Numerical Optimal Control of Nonlinear Systems

Abstract

In this chapter, we focus on numerically solving the constrained finite horizon nonlinear optimal control problems occurring in each iterate of the NMPC procedure. To this end, we first state 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.

Problems

• 1. Consider the constraint set $\varOmega = \{ (z_1, z_2, z_3) \mid z_3^2 \ge z_1^2 + z_2^2 \}$. Show that while $\varOmega$ is a set with nonsmooth boundary, the set itself is described by a smooth function.

• 2. One can trivially construct an example of a feasible set $\varOmega$ and a feasible point $z^\star$ at which the LICQ (see Definition 12.14) is satisfied but the constraints are nonlinear. Prove whether or not the reverse situation holds, i.e., the active constraints are linear but the LICQ is not satisfied.

• 3. Consider the constraints

  $$\begin{aligned} z_2 \le z_1^3 \qquad \text {and} \qquad z_2 \ge 0. \end{aligned}$$

  Show that at $z = (0, 0)$ we have $T_\varOmega (z) \subsetneq \mathscr {F}(z)$.

• 4. Similar to the idea of Condensing in Sect. 12.4 suppose that $s_j$ is a vector of dimension d allowing to get rid of the index function $\iota (\cdot )$ and that the index function $\varsigma (\cdot ): \{0, \ldots , r_s - 1\} \rightarrow \{0, \ldots , N\}$ is strictly monotonically increasing with $\varsigma (0) = 0$. Consider the corresponding linearized continuity condition

  $$\begin{aligned} S(z) + \nabla _z S(z)^\top \varDelta s = 0 \end{aligned}$$

  (12.75)

  where again $\varDelta s$ denotes the part of the search direction d corresponding to the shooting nodes. Show that $\varDelta s_1$, $\ldots$, $\varDelta s_{r_s - 1}$ can be computed from $\varDelta s_0$ which is the search direction for the shooting node corresponding to the initial value. Hint: Reformulate the continuity condition

  $$\begin{aligned} \left[ s_{j + 1} - f \left( x_u(\varsigma (j + 1) - 1, x_0), u(\varsigma (j + 1) - 1)) \right) \right] _{j \in \{0, \ldots , r_s - 1\}} = 0, \end{aligned}$$

  (12.76)

  to get rid of all state vectors $x_u(k, x_0)$ which are not shooting node vectors, i.e., for which $\varsigma (j) \not = k$.

• 5. Formulate the optimization problem in standard form for problem (OCP$_\text {N,e}^\text {n}$) using

  • (a) recursive elimination and an additional endpoint constraint $\mathbb {X}_0(n) = \{ x_*\}$.

  • (b) multiple shooting with shooting vector $s_N \in \mathbb {R}^d$ with initial value $s_N = x_*$ inducing the constraints

    $$\begin{aligned} S(z) = \left[ s_N - f \left( x_u(N - 1, x_0), u(N - 1)) \right) \right] = 0. \end{aligned}$$

  Using either of the presented optimization methods, show that the iterates will not coincide in general.