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Stabilization of strictly pre-dissipative nonlinear receding horizon control by terminal costs

Lars Grüne, Mario Zanon

TL;DR

This work tackles stabilization of receding horizon control for nonlinear systems under strict pre-dissipativity by showing that a carefully designed terminal cost can restore stability without terminal constraints. By reformulating the problem with a rotated stage cost $L(x,u)$ and a storage function $\lambda$, the authors prove that if $V^{\mathrm{f}}+\lambda$ is bounded below, semiglobal practical asymptotic stability is achieved; if this sum is positive semidefinite, true asymptotic stability follows (and can be guaranteed with a linear $\rho$). In the linear-quadratic setting, $\lambda(x)$ can take a quadratic form and the conditions simplify to relations between $P^{\mathrm{f}}$, $\Lambda$, and the equilibrium, with terminal costs like $V^{\mathrm{f}}(x)=a x^2$, $a>0$, stabilizing the closed loop. An illustrative nonlinear example demonstrates how the choice of terminal cost transitions the closed-loop behavior from instability to practical, and then true, stability, highlighting the practical relevance of the proposed terminal-cost design for RP MPC.

Abstract

It is known that receding horizon control with a strictly pre-dissipative optimal control problem yields a practically asymptotically stable closed loop when suitable state constraints are imposed. In this note we show that alternatively suitably bounded terminal costs can be used for stabilizing the closed loop.

Stabilization of strictly pre-dissipative nonlinear receding horizon control by terminal costs

TL;DR

This work tackles stabilization of receding horizon control for nonlinear systems under strict pre-dissipativity by showing that a carefully designed terminal cost can restore stability without terminal constraints. By reformulating the problem with a rotated stage cost and a storage function , the authors prove that if is bounded below, semiglobal practical asymptotic stability is achieved; if this sum is positive semidefinite, true asymptotic stability follows (and can be guaranteed with a linear ). In the linear-quadratic setting, can take a quadratic form and the conditions simplify to relations between , , and the equilibrium, with terminal costs like , , stabilizing the closed loop. An illustrative nonlinear example demonstrates how the choice of terminal cost transitions the closed-loop behavior from instability to practical, and then true, stability, highlighting the practical relevance of the proposed terminal-cost design for RP MPC.

Abstract

It is known that receding horizon control with a strictly pre-dissipative optimal control problem yields a practically asymptotically stable closed loop when suitable state constraints are imposed. In this note we show that alternatively suitably bounded terminal costs can be used for stabilizing the closed loop.

Paper Structure

This paper contains 11 sections, 3 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Receding Horizon Control
  4. Strict dissipativity
  5. Main results
  6. Results for positive semidefinite $V^{\mathrm{f}}+\lambda$
  7. Results for $V^{\mathrm{f}}+\lambda$ bounded from below
  8. Relation to the required supply
  9. Consequences for the linear quadratic case
  10. Illustrative example
  11. Conclusions

Key Result

Lemma 3.1

Consider the RH-OCP eq:empc for an arbitrary finite horizon $N$. Assume strict pre-dissipativity with storage function $\lambda$. Then the problem with stage cost $\ell$ and terminal cost $V^{\mathrm{f}}$ has the same optimal trajectories $x_k^\star$ and control sequences $u_k^\star$ as the RH-OCP p

Figures (3)

  • Figure 1: Difference between closed-loop solution and $\bar{x}^\star$ with $V^{\mathrm f} \equiv 0$
  • Figure 2: Difference between closed-loop solution and $\bar{x}^\star$ with $V^{\mathrm f}(x)=2x^2$ and optimization horizon $N=3$ (top) and $N=5$ (bottom)
  • Figure 3: Difference between closed-loop solution and $\bar{x}^\star$ with $V^{\mathrm f}(x)=2x^2-2\nu x$ and optimization horizon $N=3$. After time step $k=12$ the solution deteriorates because of roundoff errors.

Theorems & Definitions (7)

  • Definition 2.1
  • Example 2.2
  • Lemma 3.1
  • Definition 3.2
  • Theorem 3.5
  • Remark 3.6
  • Theorem 3.9