Stability analysis of switched systems using variational principles: An introduction

Margaliot, Michael

doi:10.1016/j.automatica.2006.06.020

Cited by 238 publications

(131 citation statements)

References 58 publications

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“…Our work is motivated by the variational analysis of continuous-time switched systems (see Margaliot (2006); Margaliot and Branicky (2009); Sharon and Margaliot (2007); Margaliot and Liberzon (2006)). This approach was also extended to analyze discrete-time switched systems (see Barabanov (2005); Monovich and Margaliot (2011a,b) and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Controllability of Boolean control networks via the Perron–Frobenius theory

2012

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Boolean control networks (BCNs) are recently attracting considerable interest as computational models for genetic and cellular networks. Addressing control-theoretic problems in BCNs may lead to a better understanding of the intrinsic control in biological systems, as well as to developing suitable protocols for manipulating biological systems using exogenous inputs. We introduce two definitions for controllability of a BCN, and show that a necessary and sufficient condition for each form of controllability is that a certain nonnegative matrix is irreducible or primitive, respectively. Our analysis is based on a result that may be of independent interest, namely, a simple algebraic formula for the number of different control sequences that steer a BCN between given initial and final states in a given number of time steps, while avoiding a set of forbidden states.

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Section: Introductionmentioning

confidence: 99%

Controllability of Boolean control networks via the Perron–Frobenius theory

2012

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“…This connectsṼ and the first integral of the dynamics x(k + 1) =Ã 1 x(k). Note that the solution of the HJB equation for planar continuoustime linear switched systems is a concatenation of first integrals of the two subsystems (Margaliot and Langholz (2003); Margaliot (2006)). We believe that this idea may be useful in constructing a Barabanov norm for more general classes of discrete-time switched systems.…”

Section: =ṽ (ã 1 X)mentioning

confidence: 99%

“…This variational approach for analyzing GUAS was pioneered by E. S. Pyatntisky in the context of the celebrated absolute stability problem (Pyatnitskii (1970(Pyatnitskii ( , 1971), and proved to be quite useful in the analysis of continuous-time switched systems; see the survey papers (Barabanov (2005); Margaliot (2006)) as well as the more recent papers (Sharon and Margaliot (2007); Margaliot and Branicky (2009)), and the related work in (Boscain (2002); Balde and Boscain (2008); Balde et al (2009)). Recently, the variational approach was also used to analyze the stability of discrete-time linear switched systems (Monovich and Margaliot (2011a,b)).…”

Section: Introductionmentioning

confidence: 99%

Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems

Teichner

Margaliot

2012

Automatica

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We consider the stability under arbitrary switching of a discrete-time linear switched system. A powerful approach for addressing this problem is based on studying the "most unstable" switching law (MUSL). If the solution of the switched system corresponding to the MUSL converges to the origin, then the switched system is stable for any switching law. The MUSL can be characterized using optimal control techniques. This variational approach leads to a Hamilton-Jacobi-Bellman equation describing the behavior of the switched system under the MUSL. The solution of this equation is sometimes referred to as a Barabanov norm of the switched system. Although the Barabanov norm was studied extensively, it seems that there are few examples where it was actually computed in closed-form. In this paper, we consider a special class of positive planar discretetime linear switched systems and provide a closed-form expression for a corresponding Barabanov norm and a MUSL. The unit circle in this norm is a parallelogram.

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“…The references mentioned above focused on deterministic switched systems. Stochastic switched systems have recently been the focus of attention in control engineering such as [13], [14]. The stability of stochastic nonlinear systems with state-dependent switching was investigated in [15], where the switching laws can be viewed as a closed-loop control.…”

Section: Introductionmentioning

confidence: 99%