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Model Predictive Control for T-S Fuzzy Markovian Jump Systems Using Dynamic Prediction Optimization

Bin Zhang

TL;DR

This paper investigates the model predictive control problem for the constrained discrete-time Takagi-Sugeno fuzzy Markovian jump systems (FMJSs) under imperfect premise matching rules and proposes a set of mode-dependent state feedback fuzzy controllers within the frame of dynamic prediction optimizing (DPO)-MPC.

Abstract

In this paper, the model predictive control (MPC) problem is investigated for the constrained discrete-time Takagi-Sugeno fuzzy Markovian jump systems (FMJSs) under imperfect premise matching rules. To strike a balance between initial feasible region, control performance, and online computation burden, a set of mode-dependent state feedback fuzzy controllers within the frame of dynamic prediction optimizing (DPO)-MPC is delicately designed with the perturbation variables produced by the predictive dynamics. The DPO-MPC controllers are implemented via two stages: at the first stage, terminal constraints sets companied with feedback gain are obtained by solving a ``min-max'' problem; at the second stage, and a set of perturbations is designed felicitously to enlarge the feasible region. Here, dynamic feedback gains are designed for off-line using matrix factorization technique, while the dynamic controller state is determined for online over a moving horizon to gradually guide the system state from the initial feasible region to the terminal constraint set. Sufficient conditions are provided to rigorously ensure the recursive feasibility of the proposed DPO-MPC scheme and the mean-square stability of the underlying FMJS. Finally, the efficacy of the proposed methods is demonstrated through a robot arm system example.

Model Predictive Control for T-S Fuzzy Markovian Jump Systems Using Dynamic Prediction Optimization

TL;DR

This paper investigates the model predictive control problem for the constrained discrete-time Takagi-Sugeno fuzzy Markovian jump systems (FMJSs) under imperfect premise matching rules and proposes a set of mode-dependent state feedback fuzzy controllers within the frame of dynamic prediction optimizing (DPO)-MPC.

Abstract

In this paper, the model predictive control (MPC) problem is investigated for the constrained discrete-time Takagi-Sugeno fuzzy Markovian jump systems (FMJSs) under imperfect premise matching rules. To strike a balance between initial feasible region, control performance, and online computation burden, a set of mode-dependent state feedback fuzzy controllers within the frame of dynamic prediction optimizing (DPO)-MPC is delicately designed with the perturbation variables produced by the predictive dynamics. The DPO-MPC controllers are implemented via two stages: at the first stage, terminal constraints sets companied with feedback gain are obtained by solving a ``min-max'' problem; at the second stage, and a set of perturbations is designed felicitously to enlarge the feasible region. Here, dynamic feedback gains are designed for off-line using matrix factorization technique, while the dynamic controller state is determined for online over a moving horizon to gradually guide the system state from the initial feasible region to the terminal constraint set. Sufficient conditions are provided to rigorously ensure the recursive feasibility of the proposed DPO-MPC scheme and the mean-square stability of the underlying FMJS. Finally, the efficacy of the proposed methods is demonstrated through a robot arm system example.
Paper Structure (19 sections, 6 theorems, 60 equations, 5 figures, 1 table)

This paper contains 19 sections, 6 theorems, 60 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Formulation and Preliminaries
  3. System Model
  4. DPO-Based Control Law Design
  5. Preliminaries
  6. Optimization Problem in Terminal Constraint Set
  7. Control Law in Terminal Constraint Set
  8. Terminal Cost Function
  9. Performance Optimization in Terminal Constraint Set
  10. Perturbation Variable Design
  11. Maximizing the Initial Feasible Region
  12. The Online Optimization Problem of Controller State $\eta_{s}$
  13. Stability Analysis and Algorithm
  14. Illustrative example
  15. Example
  16. ...and 4 more sections

Key Result

Lemma 1

Let the $S$ and $R$ be given positive matrix. Assume that there exist matrices $W_{\imath,\hbar}>0$, a scalar $\sigma>0$, a set of matrices $\tilde{K}_{\imath, \upsilon}$ and invertible matrices $G_{\imath,\upsilon}$ , such that the following inequalities holds: where Then, condition eq:3-4-4 is satisfied, and the mode-dependent controller gain is calculated by

Figures (5)

  • Figure 1: Comparison of initial feasible region before and after expansion
  • Figure 2: The system state evolution $x_s$ with three different MPC strategy. (Closed-loop evolution for 100 times different experiments.)
  • Figure 3: The system states evolution $x_s$ without control.
  • Figure 4: A possible sequence of system modes
  • Figure 5: The evolution of control inputs. (Closed-loop evolution for 100 times different experiments.)

Theorems & Definitions (16)

  • Remark 1
  • Definition 1
  • Definition 2
  • Lemma 1
  • Proof 1
  • Remark 2
  • Lemma 2
  • Remark 3
  • Remark 4
  • Lemma 3
  • ...and 6 more