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In this note, we introduce the concepts of best case and worst-case convergence rates for switched linear systems. Their connections to the problems of stability and stabilization are briefly discussed. For simultaneously triangularizable switched systems, explicit formulas are presented for the convergence rates in terms of the system eigenvalues. Index Terms--Convergence rate, simultaneous triangularization, switched linear systems, switching path, stability, stabilization.

In this note, the problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems whose system matrices [A.sub.1] and [A.sub.2] are in companion form is considered. It is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product [A.sub.1] [A.sub.2] does not have an eigenvalue that is real and negative. Examples are presented to illustrate the result. Index Terms--Quadratic stability, stability theory, switched linear systems.

This paper presents the stability and performance conditions for discrete-time constrained switched linear systems under arbitrary switching. Design methods for such systems, which constrain the set of admissible switching sequences to attain stability and a desired performance level, are also presented. The key procedures presented in this paper are based on classic convex optimisation and graph theory results. Simulations and numerical examples are included to illustrate the main features of the developed theory. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm for estimating parameters of a source term of a linear advection-difiusion equation with an uncertain advection-velocity ffeld. First, we apply a minimax state estimation technique in order to reduce uncertainty introduced by the coeffcients. Then we design a source localization algorithm which uses the state estimator as a model and estimates the parameters of the source term given incomplete and noisy data. The principal novelty of the proposed algorithm is in that it is robust with respect to uncertainty in advection coeffcients. The localization algorithm is sequential; that is, it updates both the state estimate and the source estimate once a new observation arrives. To demonstrate the effcacy of the proposed algorithm, we present a numerical example of source localization in two spatial dimensions for the advectiondominated transport of a nonreactive pollutant emanating from a point source. [ABSTRACT FROM AUTHOR]

In this paper, we present characterisations of linear, shift-invariant, discrete-time systems that exhibit mixtures of small gain-type properties and positive real-type behaviours in a certain manner. These “mixed” systems are already fairly well characterised in the continuous-time domain, but the widespread adoption of digital controllers makes it necessary to verify whether commonly used discretisation procedures preserve the characteristic of “mixedness”. First, we analyse the effects of classical discretisation methods on the “mixed” property using Nyquist methods. A frequency domain feedback stability result is then presented. Finally, we develop a spectral-based characterisation of “mixed” discrete-time systems which provides a practical computational test that can also be applied to the MIMO case. Several examples validate the developed theory. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we consider the asymptotic stability of a class of discrete-time switching linear systems, where each of the constituent subsystems is Schur stable. We first present an example to motivate our study, which illustrates that the bilinear transform does not preserve the stability of a class of switched linear systems. Consequently, continuous time stability results cannot be transformed to discrete time analogs using this transformation. We then present a subclass of discrete-time switching systems that arise frequently in practical applications. We prove that global attractivity for this subclass can be obtained without requiring the existence of a common quadratic Lyapunov function (CQLF). Using this result, we present a synthesis procedure to construct switching stabilizing controllers for an automotive control problem, which is related to the stabilization of a vehicle’s roll dynamics subject to switches in the center of gravitys (CG) height. [Copyright &y& Elsevier]

Abstract: In this paper we derive a necessary and sufficient condition for the existence of a diagonal common quadratic Lyapunov function (CQLF) for a pair of positive linear time-invariant (LTI) systems. [Copyright &y& Elsevier]

We present a result on the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems. We show that this result characterizes, generalizes and provides new perspectives on several well-known stability results. [ABSTRACT FROM AUTHOR]

A sufficient condition for the existence of a Lyapunov function of the form V(x) = xT xP, P = PT > 0, P ∈ Rn×n, for the stable linear time invariant systems x˙ = iA x, iA ∈ Rn×n, iA ∈ A ≜ {A1, …, mA}, is that the matrices iA are Hurwitz, and that a non‐singular matrix T exists, such that iT A T−1, i ∈ {1, …, m}, is upper triangular [Mori, Y., Mori, T., & Kuroe, Y., Proceedings of Electronic Information and Systems Conference (1996); Proceedings of 36th Conference on Decision and Control, (1997); Liberzon, D., Hespanha, J.P., & Morse, S., Technical Report, Laboratory for Control Science and Engineering, Yale University, (1998); Shorten, R. & Narendra, K., Proceedings of Conference on Decision and Control, (1998)]. The existence of such a function, referred to as a common quadratic Lyapunov function (CQLF), is sufficient to guarantee the exponential stability of the switching system x˙ = A(t)x, A(t) ∈ A. In this paper we investigate the stability properties of a related class of switching systems. We consider sets of matrices A, where no single matrix T exists that simultaneously transforms each iA ∈ A to upper triangular form, but where a set of non‐singular matrices i jT exist such that the matrices {i jiT A i jT−1, i jjT A i jT−1}, i, j ∈ {1, …, m}, are upper triangular. We show that, for a special class of such systems, the origin of the switching system x˙ = A(t)x, A(t) ∈ A, is globally attractive. A novel technique is developed to derive this result, and the applicability of this technique to more general systems is discussed towards the end of the paper. [ABSTRACT FROM PUBLISHER]

Abstract: “Mixedness” is a property that captures elements of the notions of passivity and small gain. In the frequency domain, a linear, time-invariant system is called “mixed” if, over some frequency bands, it is strictly passive and, over the remaining frequencies, it has a gain of less than one; there exist no frequencies over which the system has neither of the notions of these properties associated with it. In this paper, a test is developed for determining whether or not a linear, time-invariant system is “mixed”. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the existence of quadratic Lyapunov functions for certain types of switched linear systems. Given a partition of the state-space, a set of matrices (linear dynamics), and a matrix-valued function constructed by associating these matrices with regions of the state-space in a manner governed by the partition, we ask whether there exists a positive definite symmetric matrix such that is negative definite for all . For planar systems, necessary and sufficient conditions are given. Extensions for higher order systems are also presented. [Copyright &y& Elsevier]

Abstract: In this note we consider the stability properties of a system class that arises in the control design problem of switched linear systems. The control design we are studying is based on a classical pole-placement approach. We analyse the stability of the resulting switched system and develop analytic conditions which reduce the complexity of the stability problem. We further consider two special cases for which strongly simplified conditions are obtained that support the analytic controller design. [Copyright &y& Elsevier]

In this contribution we show that under some circumstances the Kalman–Yakubovich–Popov (KYP) lemma can be applied for testing quadratic stability of multi-mode switched linear systems. Based on this observation, we introduce several constructive algorithms which enable us to associate a transfer function matrix to a switched linear system given in general form. The strict positive realness test of the resulting transfer function matrix can be easily achieved by available known algebraic conditions. By means of some examples we illustrate the step-by-step implementation of these algorithms. [ABSTRACT FROM AUTHOR]

In this paper we give necessary and sufficient conditions for weak and strong quadratic stability of a class of switched linear systems consisting of two subsystems, associated with symmetric transfer function matrices. These conditions can simply be tested by checking the eigenvalues of the product of two subsystem matrices. This result is an extension of the result by Shorten and Narendra for strong quadratic stability, and the result by Shorten et al. on weak quadratic stability for switched linear systems. Examples are given to illustrate the usefulness of our results. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive stability results for large-scale interconnections of “mixed” linear, time-invariant systems using classical Nyquist arguments. We compare our results with Moylan and Hill (1978) . Our results indicate that, if one relaxes assumptions on the subsystems in an interconnection from assumptions of passivity or small gain to assumptions of “mixedness,” then the Moylan and Hill-like conditions on the interconnection matrix become more stringent. Finally, we explore a condition for the stability of large-scale, time-varying interconnections of strictly positive real systems. This condition mirrors the condition obtained in  for time-invariant interconnections and is thus an extension of this work. [Copyright &y& Elsevier]