﻿ 离散系统有限时间稳定预测控制 Finite Time Stable Predictive Control for Discrete Systems

Operations Research and Fuzziology
Vol. 09  No. 01 ( 2019 ), Article ID: 28283 , 8 pages
10.12677/ORF.2019.91002

Finite Time Stable Predictive Control for Discrete Systems

Shuang Zhang, Xiaohua Liu

Ludong University, Yantai Shandong

Received: Dec. 8th, 2018; accepted: Dec. 21st, 2018; published: Dec. 28th, 2018

ABSTRACT

The finite-time stable predictive control problem for a class of discrete-time linear time-invariant systems is studied. The concept of finite-time stability is introduced into predictive control, and the relationship between finite-time stability and predictive control is established by the state feedback. The minimization optimization problem in finite time domain is transformed into a constrained problem with linear matrix inequalities by constructing the Lyapunov function. The sufficient conditions for the existence of state feedback control rate are given by using linear matrix inequalities. The numerical simulation results show the effectiveness of the proposed method.

Keywords:Model Predictive Control, Finite-Time Stability, State Feedback, Linear Matrix Inequality

Copyright © 2019 by authors and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

2. 问题描述

$\left\{\begin{array}{l}x\left(k+1\right)=Ax\left(k\right)+Bu\left(k\right),x\left(0\right)={x}_{0}\\ y\left(k+1\right)=Cx\left(k\right)\end{array}$ (1)

$\underset{u\left(k\right)}{\mathrm{min}}{J}_{N}\left(k\right)=\underset{u\left(k\right)}{\mathrm{min}}\underset{i=0}{\overset{N-1}{\sum }}{x}^{T}\left(k+i|k\right){Q}_{1}x\left(k+i|k\right)+{u}^{T}\left(k+i|k\right){Q}_{2}u\left(k+i|k\right)$ (2)

$u\left(k\right)=Kx\left(k\right)$ (3)

$x\left(k+1\right)=\left(A+BK\right)x\left(k\right)$ (4)

$V\left(x\left(k+i+1|k\right)\right)-V\left(x\left(k+i|k\right)\right)\le -\left({x}^{T}\left(k+i|k\right){Q}_{1}x\left(k+i|k\right)+{u}^{T}\left(k+i|k\right){Q}_{2}u\left(k+i|k\right)\right)$

${J}_{N}\left(k\right)\le V\left(x\left(k|k\right)\right)$ (5)

$x\left(k+1\right)=Ax\left(k\right),k\in {N}^{+}$ (6)

1) 系统(6)关于 $\left({\delta }_{x},\epsilon ,R,N\right)$ 是有限时间稳定的

2) 对于 $\forall k\in \left\{1,\cdots ,N\right\}$${\left({A}^{T}\right)}^{k}R{A}^{k}<\frac{{\epsilon }^{2}}{{\delta }_{x}^{2}}R$ 成立

3) 对于 $\forall k\in \left\{1,\cdots ,N\right\}$ ，若 ${P}_{k}\left(k\right)=R$${P}_{k}\left(h\right)={A}^{T}{P}_{k}\left(h+1\right)A,h\in \left\{0,1,\cdots ,k-1\right\}$

${P}_{k}\left(0\right)<\frac{{\epsilon }^{2}}{{\delta }_{x}^{2}}R$

4) 对于 $\forall k\in \left\{1,\cdots ,N\right\}$ ，存在对称矩阵值函数 ${P}_{k}\left(\cdot \right):h\in \left\{0,1,\cdots ,k\right\}↦{P}_{k}\left(h\right)\in {\Re }^{n×n}$

$S\left(x\right)=\left[\begin{array}{cc}{S}_{11}\left(x\right)& {S}_{12}\left(x\right)\\ {S}_{21}\left(x\right)& {S}_{22}\left(x\right)\end{array}\right]$ ，其中 ${S}_{11}\left(x\right)$ 是方阵。

1) $S\left(x\right)<0$

2) ${S}_{11}\left(x\right)<0$${S}_{22}\left(x\right)-{S}_{12}^{T}\left(x\right){S}_{11}^{-1}\left(x\right){S}_{12}\left(x\right)<0$

3) ${S}_{22}\left(x\right)<0$${S}_{11}\left(x\right)-{S}_{12}\left(x\right){S}_{22}^{-1}\left(x\right){S}_{12}^{T}\left(x\right)<0$

3. 有限时间稳定状态反馈

$\left(\begin{array}{cc}\begin{array}{l}-\gamma Q\\ AQ+BL\end{array}& \begin{array}{l}{\left(AQ+BL\right)}^{T}\\ -Q\end{array}\end{array}\right)<0$ (7)

$\frac{{\lambda }_{\mathrm{max}}\left(\stackrel{˜}{Q}\right)}{{\lambda }_{\mathrm{min}}\left(\stackrel{˜}{Q}\right)}<\frac{1}{{\gamma }^{N}}\frac{{\epsilon }^{2}}{{\delta }_{x}^{2}}$ (8)

$E={\left[{\left({C}_{1}A\right)}^{T}{\left({C}_{2}A\right)}^{T}\cdots {\left({C}_{r}A\right)}^{T}\right]}^{T}$

$S={\left[{S}_{1}^{T}{S}_{2}^{T}\cdots {S}_{r}^{T}\right]}^{T}$

${S}_{j}={C}_{j}B$$\left(j=1,\cdots ,r\right)$

$\stackrel{˜}{Q}={R}^{-\frac{1}{2}}Q{R}^{-\frac{1}{2}}$

$\begin{array}{c}{\gamma }^{k}V\left(x\left(0\right)\right)={\gamma }^{N}\left[x{\left(0\right)}^{T}Qx\left(0\right)\right]\\ \le {\gamma }^{N}\left[{\lambda }_{\mathrm{max}}\left(\stackrel{˜}{Q}\right)x{\left(0\right)}^{T}Rx\left(0\right)\right]\\ \le {\gamma }^{N}\left[{\lambda }_{\mathrm{max}}\left(\stackrel{˜}{Q}\right){\delta }_{x}^{2}\right]\end{array}$ (10)

$\begin{array}{c}V\left(x\left(k\right)\right)=x{\left(k\right)}^{T}Qx\left(k\right)\\ \ge {\lambda }_{\mathrm{min}}\left(\stackrel{˜}{Q}\right)x{\left(k\right)}^{T}Rx\left(k\right)\end{array}$ (11)

$x{\left(k\right)}^{T}Rx\left(k\right)\le \frac{{\lambda }_{\mathrm{max}}\left(\stackrel{˜}{Q}\right)}{{\lambda }_{\mathrm{min}}\left(\stackrel{˜}{Q}\right)}{\gamma }^{N}{\delta }_{x}^{2}<{\epsilon }^{2}$

$Q={P}^{-1}$ ，我们有 ${\stackrel{^}{A}}^{T}Q{}^{-1}\stackrel{^}{A}-\gamma {Q}^{-1}<0$

$\left(\begin{array}{cc}\begin{array}{l}-\gamma {Q}^{-1}\\ \stackrel{^}{A}\end{array}& \begin{array}{l}{\stackrel{^}{A}}^{T}\\ -Q\end{array}\end{array}\right)$

$\left(\begin{array}{cc}-\gamma Q& Q{\stackrel{^}{A}}^{T}\\ \stackrel{^}{A}Q& -Q\end{array}\right)=\left(\begin{array}{cc}-\gamma Q& Q{\stackrel{^}{A}}^{T}\\ \stackrel{^}{A}Q& -Q\end{array}\right)=\left(\begin{array}{cc}-\gamma Q& {\left(AQ+BKQ\right)}^{T}\\ \left(AQ+BKQ\right)& -Q\end{array}\right)$

$kQ=L$ ，同时 $Q={k}_{1}T$ ，此时有 $k=L{\left({k}_{1}T\right)}^{-1}$ ，即为所求的增益矩阵。

4. 仿真案例

$A=\left[\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 1& 0\\ 0& 0& 0& 1\\ 1& 0& -1& 0\end{array}\right]$$B=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right]$$C=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]$

Figure 1. States of the closed-loop system

Figure 2. The trajectory of the control input

Figure 3. Output of the closed-loop system

Figure 4. The trajectory of x’Rx

Figure 5. Performance index change

5. 总结

Finite Time Stable Predictive Control for Discrete Systems[J]. 运筹与模糊学, 2019, 09(01): 6-13. https://doi.org/10.12677/ORF.2019.91002

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