Distributed NMPC

Abstract

For large-scale systems such as street traffic, cyber- physical production systems or energy grids on an operational level, the MPC approach introduced in Chap. 3 is typically inapplicable in real time. Moreover, communication restrictions or privacy considerations may render the centralized solution of the optimal control problem in each step of the NMPC scheme impossible. To cope with these issues, the optimal control problem is split into subproblems, which are simpler to solve but may be linked by dynamics, cost functions or constraints. As the examples indicate, each subproblem may be seen as an independent unit. If these units are not coordinated, i.e., if there exists no data exchange and if inputs from connected units are considered as disturbances, the problem is referred to as decentralized. Including communication, the problem is called distributed and can again be split into subclasses of cooperative and noncooperative control. Within this chapter, we impose the assumption of flawless communication to analyze both stability and performance of the overall system for the distributed case. Additionally, we briefly sketch how to analyze the robustness of the distributed setting. Last, we discuss basic coordination methods on the tactical control layer to solve the distributed problem and relate these methods to our stability results.

Problems

• 1. Reconsider Example 9.11 with dynamics

  $$\begin{aligned} x(n + 1) = \begin{pmatrix} x^1(n + 1) \\ x^2(n + 1) \end{pmatrix} = \begin{pmatrix} 1 &{} 0 \\ 0 &{} 1 \end{pmatrix} \begin{pmatrix} x^1(n) \\ x^2(n) \end{pmatrix} + \begin{pmatrix} u^1(n) \\ u^2(n) \end{pmatrix} =: f(x(n), u(n)). \end{aligned}$$

  constraints $|x^1 - x^2| \ge \delta > 0$ and costs

  $$\begin{aligned} J_N(x_0, u(\cdot )) =&\sum _{k= 0}^{N- 1} \left( \sum _{{p}= 1}^{P} \Vert x^{p}_{u^p}(k, x^p_0, i^{p}) - x_{*}^{p}\Vert ^2 + \Vert u^{p}(k) \Vert ^2 \right) \\&\qquad + \sum _{{p}= 1}^{P} \Vert x^{p}_{u^p}(N, x^p_0, i^{p}) - x_{*}^{p}\Vert ^2 \end{aligned}$$

  with $x_0^1 \not = x_0^2$ and $x_{*}^1 = x_{*}^2$. Show that Assumption 9.32 holds and explain why Theorem 9.33 still fails to apply.

• 2. Consider the inverted pendulum from Example 2.10 with dynamics

  $$\begin{aligned} \dot{x}_1(t)&= x_2(t) \\ \dot{x}_2(t)&= u(t) \\ \dot{x}_3(t)&= x_4(t) \\ \dot{x}_4(t)&= -\frac{g}{l}\sin (x_3(t)) - u(t) \cos (x_3(t)) - \frac{k_L}{l} x_4(t) | x_4(t) | - k_R \text {sgn}(x_4(t)) \end{aligned}$$

  Extending the single pendulum to a ${p}$ pendulum (for each pendulum the tip is attached to the end of the previous pendulum) and separating cart and each pendulum into subsystems, show that the Algorithm of Richards and How can only be applied for one order of subsystems.

• 3. Consider two cars modeled as points in the plane with dynamics

  $$\begin{aligned} x^p(n+1) = x^p(n) + u^{p}(n) \end{aligned}$$

  where $x^p(n) \in \mathbb {R}^2$ and $u^{p}(n) \in [-\delta , \delta ] \subset \mathbb {R}^2$. Assume that the constraints are given by $\Vert x^1(n) - x^2(n) \Vert _2 \ge 1$ and the cost function is defined via

  $$\begin{aligned} J^{p}_N(x^p, u^{p}) = \sum _{k= 0}^{N- 1} \Vert x^{p}_{u^p}(k; x^p_0) - x_{*}^{p}\Vert ^2 \end{aligned}$$

  $\Vert x_{*}^1 - x_{*}^2 \Vert _2 \ge 1$. Given $\delta = 1$, what is the minimal horizon $N$ such that for each ${p}$ we observe a decrease in $J^{p}_N$ for all feasible initial values? Does the minimal horizon change for smaller $\delta$?