19 research outputs found
Stabilization of Discrete-Time Planar Switched Linear Systems with Impulse
We study the stabilization problem of discrete-time planar switched linear systems with impulse. When all subsystems are controllable, based on an explicit estimation on the state transition matrix, we establish a sufficient condition such that the switched impulsive system is stabilizable under arbitrary switching signal with given switching frequency. When there exists at least one uncontrollable
subsystem, a sufficient condition is also given to guarantee the stabilization of the switched impulsive system under appropriate switching signal
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
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
Stability of switched linear differential systems
We study the stability of switched systems where the dynamic modes are
described by systems of higher-order linear differential equations not
necessarily sharing the same state space. Concatenability of trajectories at
the switching instants is specified by gluing conditions, i.e. algebraic
conditions on the trajectories and their derivatives at the switching instant.
We provide sufficient conditions for stability based on LMIs for systems with
general gluing conditions. We also analyse the role of positive-realness in
providing sufficient polynomial-algebraic conditions for stability of two-modes
switched systems with special gluing conditions
Formal synthesis of stabilizing controllers for periodically controlled linear switched systems
In this paper, we address the problem of synthesizing periodic switching controllers for stabilizing a family of linear systems. Our broad approach consists of constructing a finite game graph based on the family of linear systems such that every winning strategy on the game graph corresponds to a stabilizing switching controller for the family of linear systems. The construction of a (finite) game graph, the synthesis of a winning strategy and the extraction of a stabilizing controller are all computationally feasible. We illustrate our method on an example
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)T P +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
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
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