In this chapter, we introduce the nonlinear model predictive control algorithm in a rigorous way. We start by defining a basic NMPC algorithm for constant reference and continue by formalizing state and control constraints. Viability (or weak forward invariance) of the set of state constraints is introduced and the consequences for the admissibility of the NMPC feedback law are discussed. After having introduced NMPC in a special setting, we describe various extensions of the basic algorithm, considering time varying reference solutions, terminal constraints and costs and additional weights. Finally, we investigate the optimal control problem corresponding to this generalized setting and prove several properties, most notably the dynamic programming principle.The Basic NMPC AlgorithmAs already outlined in the introductory Chap. 1, the idea of the NMPC scheme is as follows: at each sampling instant n we optimize the predicted future behavior of the system over a finite time horizon k = 0, . . . , N − 1 of length N ≥ 2 and use the first element of the resulting optimal control sequence as a feedback control value for the next sampling interval. In this section we give a detailed mathematical description of this basic idea for a constant reference x ref ≡ x * ∈ X. The time varying case as well as several other variants will then be presented in Sect. 3.3.A prerequisite for being able to find a feedback law which stabilizes the system at x * is that x * is an equilibrium of the nominal closed-loop system (2.5), i.e., x * = f (x * , μ(x * ))-this follows immediately from Definition 2.14 with g(x) = f (x, μ(x)). A necessary condition for this is that there exists a control value u * ∈ U withwhich we will assume in the sequel. The cost function to be used in our optimization should penalize the distance of an arbitrary state x ∈ X to x * . In addition, it is often

Nonlinear Model Predictive Control

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