Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results

Lin, Hai; Antsaklis, Panos J.

doi:10.1109/tac.2008.2012009

Cited by 2,413 publications

(1,422 citation statements)

References 90 publications

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“…This choice has a main motivation. Indeed, as reviewed in [19], the existence of a common quadratic Lyapunov function is a sufficient condition for global uniform asymptotic stability of a switching system. On the other hand, asymptotic stability under zero input of a nonlinear system implies local input-to-state stability, as was shown in [31].…”

Section: Disturbance Decoupling With State Stabilitymentioning

confidence: 99%

“…As is well-known [18,19], different definitions of stability apply to switching systems without inputs, ranging from quadratic stability to exponential stability or asymptotic stability. Moreover, a switching system may not enjoy a certain stability property under arbitrary switching, but may have this property when it is ruled by somehow restricted switching signals, such as those satisfying a minimum dwell-time or average dwell-time: i.e., the time or the average time between two consecutive switches is not smaller than a constant [20,21].…”

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confidence: 99%

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Stability issues in disturbance decoupling for switching linear systems

Zattoni

2015

2015 Proceedings of the Conference on Control and Its Applications

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Disturbance decoupling -i.e., the problem of making the output of a dynamical system insensitive to undesired inputs -is a classical problem of control theory and a main concern in control applications. Hence, it has been solved for many classes of dynamical systems, considering both structural and stability requirements. As to decoupling in linear switching systems, several definitions of stability apply. The aim of this contribution is investigating different decoupling problems with progressively more stringent stability requirements: from structural decoupling to decoupling with local input-to-state stability. A convex procedure for the computation of the switching compensator is presented, based on the fact that quadratic stability under arbitrary switching guarantees global uniform asymptotic stability and the latter implies local input-to-state stability. Measurable and inaccessible disturbances are considered in a unified setting. The work is focused on discrete-time systems, although all the results hold for continuous-time systems as well, with the obvious modifications.1 Introduction. Disturbance decoupling, which is a main issue in control system design, was first solved for linear systems in the late sixties, within the geometric approach (see, e.g., [1,2] and the references therein). Since then, the problem has been reformulated for different classes of dynamical systems and, due to the peculiarities of each context, it still attracts a fair amount of research effort: e.g., nonlinear systems have recently been considered in [3], descriptor systems and systems over rings in [4], timedelay systems in [5], linear parameter varying systems in [6,7].Lately, disturbance decoupling has been tackled for switching linear systems, that are dynamical systems described by a set of linear time-invariant systemseach of them modeling a different mode of operationand a switching signal, designating the active mode at each time instant [8][9][10]. Indeed, when switching linear systems are involved, the problem of disturbance decoupling lends itself to a number of formulations, character-

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Section: Disturbance Decoupling With State Stabilitymentioning

confidence: 99%

mentioning

confidence: 99%

Stability issues in disturbance decoupling for switching linear systems

Zattoni

2015

2015 Proceedings of the Conference on Control and Its Applications

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“…Stability is a fundamental problem for switched systems, which has been extensively investigated and fruitful results have been reported [1][2][3][4][5][6][7][8][9][10][11]. However, most results focus on switched continuous-time systems.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, unstable subsystems may inevitably appear and, in some situations, cannot be avoided. Thus, designing switching laws to stabilize a switched system whose subsystems are all unstable is one of the most serious challenges for switched systems [2,4,8,12]. This problem has been widely studied [13][14][15][16][17], and most designs are state-dependent switching strategies such as the min-projection strategy and largest region function approach.…”

Section: Introductionmentioning

confidence: 99%

Exponential Stabilization of Switched Discrete‐Time Systems with All Unstable Modes

2017

Asian Journal of Control

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This paper studies the exponential stabilization of switched discrete-time systems whose subsystems are unstable. A new sufficient condition for the exponential stability of the class of systems is proposed. The result obtained is based on the determination of a lower bound of the maximum dwell time by virtue of the multiple Lyapunov functions method. The key feature is that the given stability condition does not need the value of the Lyapunov function to uniformly decrease at every switching instant. An example is provided to illustrate the effectiveness of the proposed result.

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“…The issue of the stability and stabilization of switched systems, which has been clearly formalized in [10], has generated many contributions to the control theory literature (see [11,12,13,14,15,16] and references therein). Mainly two frameworks should be distinguished related to the nature of the switching law.…”

Section: Introductionmentioning

confidence: 99%

Min-switching local stabilization for discrete-time switching systems with nonlinear modes

Jungers

Gonzaga

Daafouz

2013

Nonlinear Analysis: Hybrid Systems

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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.

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Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results

Cited by 2,413 publications

References 90 publications

Stability issues in disturbance decoupling for switching linear systems

Stability issues in disturbance decoupling for switching linear systems

Exponential Stabilization of Switched Discrete‐Time Systems with All Unstable Modes

Min-switching local stabilization for discrete-time switching systems with nonlinear modes

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