Quadratic Stability and Singular SISO Switching Systems

Abstract

Abstract-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 , , 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 ( ). 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.Index Terms-LTI systems.

RETRACTED: Synchronization analysis of linearly coupled singular systems

“…Singular systems are very complicated since one do not know whether the solution exists, not to mention of the stability or synchronization of the solution for the system regularity and impulse elimination. In fact, singular systems give a more general description of physical systems than the normal one, and many studies have extended concepts and results from the normal systems theory into the realm of singular systems [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. In [23], the synchronization problem of linearly coupled singular systems with the coupling matrix assumed to be symmetric and irreducible is investigated and a sufficient condition for globally asymptotic synchronization is derived based on the Lyapunov stability theory.…”

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

“…In particular, Theorem 2.1 provides a simple spectral characterization of strictly positive real transfer functions. Due to space limitations, these results are all given without proof; proofs are available in the journal submission [5] …”

“…This follows from the fact thatd = H(0) and H(0)+H(0) * > 0 since H is SPR. Now we give the aforementioned spectral characterisation of strict positive realness [5]. ) has no negative real eigenvalues and exactly one zero eigenvalue.…”

A geometrical treatment for obtaining necessary and sufficient conditions for joint quadratic Lyapunov function existence for state-dependent, switched systems: A two-dimensional case

“…Refer to [13], [14] for proof. Note that (4) resembles the kinds of singularity conditions that arise for the CQLF problem in two dimensions.…”