Control of uncertain and parameter-variant piecewise linear systems

This paper introduces a method for controller design of uncertain and parameter-variant piecewise linear systems. The idea is to specify a desired performance, which is represented by a nominal system. Around this nominal system a compact set of systems is obtained which will be robustly stable against switching among members of this set: such a set of systems is then called the stable switched set. The paper shows that obtaining the stable switched set is a signomial program. It is shown that upper bounds on signomial programs can be easily obtained, which are used to compute the stable switched set. A sufficient condition is given for the existence of a common state feedback controller that stabilizes a number of uncertain and parameter-variant linear systems on a stable switched set. The paper proposes a synthesis procedure in terms of a constrained convex optimization problem that places the uncertain and parameter-variant systems optimally close to the desired nominal system, using one common state feedback controller. An extension is shown for the case that there exists no common state feedback controller. The synthesis framework is then applied to a simple example to demonstrate the procedure. It is shown that real systems, like active suspension of a car, transform natural into uncertain and parameter-variant piecewise linear systems.