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Solution of output feedback μ controller based on LMI
LI Zhe , GAO Yuanlou , LI Peilin
School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Received: 2015-08-31; Accepted: 2015-11-27; Published online: 2016-01-05 10:42
Corresponding author. GAO Yuanlou, Tel.:010-82339757, E-mail:gaoyuanlou@263.net
Abstract: D-K iteration is the main implementation method of structured singular value (μ), which has problems of strict solution conditions and poor system suitability. Aimed at overcoming limitation shortage of D-K iteration application, the linear matrix inequality (LMI) was proposed to improve D-K iteration, which uses Schur's lemma and bounded real lemma to get LMI criterion of upper boundary of structured singular value, and elimination method was developed to obtain H controller in output feedback system. Based on improvements related to LMI, D-K iteration was adopted to solve μ controller in output feedback system, which avoids solution limitation of Riccati equation and influence by selection quality of some uncertain parameters, enhances its applicability in general system, and improves the solution efficiency of controller in output feedback system. Numerical results show that this method gets not only robust stability but also robust performance superior to the traditional D-K iteration of output feedback system.
Key words: structured singular value     linear matrix inequality (LMI)     output feedback     controller     robust performance

1 预备知识 1.1 参数摄动模型

μ问题可描述成:对给定不确定量集合Δ, γR+及标称系统G(s)∈H空间。

 (1)

 (2)

 (3)
 图 1 输出反馈系统模型 Fig. 1 Output feedback system model

 (4)

 (5)
1.2 矩阵不等式

1.2.1 Schur引理[11]

 (6)

1) S < 0

2) S11 < 0, S22 -S12 TS11 -1S12 < 0

3) S22 < 0, S11 -S12 S22 -1S12 T < 0

 (7)

 (8)

1.2.2 有界实引理

γ>0, 系统G(s)的状态空间矩阵为(A, B, C, E), 则以下条件等价：

1) 系统渐近稳定, 且||G(s)|| < γ

2) 存在一个对称矩阵P>0, 使得

 (9)
2 D-K迭代的LMI表述 2.1 D求解

 (10)

[12]。式(10)中:Di *为矩阵Di的共轭转置; Fl(G, K)表示系统G与控制器K形成闭环系统的下线性变换。

 (11)

 (12)

 (13)

 (14)
 (15)
 (16)
 (17)

2.2 输出反馈控制器K求解

2.2.1 输出反馈的LMI表述

H控制器u=Ky应用到系统后得到闭环系统为

 (18)

 (19)

2.2.2 消元法

1) 求满足下列条件的矩阵XY

 (20)
 (21)
 (22)

2) 求满足X-Y-1=X1X1T的矩阵X1Rn×nk, 其中nkXY－1的秩；再通过式(23)构造Xcl, 文献[9]已证明, 只要1)中不等式条件满足且要设计的控制器维数大于等于系统状态量即nkn, 就总能找到满足要求的Xcl:

 (23)

 (24)

3) 将式(23)、式(24)代入式(19)得到如式(25)的等价表述:

 (25)

3 基于LMI的D-K迭代步骤

1) 初始化K, 求解满足

 (26)

2) 根据最小化||DMD－1||D(s)矩阵, 设计求解满足min||DMD－1||的控制器K, 即

3) 代入K状态参数, 返回1)继续求解D, 重复迭代直到K满足要求, 得到最优化的μ控制器。

4 数值仿真

 图 2 参数摄动对象系统模型 Fig. 2 Parameter perturbation model of object system

 图 3 μ/文献[10]系统结构奇异值曲线 Fig. 3 Structured singular value curves of μ/Ref.[10] system

 (27)

 (28)
 图 4 μ控制器幅频特性 Fig. 4 Amplitude-frequency characteristic of μ controller

μ控制器的鲁棒性能进行对比, 对应参数摄动结构的系统矩阵结构奇异值曲线如图 5所示。可知μ控制器系统比PID控制器系统的最大奇异值更小, 曲线过渡也更加平稳, 鲁棒稳定性更好。图 3所示采用文献[10]处理方法求解出的控制器系统与μ控制系统结构奇异值曲线对比, 可知虽然Riccati方程求解条件满足, 但是由于系统矩阵处理后存在一定程度失真, 导致控制器鲁棒稳定性变差, 无法得到最优的鲁棒控制器。

 图 5 μ/PID系统结构奇异值曲线 Fig. 5 Structured singular value curves of μ/PID system

 图 6 μ/PID系统阶跃响应 Fig. 6 Step response of μ/PID system
 图 7 干扰输入μ/PID系统输出 Fig. 7 Output of input disturbance μ/PID system
5 结论

1) 综合了LMI方法适用性广以及D-K算法交替凸优化的优点。

2) 相对于文献[7-8]中的控制器由于无需状态观测器更具有工程实用性。

3) 相较于文献[15]中方法得到的PID控制器具有较好的动态性能以及较好的鲁棒性能; 相较于文献[10]方法解出的μ控制器具有较好的鲁棒稳定性和鲁棒性能。

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文章信息

LI Zhe, GAO Yuanlou, LI Peilin

Solution of output feedback μ controller based on LMI

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(10): 2231-2237
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0556