Balanced truncation for linear switched systems

In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems. This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems. Specifically in this paper, we provide a bound on the approximation error in L2 norm for continuous-time and l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of bal- anced truncation depends only on the input-output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2 and l2 gains depend only on the input-output map.Comment: We have corrected a number of typos and inconsistencies. In addition, we added new results in Theorem

Analysis of switched and hybrid systems - beyond piecewise quadratic methods

This paper presents a method for stability analysis of switched and hybrid systems using polynomial and piecewise polynomial Lyapunov functions. Computation of such functions can be performed using convex optimization, based on the sum of squares decomposition of multivariate polynomials. The analysis yields several improvements over previous methods and opens up new possibilities, including the possibility of treating nonlinear vector fields and/or switching surfaces and parametric robustness analysis in a unified way

Cooperation of Multi-agent Systems

University of Technology Sydney. Faculty of Engineering and Information Technology.The cooperation of multi-agent systems represents that a group of agents complete the common tasks that are difficult or impossible for an individual agent or a single system to finish. The cooperation of multi-agent systems has received considerable attention and has been widely studied over the past few years. The consensus is the basis of the cooperation of multi-agent systems. Researching the cooperation of multi-agent systems probably involves a large number of theories, such as graph theories and stability theories of switched systems. In this thesis, we study consensus, tracking control, and containment control under a fixed graph. For a time-invariant network of multi-agent systems, its topological characteristic can be described by a fixed graph. For the time-variant network, we model the multi-agent systems by switched systems in this thesis and get some new conclusions. We transform the consensus problems into stability problems first and then get some new conclusions by the aid of conventional methods. For traditional approaches, there are many limitations. Many of the involved theories also need to be further improved. For example, the stability problems of switched systems have not yet been thoroughly resolved. Therefore, we also do much research work on stability analysis of switched systems. For switched systems, we propose some novel theories and methods to reduce the limitations of conventional stability analysis methods. We propose some sequence-based methods to resolve the stability problems and get some new results. It is proved that the switched systems are globally uniformly asymptotically stable when the sequence-based mode-dependence average dwell time satisfies the conditions deduced by this thesis. We use the proposed stability theories and methods to analyse the consensus of multi-agent systems under switched systems and get some new results. A proposed model transformation can transform consensus problems into stability problems. We get some novel results. In this thesis, we first introduce the background and then give a brief literature review on the stability analysis of switched systems and the cooperation of multi-agent systems. After that, Chapter 3 addresses multi-agent systems under a fixed topology. The research on the stability of switched systems is presented in Chapter 4. A consensus of second-order multi-agent systems under switched topologies based on the sequence is studied in Chapter 5. The research plan, progress, and our publications on this research topic are also shown in this report. Finally, we summarize my past work and give a conclusion

Common Lyapunov Function Based on Kullback–Leibler Divergence for a Switched Nonlinear System

Many problems with control theory have led to investigations into switched systems. One of the most urgent problems related to the analysis of the dynamics of switched systems is the stability problem. The stability of a switched system can be ensured by a common Lyapunov function for all switching modes under an arbitrary switching law. Finding a common Lyapunov function is still an interesting and challenging problem. The purpose of the present paper is to prove the stability of equilibrium in a certain class of nonlinear switched systems by introducing a common Lyapunov function; the Lyapunov function is based on generalized Kullback–Leibler divergence or Csiszár's I-divergence between the state and equilibrium. The switched system is useful for finding positive solutions to linear algebraic equations, which minimize the I-divergence measure under arbitrary switching. One application of the stability of a given switched system is in developing a new approach to reconstructing tomographic images, but nonetheless, the presented results can be used in numerous other areas

Analysis and synthesis of randomly switched systems with known sojourn probabilities

In this paper, a new approach is proposed and investigated for the stability analysis and stabilizing controller design of randomly switched linear discrete systems. The approach is based on sojourn probabilities and it is assumed that these probabilities are known a prior. A new Lyapunov functional is constructed and two main theorems are proved in this paper. Theorem 1 gives a sufficient condition for a switched system with known sojourn probabilities to be mean square stable. Theorem 2 gives a sufficient condition for the design of a stabilizing controller. The applications of these theorems and the corresponding corollary and lemma are demonstrated by three numerical examples. Finally, some future research is proposed

Novel characterizations for switched nonlinear systems with average dwell time: further findings

It is well known, present day theory of switched systems is largely based on assuming certain small but finite time interval termed average dwell time. Thus it appears dominantly characterized by some slow sw itching condition with average dwell time satisfying a certain lower bound, which implies a constraint nonetheless. In cases of nonlinear systems there may well appear non-expected comple xity phenomena of particularly different nature when switching becomes no longer. A fast switching condition with average dwell time satisfying an upper bound is explored and established. A comparison analysis of these innovated characterization s via slightly different overview yielded new results on the tran sient behaviour of switched nonlinear systems, while preserving the system stability. The multiple-Lyapunov functions approac h is used in the analysis and switched systems framework is extended shading new light on the underlying, switching caused system complexities