A result on common quadratic Lyapunov functions

A result on common quadratic Lyapunov functions

In this paper we define strong and weak common quadratic Lyapunov functions (CQLF’s) for sets of linear time-invariant (LTI) systems. We show that the simultaneous existence of a weak CQLF of a special form, and the non-existence of a strong CQLF, for a pair of LTI systems, is characterised by easily verifiable algebraic conditions. These conditions are found to play an important role in provin...

On quadratic Lyapunov functions

On common linear/quadratic Lyapunov functions for switched linear systems

Using duality and complementarity ideas, and Z-transformations, in this article, we discuss equivalent ways of describing the existence of common linear/quadratic Lyapunov functions for switched linear systems. In particular, we extend a recent result of Mason-Shorten on positive switched system with two constituent linear time-invariant systems to an arbitrary finite system.

Preservation of common quadratic Lyapunov functions and Padé approximations

It is well known that the bilinear transform, or first order diagonal Padé approximation to the matrix exponential, preserves quadratic Lyapunov functions between continuous-time and corresponding discrete-time linear time invariant (LTI) systems, regardless of the sampling time. It is also well known that this mapping preserves common quadratic Lyapunov functions between continuous-time and di...

On common quadratic Lyapunov functions for stable discrete-time LTI systems

This paper deals with the question of the existence of weak and strong common quadratic Lyapunov functions (CQLFs) for stable discrete-time linear timeinvariant (LTI) systems. The main result of the paper provides a simple characterisation 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 disc...