On the quadratic stability of switched interval systems: Preliminary results

In this paper we present some preliminary results on the quadratic stability of switched systems with uncertain parameters. We show that the quadratic stability of a class of switched uncertain systems may be readily verified using simple algebraic conditions. Examples are presented to demonstrate the efficacy of our techniques

Some results on quadratic stability of switched systems with interval uncertainty

In this paper we present some results on the quadratic stability of switched systems with uncertain parameters. We show that the quadratic stability of a class of switched uncertain systems may be readily veried using simple algebraic conditions

Some results on quadratic stability of switched systems with interval uncertainty

In this paper we present some results on the quadratic stability of switched systems with uncertain parameters. We show that the quadratic stability of a class of switched uncertain systems may be readily veried using simple algebraic conditions

A Unifying Framework for the Circle Criterion and Other Quadratic Stability Criteria

We present a result on the existence of a common quadratic Lyapunov function for a pair of linear timeinvariant systems. We show that this result characterises, generalises, and provides new perspectives on several well-known stability results. In particular, new time-domain formulations of the Circle Criterion and Meyer’s extension of the KYP lemma are presented

The geometry of convex cones associated with the Lyapunov inequality and the common Lyapunov function problem

In this paper, the structure of several convex cones that arise in the study of Lyapunov functions is investigated. In particular, the cones associated with quadratic Lyapunov functions for both linear and non-linear systems are considered, as well as cones that arise in connection with diagonal and linear copositive Lyapunov functions for positive linear systems. In each of these cases, some technical results are presented on the structure of individual cones and it is shown how these insights can lead to new results on the problem of common Lyapunov function existence

The geometry of convex cones associated with the Lyapunov inequality and the common Lyapunov function problem

In this paper, the structure of several convex cones that arise in the study of Lyapunov functions is investigated. In particular, the cones associated with quadratic Lyapunov functions for both linear and non-linear systems are considered, as well as cones that arise in connection with diagonal and linear copositive Lyapunov functions for positive linear systems. In each of these cases, some technical results are presented on the structure of individual cones and it is shown how these insights can lead to new results on the problem of common Lyapunov function existence

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

On the Kalman-Yakubovich-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

Topics in Automotive Rollover Prevention: Robust and Adaptive Switching Strategies for Estimation and Control

The main focus in this thesis is the analysis of alternative approaches for estimation and control of automotive vehicles based on sound theoretical principles. Of particular importance is the problem rollover prevention, which is an important problem plaguing vehicles with a high center of gravity (CG). Vehicle rollover is, statistically, the most dangerous accident type, and it is difficult to prevent it due to the time varying nature of the problem. Therefore, a major objective of the thesis is to develop the necessary theoretical and practical tools for the estimation and control of rollover based on robust and adaptive techniques that are stable with respect to parameter variations. Given this background, we first consider an implementation of the multiple model switching and tuning (MMST) algorithm for estimating the unknown parameters of automotive vehicles relevant to the roll and the lateral dynamics including the position of CG. This results in high performance estimation of the CG as well as other time varying parameters, which can be used in tuning of the active safety controllers in real time. We then look into automotive rollover prevention control based on a robust stable control design methodology. As part of this we introduce a dynamic version of the load transfer ratio (LTR) as a rollover detection criterion and then design robust controllers that take into account uncertainty in the CG position. As the next step we refine the controllers by integrating them with the multiple model switched CG position estimation algorithm. This results in adaptive controllers with higher performance than the robust counterparts. In the second half of the thesis we analyze extensions of certain theoretical results with important implications for switched systems. First we obtain a non-Lyapunov stability result for a certain class of linear discrete time switched systems. Based on this result, we suggest switched controller synthesis procedures for two roll dynamics enhancement control applications. One control design approach is related to modifying the dynamical response characteristics of the automotive vehicle while guaranteeing the switching stability under parametric variations. The other control synthesis method aims to obtain transient free reference tracking of vehicle roll dynamics subject to parametric switching. In a later discussion, we consider a particular decentralized control design procedure based on vector Lyapunov functions for simultaneous, and structurally robust model reference tracking of both the lateral and the roll dynamics of automotive vehicles. We show that this controller design approach guarantees the closed loop stability subject to certain types of structural uncertainty. Finally, assuming a purely theoretical pitch, and motivated by the problems considered during the course of the thesis, we give new stability results on common Lyapunov solution (CLS) existence for two classes of switching linear systems; one is concerned with switching pair of systems in companion form and with interval uncertainty, and the other is concerned with switching pair of companion matrices with general inertia. For both problems we give easily verifiable spectral conditions that are sufficient for the CLS existence. For proving the second result we also obtain a certain generalization of the classical Kalman-Yacubovic-Popov lemma for matrices with general inertia