uni-frankfurt.de 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.http://www.math.uni-frankfurt.de/~gruene/papers/iss.html

***************************************************************************** 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. E-MAIL: jjar@xanum.uam.mx, rsua@xanum.uam.mx. 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. FTP ACCESS: ftp.mat.uam.mx/pub/ecdiff/rsuarez/Robust.ps

******************************************************************************* NCA-5-1-980604 TITLE: Energy--based Lyapunov functions for forced Hamiltonian systems with dissipation AUTHORS: B. Maschke, R. Ortega and A. J. van der Schaft EMAIL: rortega@lss.supelec.fr 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.http://www.supelec.fr/invi/lss/fr/personnels/ortega/Bienvenue.htm

***************************************************************************** NCA 5-2-980604 TITLE: Necessary and sufficient conditions for passivity of the LuGre friction model AUTHORS:N. Barabanov and R. Ortega EMAIL: rortega@lss.supelec.fr 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.http://www.supelec.fr/invi/lss/fr/personnels/ortega/Bienvenue.htm

***************************************************************************** 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: fabian@math.uni-bremen.de 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.http://www.math.uni-bremen.de/~fabian/work/Archive/RobLoc.ps

***************************************************************************** NCA 5-4-980617 TITLE: Towards the supervisory control of uncertain nonholonomic systems AUTHORS: J. HESPANHA, D. LIBERZON, A. S. MORSE (Yale University) EMAIL: joao.hespanha@yale.edu, liberzon@sysc.eng.yale.edu, morse@sysc.eng.yale.edu 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 algorithms.http://giskard.eng.yale.edu/cvc/staff/liberzon/publications.html

****************************************************************************** NCA-6-1-980715 TITLE: Overview of complexity and decidability results for three classes of elementary nonlinear systems AUTHORS: Vincent D. Blondel and John N. Tsitsiklis EMAIL: vblondel@ulg.ac.be 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 are sought. 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 for which polynomial time algorithms are known.http://www.ulg.ac.be/mathsys/blondel/papers/

***************************************************************************** NCA 6-2-980722 TITLE: A characterization of integral input to state stability AUTHORS: D. Angeli, E.D. Sontag, Y. Wang ABSTRACT: 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/iiss-asw.ps.gz

***************************************************************************** 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.)http://www.math.rutgers.edu/~sontag/FTP_DIR/vc-expo.ps.gz

************************************************************************* 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/qiao-scl.ps.gz

************************************************************************* NCA 6-5-980722 TITLE: A learning result for continuous-time recurrent neural networks AUTHOR: E.D. Sontag ABSTRACT: 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/recur-learn.ps.gz

************************************************************************* 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 mentioned.http://www.math.rutgers.edu/~sontag/FTP_DIR/recur-survey-book.ps.gz

************************************************************************* 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 EMAIL: zl5y@virginia.edu 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/sat-dt-iss.ps.gz

************************************************************************** NCA 7-1-980908 TITLE: Regularity of Solutions of Burgers' Equation with Globally Stabilizing Nonlinear Boundary Feedback AUTHORS: A. Balogh and M. Krstic EMAIL: krstic@ucsd.edu 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 controller.http://www-ames.ucsd.edu/research/krstic/

************************************************************************** NCA 7-2-980908 TITLE: Global Boundary Stabilization of the Korteweg-de Vries-Burgers Equation AUTHORS: W.-J. Liu and M. Krstic EMAIL: krstic@ucsd.edu 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.http://www-ames.ucsd.edu/research/krstic/

************************************************************************* NCA 7-3-980908 TITLE: Stability Enhancement by Boundary Control in the Kuramoto-Sivashinsky Equation AUTHORS: W.-J. Liu and M. Krstic EMAIL: krstic@ucsd.edu 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.http://www-ames.ucsd.edu/research/krstic/

************************************************************************* 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. EMAIL: liberzon@sysc.eng.yale.edu, sontag@control.rutgers.edu, ywang@control.math.fau.eduhttp://pantheon.yale.edu/~dml33 http://www.math.rutgers.edu/~sontag/FTP_DIR/clf-iiss.ps.gz

************************************************************************* 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 EMAIL: praly@cas.ensmp.fr 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.http://cas.ensmp.fr/~praly/Publications/In-Dist-Dissipativity.ps.gz

************************************************************************** NCA 8-2-981026 TITLE: Stability and stabilization: Discontinuities and the effect of disturbances AUTHOR: Eduardo D. Sontag EMAIL: sontag@control.rutgers.edu 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 ISShttp://www.math.rutgers.edu/~sontag/FTP_DIR/stab-survey98.ps.gz

************************************************************************* NCA 8-3-981026 TITLE: A polynomial-time algorithm for state equivalence in hybrid PL systems AUTHORS: Bhaskar DasGupta and Eduardo D. Sontag EMAIL: bhaskar@crab.rutgers.edu, sontag@control.rutgers.edu 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/pl-decide.ps.gz

************************************************************************* NCA 8-4-981113 TITLE: A Survey of Computational Complexity Results in Systems and Control AUTHORS: Vincent D. Blondel and John N. Tsitsiklis EMAIL: vblondel@ulg.ac.be 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.http://www.ulg.ac.be/mathsys/blondel/publications.html

******************************************************************************* NCA 9-1-981126 TITLE: On the rate of convergence of infinite horizon discounted optimal value functions AUTHOR: L. GRUENE and F. WIRTH Universita di Roma "La Sapienza", Italy and J.W. Goethe-Universitaet Frankfurt a.M., Germany, Universitaet Bremen, Germany EMAIL: gruene@math.uni-frankfurt.de, fabian@math.uni-bremen.de 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.http://www.math.uni-frankfurt.de/~gruene/papers/rate.html or: http://www.math.uni-bremen.de/zetem/Berichte/report9806.ps.gz

************************************************************************** NCA 9-2-981203 TITLE: HOMOGENEOUS STATE FEEDBACK STABILIZATION OF HOMOGENEOUS CONTROL SYSTEMS AUTHOR: L. GRUENE Universita di Roma "La Sapienza", Italy and J.-W. Goethe-Universitaet Frankfurt a.M., Germany EMAIL: gruene@math.uni-frankfurt.de KEYWORDS: Homogeneous system, state feedback stabilization, control Lyapunov functions, Lyapunov exponents ABSTRACT: 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.http://www.math.uni-frankfurt.de/~gruene/papers/hom.html

************************************************************************* 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 EMAIL: gruene@math.uni-frankfurt.de sontag@control.rutgers.edu fabian@math.uni-bremen.de 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/gas2ges.ps.gz

************************************************************************** NCA 10-1-990105 TITLE: Benchmark problems in stability and design of switched systems AUTHORS: Daniel Liberzon and A. Stephen Morse (Yale University) EMAIL: liberzon@sysc.eng.yale.edu ABSTRACT: A switched system is a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule that governs the switching 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.http://pantheon.yale.edu/~dml33/survey.ps

************************************************************************** NCA 10-2-99-0110 TITLE: Notions of Input to Output Stability AUTHORS: Eduardo Sontag and Yuan Wang EMAIL: sontag@gauss.rutgers.edu 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 state. 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/ios.ps.gz

************************************************************************* 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 EMAIL: sontag@gauss.rutgers.edu 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/ios.ps.gz

************************************************************************* 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, Russia EMAIL: sachkov@sys.botik.ru 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 References Indexhttp://www.botik.ru/PSI/CPRC/sachkov/obzor.dvi and http://www.botik.ru/PSI/CPRC/sachkov/obzor.ps

************************************************************************* NCA 10-5-990129 TITLE: Supervisory Control of Integral-Input-to-State Stabilizing Controllers AUTHORS: Joao P. Hespanha and A. Stephen Morse EMAIL: hespanha@eecs.berkeley.edu, morse@sysc.eng.yale.edu 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.http://robotics.eecs.berkeley.edu/~hespanha/published.html

************************************************************************** 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 E-MAIL: rortega@lss.supelec.fr 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 computation. MUST REQUEST BY EMAIL ************************************************************************* NCA 11-2-990226 TITLE: Forward completeness, unboundedness observability, and their Lyapunov characterizations AUTHORS: David Angeli and Eduardo Sontag EMAIL: sontag@control.rutgers.edu 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 inequalities. 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.http://www.math.rutgers.edu/~sontag/FTP_DIR/uo.ps.gz

************************************************************************* NCA 11-3-990304 TITLE: Disturbance attenuation with bounded controls AUTHORS: Daniel Liberzon (Yale University) EMAIL: liberzon@sysc.eng.yale.edu ABSTRACT: We consider the problem of achieving disturbance attenuation in the ISS and integral-ISS sense for nonlinear systems using 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.http://pantheon.yale.edu/~dml33/bounded.ps

************************************************************************** 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 EMAIL: julio@ee.newcastle.edu.au 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 system.ftp://eebrett.newcastle.edu.au:2121/pub/Reports/EE99007.ps.gz, ftp://eebrett.newcastle.edu.au:2121/pub/Reports/EE99007.pdf.gz

************************************************************************** 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 E-MAIL: townley@maths.ex.ac.uk 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 measure. 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.http://www.maths.ex.ac.uk/~townley/inv_man.ps>

************************************************************************** NCA 12-3-992204 TITLE: Remarks on continuous feedback AUTHORS: E. Sontag and H.J. Sussmann E-mail: sontag@math.rutgers.edu 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.http://www.math.rutgers.edu/~sontag/papers.html

************************************************************************** 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. EMAIL: teel@ece.ucsb.edu praly@cas.ensmp.fr 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. FTP/WWW ACCESS:ftp://ftp-ccec.ece.ucsb.edu/pub/teel/submitted/converse.ps.gz

************************************************************************* NCA 13-2-990701 TITLE: A note on stability of arbitrarily switched homogeneous systems AUTHOR: David Angeli (DSI, Firenze) E-MAIL: angeli@dsi.unifi.ithttp://www.dsi.unifi.it/~angeli/papers/sclhomo.ps

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) E-MAIL: angeli@dsi.unifi.ithttp://www.dsi.unifi.it/~angeli/papers/scliiss.ps

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 Systems 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 EMAIL: atassiah@seidel.ece.ucsb.edu 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:http://www-ccec.ece.ucsb.edu/~atassiah/FTP-DIR/papers.html

************************************************************************ NCA 14-1-990813 TITLE: Stability Enhancement by Boundary Control in 2D Channel Flow AUTHORS: Andras Balogh, Weijiu Liu and Miroslav Krstic EMAIL: krstic@ucsd.edu 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: http://www-mae.ucsd.edu/research/krstic/krstic/papers/2dns.ps.gz *************************************************************** NCA 14-2-990813 TITLE: Adaptive Control of Burgers' Equation with Unknown Viscosity AUTHORS: Weijiu Liu and Miroslav Krstic EMAIL: krstic@ucsd.edu 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: http://www-mae.ucsd.edu/research/krstic/krstic/papers/adaptiveburgers.ps *************************************************************** NCA 14-3-990813 TITLE: Backstepping Boundary Control of Burgers' Equation with Actuator Dynamics AUTHORS: Weijiu Liu and Miroslav Krstic EMAIL: krstic@ucsd.edu 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: http://www-mae.ucsd.edu/research/krstic/krstic/papers/burg-bkst.ps *************************************************************** 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,morari@aut.ee.ethz.ch, v.dua,e.pistikopoulos@ic.ac.uk 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. ftp://aut.ethz.ch/pub/publications/AUTAr-99-33.pdf ftp://aut.ethz.ch/pub/publications/AUTAr-99-34.ps *************************************************************** 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. E-mail: doug@icu.strath.ac.uk 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. http://www.icc.strath.ac.uk/~doug/1999-1.zip *************************************************************** NCA 14-6-992511 TITLE: Uniting Local and Global Controllers with Robustness to Vanishing Noise AUTHORS: Christophe Prieur, Laboratoire d'analyse numerique, Universite Paris Sud Laurent Praly, CAS, Ecole des Mines EMAIL: praly@cas.ensmp.fr KEYWORDS: nonlinear stabilization, hybrid control, time-varying control, disturbance,measurement errors, generalized trajectories, Krasovskii's solutions ABSTRACT: 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. FTP/WWW ACCESS: http://cas.ensmp.fr/Publications/Download/Regulation/Loc_glob_Paper_MCSS.ps. gz *************************************************************** NCA 14-7-991217 TITLE: A note on periodic output feedback stabilization AUTHORS: Luc Moreau and Dirk Aeyels, Universiteit Gent EMAIL: lmoreau@ensmain.rug.ac.be 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. WWW ACCESS: http://ensmain.rug.ac.be/~lmoreau/pub_lm.html *************************************************************** NCA 14-8-000217 TITLE: Input-output-to-state stability AUTHORS: Mikhail Krichman, Eduardo D. Sontag, Yuan Wang EMAIL: sontag@gauss.rutgers.edu KEYWORDS: detectability - norm observers - Lyapunov functions - ISS 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. FTP/WWW ACCESS: http://www.math.rutgers.edu/~sontag/FTP_DIR/ioss.ps.gz http://www.math.rutgers.edu/~sontag/FTP_DIR/ioss.pdf *************************************************************** 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 EMAIL: sontag@gauss.rutgers.edu 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. FTP/WWW ACCESS: http://www.math.rutgers.edu/~sontag/FTP_DIR/chem.ps.gz http://www.math.rutgers.edu/~sontag/FTP_DIR/chem.pdf *************************************************************** NCA 14-10-000217 TITLE: Further equivalences and semiglobal versions of integral input to state stability AUTHORS: David Angeli, Eduardo D. Sontag, Yuan Wang EMAIL: sontag@gauss.rutgers.edu KEYWORDS: input to state stability - Lyapunov methods - system gains 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. FTP/WWW ACCESS: http://www.math.rutgers.edu/~sontag/FTP_DIR/iiss-new.ps.gz http://www.math.rutgers.edu/~sontag/FTP_DIR/iiss-new.pdf *************************************************************** 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 EMAIL: sontag@gauss.rutgers.edu 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. FTP/WWW ACCESS: http://www.math.rutgers.edu/~sontag/FTP_DIR/hoo.ps.gz http://www.math.rutgers.edu/~sontag/FTP_DIR/hoo.pdf *************************************************************** NCA 14-12-000217 TITLE: Neural systems as nonlinear filters AUTHORS: Wolfgang Maass and Eduardo D. Sontag EMAIL: sontag@gauss.rutgers.edu KEYWORDS: neural networks - spiking nets - Volterra series - filters 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. FTP/WWW ACCESS: http://www.math.rutgers.edu/~sontag/FTP_DIR/spiking.ps.gz http://www.math.rutgers.edu/~sontag/FTP_DIR/spiking.pdf *************************************************************** NCA 14-13-000217 TITLE: Universal formulas for feedback stabilization with respect to Minkowski balls AUTHORS: Michael Malisoff and Eduardo D. Sontag EMAIL: sontag@gauss.rutgers.edu 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. FTP/WWW ACCESS: http://www.math.rutgers.edu/~sontag/FTP_DIR/minkowski.ps.gz http://www.math.rutgers.edu/~sontag/FTP_DIR/minkowski.pdf

************************************************************************ 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 EMAIL: camilli@axcasp.caspur.it gruene@math.uni-frankfurt.de fabian@math.uni-bremen.de 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. FTP/WWW ACCESS: http://www.math.uni-frankfurt.de/~gruene/papers/zubov.html http://www.math.uni-bremen.de/~fabian/work/Archive/zubov.ps.gz http://www.math.uni-bremen.de/~fabian/work/Archive/zubov.pdf *************************************************************** 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 EMAIL: camilli@axcasp.caspur.it gruene@math.uni-frankfurt.de fabian@math.uni-bremen.de 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. FTP/WWW ACCESS: http://www.math.uni-frankfurt.de/~gruene/papers/regul.html http://www.math.uni-bremen.de/zetem/Berichte/report0006.ps.gz *************************************************************** NCA 15-3-002303 TITLE: Zubov's method for perturbed differential equations AUTHORS: Fabio Camilli, Universita de l'Aquila, Italy Lars Gruene, J.-W. Goethe-Universitaet, Frankfurt a.M., Germany Fabian Wirth, Universitaet Bremen, Germany EMAIL: camilli@axcasp.caspur.it gruene@math.uni-frankfurt.de fabian@math.uni-bremen.de KEYWORDS: asymptotically stable compact set, robust domain of attraction, 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. FTP/WWW ACCESS: http://www.math.uni-frankfurt.de/~gruene/papers/zubov_mtns.html http://www.math.uni-bremen.de/~fabian/work/Archive/mtnszubov.ps.gz http://www.math.uni-bremen.de/~fabian/work/Archive/mtnszubov.pdf *************************************************************** NCA 15-4-002303 TITLE: Computing control Lyapunov functions via a Zubov type algorithm AUTHORS: Lars Gruene, J.-W. Goethe-Universitaet, Frankfurt a.M., Germany Fabian Wirth, Universitaet Bremen, Germany EMAIL: gruene@math.uni-frankfurt.de fabian@math.uni-bremen.de KEYWORDS: asymptotic controllability, domain of asymptotic controllability, control Lyapunov function, Zubov's method, computational approach 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. FTP/WWW ACCESS: http://www.math.uni-frankfurt.de/~gruene/papers/clf_cdc.html http://www.math.uni-bremen.de/~fabian/work/Archive/cdcclf.ps.gz http://www.math.uni-bremen.de/~fabian/work/Archive/cdcclf.pdf *************************************************************** 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 EMAIL: gruene@math.uni-frankfurt.de kloeden@math.uni-frankfurt.de KEYWORDS: affine control systems, Taylor expansion, Taylor schemes, 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. FTP/WWW ACCESS: http://www.math.uni-frankfurt.de/~gruene/papers/affine.html *************************************************************** NCA 15-6-002303 TITLE: Attractors under perturbation and discretization AUTHORS: Lars Gruene, J.-W. Goethe-Universitaet, Frankfurt a.M., Germany EMAIL: gruene@math.uni-frankfurt.de 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. FTP/WWW ACCESS: http://www.math.uni-frankfurt.de/~gruene/papers/att_cdc.html *************************************************************** NCA 15-7-000327 TITLE: On controllability of the real shifted inverse power iteration. AUTHORS: Uwe Helmke, Universitaet Wuerzburg, Germany Fabian Wirth, Universitaet Bremen, Germany EMAIL: helmke@mathematik.uni-wuerzburg.de fabian@math.uni-bremen.de 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. FTP/WWW ACCESS: http://www.math.uni-bremen.de/~fabian/work/Archive/reshift.ps.gz http://www.math.uni-bremen.de/~fabian/work/Archive/reshift.pdf *************************************************************** NCA 15-8-000503 TITLE: Stabilization of Rotational Motion with Application to Spacecraft Attitude Control AUTHOR: Rafal Wisniewski EMAIL: raf@control.auc.dk 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 avoidance. WWW access: http://www.control.auc.dk/~raf/Papers/HamSat.ps *************************************************************** NCA 15-9-000516 TITLE: Adaptive grid generation for evolutive Hamilton-Jacobi-Bellman equations AUTHORS: Lars Gruene, J.-W. Goethe-Universitaet, Frankfurt a.M., Germany EMAIL: gruene@math.uni-frankfurt.de 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. FTP/WWW ACCESS: http://www.math.uni-frankfurt.de/~gruene/papers/evol.html *************************************************************** NCA 15-10-000615 TITLE: Minimum-phase nonlinear systems: a new definition AUTHORS: D. Liberzon, A. S. Morse, E. D. Sontag EMAIL: daniel.liberzon@yale.edu 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. WWW access: http://pantheon.yale.edu/~dml33/minphase.ps