A new class of Lyapunov functions for nonstandard switching systems: The stability analysis problem (original) (raw)
Related papers
2014
This paper addresses the estimation of the domain of attraction for discrete-time nonlinear systems where the vector field is subject to changes. Firstly, the paper considers the case of switched systems, where the vector field is allowed to arbitrarily switch among the elements of a finite family. Secondly, the paper considers the case of hybrid systems, where the state space is partitioned into several regions described by polynomial inequalities, and the vector field is defined on each region independently from the other ones. In both cases, the problem consists of computing the largest sublevel set of a Lyapunov function included in the domain of attraction. An approach is proposed for solving this problem based on convex programming, which provides a guaranteed inner estimate of the sought sublevel set. The conservatism of the provided estimate can be decreased by increasing the size of the optimization problem. Some numerical examples illustrate the proposed approach.
Stabilization and performance analysis for a class of switched systems
2004
This paper investigates stability and control design problems with performance analysis for discrete-time switched linear systems. The switched Lyapunov function method is combined with Finsler's Lemma to generate various tests in the enlarged space containing both the state and its time difference, allowing extra degree of freedom for stability analysis and control design. Two performance measures being considered are the decay rate and the inputoutput performance. A new LMI based stability test for the existence of switched Lyapunov functions is first developed. If a switched Lyapunov function exists, asymptotic stability of the switched system also implies its exponential stability. An LMI optimization problem is then formulated to find a bound on the decay rate of the system. To attain the bound, state feedback control gains are designed. Using the same framework and the well-known S-procedure, a generalized sufficient LMI condition is obtained which guarantees a γperformance of the closed-loop switched systems subject to input disturbances.
Min-switching local stabilization for discrete-time switching systems with nonlinear modes
Nonlinear Analysis: Hybrid Systems, 2013
This paper deals with the discrete-time switched Lur'e problem in finite domain. The aim is to provide a stabilization inside an estimate of the origin's basin of attraction, large as possible, via a suitable switching rule. The design of this switching rule is based on the min-switching policy and can be induced by sufficient conditions given by Lyapunov-Metzler inequalities. Nevertheless instead of intuitively considering a switched quadratic Lyapunov function for this approach, a suitable switched Lyapunov function including the modal nonlinearities is proposed. The assumptions required to characterize the nonlinearities are only mode-dependent sector conditions, without constraints related to the slope of the nonlinearities. An optimization problem is provided to allow the maximization of the size of the basin of attraction estimate-which may be composed of disconnected sets-under the stabilization conditions. A numerical example illustrate the efficiency of our approach and emphasize the specificities of our tools.
Switching and stability properties of conewise linear systems
ESAIM: Control, Optimisation and Calculus of Variations, 2010
Being a unique phenomenon in hybrid systems, mode switch is of fundamental importance in dynamic and control analysis. In this paper, we focus on global long-time switching and stability properties of conewise linear systems (CLSs), which are a class of linear hybrid systems subject to state-triggered switchings recently introduced for modeling piecewise linear systems. By exploiting the conic subdivision structure, the "simple switching behavior" of the CLSs is proved. The infinitetime mode switching behavior of the CLSs is shown to be critically dependent on two attracting cones associated with each mode; fundamental properties of such cones are investigated. Verifiable necessary and sufficient conditions are derived for the CLSs with infinite mode switches. Switch-free CLSs are also characterized by exploring the polyhedral structure and the global dynamical properties. The equivalence of asymptotic and exponential stability of the CLSs is established via the uniform asymptotic stability of the CLSs that in turn is proved by the continuous solution dependence on initial conditions. Finally, necessary and sufficient stability conditions are obtained for switch-free CLSs. optimization, systems/control, and robotics, due to their wide applications in the modeling of nonsmooth physical systems and dynamic optimization in operations research. See the two survey articles and the research papers as well as the references therein for various issues and results arising from many theoretical and applied problems.
Stability analysis for a class of nonlinear switched systems
Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304), 2000
In the present paper, we study several qualitative properties of a class of nonlinear switched systems under certain switching laws. First, we show that if all the subsystems are linear time-invariant and the system matrices are commutative componentwise and stable, then the entire switched system is globally exponentially stable under arbitrary switching laws. Next, we study the above linear switched systems with certain nonlinear perturbations, which can be either vanishing or non-vanishing. Under reasonable assumptions, global exponential stability is established for these systems. We further study the stability and instability properties, under certain switching laws, for switched systems with commutative subsystem matrices that may be unstable. Results for both continuous-time and discrete-time cases are presented.
Stability Criteria via Common Non-strict Lyapunov Matrix for Discrete-time Linear Switched Systems
2011
Let S = {S 1 , S 2 } ⊂ R d×d have a common, but not necessarily strict, Lyapunov matrix (i.e. there exists a symmetric positive-definite matrix P such that P − S T k PS k ≥ 0 for k = 1, 2). Based on a splitting theorem of the state space R d (Dai, Huang and Xiao, arXiv:1107.0132v1[math.PR]), we establish several stability criteria for the discrete-time linear switched dynamics x n = S σ n · · · S σ 1 (x 0 ), x 0 ∈ R d and n ≥ 1 governed by the switching signal σ : N → {1, 2}. More specifically, let ρ(A) stand for the spectral radius of a matrix A ∈ R d×d , then the outline of results obtained in this paper are: (1) For the case d = 2, S is absolutely stable (i.e., S σ n · · · S σ 1 → 0 driven by all switching signals σ) if and only if ρ(S 1 ), ρ(S 2 ) and ρ(S 1 S 2 ) all are less than 1; (2) For the case d = 3, S is absolutely stable if and only if ρ(A) < 1 ∀A ∈ {S 1 , S 2 } ℓ for ℓ = 1, 2, 3, 4, 5, 6, and 8. This further implies that for any S = {S 1 , S 2 } ⊂ R d×d with the generalized spectral radius ρ(S) = 1 where d = 2 or 3, if S has a common, but not strict in general, Lyapunov matrix, then S possesses the spectral finiteness property.
Analysis and control of switched linear systems via modified Lyapunov-Metzler inequalities
International Journal of Robust and Nonlinear Control, 2014
This paper addresses analysis and switching control problems of continuous/discrete-time switched linear systems. A particular class of matrix inequalities, the so-called Lyapunov-Metzler inequalities, will be modified to provide conditions for stability analysis and output feedback control synthesis under a relaxed min-switching logic. The switching rule combined with switching output feedback controllers will be designed to stabilize the switched system and satisfy a prespecified L 2 (`2) gain performance. The proposed analysis and switching control approach could refrain frequent switches commonly observed in minswitching based designs. The effectiveness of the proposed approach will be illustrated through numerical examples.
Stability analysis of discrete-time LPV switched systems
IFAC-PapersOnLine, 2020
This paper addresses the stability problem for discrete-time switched systems under autonomous switching. Each mode of the switched system is modeled as a Linear Parameter Varying (LPV) system, the time-varying parameters can vary arbitrarily fast and are represented in a polytopic form. The Lyapunov theory is employed to get new conditions in the form of parameter-dependent LMIs. The constructed Lyapunov function takes advantage of using an augmented state vector with shifted states in its construction. In this sense, the Lyapunov function employed in this paper can be viewed as a discrete-time LPV switched Lyapunov function. Numerical experiments illustrate the efficacy of the technique in providing stability certificates.
On the -stabilization of switched nonlinear systems via state-dependent switching rule
Applied Mathematics and Computation, 2010
This paper considers switching stabilization of some general nonlinear systems. Assuming certain properties of a convex linear combination of the nonlinear vector fields, two ways of generating stabilizing switching signals are proposed, i.e., the minimal rule and the generalized rule, both based on a partition of the time-state space. The main theorems show that the resulting switched system is globally uniformly asymptotically stable and globally uniformly exponentially stable, respectively. It is shown that the stabilizing switching signals do not exhibit chattering, i.e., two consecutive switching times are separated by a positive amount of time. In addition, under the generalized rule, the switching signal does not exhibit Zeno behavior (accumulation of switching times in a finite time). Stability analysis is performed in terms of two measures so that the results can unify many different stability criteria, such as Lyapunov stability, partial stability, orbital stability, and stability of an invariant set. Applications of the main results are shown by several examples, and numerical simulations are performed to both illustrate and verify the stability analysis.
A Characterization of Lyapunov Inequalities for Stability of Switched Systems
2016
We study stability criteria for discrete-time switched systems and provide a meta-theorem that characterizes all Lyapunov theorems of a certain canonical type. For this purpose, we investigate the structure of sets of LMIs that provide a sufficient condition for stability. Various such conditions have been proposed in the literature in the past fifteen years. We prove in this note that a family of languagetheoretic conditions recently provided by the authors encapsulates all the possible LMI conditions, thus putting a conclusion to this research effort. As a corollary, we show that it is PSPACE-complete to recognize whether a particular set of LMIs implies stability of a switched system. Finally, we provide a geometric interpretation of these conditions, in terms of existence of an invariant set.