Nonlinear Model Predictive Control

In this introduction, we present the basics of NMPC in an informal way. In particular, we introduce the central idea of iterative optimal control on a moving finite horizon. We provide a brief history of NMPC and MPC, explain the organization of the material in this book and mention some topics which are not covered.

We introduce the class of systems treated in the book—nonlinear discrete time control systems on metric spaces—and illustrate them by several examples. Moreover, we discuss sampled data systems as an important special case. Afterwards, we introduce the necessary background material from Lyapunov stability theory for discrete time and sampled data systems which will be needed for the stability analysis of NMPC schemes.

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.

In this chapter we give an introduction to nonlinear infinite horizon optimal control. The dynamic programming principle as well as several consequences of this principle are proved. One of the main results of this chapter is that the infinite horizon optimal feedback law asymptotically stabilizes the system and that the infinite horizon optimal value function is a Lyapunov function for the closed loop system. Motivated by this property we formulate a relaxed version of the dynamic programming principle, which allows to prove stability and suboptimality results for nonoptimal feedback laws and without using the optimal value function. A practical version of this principle is provided, too. These results will be central in the following chapters for the stability and performance analysis of NMPC algorithms. For the special case of sampled-data systems we finally show that for suitable integral costs asymptotic stability of the continuous time sampled data closed loop system follows from the asymptotic stability of the associated discrete time system.

In this chapter we present a comprehensive stability and suboptimality analysis for NMPC schemes with stabilizing terminal constraints. Both endpoint constraints as well as regional constraints plus Lyapunov function terminal cost are covered. We show that viability of the state constraint set can be replaced by viability of the terminal constraint set in order to ensure feasibility of the NMPC optimal control problem along the closed loop trajectories. The “reversing of monotonicity” of the finite time optimal value functions is proved and used in order to apply the relaxed dynamic programming framework introduced in the previous chapter. Using this framework, stability, suboptimality (i.e., estimates about the infinite horizon performance of the NMPC closed loop system) and inverse optimality results are proved.

In this chapter we present a stability and suboptimality analysis for NMPC schemes without stabilizing terminal constraints. After defining the setting and presenting motivating examples we introduce an asymptotic controllability assumption and give a detailed derivation of stability and performance estimates based on this assumption and the relaxed dynamic programming framework introduced before. We show that our stability criterion is tight for the class of systems satisfying the controllability assumption and give conditions under which the level of suboptimality and a bound on the optimization horizon length needed for stability can be explicitly computed from the parameters in the controllability condition. As a spinoff we recover the well known result that—under suitable conditions—stability of the NMPC closed loop can be expected if the optimization horizon is sufficiently large. We further deduce qualitative properties of the running cost which lead to stability with small optimization horizons and illustrate by means of two examples how these criteria can be used even if the parameters in the controllability assumption cannot be evaluated precisely. Finally, we give weaker conditions under which semiglobal and semiglobal practical stability of the NMPC closed loop can be ensured.

The results developed so far in this book can be extended in many ways. In this chapter we present a selection of possible variants and extensions. Some of these introduce new combinations of techniques developed in the previous chapters, others relax some of the previous assumptions in order to obtain more general results or strengthen assumptions in order to derive stronger results. Several sections contain algorithmic ideas which can be added on top of the basic NMPC schemes from the previous chapters. Parts of this chapter contain results which are somewhat preliminary and are thus subject to further research. Some sections have a survey like style and, in contrast to the other chapters of this book, proofs are occasionally only sketched with appropriate references to the literature.

In this chapter we consider two different but related issues. In the first part we discuss the feasibility problem, i.e., that the nominal NMPC closed loop solutions remain inside a set on which the finite horizon optimal control problems defining the NMPC feedback law are feasible. We formally define the property of recursive feasibility and explain why the assumptions of the previous chapters, i.e., viability of the state constraint set or of the terminal constraint set ensure this property. Then we present two ways to relax the viability assumption on the state constraint set in the case that no terminal constraints are used. After a comparative discussion on NMPC schemes with and without stabilizing terminal constraints, we start with the second part of the chapter in which robustness of the closed loop under additive perturbations and measurement errors is investigated. Here robustness concerns both feasibility and admissibility as well as stability of the closed loop. We provide different assumptions and resulting NMPC schemes for which we can rigorously prove such robustness results and also discuss examples which show that in general robustness may fail to hold.

This chapter is particularly devoted to sampled data systems, which need to be discretized in order to be able to solve the optimal control problem within the NMPC algorithm numerically. We present suitable methods, discuss the convergence theory for one step methods and give an introduction into step size control algorithms. Furthermore, we explain how these methods can be integrated into NMPC algorithms, investigate how the numerical errors affect the stability of the NMPC controller derived from the numerical model and show which kind of robustness is needed in order to ensure a practical kind of stability.

In this chapter, we present methods for the numerical solution of the constrained finite horizon nonlinear optimal control problems which occurs in each iterate of the NMPC procedure. To this end, we first discuss standard discretization techniques to obtain a nonlinear optimization problem in standard form. Utilizing this form, we outline basic versions of the two most common solution methods for such problems, that is Sequential Quadratic Programming (SQP) and Interior Point Methods (IPM). Furthermore, we investigate interactions between the differential equation solver, the discretization technique and the optimization method and present several NMPC specific details concerning the warm start of the optimization routine. Finally, we discuss NMPC variants relying on inexact solutions of the finite horizon optimal control problem.