Stabilization Criterion for A Class of Interval Fractional-order Systems Using Fractional-order PIλ Controllers
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摘要: 针对含有一个分数阶项的区间分数阶被控对象，提出了采用分数阶PIλ控制器的闭环系统可镇定性判定准则.将闭环系统的特征函数分解为扰动函数和标称函数，给出了扰动函数值集顶点的构造方法.根据被控对象分数阶阶次和控制器的阶次，研究了值集形状是否切换和切换频率的计算方法.此外，给出了测试频率区间的上下界，以实现在有限频率区间内判定闭环系统值集与原点的位置关系.在假设值集顶点函数在测试频率区间内不为零和闭环标称系统稳定的情况下，以解析的方式提出了采用分数阶PIλ控制器闭环系统的可镇定性判定准则.最后，通过对数值算例的可镇定性分析，验证了提出的判定准则的有效性.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.
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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
Fig. 1 Value sets of characteristic function $F({\rm j}\omega)$ for $\omega=0.4, 0.8, 1.2, \omega_0, 2, 3$
图 2 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
Fig. 2 Curves of vertex functions $|F^j_i|$, $i=1, 2, \cdots, L_j$, $j=1, 2, 3$, within $\omega\in\Omega'_j$
图 5 当$\omega=1, 1.2, 1.4, \omega_0, 1.8, 2$时, 特征函数$F({\rm j}\omega)$的值集
Fig. 5 Value sets of characteristic function $F({\rm j}\omega)$ for $\omega=1, 1.2, 1.4, \omega_0, 1.8, 2$
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
Fig. 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 7 当$\omega=\omega_{4, 4}^1$时, 扩大不确定区间后的$F(\omega_{4, 4}^1{\rm j})$对应的值集
Fig. 7 Value set of $F(\omega_{4, 4}^1{\rm j})$ at $\omega=\omega_{4, 4}^1$ for enlarged interval case
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