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Showing papers on "Lyapunov function published in 1999"


Book
22 Jun 1999
TL;DR: In this article, the authors compare Linear vs. Nonlinear Control of Differential Geometry with Linearization by State Feedback (LSF) by using Linearization and Geometric Non-linear Control (GNC).
Abstract: 1 Linear vs. Nonlinear.- 2 Planar Dynamical Systems.- 3 Mathematical Background.- 4 Input-Output Analysis.- 5 Lyapunov Stability Theory.- 6 Applications of Lyapunov Theory.- 7 Dynamical Systems and Bifurcations.- 8 Basics of Differential Geometry.- 9 Linearization by State Feedback.- 10 Design Examples Using Linearization.- 11 Geometric Nonlinear Control.- 12 Exterior Differential Systems in Control.- 13 New Vistas: Multi-Agent Hybrid Systems.- References.

1,925 citations


Journal ArticleDOI
TL;DR: In this paper, a new robust stability condition for uncertain discrete-time systems with convex polytopic uncertainty is given, which enables to check stability using parameter-dependent Lyapunov functions which are derived from LMI conditions.

1,456 citations


Journal ArticleDOI
TL;DR: The approach exploits the gain-scheduling nature of fuzzy systems and results in stability conditions that can be verified via convex optimization over linear matrix inequalities, and special attention is given to the computational aspects of the approach.
Abstract: Presents an approach to stability analysis of fuzzy systems. The analysis is based on Lyapunov functions that are continuous and piecewise quadratic. The approach exploits the gain-scheduling nature of fuzzy systems and results in stability conditions that can be verified via convex optimization over linear matrix inequalities. Examples demonstrate the many improvements over analysis based on a single quadratic Lyapunov function. Special attention is given to the computational aspects of the approach and several methods to improve the computational efficiency are described.

775 citations


Journal ArticleDOI
TL;DR: In this article, a sucient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices is presented, and it is shown that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function.

670 citations


Journal ArticleDOI
TL;DR: The focus of this paper is to prove a converse Lyapunov theorem for this class of systems in which the dynamics at any instant in time will follow one of a fixed set of vector fields.
Abstract: The authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields. They allow switching between members of the family of vector fields to be completely random. The focus of this paper is to prove a converse Lyapunov theorem for this class of systems.

572 citations


Journal ArticleDOI
TL;DR: A Lyapunov-based stabilizing control design method for uncertain nonlinear dynamical systems using fuzzy models is proposed, finding sufficient conditions for stability and stabilizability of fuzzy models using fuzzy state feedback controllers.
Abstract: A Lyapunov-based stabilizing control design method for uncertain nonlinear dynamical systems using fuzzy models is proposed. The controller is constructed using a design model of the dynamical process to be controlled. The design model is obtained from the truth model using a fuzzy modeling approach. The truth model represents a detailed description of the process dynamics. The truth model is used in a simulation experiment to evaluate the performance of the controller design. A method for generating local models that constitute the design model is proposed. Sufficient conditions for stability and stabilizability of fuzzy models using fuzzy state feedback controllers are given. The results obtained are illustrated with a numerical example involving a four-dimensional nonlinear model of a stick balancer.

526 citations


Journal ArticleDOI
TL;DR: In this paper a nonlinear observer is derived and is proven to be passive and GES, and the number of tuning parameters is reduced to a minimum by using passivity theory, which results in a simple and intuitive tuning procedure.

494 citations


Journal ArticleDOI
TL;DR: The cyclic low-rank Smith method is presented, which is an iterative method for the computation of low- rank approximations to the solution of large, sparse, stable Lyapunov equations, and a heuristic for determining a set of suboptimal alternating direction implicit (ADI) shift parameters is proposed.
Abstract: In this paper we present the cyclic low-rank Smith method, which is an iterative method for the computation of low-rank approximations to the solution of large, sparse, stable Lyapunov equations. It is based on a generalization of the classical Smith method and profits by the usual low-rank property of the right-hand side matrix. The requirements of the method are moderate with respect to both computational cost and memory. Furthermore, we propose a heuristic for determining a set of suboptimal alternating direction implicit (ADI) shift parameters. This heuristic, which is based on a pair of Arnoldi processes, does not require any a priori knowledge on the spectrum of the coefficient matrix of the Lyapunov equation. Numerical experiments show the efficiency of the iterative scheme combined with the heuristic for the ADI parameters.

432 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that a finite-dimensional continuous-time system is forward complete if solutions exist globally, for positive time, in a necessary and sufficient manner by means of smooth scalar growth inequalities.

389 citations


Journal ArticleDOI
TL;DR: In this article, the stability and stabilizability properties of systems with discontinuous righthand side (with solutions intended in Filippov's sense) by means of locally Lipschitz continuous and regular Lyapunov functions were studied.
Abstract: We study stability and stabilizability properties of systems with discontinuous righthand side (with solutions intended in Filippov's sense) by means of locally Lipschitz continuous and regular Lyapunov functions. The stability result is obtained in the more general context of differential inclusions. Concerning stabilizability, we focus on systems affine with respect to the input: we give some sufficient conditions for a system to be stabilized by means of a feedback law of the Jurdjevic-Quinn type.

383 citations


Journal ArticleDOI
TL;DR: Asymptotic stability of a class of linear equations with arbitrary discrete and distributed delays is investigated and the approach of deriving various Riccati equations using the direct Lyapunov method is proposed.
Abstract: Asymptotic stability of a class of linear equations with arbitrary discrete and distributed delays is investigated. Both delay-independent and delay-dependent stability conditions are formulated in terms of existence of positive definite solutions to Riccati matrix equations. The approach of deriving various Riccati equations using the direct Lyapunov method is proposed.

Journal ArticleDOI
TL;DR: In this article, a Lyapunov function is used to determine conditions to guarantee input-to-state stability (ISS) which also ensures global asymptotic stability (GAS).
Abstract: In this paper a novel approach to assess the stability of dynamic neural networks is presented. Using a Lyapunov function, we determine conditions to guarantee input-to-state stability (ISS) which also ensures global asymptotic stability (GAS). The applicability of these conditions is illustrated by two examples.

Journal ArticleDOI
TL;DR: This paper presents necessary and sufficient conditions to test for quadratic stabilizability and for robust stabilizing control laws for switched controller systems and algorithms which can be used to construct appropriately stabilized control laws.

Proceedings ArticleDOI
02 Jun 1999
TL;DR: A new parametrization is proposed for the modeling of control effector failures in flight control applications that derives stable multiple model adaptive reconfigurable control algorithms for the most complex case when one of the effectors undergoes float, lock-in-place or hard-over failure, while all others lose effectiveness.
Abstract: We propose a new parametrization for the modeling of control effector failures in flight control applications. The failures include float, lock-in-place, hard-over, and loss of effectiveness. It is shown that the resulting representation leads naturally to a multiple model formulation of the corresponding control problem that can be solved using a multiple model adaptive reconfigurable control approach. We derive stable multiple model adaptive reconfigurable control algorithms for the most complex case when one of the effectors undergoes float, lock-in-place or hard-over failure, while all others lose effectiveness. The stability of the overall reconfigurable control system is demonstrated using the Lyapunov method and the separation between identification and control arising in the context of indirect adaptive control. The approach is illustrated through numerical simulations of the F-18 aircraft carrier landing manoeuvre.

Book
01 Jan 1999
TL;DR: First order linear differential equations Two-dimensional systems and state space Higher dimensional linear systems Approximation & Simulation Nonlinear dynamics & bifurcations Computation by excitatory & inhibitory networks Nonlinear oscillations.
Abstract: First order linear differential equations Two-dimensional systems and state space Higher dimensional linear systems Approximation & Simulation Nonlinear dynamics & bifurcations Computation by excitatory & inhibitory networks Nonlinear oscillations Action potentials & limit cycles Neural adaptation & bursting Neural Chaos Synapses & synchrony Swimming & traveling waves Lyapunov functions & memory Diffusion & dendrites Nonlinear dynamics & brain function Appendix.

Journal ArticleDOI
TL;DR: It is shown that the decreasing Lyapunov function condition leads to a linear matrix inequality (LMI) problem, which points out the connection between a good convergence behavior of the EKO and the instrumental matrices R/ sub k/ and Q/sub k/.
Abstract: The authors show how the extended Kalman filter, used as an observer for nonlinear discrete-time systems or extended Kalman observer (EKO), becomes a useful state estimator when the arbitrary matrices, namely R/sub k/ and Q/sub k/, are adequately chosen. As a first step, we use the linearization technique given by Boutayed et al. (1997), which consists of introducing unknown diagonal matrices to take the approximation errors into account. It is shown that the decreasing Lyapunov function condition leads to a linear matrix inequality (LMI) problem, which points out the connection between a good convergence behavior of the EKO and the instrumental matrices R/sub k/ and Q/sub k/. In order to satisfy the obtained LMI, a particular design of Q/sub k/ is given. High performances of the proposed technique are shown through numerical examples under the worst conditions.

Journal ArticleDOI
TL;DR: This paper presents the derivation, simulation, and implementation of a nonlinear tracking control law for a hydraulic servosystem that provides for exponentially stable force trajectory tracking and is extended to provide position tracking.
Abstract: This paper presents the derivation, simulation, and implementation of a nonlinear tracking control law for a hydraulic servosystem. An analysis of the nonlinear system equations is used in the derivation of a Lyapunov function that provides for exponentially stable force trajectory tracking. This control law is then extended to provide position tracking. The proposed controller is simulated and then implemented on an experimental hydraulic system to test the limits of its performance and the realistic effects of friction.

Proceedings ArticleDOI
07 Dec 1999
TL;DR: A Lyapunov-based theory for asynchronous dynamical systems is presented and it is shown how LyAPunov functions and controllers can be constructed for such systems by solving linear matrix inequality (LMI) and bilinear matrix equality (BMI) problems.
Abstract: We consider dynamical systems which are driven by external "events" that occur asynchronously. It is assumed that the event rates are fixed, or at least they can be bounded on any time period of length T. Such systems are becoming increasingly important in control due to the very rapid advances in digital systems, communication systems, and data networks. Examples of asynchronous systems include, control systems in which signals are transmitted over an asynchronous network, parallelized numerical algorithms, and queuing networks. We present a Lyapunov-based theory for asynchronous dynamical systems and show how Lyapunov functions and controllers can be constructed for such systems by solving linear matrix inequality (LMI) and bilinear matrix inequality (BMI) problems. Examples are also presented that demonstrate the effectiveness of the approach in analyzing practical systems.

Dissertation
01 Jan 1999
TL;DR: In this paper, a closed-loop hybrid system structure consisting of a hybrid plant and a hybrid controller is proposed, which is suitable for describing the essential dynamics of a fairly large class of physical systems in control engineering applications.
Abstract: Many physical systems today are modeled by interacting continuous and discrete event systems. Such hybrid systems contain both continuous and discrete states that influence the dynamic behavior. There has been an increasing interest in these types of systems during the last decade, mostly due to the growing use of computers in the control of physical plants but also as a result of the hybrid nature of physical processes. Hybrid system models, suitable for describing the essential dynamics of a fairly large class of physical systems in control engineering applications, are proposed in this thesis. The continuous dynamics is described by differential equations whose evolution depends on continuous states and inputs as well as discrete states. The discrete dynamics is modeled by discrete event systems dependent on discrete and continuous states and inputs. It is shown that hybrid systems can be constructed by modular decompositions. A closed-loop hybrid system structure consisting of an open-loop hybrid plant and a hybrid controller, suitable for the description of control engineering systems, is proposed. Stability is one of the most important properties of dynamic systems. A large portion of this thesis is focused on conditions ensuring stability of hybrid systems. The stability results are extensions of Lyapunov theory where the existence of an abstract energy function satisfying certain properties verifies stability. It is shown how the search for such functions can be formulated as linear matrix inequality (LMI) problems, where solutions can be found by computerized methods. Stability robustness dealing with the possibility to guarantee stability despite the presence of model uncertainties is also treated. A large number of examples illustrating different approaches is given. There are many controller structures in the industry consisting of local controllers and the design task is to decide the appropriate switching among these. A related problem occurs in the case of having discrete actuators in continuous processes. The design part of this thesis addresses the problem of how to switch between different continuous vector fields guaranteeing stability of the closed-loop system.

Journal ArticleDOI
TL;DR: In this article, the authors focus on mixed delay-independent/delay-dependent asymptotic stability problems of a class of linear systems described by delay-differential equations involving several constant but unknown delays and give sufficient conditions for characterizing unbounded stability regions in the delays parameter space.
Abstract: This paper focuses on 'mixed' delay-independent/delay-dependent asymptotic stability problems of a class of linear systems described by delay-differential equations involving several constant but unknown delays. We give some sufficient conditions for characterizing unbounded stability regions in the delays parameter space. The proposed approach makes use of some appropriate Liapunov-Krasovskii functionals, and the results obtained are expressed in terms of matrix inequalities. We also discuss several ways to construct such analytic functionals. These results allow us to recover (or to improve) as limit cases previous delay-independent or/and delay-dependent conditions from the control literature.

Journal ArticleDOI
TL;DR: In this article, a general model of hybrid dynamical systems, which is defined on an arbitrary metric space, is defined. And a variety of Lyapunov and Lagrange stability results are established and made public in a scattering of publications and workshop records.
Abstract: Dramatic progress in computing capabilities has resulted in the synthesis and implementation of increasingly complex dynamical systems. Since such systems frequently exhibit simultaneously several kinds of dynamic behavior in different parts of the system, they are referred to as hybrid dynamical systems. Such systems frequently defy traditional modeling and analysis techniques since the different system components may evolve along different notions of "time", including real time, discrete time, and discrete events. Most investigations of such systems to date involve ad hoc models and tailor-made analysis results. In some recent work, however, a general model has been proposed which contains most of the different classes of hybrid dynamical systems considered in the literature as special cases. At the core of this general model of hybrid dynamical system, which is defined on an arbitrary metric space, is a notion of generalized time. For this class of general hybrid dynamical systems, a variety of Lyapunov and Lagrange stability results have been established and made public in a scattering of publications and workshop records. Our objective in this paper is to present a unified overview of the more important aspects of this work.

Journal ArticleDOI
TL;DR: The main focus of the paper is on the presentation of a second method, which extends the applicability of stable adaptive fuzzy control to a broader class of nonlinear plants; this is achieved by an improved controller structure adopted from the neural network domain.
Abstract: Stable adaptive fuzzy control is a self-tuning concept for fuzzy controllers that uses a Lyapunov-based learning algorithm, thus guaranteeing stability of the system plant-controller-learning algorithm and convergence of the plant output to a given reference signal. In the paper, two new methods for stable adaptive fuzzy control are presented. The first method is an extension of an existing concept: it is shown that a major drawback of that concept, the necessity for new adaptation at every change of the reference signal, can be avoided by a simple modification. The main focus of the paper is on the presentation of a second method, which extends the applicability of stable adaptive fuzzy control to a broader class of nonlinear plants; this is achieved by an improved controller structure adopted from the neural network domain. Performance and limitations of the proposed methods, as well as some practical design aspects, are discussed and illustrated with simulation results.

Posted Content
TL;DR: In this article, necessary and sufficient characterizations of several notions of input to output stability are presented, and the results given here extend their validity to the case when the output, but not necessarily the entire internal state, is being regulated.
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. The work is related to partial stability of differential equations (the particular case which arises when there are no external inputs).

Posted Content
TL;DR: This work establishes a complete equivalence between the IOSS property and the existence of a certain type of smooth Lyapunov function, and obtains characterizations of nonlinear detectability in terms of relative stability and of finite-energy estimates.
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.

Journal ArticleDOI
TL;DR: In this article, the existence of smooth Lyapunov functions is shown to imply the presence of feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances.
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 (or robust) 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 (in general, discontinuous) feedback stabilizers which are robust with respect to small measurement errors and small additive external disturbances. ∗Supported in part by Russian Fund for Fundamental Research Grant 96-01-00219 and by the Rutgers Center for Systems and Control (SYCON). Work done while visiting Rutgers University, Mathematics Department. On leave from Steklov Institute of Mathematics, Moscow 117966, Russia †Supported in part by US Air Force Grant AFOSR-94-0293

Journal ArticleDOI
TL;DR: A general comparison theory is developed for the class of hybrid dynamical systems considered herein, making use of stability preserving mappings, and it is shown how these results can be applied to establish some of the Principal Lyapunov Stability Theorems.

Journal ArticleDOI
TL;DR: This work investigates the numerical solution of the stable generalized Lyapunov equation via the sign function method and considers some modifications and discusses how to solve generalized LyAPunov equations with semidefinite constant term for the Cholesky factor.
Abstract: We investigate the numerical solution of the stable generalized Lyapunov equation via the sign function method This approach has already been proposed to solve standard Lyapunov equations in several publications The extension to the generalized case is straightforward We consider some modifications and discuss how to solve generalized Lyapunov equations with semidefinite constant term for the Cholesky factor The basic computational tools of the method are basic linear algebra operations that can be implemented efficiently on modern computer architectures and in particular on parallel computers Hence, a considerable speed-up as compared to the Bartels–Stewart and Hammarling methods is to be expected We compare the algorithms by performing a variety of numerical tests

Journal ArticleDOI
TL;DR: In this article, robust stability conditions for continuous-time systems were extended to the discrete-time case and expressed in terms of Linear Matrix Inequalities (LMI), being thus simply and efficiently computable.

Journal ArticleDOI
TL;DR: The authors analyze the problem of synthesizing a state feedback control for the class of uncertain continuous-time linear systems affected by time-varying memoryless parametric uncertainties and proves that such a Lyapunov function can always be obtained by "smoothing" a polyhedral function for which construction algorithms are available.
Abstract: The authors analyze the problem of synthesizing a state feedback control for the class of uncertain continuous-time linear systems affected by time-varying memoryless parametric uncertainties. They consider as candidate Lyapunov functions the elements of the class /spl Sigma//sub p//sup z/ which is formed by special homogeneous positive definite functions. They show that this class is universal in the sense that a Lyapunov function exists if and only if there exists a Lyapunov function in /spl Sigma//sub p//sup z/. They prove this result in a constructive way, showing that such a Lyapunov function can always be obtained by "smoothing" a polyhedral function for which construction algorithms are available. The authors show that unlike the polyhedral Lyapunov functions, these functions allow us to derive explicit formulas for the stabilizing controller.

DOI
01 Jan 1999
TL;DR: In this paper, global properties of PI velocity and PID position control are analyzed using a passivity and Lyapunov-based approach, and these linear control laws are then extended by nonlinear components based on the friction model considered.
Abstract: High-precision tracking requires excellent control of slow motion and positioning. Recent advances have provided dynamic friction models that represent almost all experimentally observed properties of friction. The state space formulation of these new mathematical descriptions has the property that the state derivatives are continuous functions. This enables the application of established theories for nonlinear systems. The existence of locally stable fixed points does not imply for nonlinear systems the absence of limit cycles (periodic orbits) or unstable solutions. Therefore, global properties of PI velocity and PID position control are analyzed using a passivity and Lyapunov based approach. These linear control laws are then extended by nonlinear components based on the friction model considered. The applications presented in this work are in the domains of mechatronics and machine-tools.