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In this paper we propose a state estimation method for linear parabolic partial differential equations (PDE) that accounts for errors in the model, truncation, and observations. It is based on an extension of the Galerkin projection method. The extended method models projection coefficients, representing the state of the PDE in some basis, by means of a differential-algebraic equation (DAE). The original estimation problem for the PDE is then recast as a state estimation problem for the constructed DAE using a linear continuous minimax filter. We construct a numerical time integrator that preserves the monotonic decay of a nonstationary Lyapunov function along the solution. To conclude, we demonstrate the efficacy of the proposed method by applying it to the tracking of a discharged pollutant slick in a two-dimensional fluid. [ABSTRACT FROM AUTHOR]

A sufficient condition for the stability of large-scale interconnections ofNlinear time-variant systems is presented. Such a condition represents important extensions to passivity criteria and ensures stability by means of the existence of a positive definite (full-block) matrixPwhich is a common solution to Lyapunov equations involving a diagonal stacking of theNsystems and the interconnection structure matrix. An experimental methodology for the verification of the sufficient condition also is proposed, based on evolutionary computation techniques. Applications of the new stability results are provided through illustrative examples, which are developed using particle swarm optimisation and genetic algorithms. [ABSTRACT FROM AUTHOR]

In this article, a general notion of common diagonal Lyapunov matrix is formulated for a collection of n × n matrices A 1, … , A s , and cones k 1, … , k s in ℝ n . Necessary and sufficient conditions are derived for the existence of a common diagonal Lyapunov matrix in this setting. The conditions are similar to and extend the well-known criteria for the case s = 1, k 1 = ℝ n . [ABSTRACT FROM AUTHOR]

Abstract: We revisit the theorem of Barker, Berman and Plemmons on the existence of a diagonal quadratic Lyapunov function for a stable linear time-invariant (LTI) dynamical system [G.P. Barker, A. Berman, R.J. Plemmons, Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra 5(3) (1978) 249–256]. We use recently derived results to provide an alternative proof of this result and to derive extensions. [Copyright &y& Elsevier]

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]

The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area. [ABSTRACT FROM AUTHOR]

Abstract: The condition that a finite collection of stable matrices {A 1,…, A M } has no common quadratic Lyapunov function (CQLF) is formulated as a hierarchy of singularity conditions for block matrices involving a number of unknown parameters. These conditions are applied to the case of two stable 3×3 matrices, where they are used to derive necessary and sufficient conditions for the non-existence of a CQLF. [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]

This paper deals with the question of the existence of weak and strong common quadratic Lyapunov functions (CQLFs) for stable discrete‐time linear time‐invariant (LTI) systems. The main result of the paper provides a simple characterization of pairs of such systems for which a weak CQLF of a given form exists but for which no strong CQLF exists. An application of this result to second‐order discrete‐time LTI systems is presented. [ABSTRACT FROM PUBLISHER]

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]

We consider the asymptotic stability of the time‐varying dynamic system Ω : x˙ = A(t)x, A(t) ∈ Rn × n, A(t) ∈ A = {A1, …, Am}, where Ai is Hurwitz and where a set of non‐singular matrices Ti j exist such that any pair of matrices {Ti j Ai Ti j−1, Ti j Aj Ti j−1}, i, j ∈ {1, …, m}, are upper triangular. Switching systems of this form are referred to as pairwise triangularizable switching systems. It can be established that (a) pairwise triangularizability is not sufficient to guarantee the existence of a common quadratic Lyapunov function for the linear time‐invariant dynamic systems ΣAi : x˙ = Ai x; (b) additional conditions can be specified which guarantee asymptotic stability of the switching system Ω. In this paper we also show that pairwise triangularizability is not even sufficient to guarantee asymptotic stability of the switching system Ω. We also show that the method of proof of stability in (b), which does not assume the existence of a common quadratic Lyapunov function, can be used to prove the asymptotic stability of more general switching systems (systems that are not pairwise triangularizable). Finally, we show that our results can be used as the basis for the design of practical control systems; namely, for the design of an automobile speed switched controller with guaranteed stability properties. [ABSTRACT FROM PUBLISHER]

In this paper we present sufficient conditions for the stability of a number of classes of time-varying systems. These results are of interest for two reasons. Firstly, the conditions presented take the form of matrix pencil eigenvalue criteria, and are therefore co-ordinate independent and easily verifiable. Secondly, we show a direct relationship between the form of our criteria, and the choice of Lyapunov function used to demonstrate stability (quadratic or unic). These results indicate the importance of the region of the complex plane known as the 45°-Region. [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: In this note, a result is presented that may be considered an extension of the classical Kalman–Yakubovich–Popov (KYP) lemma. Motivated by problems in the design of switched systems, we wish to infer the existence of a quadratic Lyapunov function (QLF) for a nonlinear system in the case where a matrix defining one system is a rank-1 perturbation of the other and where switching between the systems is orchestrated according to a conic partitioning of the state space . We show that a necessary and sufficient condition for the existence of a QLF reduces to checking a single constraint on a sum of transfer functions irrespective of problem dimension. Furthermore, we demonstrate that our conditions reduce to the classical KYP lemma when the conic partition of the state space is , with the transfer function condition reducing to the condition of Strict Positive Realness. [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 paper we derive necessary and sufficient conditions for the existence of a common linear co-positive Lyapunov function for a finite set of linear positive systems. Both the state dependent and arbitrary switching cases are considered. Our results reveal an interesting characterisation of “linear” stability for the arbitrary switching case; namely, the existence of such a linear Lyapunov function can be related to the requirement that a number of extreme systems are Metzler and Hurwitz stable. Examples are given to illustrate the implications of our results. [Copyright &y& Elsevier]

We demonstrate a novel application of nonlinear systems in the design of pattern classification systems. We show that pattern classification systems can be designed based upon training algorithms designed to control the qualitative behaviour of a nonlinear system. Our paradigm is illustrated by means of a simple chaotic system—the Baker''s map. Algorithms for training the system are presented and examples are given to illustrate the operation and learning of the system for pattern classification tasks. [Copyright &y& Elsevier]

In this paper we first derive some general Lyapunov based stability conditions for switched linear descriptor systems and illustrate them with a simple mechanical system. Next we develop simple conditions for specific classes of systems. In particular, we consider the important case when switching between a standard system and an index one descriptor system, and systems where switching occurs between an index one and an index two descriptor system. Examples are given to illustrate these results. [ABSTRACT FROM AUTHOR]