Stabilization of orthogonal piecewise linear systems using piecewise linear Lyapunov-like functions

In this paper the problem of asymptotic stabilization of orthogonal piecewise linear (OPL) systems is considered. OPL systems arise when the state-space is partitioned into a number of distinct regions by hyperplanes orthogonal to the states. Different regions possess different continuous linear affine dynamics. Such systems are a special class of hybrid systems and can be thought of as approximations to nonlinear systems. A rule-based control design is proposed that asymptotically stabilises the system, and Lyapunov-like techniques are used for the determination of the control laws, implemented as different static state feedback designs for different regions. In particular, special types of piecewise linear Lyapunov functions are introduced, which allow independent local design and offer stability of all possible sliding modes. Optimized control solutions are achieved using simple linear programming algorithms