﻿ 多变量时滞非方系统的分数阶Smith预估控制
 控制与决策  2019, Vol. 34 Issue (6): 1331-1337 0

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ZHAO Zhi-cheng, XU Na, ZHANG Jing-gang. Fractional order Smith predictor control for non-square systems with time-delay[J]. Control and Decision, 2019, 34(6): 1331-1337. DOI: 10.13195/j.kzyjc.2017.1637.
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### 文章历史

Fractional order Smith predictor control for non-square systems with time-delay
ZHAO Zhi-cheng , XU Na , ZHANG Jing-gang
School of Electronic Information Engineering, Taiyuan University of Science and Technology, Taiyuan 030024, China
Abstract: A fractional order Smith predictive control approach based on the inverted decoupling is proposed for non-square systems with time-delay. Firstly, inverted decoupling is extended into m × n non-square systems. The design method of the inverted decoupling matrix is proposed. At the same time, to ensure the decoupling matrix stable and regular, the realizability conditions and the compensation method of the controlled object are provided. Then, we design a fractional order Smith predictive controller for decoupled signal-loop systems. The design method of the fractional order controller is simplitied using the equivalence relation between the IMC(internal model control) and the Smith predictive control. Furthermore, we propose a tuning methodology for controller parameters based on the maximum sensitivity. Finally, the typical Shell standard control problem is studied to verify the effectiveness of the proposed method. The simulation results show that the proposed method is not only simple in design and easy to implement, but also convenient in parameter tuning, and has a better tracking performance, disturbance rejection property and robustness.
Keywords: non-square processes    inverted decoupling    fractional order control    Smith predictor control    internal model control    maximum sensitivity
0 引言

1 多变量Smith预估控制结构

 图 1 Smith预估控制结构

 (1)
 (2)

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2 反向解耦方法 2.1 m× n非方系统的反向解耦

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 图 2 反向解耦结构

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n-m=1时, 有

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n-m>1时, 有

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2.2 解耦矩阵可实现性

1) 解耦矩阵中不允许出现超前环节, 因此, gii(s)的时滞时间必须为第i行元素中最小的值, 即

 (26)

2) 解耦矩阵必须为正则的, 各个元素的相对增益γii必须大于或等于0, 即gii(s)的相对增益必须为第i行中最小的, 有

 (27)

3) 当传递函数矩阵含有右半平面的零点时, 解耦矩阵有可能会含有右半平面的极点, 因此, gii(s)须含有i行所有的右半平面零点且阶次ϕij最小, 即

 (28)

gii(s)不满足式(26)~(28)时, 需要添加一个矩阵N(s)构建新的被控对象Gn(s)=G(s)N(s), 其中N(s)为对角矩阵, 即

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3 分数阶Smith预估控制器的设计 3.1 分数阶控制器设计

 图 3 Smith预估控制的等价内模控制结构

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F(s)为m× m低通分数阶滤波器矩阵, 即

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α=1时, Gc(s)为PI控制器; 当0 < α < 1时, Gc(s)为ID控制器; 当1 < α < 2时, Gc(s)为Ⅱ控制器.

3.2 分数阶控制器参数整定

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 图 4 最大灵敏度的几何解释

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s=jω代入式(38), 可得

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 图 5 参数η与Ms关系图

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4 仿真结果分析

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G(s)含有2个被控变量y1y2和3个操作变量u1u2u3.假设模型完全匹配, 则根据式(7)可知解耦后的广义被控对象为

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 图 6 标称情况下y1的阶跃响应
 图 7 标称情况下y2的阶跃响应
 图 8 输入扰动下y1的阶跃响应
 图 9 输入扰动下y2的阶跃响应

 图 10 摄动情况下y1的阶跃响应
 图 11 摄动情况下y2的阶跃响应

5 结论

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