Kai Wulff

Quadratic Stability and Singular SISO Switching Systems

In this note, we consider 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 are of the form A, A-ghT, and where one of the matrices is singular. A necessary and sufficient condition for the existence of such a function is given in terms of the spectrum of the product A(A-ghT) . The technical note also contains a spectral characterization of strictly positive real transfer functions which are strictly proper. Examples are presented to illustrate our results.

A Note on Spectral Conditions for Positive Realness of Transfer Function Matrices

Necessary and sufficient conditions for positive realness of general transfer function matrices are derived. The conditions are expressed in terms of eigenvalues of matrix functions of the state matrices representation of the LTI system. Illustrative numerical examples are provided.

Brief paper: A control design method for a class of switched linear systems

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.

Brief paper: A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control

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 vehicles roll dynamics subject to switches in the center of gravitys (CG) height.

On the 45° -Region and the uniform asymptotic stability of classes of second order parameter-varying and switched systems

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.

On spectral conditions for positive realness of transfer function matrices

Necessary and sufficient conditions for positive realness of general transfer function matrices are derived. These conditions can be checked using eigenvalue solvers for both proper and strictly proper transfer function matrices.

A quadratic stability result for singular switched systems with application to anti-windup control

In this note we consider 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 are of the form A, A-ghT, and where one of the matrices is singular. We then apply this result in a study of a feedback system with a saturating actuator

Convex cones, lyapunov functions, and the stability of switched linear systems

Recent research on switched and hybrid systems has resulted in a renewed interest in determining conditions for the existence of a common quadratic Lyapunov function for a finite number of stable LTI systems. While efficient numerical solutions to this problem have existed for some time, compact analytical conditions for determining whether or not such a function exists for a finite number of systems have yet to be obtained. In this paper we present a geometric approach to this problem. By making a simplifying assumption we obtain a compact time-domain condition for the existence of such a function for a pair of LTI systems. We show a number of new and classical Lyapunov results can be obtained using our framework. In particular, we demonstrate that our results can be used to obtain compact time-domain versions of the SISO Kalman-Yacubovich-Popov lemma, the Circle Criterion, and stability multiplier criteria. Finally, we conclude by posing a number of open questions that arise as a result of our approach.

Certainty equivalence adaptation combined with super-twisting sliding-mode control

ABSTRACT In this paper, a Lyapunov-based control concept is presented that combines variable structure and adaptive control. The considered system class consists of nonlinear single input systems which are affected by matched structured and unstructured uncertainties. Resorting to the certainty equivalence principle, the controller exploits advantages of both the sliding-mode and the adaptive control methodology. It is demonstrated that the gains of the discontinuous control action may be reduced remarkably when compared with pure sliding-mode-based approaches. The efficiency of the presented concept is demonstrated in detail, using results of numerical simulations.

Stabilization of Switched Linear Differential Algebraic Equations and Periodic Switching

We investigate the stabilizability of switched linear systems of differential-algebraic equations (DAEs). For such systems we introduce a parameterized family of switched ordinary differential equations that approximate the dynamic behavior of the switched DAE. A necessary and sufficient criterion for the stabilizability of a switched DAE system using time-dependent switching is obtained in terms of these parameterized approximations. Furthermore, we provide conditions for the stabilizability of switched DAEs via fast switching as well as using solely the consistency projectors of the constituent systems. The stabilization of switched DAEs with commuting vector fields is also analyzed.