非齐次马尔可夫LPV系统的H∞异步控制

doi:10.3969/j.issn.1001-506X.2020.05.24

摘要/Abstract

摘要：

针对存在外界干扰的情况,对一类离散时间非齐次马尔可夫线性变参数(linear parameter-varying, LPV)系统的H_∞异步控制问题进行了研究。由于实际非齐次马尔可夫系统的系统模态难以直接获得,考虑了控制器模态和系统模态不一致的异步问题,并用一个非齐次隐马尔可夫模型建模。非齐次马尔可夫系统和非齐次隐马尔可夫链的时变特性分别由分段齐次状态转移概率矩阵和分段齐次条件概率矩阵描述,且受同一个高层齐次马尔可夫链支配。针对上述问题,完成了对闭环非齐次马尔可夫LPV系统的稳定性分析、H_∞性能分析和H_∞异步控制器设计,使系统在受到外界干扰和控制器模态与系统模态存在异步现象的情况下满足给定的H_∞性能指标。最后通过对直流电机的仿真实验验证了利用非齐次信息得到理论的有效性和优越性。

关键词: 非齐次马尔可夫跳跃系统, 非齐次隐马尔可夫链, 时变转移概率, 分段齐次转移概率, H_∞异步控制

Abstract:

This paper investigates the problem of asynchronous H_∞ control of a class of discrete-time non-homogeneous Markov jump linear parameter-varying (LPV) systems with external disturbances. Considering that actual system modes of non-homogeneous Markov system are difficult to acquire directly, a non-homogeneous hidden Markov model is adopted to describe the asynchronous phenomenon between controller modes and system modes. The time-varying properties of non-homogeneous Markov jump systems and non-homogeneous hidden Markov chains are described by piecewise homogeneous transition probability matrices and piecewise homogeneous conditional probability matrices, respectively, and they are governed by the same higher-level homogeneous Markov chain. Based on the above-mentioned issues, this paper accomplishes the stability analysis, H_∞ performance analysis and asynchronous H_∞ controller design of the closed-loop system with external disturbance input and asynchronous phenomenon to guarantee the system with the achieved H_∞ performance index. Finally, a practical example of a direct current motor is provided to illustrate the validity and superiority of the developed theoretical results allowing for non-homogeneous information.

Key words: non-homogeneous Markov jump system (MJS), non-homogeneous hidden Markov chain, time-varying transition probability, piecewise homogeneous transition probability, asynchronous H_∞ control

中图分类号:

TP273

孙维阳, 刘雨. 非齐次马尔可夫LPV系统的H_∞异步控制[J]. 系统工程与电子技术, 2020, 42(5): 1152-1159.

Weiyang SUN, Yu LIU. Asynchronous H_∞ control for a class of non-homogeneous Markov jump LPV systems[J]. Systems Engineering and Electronics, 2020, 42(5): 1152-1159.

图/表 5

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图1

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系统模态	a_{i, 11}	a_{i, 12}	a_{i, 21}	a_{i, 22}	a_{i, 31}	a_{i, 33}	b_{i, 1}	b_{i, 2}
i=1	-0.479 9	5.154 6	-3.816 3	14.473 2	0.139 9	-0.925 6	5.870 6	15.501 1
i=2	-1.602 6	9.163 2	-0.591 9	3.031 7	0.074 1	0.074 1	10.285 1	2.228 3
i=3	0.634 6	0.917 8	-0.505 7	2.481 2	0.386 6	0.098 2	0.787 5	1.530 3

状态转移概率矩阵	Θ^Ⅰ	Θ^Ⅱ	Θ^Ⅲ	Θ^Ⅳ	Θ^Ⅴ
Θ	$\left[\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right]$	$\left[\begin{array}{cc}0.3 & 0.7 \\ 0.8 & 0.2\end{array}\right]$	$\left[\begin{array}{cc}0.5 & 0.5 \\ 0.5 & 0.5\end{array}\right]$	$\left[\begin{array}{ll}0.7 & 0.3 \\ 0.2 & 0.8\end{array}\right]$	$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

条件转移概率矩阵	{Λ₁^Ⅰ, Λ₂^Ⅰ}	{Λ₁^Ⅱ, Λ₂^Ⅱ}	{Λ₁^Ⅲ, Λ₂^Ⅲ}	{Λ₁^Ⅳ, Λ₂^Ⅳ}	{Λ₁^Ⅴ, Λ₂^Ⅴ}
Λ₁	$\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$	$\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0.1 & 0.05 & 0.85\end{array}\right]$	$\left[\begin{array}{ccc}0.85 & 0.1 & 0.05 \\ 0 & 1 & 0 \\ 0.1 & 0.05 & 0.85\end{array}\right]$	$\left[\begin{array}{ccc}0.85 & 0.1 & 0.05 \\ 0.02 & 0.97 & 0.01 \\ 0.1 & 0.05 & 0.85\end{array}\right]$	$\left[\begin{array}{ccc}0.25 & 0.23 & 0.52 \\ 0.21 & 0.28 & 0.51 \\ 0.32 & 0.41 & 0.27\end{array}\right]$
Λ₂	$\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$	$\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0.1 & 0.1 & 0.8\end{array}\right]$	$\left[\begin{array}{ccc}0.9 & 0.05 & 0.05 \\ 0 & 1 & 0 \\ 0.1 & 0.1 & 0.8\end{array}\right]$	$\left[\begin{array}{lll}0.9 & 0.05 & 0.05 \\ 0.07 & 0.9 & 0.03 \\ 0.1 & 0.1 & 0.8\end{array}\right]$	$\left[\begin{array}{ccc}0.42 & 0.23 & 0.35 \\ 0.28 & 0.33 & 0.39 \\ 0.39 & 0.4 & 0.21\end{array}\right]$

γ_opt	{Λ₁^Ⅰ, Λ₂^Ⅰ}	{Λ₁^Ⅱ, Λ₂^Ⅱ}	{Λ₁^Ⅲ, Λ₂^Ⅲ}	{Λ₁^Ⅳ, Λ₂^Ⅳ}	{Λ₁^Ⅴ, Λ₂^Ⅴ}
Θ^Ⅰ	0.173	0.177	0.263	0.368	1.214
Θ^Ⅱ	0.186	0.192	0.288	0.403	1.359
Θ^Ⅲ	0.195	0.203	0.308	0.422	1.363
Θ^Ⅳ	0.253	0.265	0.388	0.505	1.428
Θ^Ⅴ	1.438	1.499	2.406	2.942	无解

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