Feasibility and Robustness

Abstract

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 conditions, 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 the feasibility and admissibility as well as the 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.

Problems

• 1. Consider the feasible set $\mathscr {F}_N$ for a constraint set $\mathbb {X}\subset X$ and an optimization horizon $N\in \mathbb {N}$ according to Definition 7.2. Assume that for a point $x\in \mathbb {X}$ and some $K\in \mathbb {N}$ there exists an admissible control sequence $u\in \mathbb {U}^K(x)$ with $x_u(K,x)\in \mathscr {F}_N$. Prove that $x\in \mathscr {F}_{N+K}$ holds.

• 2. Consider a symmetric matrix $Q\in \mathbb {R}^{n\times n}$ and a constant $C>0$ such that the inequality $|x^\top Q y | \le C\Vert x\Vert \Vert y\Vert$ holds for all $x,y\in \mathbb {R}^n$. Let $\rho >0$ be given and consider the set $A=\overline{\mathscr {B}}_\rho (0)$.

  • (a) Show that

    $$\begin{aligned} \omega (r) = 2C\rho \, r \end{aligned}$$

    is a modulus of continuity of the function $W(x)=x^\top Q x$ on A.

  • (b) Compute a modulus of continuity of the function $W(x)=(x^\top Q x)^2$ on A.

• 3. Verify the following facts that have been used in Example 7.31.

  • (a) For $x\in \mathbb {R}^2$ with $x_2>0$ and $u\in \mathbb {R}$ with $u<0$ the step $x^+=f(x,u)$ defines a clockwise movement.

  • (b) For all $c\in (0,1)$, all circles $S_r$ with $r > r_c = c/\sqrt{1-c^2}$ and all points $x\in S_{r}\cap \mathbb {X}$ with $x_2 > r$ and $x_1=c$ the relation $f(x,-1)\not \in \mathbb {X}$ holds. Use this fact to conclude that for all initial values $x\in S_{r}\cap \mathbb {X}$ with $x_2 > r$ it is not possible to move clockwise toward 0.

  • (c) For all $c<1/2$ and $y'\in Y'$ with $\varepsilon >0$ sufficiently small the inequality $\ell (f(y',-1)) > \ell (y')$ holds.