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25
Analysis of discrete–time linear switched systems: A variational approach
 SIAM J. Control Optim
, 2011
"... A powerful approach for analyzing the stability of continuous–time switched systems is based on using tools from optimal control theory to characterize the “most unstable ” switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the speci ..."
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Cited by 10 (6 self)
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A powerful approach for analyzing the stability of continuous–time switched systems is based on using tools from optimal control theory to characterize the “most unstable ” switching law. This reduces the problem of determining stability under arbitrary switching to analyzing stability for the specific “most unstable ” switching law. More generally, this so–called variational approach was successfully applied to derive nice–reachability–type results for both linear and nonlinear continuous–time switched systems. Motivated by this, we develop in this paper an analogous approach for discrete–time linear switched systems. We derive and prove a necessary condition for optimality of the “most unstable ” switching law. This yields a type of discrete–time maximum principle (MP). We demonstrate using an example that this MP is in fact weaker than its continuous–time counterpart. To overcome this, we introduce the auxiliary system of a discrete–time linear switched system, and show that regularity properties of time–optimal controls for the auxiliary system imply nice–reachability results for the original discrete– time linear switched system. Using this approach, we derive several new Lie–algebraic conditions guaranteeing nice–reachability results. These results, and their proofs, turn out to be quite different from their continuous–time counterparts.
MinimumTime Control of Boolean Networks
, 2012
"... Boolean networks (BNs) are discretetime dynamical systems with Boolean statevariables. BNs are recently attracting considerable interest as computational models for biological systems and, in particular, as models of gene regulating networks. Boolean control networks (BCNs) are Boolean networks wi ..."
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Cited by 5 (3 self)
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Boolean networks (BNs) are discretetime dynamical systems with Boolean statevariables. BNs are recently attracting considerable interest as computational models for biological systems and, in particular, as models of gene regulating networks. Boolean control networks (BCNs) are Boolean networks with Boolean inputs. We consider the problem of steering a BCN from a given state to a desired state in minimal time. Using the algebraic statespace representation (ASSR) of BCNs we derive several necessary conditions, stated in the form of maximum principles (MPs), for a control to be timeoptimal. In the ASSR every state and input vector is a canonical vector. Using this special structure yields an explicit statefeedback formula for all timeoptimal controls. To demonstrate the theoretical results, we develop a BCN model for the genetic switch controlling the lambda phage development upon infection of a bacteria. Our results suggest that this biological switch is designed in a way that guarantees minimal time response to important environmental signals.
Explicit Construction of a Barabanov Norm for a Class of Positive Planar Discrete–Time Linear Switched Systems ⋆
"... 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, the ..."
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Cited by 4 (0 self)
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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 discrete– time 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.
Extremal norms for positive linear inclusions
"... We consider the joint spectral radius of sets of matrices for discrete or continuous positive linear inclusions and study associated extremal norms. We show that under a matrixtheoretic notion of irreducibility there exist absolute extremal norms. This property is used to extend regularity result ..."
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Cited by 4 (3 self)
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We consider the joint spectral radius of sets of matrices for discrete or continuous positive linear inclusions and study associated extremal norms. We show that under a matrixtheoretic notion of irreducibility there exist absolute extremal norms. This property is used to extend regularity results for the joint spectral radius. In particular, we see that in the case of positive systems irreducibility in the sense of nonnegative matrices, which is weaker than the usual representation theoretic concept, is sufficient for local Lipschitz properties of the joint spectral radius.
Delayindependent stability of switched linear systems with unbounded timevarying delays
 2012, Article ID 560897
"... This paper is focused on delayindependent stability analysis for a class of switched linear systems with timevarying delays that can be unbounded. When the switched system is not necessarily positive, we first establish a delayindependent stability criterion under arbitrary switching signal by u ..."
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Cited by 4 (0 self)
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This paper is focused on delayindependent stability analysis for a class of switched linear systems with timevarying delays that can be unbounded. When the switched system is not necessarily positive, we first establish a delayindependent stability criterion under arbitrary switching signal by using a new method that is different from the methods to positive systems in the literature. We also apply this method to a class of timevarying switched linear systems with mixed delays.
MATHER SETS FOR SEQUENCES OF MATRICES AND APPLICATIONS TO THE STUDY OF JOINT SPECTRAL RADII
, 2011
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On the stability of continuoustime positive switched systems with rank one difference. submitted to Control & Cybernetics, (special issue for the 80th birthday of T. Kaczorek
, 2012
"... AbstractContinuoustime positive systems, switching among p subsystems, are introduced, and a complete characterization for the existence of a common linear copositive Lyapunov function for all the subsystems is provided. In particular, the existence of such a Lyapunov function is related to the e ..."
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Cited by 2 (2 self)
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AbstractContinuoustime positive systems, switching among p subsystems, are introduced, and a complete characterization for the existence of a common linear copositive Lyapunov function for all the subsystems is provided. In particular, the existence of such a Lyapunov function is related to the existence of common quadratic copositive Lyapunov functions. When the subsystems are obtained by applying different feedback control laws to the same continuoustime singleinput positive system, the above characterization leads to a very easy checking procedure.
STABILITY CRITERIA FOR SIS EPIDEMIOLOGICAL MODELS UNDER SWITCHING POLICIES
"... (Communicated by Pierre Magal) Abstract. We study the spread of disease in an SIS model for a structured population. The model considered is a timevarying, switched model, in which the parameters of the SIS model are subject to abrupt change. We show that the joint spectral radius can be used as a ..."
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(Communicated by Pierre Magal) Abstract. We study the spread of disease in an SIS model for a structured population. The model considered is a timevarying, switched model, in which the parameters of the SIS model are subject to abrupt change. We show that the joint spectral radius can be used as a threshold parameter for this model in the spirit of the basic reproduction number for timeinvariant models. We also present conditions for persistence and the existence of periodic orbits for the switched model and results for a stochastic switched model. 1. Introduction. In this paper
Generating Functions of Switched Linear Systems: Analysis, Computation, and Stability Applications
"... AbstractIn this paper, a unified framework is proposed to study the exponential stability of discretetime switched linear systems, and more generally, the exponential growth rates of their trajectories, under three types of switching rules: arbitrary switching, optimal switching, and random switc ..."
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AbstractIn this paper, a unified framework is proposed to study the exponential stability of discretetime switched linear systems, and more generally, the exponential growth rates of their trajectories, under three types of switching rules: arbitrary switching, optimal switching, and random switching. It is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong, the weak, and the mean generating functions, respectively. In particular, necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence. Various properties of the generating functions are established and their relations are discussed. Algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples.