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Constrained finite-time stabilization by model predictive control: an infinite control horizon framework

Bing Zhu, Xiaozhuoer Yuan, Zewei Zheng, Zongyu Zuo

TL;DR

It is proved that the proposed finite-time MPC guarantees finite-time stabilization performance once the state trajectory enters the predefined terminal set and is shown to be equivalently implementable as a finite-horizon MPC with a terminal cost, thereby ensuring computational tractability.

Abstract

Existing results on finite-time model predictive control (MPC) often rely on terminal equality constraint, switching inside one-step region, or terminal cost with short control horizon, leading to limited initial feasibility. This paper proposes an infinite-horizon Model Predictive Control (MPC) framework for the constrained finite-time stabilization of discrete-time systems, overcoming limitations found in existing finite-time MPC results. The proposed framework is built upon a terminal cost strategy, but expands it by replacing the short-horizon terminal cost with the sum of stage costs over an infinite control horizon. This design choice significantly enlarges the initial feasibility region and avoids the need for terminal equality constraints or switching strategies during implementation. It is proved that the proposed finite-time MPC guarantees finite-time stabilization performance once the state trajectory enters the predefined terminal set. The infinite-horizon finite-time MPC is shown to be equivalently implementable as a finite-horizon MPC with a terminal cost, thereby ensuring computational tractability. The proposed finite-time MPC is systematically extended and shown to be applicable to both constrained multi-input linear systems and a class of constrained nonlinear systems that are feedback linearizable.

Constrained finite-time stabilization by model predictive control: an infinite control horizon framework

TL;DR

It is proved that the proposed finite-time MPC guarantees finite-time stabilization performance once the state trajectory enters the predefined terminal set and is shown to be equivalently implementable as a finite-horizon MPC with a terminal cost, thereby ensuring computational tractability.

Abstract

Existing results on finite-time model predictive control (MPC) often rely on terminal equality constraint, switching inside one-step region, or terminal cost with short control horizon, leading to limited initial feasibility. This paper proposes an infinite-horizon Model Predictive Control (MPC) framework for the constrained finite-time stabilization of discrete-time systems, overcoming limitations found in existing finite-time MPC results. The proposed framework is built upon a terminal cost strategy, but expands it by replacing the short-horizon terminal cost with the sum of stage costs over an infinite control horizon. This design choice significantly enlarges the initial feasibility region and avoids the need for terminal equality constraints or switching strategies during implementation. It is proved that the proposed finite-time MPC guarantees finite-time stabilization performance once the state trajectory enters the predefined terminal set. The infinite-horizon finite-time MPC is shown to be equivalently implementable as a finite-horizon MPC with a terminal cost, thereby ensuring computational tractability. The proposed finite-time MPC is systematically extended and shown to be applicable to both constrained multi-input linear systems and a class of constrained nonlinear systems that are feedback linearizable.
Paper Structure (17 sections, 4 theorems, 69 equations, 5 figures, 1 algorithm)

This paper contains 17 sections, 4 theorems, 69 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notation
  4. Existing result on finite-time MPC
  5. Problem statement
  6. Main results
  7. Finite-time MPC with infinite control horizon
  8. Implementation via finite-horizon MPC
  9. Extension to multi-input linear systems
  10. Extension to constrained nonlinear systems
  11. Numerical examples
  12. Finite-time MPC for single input linear systems
  13. Finite-time MPC for multi-input linear systems
  14. Finite-time MPC for nonlinear systems
  15. Robustness to bounded disturbance
  16. ...and 2 more sections

Key Result

Theorem 1

Consider the linear discrete-time system linear syst subject to constraints constraints. Design the cost function by cost fun, where the sum of stage costs is from $n$ to $\infty$. If the constrained optimization deadbeat opt inf is feasible initially, and the control is implemented by deadbeat opt

Figures (5)

  • Figure 1: Single-input closed-loop states and control under the proposed infinite-horizon finite-time MPC, with transient response settling in 7 steps.
  • Figure 2: Comparison of initial feasibility regions: the previous terminal-cost strategy (dashed); the proposed infinite-horizon strategy implemented via finite horizon $N=8$ (solid).
  • Figure 3: Multi-input closed-loop states and controls under the proposed infinite-horizon finite-time MPC, with transient response settling in 11 steps.
  • Figure 4: Closed-loop states and control of the nonlinear system under the proposed infinite-horizon finite-time MPC, with transient response settling in 5 steps.
  • Figure 5: Closed-loop states and control in the presence of bounded random disturbance: system states are ultimately bounded, and controls are within constraints.

Theorems & Definitions (15)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Theorem 2
  • Remark 8
  • ...and 5 more