Optimal control and analysis for constrained piecewise affine systems

One of the most important and challenging problems in control is the derivation of systematic tools for the computation of controllers for general constrained nonlinear or hybrid systems (combining continuous-valued dynamics with logic rules) that can guarantee (among others) closed-loop stability, feasibility, and optimality. The most successful modern control strategy both in theory and in practice for this class of systems is undoubtedly Receding Horizon Control (RHC), often also interchangeably called Model (Based) Predictive Control (MPC). In this thesis the focus lies on the class of constrained discrete-time piecewise affine (PWA) systems, which is equivalent to a variety of other hybrid system modeling frameworks reported in the literature. PWA systems are obtained by partitioning the extended state-input space into regions and associating with each region a different affine state update equation. This paradigm is a powerful modeling tool that can capture general nonlinearities (e.g. by local approximation), constraints, saturations, switches, discrete inputs and states, and other hybrid modeling phenomena in dynamical systems. Although PWA systems are a subclass of general nonlinear systems, most of the control theory developed for nonlinear control does not apply, due to commonly made continuity and smoothness requirements. By utilizing multi-parametric programming, the RHC problem for PWA systems can be solved off-line in order to obtain a closed-form optimal controller solution. Once the closed-form solution is computed, the resulting controller can be implemented as a simple lookup table. Thus the on-line computational effort is reduced to a simple evaluation of the lookup table at the current measured state and enables the control of complex fast sampled systems in the range of millior microseconds. The main theme of this work revolves around the efficient and systematic computation, analysis, and post-processing of closed-form, optimal, stabilizing, exact state-feedback controllers for fast sampled discrete-time PWA systems, where the cost function of the respective optimal control problem is composed of (piecewise) linear vector norms. One way to obtain the closed-form solution to the underlying constrained finite

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