Stability of piecewise affine systems through discontinuous piecewise quadratic Lyapunov functions

State-dependent switched systems characterized by piecewise affine (PWA) dynamics in a polyhedral partition of the state space are considered. Sufficient conditions on the vectors fields such that the solution crosses the common boundaries of the polyhedra are expressed in terms of quadratic inequalities constrained to the polyhedra intersections. A piece-wise quadratic (PWQ) function, not necessarily continuous, is proposed as a candidate Lyapunov function (LF). The sign conditions and the negative jumps at the boundaries are expressed in terms of linear matrix inequalities (LMIs) via cone-copositivity. A sufficient condition for the asymptotic stability of the PWA system is then obtained by finding a PWQ-LF through the solution of a set LMIs. Numerical results with a conewise linear system and an opinion dynamics model show the effectiveness of the proposed approach

An introduction to positive switched systems and their application to HIV treatment modeling

In the present work an introduction to positive switched systems is provided, along with an interesting application of this kind of systems to the biomededical area. Reflecting this twofold objective, the thesis is divided into two parts: in the first one classical theoretical aspects concerning positive switched systems are addressed by resorting to the Lyapunov function approach, while in the second part an application to the problem of drug treatment scheduling in HIV infection is presente

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

Stability and Stabilization of Positive Switched Systems with Application to HIV Treatment

HIV mutates rapidly and may develop resistance to specific drug therapies. There is no general agreement on how to optimally schedule the treatments for mitigating the effects of mutations. We examine control strategies applied to two positive switched systems models of HIV under therapy. Simulation results show that model-based control approaches may outperform the common clinical treatment recommendationsope

Asymptotic stability of piecewise affine systems with Filippov solutions via discontinuous piecewise Lyapunov functions

Asymptotic stability of continuous-time piecewise affine systems defined over a polyhedral partition of the state space, with possible discontinuous vector field on the boundaries, is considered. In the first part of the paper the feasible Filippov solution concept is introduced by characterizing single-mode Caratheodory, sliding mode and forward Zeno behaviors. Then, a global asymptotic stability result through a (possibly discontinuous) piecewise Lyapunov function is presented. The sufficient conditions are based on pointwise classifications of the trajectories which allow the identification of crossing, unreachable and Caratheodory boundaries. It is shown that the sign and jump conditions of the stability theorem can be expressed in terms of linear matrix inequalities by particularizing to piecewise quadratic Lyapunov functions and using the cone-copositivity approach. Several examples illustrate the theoretical arguments and the effectiveness of the stability result