Showing papers on "Lyapunov function published in 2002"
TL;DR: The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration and shows that the second condition is, in this case, less conservative.
Abstract: This paper addresses the problem of stability analysis and control synthesis of switched systems in the discrete-time domain. The approach followed in this paper looks at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequality-based conditions allow to check the existence of such a Lyapunov function. The first one is classical while the second is new and uses a slack variable, which makes it useful for design problems. These two conditions are proved to be equivalent for stability analysis. Investigating the static output feedback control problem, we show that the second condition is, in this case, less conservative. The reduction of the conservatism is illustrated by a numerical evaluation.
1,398 citations
TL;DR: The aim is to present a framework for understanding questions of incremental stability fully compatible with the well-known input-to-state stability approach.
Abstract: Deals with several notions of incremental stability. In other words, the focus is on stability of trajectories with respect to one another, rather than with respect to some attractor. The aim is to present a framework for understanding such questions fully compatible with the well-known input-to-state stability approach. Applications of the newly introduced stability notions are also discussed.
731 citations
TL;DR: In this paper, the authors present new synthesis procedures for discrete-time linear systems based on a recently developed stability condition which contains as particular cases both the celebrated Lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters.
Abstract: This paper presents new synthesis procedures for discrete-time linear systems. It is based on a recently developed stability condition which contains as particular cases both the celebrated Lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters. These new synthesis conditions have some nice properties: (a) they can be expressed in terms of LMI (linear matrix inequalities) and (b) the optimization variables associated with the controller parameters are independent of the symmetric matrix that defines a quadratic Lyapunov function used to test stability. This second feature is important for several reasons. First, structural constraints, as those appearing in the decentralized and static output-feedback control design, can be addressed less conservatively. Second, parameter dependent Lyapunov function can be considered with a very positive impact on the design of robust H 2 and H X control problems. Third, the design of controller with mixed ...
675 citations
TL;DR: In this paper, the ergodicity of SDEs is established by using techniques from the theory of Markov chains on general state spaces, such as that expounded by Meyn-Tweedie.
618 citations
10 Dec 2002
TL;DR: In this article, a relaxation of Lyapunov's direct method has been proposed to include systems with equality, inequality, and integral constraints, which allows robustness analysis to be performed.
Abstract: A relaxation of Lyapunov's direct method has been proposed elsewhere that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In the paper, the above technique is extended to include systems with equality, inequality, and integral constraints. This allows certain non-polynomial nonlinearities in the vector field to be handled exactly and the constructed Lyapunov functions to contain non-polynomial terms. It also allows robustness analysis to be performed. Some examples are given to illustrate how this is done.
498 citations
TL;DR: In this paper, the global tracking problem for an underactuated ship with only two propellers was studied and two constructive solutions were proposed by application of Lyapunov's direct method under sufficient conditions of persistent excitation.
443 citations
TL;DR: In this paper, the authors establish global convergence results for stochastic fictitious play for four classes of games: games with an interior ESS, zero sum games, potential games, and supermodular games.
Abstract: We establish global convergence results for stochastic fictitious play for four classes of games: games with an interior ESS, zero sum games, potential games, and supermodular games. We do so by appealing to techniques from stochastic approximation theory, which relate the limit behavior of a stochastic process to the limit behavior of a differential equation defined by the expected motion of the process. The key result in our analysis of supermodular games is that the relevant differential equation defines a strongly monotone dynamical system. Our analyses of the other cases combine Lyapunov function arguments with a discrete choice theory result: that the choice probabilities generated by any additive random utility model can be derived from a deterministic model based on payoff perturbations that depend nonlinearly on the vector of choice probabilities.
420 citations
10 Dec 2002
TL;DR: The main result is a suite of theorems about formation maintenance, task completion time, and formation velocity that moderate the requirement that, for each individual robot, there exists a control Lyapunov function.
Abstract: In this paper, the multiagent coordination problem is studied. This problem is addressed for a class of robots for which control Lyapunov functions can be found. The main result is a suite of theorems about formation maintenance, task completion time, and formation velocity. It is also shown how to moderate the requirement that, for each individual robot, there exists a control Lyapunov function. An example is provided that illustrates the soundness of the method.
407 citations
TL;DR: The technique described in this paper can be applied to polytopic uncertain systems and is based on the use of several Lyapunov functions each one corresponding to a different vertex of the uncertainty's polytope.
391 citations
TL;DR: The new approach applies to general discrete-time and continuous-time nonlinear systems with uncertainties and relies on safety properties provided by sublevel sets of equilibria-parameterized functions, which significantly broadens the class of methods available for constructing the nonlinear function that is required in the implementation of the reference governors.
386 citations
TL;DR: In this paper, a model for the dynamics of a four rotor vertical take-off and landing (VTOL) vehicle known as an X4-flyer is proposed, which incorporates the airframe and motor dynamics as well as aerodynamic and gyroscopic effects due to the rotors for quasi-stationary flight conditions.
TL;DR: In this article, a Lyapunov-Krasovskii functional is constructed for linear coupled systems of delay differential and functional equations, which corresponds directly to the descriptor form of the system and provides sufficient conditions for delay-dependent/delayindependent stability and robustness of stability with respect to small delays.
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01 Jan 2002TL;DR: In this paper, Lyapunov stability theory of differential equations and nonuniform hyperbolic theory of local manifold theory have been studied in the context of smooth hyperbolc measures.
Abstract: Introduction Lyapunov stability theory of differential equations Elements of nonuniform hyperbolic theory Examples of nonuniformly hyperbolic systems Local manifold theory Ergodic properties of smooth hyperbolc measures Bibliography Index
TL;DR: It is argued that it is sufficient to build an observer for the output tracking error of uncertain nonlinear systems to ensureUltimate boundedness of the error signals is shown through Lyapunov's direct method.
Abstract: We consider adaptive output feedback control of uncertain nonlinear systems, in which both the dynamics and the dimension of the regulated system may be unknown. However, the relative degree of the regulated output is assumed to be known. Given a smooth reference trajectory, the problem is to design a controller that forces the system measurement to track it with bounded errors. The classical approach requires a state observer. Finding a good observer for an uncertain nonlinear system is not an obvious task. We argue that it is sufficient to build an observer for the output tracking error. Ultimate boundedness of the error signals is shown through Lyapunov's direct method. The theoretical results are illustrated in the design of a controller for a fourth-order nonlinear system of relative degree two and a high-bandwidth attitude command system for a model R-50 helicopter.
TL;DR: It is shown here that the proposed neural network is stable in the sense of Lyapunov and globally convergent, globally asymptotically stable, and globally exponentially stable, respectively under different conditions.
Abstract: In this paper, we present a recurrent neural network for solving the nonlinear projection formulation. It is shown here that the proposed neural network is stable in the sense of Lyapunov and globally convergent, globally asymptotically stable, and globally exponentially stable, respectively under different conditions. Compared with the existing neural network for solving the projection formulation, the proposed neural network has a single-layer structure and is amenable to parallel implementation. Moreover, the proposed neural network has no Lipschitz condition, and, thus can be applied to solve a very broad class of constrained optimization problems that are special cases of the nonlinear projection formulation. Simulation shows that the proposed neural network is effective in solving these constrained optimization problems.
TL;DR: Lyapunov functions for classical SIR, SIRS, and SIS epidemiological models are introduced and global stability of the endemic equilibrium states of the models is established.
TL;DR: It is shown that a smooth Lyapunov function exists for a family of time-varying discrete systems if these systems are robustly globally asymptotically stable.
TL;DR: In this article, it was shown that for any compact surface, there is a residual set of continuous SL(2, R)-cocycles which either are uniformly hyperbolic or have zero exponents.
Abstract: We show that, for any compact surface, there is a residual (dense Gδ )s et of C 1 area-preserving diffeomorphisms which either are Anosov or have zero Lyapunov exponentsa.e. This result was announcedby R. Mane, but no proof was available. We also show that for any fixed ergodic dynamical system over a compact space, there is a residual set of continuous SL(2, R)-cocycles which either are uniformly hyperbolic or have zero exponents a.e.
TL;DR: A controller is developed for underactuated surface ships with only surge force and yaw moment available to globally asymptotically track a reference trajectory generated by a suitable virtual ship in a frame attached to the ship body.
Abstract: A controller is developed for underactuated surface ships with only surge force and yaw moment available to globally asymptotically track a reference trajectory generated by a suitable virtual ship in a frame attached to the ship body. The reference trajectory is allowed too be a curve including a straight line. The control development is based on Lyapunov's direct method and backstepping technique, and utilizes several properties of ship dynamics and their interconnected structure. Numerical simulations are provided to validate the effectiveness of the proposed controller.
TL;DR: By using arguments from the dissipativity theory for nonlinear systems, this paper generalizes the approach to analyze the l"2-gain of PWA systems and shows that the continuity of the Lyapunov function is not required in discrete time.
TL;DR: It is proved that the proposed robust adaptive scheme can guarantee the uniform ultimate boundedness of the closed-loop system signals.
Abstract: This paper presents a robust adaptive neural control design for a class of perturbed strict feedback nonlinear system with both completely unknown virtual control coefficients and unknown nonlinearities. The unknown nonlinearities comprise two types of nonlinear functions: one naturally satisfies the "triangularity condition" and can be approximated by linearly parameterized neural networks, while the other is assumed to be partially known and consists of parametric uncertainties and known "bounding functions." With the utilization of iterative Lyapunov design and neural networks, the proposed design procedure expands the class of nonlinear systems for which robust adaptive control approaches have been studied. The design method does not require a priori knowledge of the signs of the unknown virtual control coefficients. Leakage terms are incorporated into the adaptive laws to prevent parameter drifts due to the inherent neural-network approximation errors. It is proved that the proposed robust adaptive scheme can guarantee the uniform ultimate boundedness of the closed-loop system signals.. The control performance can be guaranteed by an appropriate choice of the design parameters. Simulation studies are included to illustrate the effectiveness of the proposed approach.
TL;DR: By Liapunov functions/functionals, sufficient conditions are obtained for general exponential stability, while by using a comparison result from the theory of monotone dynamical systems, componentwise exponential stability is discussed.
TL;DR: The direct adaptive control (DAC) has the advantage of less design effort by not using FNN to model the plant, and the free parameters of the adaptive FNN controller can be tuned on-line based on the Lyapunov synthesis approach.
Abstract: In this paper, an observer-based direct adaptive fuzzy-neural network (FNN) controller with supervisory mode for a certain class of high order unknown nonlinear dynamical system is presented. The direct adaptive control (DAC) has the advantage of less design effort by not using FNN to model the plant. By using an observer-based output feedback control law and adaptive law, the free parameters of the adaptive FNN controller can be tuned on-line based on the Lyapunov synthesis approach. A supervisory controller is appended into the FNN controller to force the state to be within the constraint set. Therefore, if the FNN controller cannot maintain the stability, the supervisory controller starts working to guarantee stability. On the other hand, if the FNN controller works well, the supervisory controller will be de-activated. The overall adaptive scheme guarantees the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded. Simulation results also show that our initial control effort is much less than those in previous works, while preserving the tracking performance.
TL;DR: It is shown that the stability of the system can be established if a piecewise Lyapunovfunction can be constructed and the function can be obtained by solving a set of linear matrix inequalities (LMIs) that is numerically feasible with commercially available software.
Abstract: Presents a stability analysis method for piecewise discrete-time linear systems based on a piecewise smooth Lyapunov function. It is shown that the stability of the system can be established if a piecewise Lyapunov function can be constructed and, moreover, the function can be obtained by solving a set of linear matrix inequalities (LMIs) that is numerically feasible with commercially available software.
TL;DR: In this article, the existence and stability of linear invariant manifolds in a network of coupled identical dynamical systems is studied, where symmetry under permutation of different units of the network is helpful to construct explicit formulae for linear invariants.
TL;DR: Performance analysis using a Lyapunov synthesis approach proves the superiority of the HA law over the direct adaptive (DA) method in terms of faster and improved tracking and parameter convergence, and this is achieved at negligible increased implementation cost or computational complexity.
Abstract: We present a combined direct and indirect adaptive control scheme for adjusting an adaptive fuzzy controller, and adaptive fuzzy identification model parameters First, using adaptive fuzzy building blocks, with a common set of parameters, we design and study an adaptive controller and an adaptive identification model that have been proposed for a general class of uncertain structure nonlinear dynamic systems We then propose a hybrid adaptive (HA) law for adjusting the parameters The HA law utilizes two types of errors in the adaptive system, the tracking error and the modeling error Performance analysis using a Lyapunov synthesis approach proves the superiority of the HA law over the direct adaptive (DA) method in terms of faster and improved tracking and parameter convergence Furthermore, this is achieved at negligible increased implementation cost or computational complexity We prove a theorem that shows the properties of this hybrid adaptive fuzzy control system, ie, bounds for the integral of the squared errors, and the conditions under which these errors converge asymptotically to zero are obtained Finally, we apply the hybrid adaptive fuzzy controller to control a chaotic system, and the inverted pendulum system
01 Aug 2002
TL;DR: A learning-based feedforward term is developed to solve a general control problem in the presence of unknown nonlinear dynamics with a known period and is utilized to achieve global asymptotic link position tracking for a robot manipulator.
Abstract: In this paper, a learning-based feedforward term is developed to solve a general control problem in the presence of unknown nonlinear dynamics with a known period. Since the learning-based feedforward term is generated from a straightforward Lyapunov-like stability analysis, the control designer can utilize other Lyapunov-based design techniques to develop hybrid control schemes that utilize learning-based feedforward terms to compensate for periodic dynamics and other Lyapunov-based approaches (e.g., adaptive-based feedforward terms) to compensate for nonperiodic dynamics. To illustrate this point, a hybrid adaptive/learning control scheme is utilized to achieve global asymptotic link position tracking for a robot manipulator.
TL;DR: It is shown how to use a new parameter dependent Lyapunov matrix procedure to determine high performance H2 robust filters by solving a linear problem constrained by linear matrix inequalities (LMIs).
Abstract: Robust filtering of linear time-invariant discrete-time uncertain systems is investigated through a new parameter dependent Lyapunov matrix procedure. Its main interest relies on the fact that the Lyapunov matrix used in stability checking does not appear in any multiplicative term with the uncertain matrices of the dynamic model. We show how to use such an approach to determine high performance H2 robust filters by solving a linear problem constrained by linear matrix inequalities (LMIs). The results encompass the previous works in the quadratic Lyapunov setting. Numerical examples illustrate the theoretical results.
27 Feb 2002
TL;DR: In this article, the Routh-Hurwitz and Schur-Cohn Criterion Nonlinear Systems and Stability Limit Sets and Invariant Manifolds Dissipative Maps Stability of Difference Equations Semicycle Analysis Dynamica Session on Semicycles Exercises InvARIANTS and RELATED LYAPUNOV FUNCTIONS Introduction Invariants for Linear Equations and Systems Invarantants and Corresponding Lyapunov Functions for Nonlinear systems Invariance under Lie Group Transformations Exercise DYNAMICS of Three-DIMENSION
Abstract: DYNAMICS OF ONE-DIMENSIONAL DYNAMICAL SYSTEMS Introduction Linear Difference Equations with Constant Coefficients Linear Difference Equations with Variable Coefficients Stability Stability in the Non-Hyperbolic Case Bifurcations Dynamica Session Symbolic Dynamics for One-Dimensional Maps Dissipative Maps and Global Attractivity Parametrisation and Poincare Functional Equation Exercises DYNAMICS OF TWO-DIMENSIONAL DYNAMICAL SYSTEMS Introduction Linear Theory Equilibrium Solutions The Riccati Equation Linearized Stability Analysis Dynamica Session Period Doubling Bifurcation Lyapunov Numbers Box Dimension Semicycle Analysis Stable and Unstable Manifold Dynamica Session on Henon's Equation Invariants Lyapunov Functions, Stability, and Invariants Dynamica Session on Lyness' Map Dissipative Maps and Systems Dynamica Session on Rational Difference Equations Area-preserving Maps and Systems Biology Applications Projects Applications in Economics Exercises SYSTEMS OF DIFFERENCE EQUATIONS, STABILITY, AND SEMICYCLES Introduction Linear Theory Stability of Linear Systems The Routh-Hurwitz and Schur-Cohn Criterion Nonlinear Systems and Stability Limit Sets and Invariant Manifolds Dissipative Maps Stability of Difference Equations Semicycle Analysis Dynamica Session on Semicycles Exercises INVARIANTS AND RELATED LYAPUNOV FUNCTIONS Introduction Invariants for Linear Equations and Systems Invariants and Corresponding Lyapunov Functions for Nonlinear Systems Invariants of Special Class of Difference Equations Applications Dynamica Session on Invariants Dynamica Session on Lyapunov Functions Invariance under Lie Group Transformations Exercises DYNAMICS OF THREE-DIMENSIONAL DYNAMICAL SYSTEMS Introduction Dynamica Session on Third Order Difference Equations Dissipative Difference Equation of Third Order Dynamica Session on Local Asymptotic Stability of Period-Two Solution Dynamica Session on Todd's Difference Equation Biology Applications Projects Exercises FRACTALS GENERATED BY ITERATED FUNCTIONS SYSTEMS Introduction Basic Definitions and Results Iterated Function System Basic Results on Iterated Functions Systems Calculation of Box Dimension for IFS Dynamica Session Exercises BIBLIOGRAPHY INDEX
TL;DR: This paper proposes the first universal controller of which the synthesis is based on Lyapunov's direct method and backstepping technique, and its result is extendible to the input-saturation when the surge and yaw velocities are considered as the controls.