Abstract

This thesis is divided into three main parts. The contribution of the first part is to present a controller synthesis method to stabilize piecewise-affine (PWA) slab systems based on invariant sets. Inspired by the theory of sliding modes, sufficient stabilization
conditions are cast as a set of Linear Matrix Inequalities (LMIs) by proper choice of an invariant set which is a target sliding surface. The method has two steps: the design of the attractive sliding surface and the design of the controller parameters. While previous approaches
to PWA controller synthesis are cast as Bilinear Matrix Inequalities (BMIs) that can, in some cases, be relaxed to LMIs at the cost of adding conservatism, the proposed
method leads naturally to a convex formulation. Furthermore, the LMIs obtained in this work have lower dimension when compared to other methods because the dimension of
the closed-loop state space is reduced.

In the second part of the thesis, it is further shown that the proposed approach is less conservative than other approaches. In other words, it will be shown that for every solution of the LMIs resulting from previous approaches, there exists a solution for the LMIs obtained from the proposed method. Furthermore, it will be shown that while previous convex controller synthesis methods have no solutions to their LMIs for some examples of PWA systems, the approach proposed in this thesis yields a solution for these examples.

The contribution of the last part of this thesis is to formulate the PWA time-delay synthesis problem as a set of LMIs. In order to do so, we first define a sliding surface, then control laws are designed to approach the specified sliding surface and ensure that the trajectories will remain on that surface. Then, using Lyapunov-Krasovskii functionals, sufficient conditions for exponential stability of the resulting reduced order system will be obtained.

Several applications such as pitch damping of a helicopter (2nd order system), rover path following example (3rd order system) and active flutter suppression (4th order system)
along with some other numerical examples are included to demonstrate the effectiveness of the approaches.