Dead-Beat Control for Polynomial Systems

This thesis contributes to a better understanding of state and output dead-beat control problems and stability of zero output constrained dynamics for the class of discrete-time polynomial systems. Dead-beat controllability is one of the fundamental notions in control theory since it establishes the existence of control laws which can achieve a desired operating regime in finite time. The class of polynomial systems that we consider is very broad. Indeed, under very mild assumptions any nonlinear input-output map can be realised by a polynomial model. Symbolic computation methods are exploited to tackle the dead-beat control problems. An algorithm for the design of minimum-time dead-beat controllers follows from our approach. In principle, the proposed method can deal with multi-input multi-output systems and bounds on controls and states can be included in a straightforward manner. The price we pay is the large computational cost, which prevent us from using this method in general. To reduce the computational requirements for our controllability tests and design methodologies a number of simpler classes of polynomial systems are considered. Mathematical tools, such as algebraic geometry, real algebraic geometry, symbolic computation and convex analysis, are exploited. In this way, a number of analytic results are obtained with which we obtain computationally feasible controllability tests and design methodologies, as well as gain some more geometric insight. Stability of zero output constrained dynamics and the related minimum phase property play an important role in output dead-beat control. The definitions found in the literature are not general enough to incorporate all behaviours that may occur in the context of polynomial systems. We revisit the definition of a minimum phase system and propose symbolic computation means to test different minimum phase properties for polynomial systems. Our results can be used for testing stability and stabilisability either by definition or by constructing Lyapunov functions.