﻿ 一类采用分数阶PI<sup>λ</sup>控制器的分数阶系统可镇定性判定准则
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 自动化学报  2017, Vol. 43 Issue (11): 1993-2002 PDF

1. 辽宁大学轻型产业学院 沈阳 110036

Stabilization Criterion for A Class of Interval Fractional-order Systems Using Fractional-order PIλ Controllers
GAO Zhe1
1. College of Light Industry, Liaoning University Shenyang 110036
Manuscript received : December 29, 2015, accepted: June 30, 2016.
Foundation Item: Supported by National Natural Science Foundation of China (61304094), Scientific Research Fund of Liaoning Provincial Education Department (L2015194, L2015198)
Recommended by Associate Editor DONG Hai-Rong
Abstract: This study proposes a stabilization criterion for interval fractional-order plants involving one fractional-order term using fractional-order PIλ controllers. The characteristic function of the closed loop system is divided into the nominal function and disturbance function, and the construction method for the vertices of the value set with respect to the disturbance function is investigated. Moreover, the upper and lower limits of the test frequency interval are offered to determine the position relationship between the origin and the value set corresponding to the closed loop system. By supposing that the vertex functions are not equal to zero within the test frequency interval and the closed loop nominal system is stable, the stabilization criterion for closed loop systems using fractional-order PIλ controllers is proposed analytically. Finally, stabilization analyses of numerical examples verify the effectiveness of the proposed criterion.
Key words: Fractional-order systems     interval uncertainties     stabilization     value set     fractional-order PIλ controllers

 $$$\label{a4} \overline{F}({\rm j}\omega)=\omega^\lambda{\rm e}^{\frac{\lambda\pi}{2}{\rm j}}(\overline{T}\omega^\alpha{\rm e}^{\frac{\alpha\pi}{2}{\rm j}}+\overline{C})+(\mathit{k}_\rm{p}\omega^\lambda{\rm e}^{\frac{\lambda\pi}{2}{\rm j}}+\mathit{k}_\rm{i})\overline{\mathit{K}}$$$ (4)
 $$$\label{a5}\begin{split} F_\Delta({\rm j}\omega)= &\omega^\lambda{\rm e}^{\frac{\lambda\pi}{2}{\rm j}}(w_T\delta_T\omega^\alpha{\rm e}^{\frac{\alpha\pi}{2}{\rm j}}+w_C\delta_C)+\\&(\mathit{k}_\rm{p}\omega^\lambda{\rm e}^{\frac{\lambda\pi}{2}{\rm j}}+\mathit{k}_\rm{i})\mathit{w_K\delta_K}\end{split}$$$ (5)

 \begin{align*}&|D_1-D_2|=|D_3-D_4|=2\omega^\lambda w_C\\ &|D_2-D_3|=|D_4-D_1|=2\omega^{\alpha+\lambda}w_T\\ &\arg(D_1-D_2)=\frac{\lambda\pi}{2}\\ &\arg(D_3-D_4)=\frac{\lambda\pi}{2}-\pi\\ &\arg(D_2-D_3)=\frac{(\lambda+\alpha')\pi}{2}+2z\pi\\ &\arg(D_4-D_1)=\frac{(\lambda+\alpha')\pi}{2}-\pi\end{align*}

 $$$\label{a6} \cot\bigg(\frac{(\lambda+\alpha')\pi}{2}-\pi\bigg)=\frac{\mathit{k}_\rm{i}+\mathit{k}_\rm{p}\omega_0^\lambda\cos(\frac{\lambda\pi}{2})}{\mathit{k}_\rm{p}\omega_0^\lambda\sin(\frac{\lambda\pi}{2})}$$$ (6)

 $$$\label{a7} \omega_0=\bigg[\frac{\mathit{k}_\rm{i}}{\mathit{k}_\rm{p}(\sin(\frac{\lambda\pi}{2})\cot(\frac{(\lambda+\alpha')\pi}{2})-\cos(\frac{\lambda\pi}{2}))}\bigg]^\frac{1}{\lambda}$$$ (7)

$\omega\in\Omega_1=(0, \omega_0)$时, 扰动函数的值集为平行六边形$S_1^\Delta$, 对应的顶点按照逆时针的顺序, 分别对应函数$G^1_1=N_1+D_2$, $G^1_2=N_1+D_3$, $G^1_3=N_1+D_4$, $G^1_4=N_2+D_4$, $G^1_5=N_2+D_1$$G^1_6=N_2+D_2.当\omega\in\Omega_2=(\omega_0, +\infty)时, 扰动函数的值集仍然为平行六边形S^\Delta_2, 对应的顶点按照逆时针的顺序, 分别对应函数G^2_1=N_1+D_3, G^2_2=N_1+D_4, G^2_3=N_1+D_1, G^2_4=N_2+D_1, G^2_5=N_2+D_2$$G^2_6=N_3+D_3$.虽然两个系统的值集都是平行六边形, 但是顶点函数的表达式不一样.当$\omega=\omega_0$时, 值集变为平行四边形$S^\Delta_3$, 对应顶点函数为$G^3_1=N_1+D_3$, $G^3_2=N_1+D_4$, $G^3_3=N_2+D_1$$G^3_4=N_2+D_2. 如果\omega_0不存在, 那么定义\Omega_1=\emptyset, \Omega_2=(0, +\infty), \Omega_3=\emptyset.当\omega\in\Omega_2时, 有\arg(N_1-N_2)\in(0, (\lambda\pi)/2)成立.也就是, 扰动函数的值集一直为平行六边形S^\Delta_2, 且顶点函数分别对应函数G^2_1=N_1+D_3, G^2_2=N_1+D_4, G^2_3=N_1+D_1, G^2_4=N_2+D_1, G^2_5=N_2+D_2$$G^2_6=N_3+D_3$.

2.2 测试频率区间

 $$$\label{a8} R_{\max}=\max\{1, \eta_1^{\frac{1}{\alpha}}\}$$$ (8)
 $$$\label{a9} R_{\min}=\min\{1, \eta_2^{\frac{1}{\lambda}}\}$$$ (9)
 $\begin{split}&\eta_1=\\ &\quad\frac{\max\{|C^-|, |C^+|\}+(\mathit{k}_\rm{p}+ \mathit{k}_\rm{i})\max\{|K^-|, |K^+|\}}{\min\{|T^-|, |T^+|\}} \end{split}\\ {\small\begin{split}&\eta_2=\\ &\quad\frac{\min\{|K^-|, |K^+|\}\mathit{k}_\rm{i}} {\max\{|T^-|, |T^+|\}\!+\!\max\{|C^-|, |C^+|\}\!+\!\max\{|K^-|, |K^+|\}\mathit{k}_\rm{p}}\end{split}}$

$\omega>1$时, $\omega^{\lambda+\alpha}>\omega^\lambda>1$成立, 那么有:

 $$$\label{a10} \begin{split} |F({\rm j}\omega)|\geq&|T|\omega^{\lambda+\alpha}-[(|C|+|\mathit{K}|\mathit{k}_\rm{p})\omega^\lambda+|\mathit{K}|\mathit{k}_\rm{i}]\geq\\ &\omega^{\lambda}[|T|\omega^\alpha-(|C|+|\mathit{K}|\mathit{k}_\rm{p}+|\mathit{K}|\mathit{k}_\rm{i})] \end{split}$$$ (10)

$0 < \omega < 1$时, $1>\omega^\lambda>\omega^{\lambda+\alpha}$成立, 那么有:

 $$$\label{a11} \begin{split} |F({\rm j}\omega)|\geq&k_{\rm i}|K|-[|T|\omega^{\lambda+\alpha}+(|C|+|\mathit{K}|\mathit{k}_\rm{p})\omega^\lambda]\geq\\ &k_{\rm i}|K|-\omega^\lambda(|T|+|C|+|\mathit{K}|\mathit{k}_\rm{p}) \end{split}$$$ (11)

2.3 闭环系统可镇定性解析判定

 $$$\label{a12} F_i^j=\sum\limits_{k=0}^{P+Q}f_{i, k}^j\omega^{k\gamma}{\rm e}^{\frac{k\gamma\pi{\rm j}}{2}}$$$ (12)

$k=0$, k=Pk=P+Q时, 系数f_{i, k}^j可以根据第2.1节的值集端点的计算方法获得, 其他情况下f_{i, k}^j=0. 根据欧拉公式, 顶点函数F_i^j可以表示为  \begin{align}\label{a13} F_i^j= &\bigg[\sum\limits_{k=0}^{P+Q}f_{i, k}^j\cos\bigg({\frac{k\gamma\pi}{2}}\bigg)\omega^{k\gamma}\bigg]+\nonumber\\ &\bigg[\sum\limits_{k=0}^{P+Q}f_{i, k}^j\sin\bigg({\frac{k\gamma\pi}{2}}\bigg)\omega^{k\gamma}\bigg]{\rm j} \end{align} (13) 定义函数  h_{i, k}^j=f_{i, k}^j\cos\bigg({\frac{k\gamma\pi}{2}}\bigg)\\g_{i, k}^j=f_{i, k}^j\sin\bigg({\frac{k\gamma\pi}{2}}\bigg) 那么顶点函数F_i^j的实部R_i^j和虚部I_i^j可以表示为  $$\label{a14} R_i^j=\sum\limits_{k=0}^{P+Q}h_{i, k}^j\omega^{k\gamma}$$ (14)  $$\label{a15} I_i^j=\sum\limits_{k=0}^{P+Q}g_{i, k}^j\omega^{k\gamma}$$ (15) 定理 2. 定义F_{L_j+1}^j=F^j_1, 假设矩阵M^j_i可逆, 对于\omega\in\Omega_j', j=1, 2, 3, i=1, 2, \cdots, L_j, L_1=L_2=6, L_3=4, 有F^j_i\neq0F^j_{i+1}\neq0成立.如果$(M^j_i)^{-1}M^j_{i+1}$在负实轴有$T_i^j$个特征值$W_{i, t}^j$, $t=1, 2, \cdots, T_i^j$, 且$T_i^j$个负特征值对应的特征向量$v_{i, t}^j=[v_{i, t, 1}^j, v_{i, t, 2}^j, \cdots, v_{i, t, 2(P+Q)}^j]^{\rm T}$满足$v_{i, t, k}^j, /v_{i, t, k-1}^j=(\omega_{i, t}^j)^\gamma$, $k=2, 3, \cdots, 2(P+Q)$, 并且$\omega_{i, t}^j\in\Omega_j'$, 那么连接函数$F^j_i$$F^j_{i+1}对应顶点的棱边(不包含函数F^j_i$$F^j_{i+1}$对应的顶点), 在$\omega^j_{i, t}$处所对应的函数$E_{i, i+1}^j$满足$E_{i, i+1}^j=0$, 其中矩阵$M_i^j$定义如下:

 $$$\label{a16} \begin{split} &M^j_i=\left[ \begin{array}{cccc} h_{i, 0}^j&h_{i, 1}^j&\cdots&h_{i, P+Q-1}^j\\ g_{i, 0}^j&g_{i, 1}^j&\cdots&g_{i, P+Q-1}^j\\ 0&h_{i, 0}^j&h_{i, 1}^j&\cdots \\ 0&g_{i, 0}^j&g_{i, 1}^j&\cdots\\ \vdots&\ddots&\ddots&\ddots\\ 0&\ldots&0&h_{i, 0}^j\\ 0&\ldots&0&g_{i, 0}^j\\ \end{array} \right.\longrightarrow\\ &\longleftarrow\left. \begin{array}{cccc} h_{i, P+Q}^j&0&\cdots&0\\ g_{i, P+Q}^j&0&\cdots&0\\ h_{i, P+Q-1}^j&h_{i, P+Q}^j&\cdots&0\\ g_{i, P+Q-1}^j&g_{i, P+Q}^j&\cdots&0\\ \ddots&\ddots&\ddots&\vdots\\ h_{i, 1}^j&\cdots&h_{i, P+Q-1}^j&h_{i, P+Q}^j\\ g_{i, 1}^j&\cdots&g_{i, P+Q-1}^j&g_{i, P+Q}^j\\ \end{array} \right]\\ \end{split}$$$ (16)

 $$$\label{a18} \frac{R_{i+1}^j+I_{i+1}^j{\rm j}}{R_{i}^j+I_{i}^j{\rm j}}=W_{i, t}^j$$$ (18)

 $$$\label{a19} R_{i+1}^j=W_{i, t}^jR_{i}^j$$$ (19)
 $$$\label{a20} I_{i+1}^j=W_{i, t}^jI_{i}^j$$$ (20)

 ${\small \begin{split}v_{i, t}^j= &[v_{i, t, 1}^j, v_{i, t, 2}^j, \cdots, v_{i, t, 2 (P+Q)}^j]^{\rm T}=\\ &[1, (\omega_{i, t}^j)^\gamma, \cdots, (\omega_{i, t}^j)^{(P+Q)\gamma}, \cdots, (\omega_{i, t}^j)^{[2(P+Q)-1]\gamma}]^{\rm T}\\ \end{split}}$

 $$$\label{a21}\begin{split} &[h_{i, 0}, h_{i, 1}, \cdots, h_{i, P+Q}, \underbrace {0, 0, \cdots, 0}_ {P+Q-1}]v_{i, t}^jW_{i, t}^j=\\&\qquad[h_{i+1, 0}, h_{i+1, 1}, \cdots, h_{i+1, P+Q}, \underbrace {0, 0, \cdots, 0}_{P+Q-1}]v_{i, t}^j\\ \end{split}$$$ (21)
 $$$\label{a22}\begin{split} &[g_{i, 0}, g_{i, 1}, \cdots, g_{i, P+Q}, \underbrace {0, 0, \cdots, 0}_{P+Q-1}]v_{i, t}^jW_{i, t}^j=\\&\qquad[g_{i+1, 0}, g_{i+1, 1}, \cdots, g_{i+1, P+Q}, \underbrace {0, 0, \cdots, 0}_{P+Q-1}]v_{i, t}^j\\ \end{split}$$$ (22)

 $$$\label{a23}\begin{split} &[0, h_{i, 0}, h_{i, 1}, \cdots, h_{i, P+Q}, \underbrace {0, 0, \cdots, 0}_{P+Q-2}]v_{i, t}^jW_{i, t}^j=\\&\qquad[0, h_{i+1, 0}, h_{i+1, 1}, \cdots, h_{i+1, P+Q}, \underbrace {0, 0, \cdots, 0}_{P+Q-2}]v_{i, t}^j\\ \end{split}$$$ (23)
 $$$\label{a24}\begin{split} &[0, g_{i, 0}, g_{i, 1}, \cdots, g_{i, P+Q}, \underbrace {0, 0, \cdots, 0}_{P+Q-2}]v_{i, t}^jW_{i, t}^j=\\&\qquad[0, g_{i+1, 0}, g_{i+1, 1}, \cdots, g_{i+1, P+Q}, \underbrace {0, 0, \cdots, 0}_{P+Q-2}]v_{i, t}^j\\ \end{split}$$$ (24)

 $$$\label{a25} M_i^jv_{i, t}^jW^j_i=M_{i+1}^jv_{i, t}^j$$$ (25)

1) $|\overline{K}/w_K|>1.$

2) $|\overline{T}/w_T|>1.$

3) 对应所有的$i=1, 2, \cdots, L_j$, $j=1, 2, 3$, $L_1=L_2=6$, $L_3=4$, 都有$T_i^j=0$成立.

1) 当$\omega=0$时, $F({\rm j}\omega)=\mathit{k}_\rm{i}K=\mathit{k}_\rm{i}(\overline{K}+\mathit{w_K\delta_K})$.如果$F({\rm j}\omega)=0$, 也就是$\overline{K}+\mathit{w_K\delta_K}=0$, 那么$\delta_K=-\overline{K}/w_K$, 因此$|\delta_K|=|\overline{K}/w_K|\leq1$.如果要求当$\omega=0$时, $F({\rm j}\omega)\neq0$成立, 也就是要求条件1成立.

2) 如果$\omega\neq0$时, $F({\rm j}\omega)$的值集不包含原点, 也就是$F({\rm j}\omega)/({\rm j}\omega)^{\lambda+\alpha}=T+[(K\mathit{k}_\rm{p}+C)({\rm j}\omega)^\lambda+K\mathit{k}_\rm{i}]/({\rm j}\omega)^{\lambda+\alpha}$的值集也不包含原点.如果$\omega\rightarrow+\infty$时, 值集包含原点那么有$T=\overline{T}+w_T\delta_T=0$, 即$|\overline{T}/w_T|\leq1$成立.因此, 如果要求当$\omega\rightarrow+\infty$时, $F({\rm j}\omega)\neq0$, 那么要求条件2成立.

3) 对于$\omega\in(0, \infty)$时, 根据定理1, 只需检验$\omega\in \Omega'_j$时, $F({\rm j}\omega)$的值集与原点的位置关系.根据定理2可知, 如果对应所有的$i=1, 2, \cdots, L_j$, $j=1, 2, 3$, $L_1=L_2=6$, $L_3=4$, 都有$T_i^j=0$成立, 那么对于所有的棱边都有$E_{i, i+1}^j\neq0$成立.原点也不会从任何棱边(不包含顶点)进入值集内部.

3 数值算例

 $$$\label{a26} P(s)=\frac{K}{Ts^{1.8}+C}$$$ (26)

 $$$\label{a27} C(s)=\mathit{k}_\rm{p}+\mathit{k}_\rm{i}\mathit{s}^{-1.2}$$$ (27)

 图 1 当$\omega=0.4, 0.8, 1.2, \omega_0, 2, 3$时, 特征函数$F({\rm j}\omega)$的值集 Figure 1 Value sets of characteristic function $F({\rm j}\omega)$ for $\omega=0.4, 0.8, 1.2, \omega_0, 2, 3$

 图 3 $F(w)$的特征值分布 Figure 3 Distribustion of eigenvalues of $F(w)$

 图 6 扩大不确定区间后的顶点函数$|F^j_i|$, $i=1, 2, \cdots, L_j$, $j=1, 2, 3$, 在$\omega\in\Omega'_j$的变化曲线 Figure 6 Curves of vertex functions $|F^j_i|$, $i=1, 2, \cdots, L_j$, $j=1, 2, 3$, within $\omega\in\Omega'_j$ for enlarged interval case

 图 8 扩大不确定区间后的$F(w)$的特征值分布 Figure 8 Curve of $D(\omega)$ for enlarged interval case

 图 9 扩大不确定区间后的$D(\omega)$变化曲线 Figure 9 Curve of $D(\omega)$ for enlarged interval case
 图 10 扩大不确定区间后的$D(\omega)$变化曲线的局部放大图 Figure 10 Zoom on curve of $D(\omega)$ for enlarged interval case

4 结论

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