一类切换线性系统的分析与控制
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摘要
本文研究了一类切换线性系统分析与控制中的若干问题，主要内容如下：
(1)利用正交变换证明了子系统状态阵为正规阵的切换线性系统可在任意切换作用下渐近稳定；针对子系统状态阵为上(下)三角阵的切换线性系统以及一类二阶切换线性系统，采用公共Lvapunov函数法证明了系统可在任意切换作用下稳定；基于滞留时间法得到了一般形式切换线性系统稳定的一个充分条件。
(2)研究了无外输入切换线性系统动态过程中的震颤现象。分析了二阶切换线性系统的收敛震颤现象与相轨迹的旋向以及相轨迹极半径增减趋势之间的关系，进而得到忽略切换延迟时二阶切换线性系统出现收敛震颤模态的充分条件。给出了切换角及一次震颤后极半径变化量的计算公式，在此基础上得到考虑切换延迟时二阶切换线性系统出现收敛震颤模态的充分条件。所得结果可用于镇定无输入切换系统。最后针对无外输入高阶切换系统，定义了震颤模态并给出收敛震颤模态的判断方法。
(3)研究了时延切换线性系统的鲁棒镇定。针对参数不确定离散时延切换线性系统，基于多重Lyapunov函数法采用状态反馈实现了任意切换作用下的鲁棒H_∞镇定，给出了迭代算法用于求解定理中的非线性矩阵不等式；采用动态输出反馈实现了离散时延切换线性系统在任意切换作用下的镇定。基于平均滞留时间法研究了循环切换序列作用下离散时延切换线性系统的状态反馈鲁棒H_∞镇定，给出了循环切换序列平均滞留时间的取值范围以及状态反馈控制律的设计方法。最后研究了状态依赖型切换规则作用下连续时间时变时延切换线性系统的鲁棒可靠控制，在最小滞留时间已知时，基于多重Lyapunov函数法设计了状态反馈控制器，使得执行机构在正常工作及出现局部故障两种情况下时延切换线性系统均可鲁棒渐近稳定。
(4)研究了单输入饱和切换系统的镇定及利用切换扩大收敛域。分别基于公共Lyapunov函数法及多重Lyapunov函数法，研究了单输入饱和切换系统的状态反馈镇定，提出了相应的切换策略以扩大收敛域。针对由两个二阶子系统组成的输入饱和切换线性系统，分析了相轨迹的运动特点并用于设计状态反馈控制律，结合所提的切换策略实现了系统的全局镇定。研究了一类概率密度函数为已知的参数不确定输入饱和切换系统在任意切换作用下的状态反馈鲁棒镇定，给出了随机梯度算法求解反馈控制律，证明了算法的收敛性。
This dissertation concerns with some issues related to the analysis and control of aclass of switched linear systems. The main contributions are summarized as follows:
(1) By using orthogonality transform, we prove that a class of switched linearsystems, of which the state matrices of subsystems are canonical matrices, are asymptoticstable under arbitrary switching rule. Based on common Lyapunov function method, aclass of second order switched linear systems and the switched linear systems of which thestate matrices of subsystems are upper (lower) triangular matrices are proved to beasymptotic stable under arbitrary switching rule. A sufficient condition of stability isobtained for general switched linear systems based on of dwell time.
(2) The chattering mode in the dynamic process of the unforced switched systems isinvestigated. The relations between the rotation directions of trajectory, the increase ordecrease the polar radius of trajectory, and the convergent chattering mode are revealed.Based on aforementioned results, sufficient conditions for chattering mode are proposed bydistinguishing between two cases: switching delay neglected and switching delayconsidered. Two formulas are shown for computing the switching angles as well as thepolar radius alteration after once chattering. The proposed results can be applied tostabilize unforced switched systems. Finally, we define the chattering mode for higherorder switched systems and presented a checking method.
(3) Robust stabilization is researched for switched systems with time delay. Firstly,by multiple Lyapunov functions method, a state feedback control strategy is proposed torobustly stabilize the discrete switched systems with time delay under arbitrary switchingrule, and an iterative algorithm is presented to solve the nonlinear matrix inequalities in theproposed theorems. The dynamic output feedback approach is established to stabilize thediscrete switched systems with time delay under arbitrary switching sequence. Then by theapproach of average dwell time, we design state feedback control law combining withcircular switching sequence to robust stabilize switched delay systems with H_∞performance, and indicate the permissive bound of circular switching sequence. Finally,robust reliable control is concerned for continuous switched systems with time varyingdelay under a state-dependent switching rule. On the premise that the minimal dwell timeof subsystems is known, state feedback control law is designed by applying multipleLyapunov functions technique to guarantee the switched delay systems to be asymptoticstable in both cases that the actuators of the switched systems work properly or partiallybreak down.
(4) Stabilization and enlargement of contract region is studied for switched linearsystems with input saturation. Based on common Lyapunov function as well as multipleLyapunov functions respectively, we investigate state feedback stabilization for theswitched liner systems with single input saturation, and propose the switching method forenlarging the contract region. Aiming at the switched systems composed by two secondorder subsystems, some characteristics of phase trajectory is analyzed and then utilized todesign state feedback control to globally stabilize the switched systems by us appropriateswitching. Finally, we research the robust stabilization of a class of switched inputsaturation systems of which the density functions of the unknown parameters are given. Analgorithm called randomized gradient algorithm is applied to solve the robust statefeedback controller. The convergence of the randomized gradient algorithm is proved.
(1)利用正交变换证明了子系统状态阵为正规阵的切换线性系统可在任意切换作用下渐近稳定；针对子系统状态阵为上(下)三角阵的切换线性系统以及一类二阶切换线性系统，采用公共Lvapunov函数法证明了系统可在任意切换作用下稳定；基于滞留时间法得到了一般形式切换线性系统稳定的一个充分条件。
(2)研究了无外输入切换线性系统动态过程中的震颤现象。分析了二阶切换线性系统的收敛震颤现象与相轨迹的旋向以及相轨迹极半径增减趋势之间的关系，进而得到忽略切换延迟时二阶切换线性系统出现收敛震颤模态的充分条件。给出了切换角及一次震颤后极半径变化量的计算公式，在此基础上得到考虑切换延迟时二阶切换线性系统出现收敛震颤模态的充分条件。所得结果可用于镇定无输入切换系统。最后针对无外输入高阶切换系统，定义了震颤模态并给出收敛震颤模态的判断方法。
(3)研究了时延切换线性系统的鲁棒镇定。针对参数不确定离散时延切换线性系统，基于多重Lyapunov函数法采用状态反馈实现了任意切换作用下的鲁棒H_∞镇定，给出了迭代算法用于求解定理中的非线性矩阵不等式；采用动态输出反馈实现了离散时延切换线性系统在任意切换作用下的镇定。基于平均滞留时间法研究了循环切换序列作用下离散时延切换线性系统的状态反馈鲁棒H_∞镇定，给出了循环切换序列平均滞留时间的取值范围以及状态反馈控制律的设计方法。最后研究了状态依赖型切换规则作用下连续时间时变时延切换线性系统的鲁棒可靠控制，在最小滞留时间已知时，基于多重Lyapunov函数法设计了状态反馈控制器，使得执行机构在正常工作及出现局部故障两种情况下时延切换线性系统均可鲁棒渐近稳定。
(4)研究了单输入饱和切换系统的镇定及利用切换扩大收敛域。分别基于公共Lyapunov函数法及多重Lyapunov函数法，研究了单输入饱和切换系统的状态反馈镇定，提出了相应的切换策略以扩大收敛域。针对由两个二阶子系统组成的输入饱和切换线性系统，分析了相轨迹的运动特点并用于设计状态反馈控制律，结合所提的切换策略实现了系统的全局镇定。研究了一类概率密度函数为已知的参数不确定输入饱和切换系统在任意切换作用下的状态反馈鲁棒镇定，给出了随机梯度算法求解反馈控制律，证明了算法的收敛性。
This dissertation concerns with some issues related to the analysis and control of aclass of switched linear systems. The main contributions are summarized as follows:
(1) By using orthogonality transform, we prove that a class of switched linearsystems, of which the state matrices of subsystems are canonical matrices, are asymptoticstable under arbitrary switching rule. Based on common Lyapunov function method, aclass of second order switched linear systems and the switched linear systems of which thestate matrices of subsystems are upper (lower) triangular matrices are proved to beasymptotic stable under arbitrary switching rule. A sufficient condition of stability isobtained for general switched linear systems based on of dwell time.
(2) The chattering mode in the dynamic process of the unforced switched systems isinvestigated. The relations between the rotation directions of trajectory, the increase ordecrease the polar radius of trajectory, and the convergent chattering mode are revealed.Based on aforementioned results, sufficient conditions for chattering mode are proposed bydistinguishing between two cases: switching delay neglected and switching delayconsidered. Two formulas are shown for computing the switching angles as well as thepolar radius alteration after once chattering. The proposed results can be applied tostabilize unforced switched systems. Finally, we define the chattering mode for higherorder switched systems and presented a checking method.
(3) Robust stabilization is researched for switched systems with time delay. Firstly,by multiple Lyapunov functions method, a state feedback control strategy is proposed torobustly stabilize the discrete switched systems with time delay under arbitrary switchingrule, and an iterative algorithm is presented to solve the nonlinear matrix inequalities in theproposed theorems. The dynamic output feedback approach is established to stabilize thediscrete switched systems with time delay under arbitrary switching sequence. Then by theapproach of average dwell time, we design state feedback control law combining withcircular switching sequence to robust stabilize switched delay systems with H_∞performance, and indicate the permissive bound of circular switching sequence. Finally,robust reliable control is concerned for continuous switched systems with time varyingdelay under a state-dependent switching rule. On the premise that the minimal dwell timeof subsystems is known, state feedback control law is designed by applying multipleLyapunov functions technique to guarantee the switched delay systems to be asymptoticstable in both cases that the actuators of the switched systems work properly or partiallybreak down.
(4) Stabilization and enlargement of contract region is studied for switched linearsystems with input saturation. Based on common Lyapunov function as well as multipleLyapunov functions respectively, we investigate state feedback stabilization for theswitched liner systems with single input saturation, and propose the switching method forenlarging the contract region. Aiming at the switched systems composed by two secondorder subsystems, some characteristics of phase trajectory is analyzed and then utilized todesign state feedback control to globally stabilize the switched systems by us appropriateswitching. Finally, we research the robust stabilization of a class of switched inputsaturation systems of which the density functions of the unknown parameters are given. Analgorithm called randomized gradient algorithm is applied to solve the robust statefeedback controller. The convergence of the randomized gradient algorithm is proved.
引文
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