Showing papers on "Lyapunov function published in 2008"

Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach

Abstract: Conventions and Notation xv Preface xxi Chapter 1. Introduction 1 Chapter 2. Dynamical Systems and Differential Equations 9 Chapter 3. Stability Theory for Nonlinear Dynamical Systems 135 Chapter 4. Advanced Stability Theory 207 Chapter 5. Dissipativity Theory for Nonlinear Dynamical Systems 325 Chapter 6. Stability and Optimality of Feedback Dynamical Systems 411 Chapter 7. Input-Output Stability and Dissipativity 471 Chapter 8. Optimal Nonlinear Feedback Control 511 Chapter 9. Inverse Optimal Control and Integrator Backstepping 557 Chapter 10. Disturbance Rejection Control for Nonlinear Dynamical Systems 603 Chapter 11. Robust Control for Nonlinear Uncertain Systems 649 Chapter 12. Structured Parametric Uncertainty and Parameter-Dependent Lyapunov Functions 719 Chapter 13. Stability and Dissipativity Theory for Discrete-Time Nonlinear Dynamical Systems 763 Chapter 14. Discrete-Time Optimal Nonlinear Feedback Control 845 Bibliography 901 Index 939

Discontinuous dynamical systems

Abstract: This article has presented an introductory tutorial on discontinuous dynamical systems. Various examples illustrate the pertinence of the continuity and Lipschitzness properties that guarantee the existence and uniqueness of classical solutions to ordinary differential equations. The lack of these properties in examples drawn from various disciplines motivates the need for more general notions than the classical one. First, we introduced notions of solution for discontinuous systems. Second, we reviewed the available tools from non- smooth analysis to study the gradient information of candidate Lyapunov functions. And, third, we presented nonsmooth stability tools to characterize the asymptotic behavior of solutions.

A Lyapunov approach to second-order sliding mode controllers and observers

Abstract: In this paper a strong Lyapunov function is obtained, for the first time, for the super twisting algorithm, an important class of second order sliding modes (SOSM). This algorithm is widely used in the sliding modes literature to design controllers, observers and exact differentiators. The introduction of a Lyapunov function allows not only to study more deeply the known properties of finite time convergence and robustness to strong perturbations, but also to improve the performance by adding linear correction terms to the algorithm. These modification allows the system to deal with linearly growing perturbations, that are not endured by the basic super twisting algorithm. Moreover, the introduction of Lyapunov functions opens many new analysis and design tools to the higher order sliding modes research area.

$\mathcal{H}_2$ Model Reduction for Large-Scale Linear Dynamical Systems

Abstract: The optimal $\mathcal{H}_2$ model reduction problem is of great importance in the area of dynamical systems and simulation. In the literature, two independent frameworks have evolved focusing either on solution of Lyapunov equations on the one hand or interpolation of transfer functions on the other, without any apparent connection between the two approaches. In this paper, we develop a new unifying framework for the optimal $\mathcal{H}_2$ approximation problem using best approximation properties in the underlying Hilbert space. This new framework leads to a new set of local optimality conditions taking the form of a structured orthogonality condition. We show that the existing Lyapunov- and interpolation-based conditions are each equivalent to our conditions and so are equivalent to each other. Also, we provide a new elementary proof of the interpolation-based condition that clarifies the importance of the mirror images of the reduced system poles. Based on the interpolation framework, we describe an iteratively corrected rational Krylov algorithm for $\mathcal{H}_2$ model reduction. The formulation is based on finding a reduced order model that satisfies interpolation-based first-order necessary conditions for $\cHtwo$ optimality and results in a method that is numerically effective and suited for large-scale problems. We illustrate the performance of the method with a variety of numerical experiments and comparisons with existing methods.

Homogeneous Approximation, Recursive Observer Design, and Output Feedback

Abstract: We introduce two new tools that can be useful in nonlinear observer and output feedback design. The first one is a simple extension of the notion of homogeneous approximation to make it valid both at the origin and at infinity (homogeneity in the bi-limit). Exploiting this extension, we give several results concerning stability and robustness for a homogeneous in the bi-limit vector field. The second tool is a new recursive observer design procedure for a chain of integrator. Combining these two tools, we propose a new global asymptotic stabilization result by output feedback for feedback and feedforward systems.

Adaptive Sliding-Mode Control for NonlinearSystems With Uncertain Parameters

Abstract: This correspondence proposes a systematic adaptive sliding- mode controller design for the robust control of nonlinear systems with uncertain parameters. An adaptation tuning approach without high- frequency switching is developed to deal with unknown but bounded system uncertainties. Tracking performance is guaranteed. System robustness, as well as stability, is proven by using the Lyapunov theory. The upper bounds of uncertainties are not required to be known in advance. Therefore, the proposed method can be effectively implemented. Experimental results demonstrate the effectiveness of the proposed control method.

A survey of linear matrix inequality techniques in stability analysis of delay systems

Abstract: Recent years have witnessed a resurgence of research interests in analysing the stability of time-delay systems. Many results have been reported using a variety of approaches and techniques. However, much of the focus has been laid on the use of the Lyapunov-Krasovskii theory to derive sufficient stability conditions in the form of linear matrix inequalities. The purpose of this article is to survey the recent results developed to analyse the asymptotic stability of time-delay systems. Both delay-independent and delay-dependent results are reported in the article. Special emphases are given to the issues of conservatism of the results and computational complexity. Connections of certain delay-dependent stability results are also discussed.

Stability Analysis of Interval Type-2 Fuzzy-Model-Based Control Systems

Abstract: This paper presents the stability analysis of interval type-2 fuzzy-model-based (FMB) control systems. To investigate the system stability, an interval type-2 Takagi-Sugeno (T-S) fuzzy model, which can be regarded as a collection of a number of type-1 T-S fuzzy models, is proposed to represent the nonlinear plant subject to parameter uncertainties. With the lower and upper membership functions, the parameter uncertainties can be effectively captured. Based on the interval type-2 T-S fuzzy model, an interval type-2 fuzzy controller is proposed to close the feedback loop. To facilitate the stability analysis, the information of the footprint of uncertainty is used to develop some membership function conditions, which allow the introduction of slack matrices to handle the parameter uncertainties in the stability analysis. Stability conditions in terms of linear matrix inequalities are derived using a Lyapunov-based approach. Simulation examples are given to illustrate the effectiveness of the proposed interval type-2 FMB control approach.

Lyapunov Matrix Equation in System Stability and Control

Abstract: Part 1 Introduction: stability of linear systems variance of linear stochastic systems quadratic performance measure book organization. Part 2 Continuous algebraic Lyapunov equation: explicit solutions solution sounds numerical solutions. Part 3 Discrete algebraic Lyapunov equation: explicit solutions bounds of solution's attributes numerical solutions. Part 4 Differential and difference Lyapunov equation: explicit solutions bounds of solution's attributes numerical solutions singularly perturbed and weakly coupled systems coupled differential equations. Part 5 Algebraic Lyapunov equation with small parameters: singularly perturbed continuous Lyapunov equation weakly coupled continuous Lyapunov equation singularly perturbed discrete systems recursive methods for weakly coupled discrete systems. Part 6 Robustness and sensitivity of the Lyapunov equation: stability robustness sensitivity of algebraic Lyapunov equation. Part 7 Iterative methods and parallel algorithms: Smith's algorithm ADI iterative method SOR iterative method parallel algorithms parallel algorithms for coupled Lyapunov equations. Part 8 Lyapunov iterations: Kleinman algorithm for Riccati equation Lyapunov iterations for jump linear systems Lyapunov iterations for Nash differential games Lyapunov iterations for output feedback control. Part 9 Concluding remarks: Sylvester equations related topics applications. Appendix: matrix inequalities.

Global synchronization of linearly hybrid coupled networks with time-varying delay

Abstract: Many real-world large-scale complex networks demonstrate a surprising degree of synchronization. To unravel the underlying mechanics of synchronization in these complex networks, a generally linearly hybrid coupled network with time-varying delay is proposed, and its global synchronization is then further investigated. Several effective sufficient conditions of global synchronization are attained based on the Lyapunov function and a linear matrix inequality (LMI). Both delay-independent and delay-dependent conditions are deduced. In particular, the coupling matrix may be nonsymmetric or nondiagonal. Moreover, the derivative of the time-varying delay is extended to any given value. Finally, a small-world network, a regular network, and scale-free networks with network size are constructed to show the effectiveness of the proposed synchronous criteria.

Coordination and Consensus of Networked Agents with Noisy Measurements: Stochastic Algorithms

Abstract: This paper considers the coordination and consensus of networked agents where each agent has noisy measurements of its neighbors’ states. For consensus seeking, we propose stochastic approximation type algorithms with a decreasing step size, and introduce the notions of mean square and strong consensus. Although the decreasing step size reduces the detrimental efiect of the noise, it also reduces the ability of the algorithm to drive the individual states towards each other. The key technique is to ensure a trade-ofi for the decreasing rate of the step size. By following this strategy, we flrst develop a stochastic double array analysis in a two agent model, which leads to both mean square and strong consensus, and extend the analysis to a class of well studied symmetric models. Subsequently, we consider a general network topology, and introduce stochastic Lyapunov functions together with the so-called direction of invariance to establish mean square consensus. Finally, we apply the stochastic Lyapunov analysis to a leader following scenario.

Global Asymptotic Stability of Recurrent Neural Networks With Multiple Time-Varying Delays

Abstract: In this paper, several sufficient conditions are established for the global asymptotic stability of recurrent neural networks with multiple time-varying delays. The Lyapunov-Krasovskii stability theory for functional differential equations and the linear matrix inequality (LMI) approach are employed in our investigation. The results are shown to be generalizations of some previously published results and are less conservative than existing results. The present results are also applied to recurrent neural networks with constant time delays.

Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems

Abstract: We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linearquadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments.

Nonquadratic Stabilization Conditions for a Class of Uncertain Nonlinear Discrete Time TS Fuzzy Models: A New Approach

Abstract: The discrete-time uncertain nonlinear models are considered in a Takagi-Sugeno form and their stabilization is studied through a non- quadratic Lyapunov function. The classical conditions consider a one- sample variation, here, the main results are obtained considering k samples variation, i.e., Deltak V(x(t)) = V(x(t + k)) - V(x(t)). The results are shown to always include the classical cases, and several examples illustrate the effectiveness of the approach.

Finite-Time Semistability and Consensus for Nonlinear Dynamical Networks

Abstract: This paper focuses on semistability and finite-time stability analysis and synthesis of systems having a continuum of equilibria. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. In this paper, we merge the theories of semistability and finite-time stability to develop a rigorous framework for finite-time semistability. In particular, finite-time semistability for a continuum of equilibria of continuous autonomous systems is established. Continuity of the settling-time function as well as Lyapunov and converse Lyapunov theorems for semistability are also developed. In addition, necessary and sufficient conditions for finite-time semistability of homogeneous systems are addressed by exploiting the fact that a homogeneous system is finite-time semistable if and only if it is semistable and has a negative degree of homogeneity. Unlike previous work on homogeneous systems, our results involve homogeneity with respect to semistable dynamics, and require us to adopt a geometric description of homogeneity. Finally, we use these results to develop a general framework for designing semistable protocols in dynamical networks for achieving coordination tasks in finite time.

A graph-theoretic approach to the method of global Lyapunov functions

Abstract: A class of global Lyapunov functions is revisited and used to resolve a long-standing open problem on the uniqueness and global stability of the endemic equilibrium of a class of multi-group models in mathematical epidemiology. We show how the group structure of the models, as manifested in the derivatives of the Lyapunov function, can be completely described using graph theory.

Output Feedback Stabilization for a Discrete-Time System With a Time-Varying Delay

Abstract: This study employs the free-weighting matrix approach to investigate the output feedback control of a linear discrete-time system with an interval time-varying delay. First, the delay-dependent stability is analyzed using a new method of estimating the upper bound on the difference of a Lyapunov function without ignoring any terms; and based on the results, a design criterion for a static output feedback (SOF) controller is derived. Since the conditions thus obtained for the existence of admissible controllers are not expressed strictly in terms of linear matrix inequalities, a modified cone complementarity linearization algorithm is employed to solve the nonconvex feasibility SOF control problem. Furthermore, the problem of designing a dynamic output feedback controller is formulated as one of designing an SOF controller. Numerical examples demonstrate the effectiveness of the method and its advantage over existing methods.

Coordinated Standoff Tracking of Moving Targets Using Lyapunov Guidance Vector Fields

Abstract: This paper presents a control structure for cooperative stand-off line-of-sight tracking of a moving target by a team of unmanned aircraft based on a Lyapunov guidance vector field that produces stable convergence to a circling limit cycle behavior. A guidance vector field is designed for a stationary target and then modified with a correction term that accounts for a moving target and constant background wind. Cooperative tracking by multiple unmanned aircraft is achieved through additional phasing, also with a Lyapunov stability analysis. Convoy protection, in which the unmanned aircraft must scout an area ahead of the moving target, is performed by extending the cooperative stand-offline-of-sight limit cycle attractor along the direction of travel. Simulation results demonstrate the behavior of the algorithms as well as the improvement that results from cooperation. Finally, simulations of a larger cooperative search, acquisition, and tracking scenario are described that illustrate the use of the cooperative standoff line-of-sight and convoy protection controllers in a realistic application.

Dynamic Output Feedback Control of Switched Linear Systems

Abstract: This paper is devoted to stability analysis and control design of switched linear systems in both continuous and discrete-time domains. A particular class of matrix inequalities, the so-called Lyapunov--Metzler inequalities, provides conditions for open-loop stability analysis and closed-loop switching control using state and output feedback. Switched linear systems are analyzed in a general framework by introducing a quadratic in the state cost determined from a series of impulse perturbations. Lower bounds on the cost associated with the optimal switching control strategy are derived from the determination of a feasible solution to the Hamilton--Jacobi--Bellman inequality. An upper bound on the optimal cost associated with a closed-loop stabilizing switching strategy is provided as well. The solution of the output feedback problem is based on the construction of a full-order linear switched filter whose state variable is used by the mechanism for the determination of the switching rule. Throughout, the theoretical results are illustrated by means of academic examples. A realistic practical application related to the optimal control of semiactive suspensions in road vehicles is reported.

A New $H_{{\bm \infty}}$ Stabilization Criterion for Networked Control Systems

Abstract: This note is concerned with robust Hinfin control of linear networked control systems with time-varying network-induced delay and data packet dropout. A new Lyapunov-Krasovskii functional, which makes use of the information of both the lower and upper bounds of the time-varying network-induced delay, is proposed to drive a new delay-dependent Hinfin stabilization criterion. The criterion is formulated in the form of a non-convex matrix inequality, of which a feasible solution can be obtained by solving a minimization problem in terms of linear matrix inequalities. In order to obtain much less conservative results, a tighter bounding for some term is estimated. Moreover, no slack variable is introduced. Finally, two numerical examples are given to show the effectiveness of the proposed design method.

Dissipative Boundary Conditions for One-Dimensional Nonlinear Hyperbolic Systems

Abstract: We give a new sufficient condition on the boundary conditions for the exponential stability of one-dimensional nonlinear hyperbolic systems on a bounded interval. Our proof relies on the construction of an explicit strict Lyapunov function. We compare our sufficient condition with other known sufficient conditions for nonlinear and linear one-dimensional hyperbolic systems.