﻿ 线性连续时间时滞系统的有限时间有界跟踪控制
 控制与决策  2019, Vol. 34 Issue (10): 2095-2104 0

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LIAO Fu-cheng, WU Ying-xue. Finite-time bounded tracking control for linear continuous systems with time-delay[J]. Control and Decision, 2019, 34(10): 2095-2104. DOI: 10.13195/j.kzyjc.2018.0190.
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### 文章历史

Finite-time bounded tracking control for linear continuous systems with time-delay
LIAO Fu-cheng , WU Ying-xue
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
Abstract: A finite-time bounded tracking control problem is investigated for a class of linear continuous systems with time-delay. Firstly, by applying a derivation method in the theory of preview control to constructing the error system with time-delay, the error signal is included in the state vector of it and then considered as the output vector. Then, the problem is transformed into an input-output finite-time stability problem of the closed-loop system of the error system, by designing a memory state feedback controller for the error system. Forthermore, based on the research methods of input-output finite-time stability and linear matrix inequation methods, the controller gains formulated in forms of linear matrix inequalities are provided by constructing a Lyapunov-Krasovskii functional. From this, a finite-time bounded tracking controller of the original system is obtained. Finally, a numerical example is given to illustrate the effectiveness and superiority of the proposed controller.
Keywords: continuous systems with time-delay    error system    preview control    input-output finite-time stability    tracking control    linear matrix inequality
0 引言

1961年, Dorato[1]首次提出了有限时间稳定的概念.随后, Kushner[2]和Weiss等[3-4]许多学者对其进行了深入研究. 2010年, Amato等[5]在前人研究的基础上, 提出了输入-输出有限时间稳定的概念, 其描述的是, 当给定系统的初始条件的界限时, 其输出信号在一段特定时间区间内始终不超过某个设定区域的特性.随后, Amato等[6]将输入-输出有限时间稳定的概念推广至脉冲系统, 给出了保证系统输入-输出有限时间稳定的充分条件, 并给出了静态输出反馈控制器的设计方法.文献[7]和文献[8]分别给出了保证脉冲系统输入-输出有限时间稳定的充分必要条件, 进一步推进了文献[6]的结果. 2014年, Amato等[9]研究了线性时变系统的输入-输出有限时间稳定问题, 并给出了在LL2型输入下的输入-输出有限时间稳定的充分条件和充分必要条件.

1 预备知识和基本概念

 (1)

 (2)

y(t)向左延拓到时间集[-τ, 0], 仍然使得y(t)=Cx(t), 则e(t)的初始条件为e(t)=Cϕ (t)-η (t), t∈ [-τ, 0].

1) S11 < 0, S22-S12TS11-1S12 < 0;

2) S22 < 0, S11-S12S22-1S12T < 0.

2 问题描述及基本假设

 (3)

3 扩大误差系统的构造

 (4)

 (5)

 (6)

 (7)

4 控制器的设计

 (8)

 (9)

 (10)
 (11)
 (12)

 (13)

 (14)

, 式(14)的[·]内的部分实际上是ξ(t)的二次型, 于是式(14)可以写为

 (15)

 (16)

 (17)

 (18)

 (19)

 (20)

 (21)

 (22)

 (23)

 (24)

 (25)
 (26)
 (27)

Φ=Φ+GK1, Gd=Gd+GK2代入上式, 得到

 (28)

Z=P-1, F=ZQZ, N=ZRZ, L1=K1Z, L2=K2Z, 则式(28)可写为

 (29)

 (30)

Ξ2 < Ξ1.因此, 若式(25)成立, 则式(29)成立, 从而式(10)成立.

 (31)

 (32)

Ad=0时, 系统(3)是一个无时滞系统, 这时只要在误差系统中取Gd=0, 上面的推导便都成立.由定理2得到的控制器仍然为(t)=K1X0(t)+K2X0(t-τ), 它可以看成是有记忆的控制器, 其中τ不再代表状态时滞, 而是一个可以根据设计需要选择的参数.另外, 还可令LMI(25)中的L2=0, 这时得到K2=0, 于是得到无记忆控制器(t)=K1X0(t).因此, 无时滞系统可以看作是本文中Ad=0时的特例.

5 数值仿真

Γ =I, c1=1, c2=2, T=10, 再取γ=0.2.

1) 干扰信号取为w(t)=0.15sin(2t).经计算得到

 (33)

(这里蕴含η (t)≡ 0), 这时r(t)满足

 图 1 目标信号为式(33)时闭环系统的输出响应

2) 干扰信号取为经计算得到

 (34)

(这里蕴含η (t)≡ 0), 这时有

 图 2 目标信号为式(34)时闭环系统的输出响应

 图 3 本文与文献[10]闭环系统输出的比较

 图 4 Ad=0时无时滞系统(3)的闭环系统输出响应
6 结论

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