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Rethinking Strict Dissipativity for Economic MPC

Mario Zanon

Abstract

Stability of economic model predictive control can be proven under the assumption that a strict dissipativity condition holds. This assumption has a clear interpretation in terms of the so-called rotated stage cost, which must have its minimum at the optimal steady state. However, contrary to dissipativity, for strict dissipativity the storage function cannot be immediately related to the value function of an optimal control problem formulated with the economic stage cost. We propose the novel concept of two-storage strict dissipativity, which requires two storage functions to satisfy dissipativity and be separated by a positive definite function. This new condition can be immediately related to optimal control by means of value functions and might be easier to verify than strict dissipativity. Furthermore, we prove that two-storage strict dissipativity is sufficient and necessary for asymptotic stability, it is related to strict dissipativity, and also to alternative approaches relying on the so-called cost-to-travel. Finally, we discuss commonly used and new terminal cost designs that guarantee asymptotic stability in the finite-horizon case.

Rethinking Strict Dissipativity for Economic MPC

Abstract

Stability of economic model predictive control can be proven under the assumption that a strict dissipativity condition holds. This assumption has a clear interpretation in terms of the so-called rotated stage cost, which must have its minimum at the optimal steady state. However, contrary to dissipativity, for strict dissipativity the storage function cannot be immediately related to the value function of an optimal control problem formulated with the economic stage cost. We propose the novel concept of two-storage strict dissipativity, which requires two storage functions to satisfy dissipativity and be separated by a positive definite function. This new condition can be immediately related to optimal control by means of value functions and might be easier to verify than strict dissipativity. Furthermore, we prove that two-storage strict dissipativity is sufficient and necessary for asymptotic stability, it is related to strict dissipativity, and also to alternative approaches relying on the so-called cost-to-travel. Finally, we discuss commonly used and new terminal cost designs that guarantee asymptotic stability in the finite-horizon case.
Paper Structure (11 sections, 19 theorems, 119 equations, 3 figures)

This paper contains 11 sections, 19 theorems, 119 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. The Infinite-Horizon Case
  4. Dissipativity and Boundedness
  5. Asymptotic Stability
  6. Finite-Horizon OCP with Terminal Cost
  7. Relationship With The Cost-To-Travel
  8. Examples
  9. Constrained Linear Quadratic MPC
  10. A Nonlinear Example
  11. Conclusions

Key Result

Proposition 2.5

Let function $V(x)$ be defined on a set $\mathcal{X}$, which is a closed subset of $\mathbb{R}^{n_x}$. If $V(\cdot)$ is continuous at the origin, $V(0) = 0$, and $V(x)$ is bounded over any bounded subset $\mathcal{\bar{X}}$ of $\mathcal{X}$, then there exists a class $\mathcal{K}$ function $\alpha(\

Figures (3)

  • Figure 1: Constrained linear quadratic example: cost rotated using different storage functions and optimal feedback law.
  • Figure 2: Top plot: $V_+(x)$, $V_-(x)$; center plot: $u_+(x)$, $u_-(f(x,u_+(x)))$, the black line highlights the case in which $u_+(x)=u_-(f(x,u_+(x)))$; bottom plot: rotated costs $L_1(x,u_+(x))$ (red) and $L_3(x,u_+(x))$ (blue) from Theorem \ref{['thm:2str_diss_implies_str_diss']}.
  • Figure 3: Top plot: convergence of $V_N$ to $V_+$ for $V_\mathrm{f}^1(x)$ (continuous lines) and $V_\mathrm{f}^2(x)$ (dashed lines) for different values of $r$; bottom plot: minimum stabilizing prediction horizon $N_\mathrm{s}$ for $V_\mathrm{f}^1(x)$ (dots) and $V_\mathrm{f}^2(x)$ (circles).

Theorems & Definitions (34)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.4: Comparison functions
  • Proposition 2.5: Rawlings2017
  • Proposition 2.6
  • Definition 2.7: Stability
  • Definition 2.8: Dissipativity
  • Remark 2.9
  • Lemma 2.10
  • Remark 2.11
  • ...and 24 more