Numerical Discretization

Abstract

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.

Problems

• 1. Prove that the solution $\varphi (t,0,x_0,u)$ of (2.6) with $t\in [0,T]$ and constant control function u satisfies the integral equation

  $$\begin{aligned} \varphi (\tau _{i + 1}; \tau _0, x_0, u) = \varphi (\tau _i; \tau _0, x_0, u) + \int \limits _{\tau _i}^{\tau _{i + 1}} f_c( \varphi (t; \tau _0, x_0, u), u ) dt. \end{aligned}$$

  for all $\tau _{i},\tau _{i+1}\in [0,T]$ with $\tau _{i+1}> \tau _i$.

• 2. Prove that the Euler and the Heun scheme satisfy the Lipschitz condition (11.5) if the vector field $f_c$ satisfies the Lipschitz condition from Assumption 2.4.

• 3. Given the control system $\dot{x}(t) = x(t) + u(t)$ with stage cost $\ell (x, u) = x^2 + u^2$.

  • (a) Consider the NMPC Algorithm 3.1 with $N = 2$ and f generated by the Euler method with $\mathscr {G}= \mathscr {T}$ for (11.14) and (11.15). Prove that the control $\mu _N(x)$ converges tends to zero as $T \rightarrow 0$ for each $x\in \mathbb {R}$.

  • (b) Consider the same situation as in (a) but with the grid

    $$\begin{aligned} \mathscr {G}:= \{ \tau _i = i {T}/{k}\,|\, i = 0,\ldots ,Nk \} \end{aligned}$$

    with $k\in \mathbb {N}$. Does the control value $\mu _N(x)$ converge if $T>0$ is fixed and k tends to infinity?

• 4. Consider the differential equation

  $$\begin{aligned} \dot{x}_1(t)&= -x_2(t) \\ \dot{x}_2(t)&= {-}x_1(t) \end{aligned}$$

  whose solution shall be used to generate a time varying reference for an NMPC algorithm.

  • (a) Using a transformation to polar coordinates, compute the analytical solution of the system.

  • (b) Show that the numerical solution of the system using Euler’s method will deviate from the analytical solution from (a) for every step size $h > 0$ and every initial value $x_0 \not = (0, 0)^\top$.

  • (c) Applying the transformation to polar coordinates, show that the occurring error from (b) can be avoided if the resulting differential equation is solved using Euler’s method.

• 5. Consider the continuous time control system

  $$\begin{aligned} \dot{x}_1(t)&= -x_2(t) + v(t) \\ \dot{x}_2(t)&= {-}x_1(t) \end{aligned}$$

  where u shall be computed via NMPC to track the (exact) time varying reference solution from Problem 4.

  • (a) Show that this system is (uniformly) asymptotically controllable in the sense of Definition 4.2 for control functions which are piecewise constant on each interval $[iT,(i+1)T)$ for arbitrary sampling time $T > 0$.

  • (b) Consider the approximate discrete time system (11.12) with $\tilde{\varphi }$ obtained from applying the Euler method with step size $h=T/k$ for arbitrary $k\in \mathbb {N}$ to the (non transformed) differential equation. Show that this approximate system is not asymptotically controllable regardless how $T>0$ and $k\in \mathbb {N}$ are chosen.

  Hint for (b): A necessary condition for asymptotic controllability is that the reference is a solution of the system.