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Stabilization of Strictly Pre-Dissipative Receding Horizon Linear Quadratic Control by Terminal Costs

Mario Zanon, Lars Grüne

Abstract

Asymptotic stability in receding horizon control is obtained under a strict pre-dissipativity assumption, in the presence of suitable state constraints. In this paper we analyze how terminal constraints can be replaced by suitable terminal costs. We restrict to the linear-quadratic setting as that allows us to obtain stronger results, while we analyze the full nonlinear case in a separate contribution.

Stabilization of Strictly Pre-Dissipative Receding Horizon Linear Quadratic Control by Terminal Costs

Abstract

Asymptotic stability in receding horizon control is obtained under a strict pre-dissipativity assumption, in the presence of suitable state constraints. In this paper we analyze how terminal constraints can be replaced by suitable terminal costs. We restrict to the linear-quadratic setting as that allows us to obtain stronger results, while we analyze the full nonlinear case in a separate contribution.
Paper Structure (18 sections, 21 theorems, 83 equations, 3 figures)

This paper contains 18 sections, 21 theorems, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Preliminary Results
  4. Cost Rotation and Pre-Stabilization
  5. Strict $(x,u)$-Pre-Dissipativity and Rotated Cost
  6. Pre-Stabilized System
  7. The Reverse Discrete-Time Algebraic Riccati Equation
  8. Positive-Semidefinite Cost Rotations
  9. Properties of the Cost-to-Go and DARE Solutions
  10. Stabilizing Terminal Costs
  11. The Antistabilizing Solution Does Exist
  12. The Antistabilizing Solution Does not Exist
  13. Convergence Implies Exponential Stability
  14. Relation With Known Results
  15. The Nonlinear Case
  16. ...and 3 more sections

Key Result

Lemma 3.3

For any finite horizon $N$ as well as for the infinite horizon problem, an LQ problem with stage cost matrix $H$ and terminal cost matrix $P_0$ yields the same feedback law $u_k=-K_N x_k$ as the rotated LQ problem with stage cost matrix $H_\Lambda$ and terminal cost matrix $P_0+\Lambda$. Moreover, t

Figures (3)

  • Figure 1: Minimum stabilizing horizon length.
  • Figure 2: Closed-loop matrix eigenvalues with $P^\mathrm{f}=10^{-4}$.
  • Figure 3: State at time $k=500$ obtained with $P^\mathrm{f}=0$ (red) and $P^\mathrm{f}=1e-4$ (blue).

Theorems & Definitions (31)

  • Example 2.1
  • Definition 3.1: Quadratic Strict $(x,u)$-Pre-Dissipativity
  • Example 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Remark 3.7
  • Proposition 3.9: Ionescu1996
  • Example 3.10
  • ...and 21 more