General Inertia and Circle Criterion

In this paper we extend the well known Kalman-Yacubovic-Popov (KYP) lemma to the case of matrices with general regular inertia. We show that the version of the lemma that was derived for the case of pairs of stable matrices whose rank difference is one, extends to the more general case of matrices with regular inertia and in companion form. We then use this result to derive an easily verifiable spectral condition for a pair of matrices with the same regular inertia to have a common Lyapunov solution (CLS), extending a recent result on CLS existence for pairs of Hurwitz matrices that can be considered as a time-domain interpretation of the famous circle criterion. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Essentially negative news about positive systems

In this paper the discretisation of switched and non-switched linear positive systems using Padé approximations is considered. Padé approximations to the matrix exponential are sometimes used by control engineers for discretising continuous time systems and for control system design. We observe that this method of approximation is not suited for the discretisation of positive dynamic systems, for two key reasons. First, certain types of Lyapunov stability are not, in general, preserved. Secondly, and more seriously, positivity need not be preserved, even when stability is. Finally we present an alternative approximation to the matrix exponential which preserves positivity, and linear and quadratic stability.     

An alternative proof of the Barker, Berman, Plemmons (BBP) result on diagonal stability and extensions

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.     

On a theorem of Redheffer concerning diagonal stability

An important problem in system theory concerns determining whether or not a given LTI system is diagonally stable. More precisely, this problem is concerned with determining conditions on a matrix A such that there exists a diagonal matrix D with positive diagonal entries (i.e. a positive diagonal matrix), satisfying A^TD+DA=-Q<0. While this problem has attracted much attention over the past half century, two results of note stand out: (i) a result based on Theorems of the Alternative derived by Barker, Berman and Plemmons; and (ii) algebraic conditions derived by Redheffer. This paper is concerned with the second of these conditions. Our principal contribution is to show that Redheffer’s result can be obtained from the Kalman-Yacubovich-Popov lemma. We then show that this method of proof leads to natural generalisations of Redheffer’s result and we use these results to derive new conditions for diagonal and Hurwitz stability for special classes of matrices.     

Adaptive disturbance transmission decoupling in active vehicle suspensions

With active vehicle suspension, one can tailor a vehicles response to load and inertial without affecting the vehicle response to road disturbances. This decoupling is achieved in [1] and [2] using a filtered combination of measured signals. These filters require exact knowledge of certain vehicle parameters including vehicle mass to achieve the desired decoupling. Here we propose a parameter adaptive version of these filters which does not require knowledge of vehicle parameters. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

State Dependent AIMD Algorithms and Consensus Problems

This papers analyses a class of nonlinear positive systems that model the dynamics of nonlinear additive-increase multiplicative-decrease (AIMD) protocols. The system class covers a range of protocols that are currently used in real communication networks, such as standard TCP, and recent proposals for congestion control protocols such as Scalable TCP. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Kalman-Yacubovich-Popov lemma and common Lyapunov solutions for matrices with regular inertia

In this paper we extend the classical Lefschetz version of the Kalman-Yacubovich-Popov (KYP) lemma to the case of matrices with general regular inertia. We then use this result to derive an easily verifiable spectral condition for a pair of matrices with the same regular inertia to have a common Lyapunov solution (CLS), extending a recent result on CLS existence for pairs of Hurwitz matrices.     

Padé Approximations of and preservation of quadratic Lyapunov functions

We investigate whether or not quadratic Lyapunov functions are preserved under Padé approximations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new methodology for the stability analysis of pairwise triangularizable and related switching systems

We consider the asymptotic stability of the time-varying dynamicsystem : = A(t)x, A(t) R^{n x n}, A(t) A= {A₁, ..., A_m}, where A_i is Hurwitz and where a set of non-singularmatrices T_{i j} exist such that any pair of matrices {T_{i j} A_iT_{i j}^–1, T_{i j} A_j T_{i j}^–1}, i, j {1, ..., m}, areupper triangular. Switching systems of this form are referredto as pairwise triangularizable switching systems. It can beestablished that (a) pairwise triangularizability is not sufficientto guarantee the existence of a common quadratic Lyapunov functionfor the linear time-invariant dynamic systems A_i : = A_i x; (b) additional conditions can be specified which guaranteeasymptotic stability of the switching system . In this paperwe also show that pairwise triangularizability is not even sufficientto guarantee asymptotic stability of the switching system .We also show that the method of proof of stability in (b), whichdoes not assume the existence of a common quadratic Lyapunovfunction, can be used to prove the asymptotic stability of moregeneral switching systems (systems that are not pairwise triangularizable).Finally, we show that our results can be used as the basis forthe design of practical control systems; namely, for the designof an automobile speed switched controller with guaranteed stabilityproperties.     

Quadratic Lyapunov functions for systems with state-dependent switching

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 A(x) 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 P such that A(x)^TP+PA(x) is negative definite for all x(t). For planar systems, necessary and sufficient conditions are given. Extensions for higher order systems are also presented.