控制系统的性能优化分析与设计限制研究
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摘要
反馈控制系统的设计限制以及网络控制系统的性能优化问题是当今科学研究的前沿和热点问题。研究系统性能极限与控制系统的内在特性，比如，非最小相位零点、不稳定极点及其参考信号等之间的关系对控制系统的设计有非常重要的参考作用。参考信号的跟踪误差的性能极限性问题已经有了比较好的结果；但是，在稳态跟踪误差为零的时候，控制信号能量的极值与控制系统本质特征及参考信号的参数关系如何?网络化控制系统结构是新一代控制系统形式的大势所趋。网络控制系统由于网络的介入，不可避免出现时滞、带宽受限、丢包、量化引起的误差、干扰信号的影响等等。在网络环境下，如何考虑网络通信引起的性能极限性问题?本文在总结前人工作的基础上，系统深入地研究了稳态跟踪误差为零时，多变量控制系统最小控制能量，并且初步考虑网络环境下的设计限制和性能优化问题。
以连续控制系统为研究对象，参考输入为单位阶跃信号，揭示了单位反馈控制系统的最小跟踪能量与被控对象的不稳定极点、控制对象稳定部分的幅值以及参考信号的输入方向向量直接的精确关系表达式。同时，证明了二自由度控制器在消除控制对象不稳定极点对系统性能指标影响的有效性。
研究多变量离散系统的最小跟踪能量的极限值。参考信号为单位阶跃序列，将进入被控对象的输入信号的范数作为性能指标，来衡量跟踪能量的大小。得到了跟踪能量的极值与被控对象单位圆外零极点以及被控对象的幅值之间的精确的关系式。同时，该最优性能指标还与参考输入信号的方向向量有直接关系。给出了二自由度控制器下的最优性能指标，并将两种情况下得到的结果进行了比较，体现了二自由度控制器在控制系统的设计与分析中的优越性。
基于脉冲切换系统模型，研究由系统的输入输出给出的性能指标的最小化问题。通过求解一系列的Riccati方程，可以求得最优控制器。主要目标是求取由系统输入和输出定义的系统的传递函数的范式的最小值。通过推导，给出了最优控制和廉价控制的性能指标式的下确界，并且得到了脉冲切换系统无源性的充分性条件，以及保证整个系统无源性的最优控制器输入。结论显示，脉冲矩阵是性能指标中的重要参量，该性能指标对于控制器的设计有参考意义。
讨论了网络环境下反馈控制系统的跟踪和干扰抑制问题。参考信号是单位阶跃信号，采用二自由度控制器。基于Youla参数化控制器，考虑从传感器到控制器受到脉冲信号的干扰，得到了能量受限下跟踪问题的性能指标的最优值。接着考虑干扰信号为高斯白噪声信号，求解了网络控制系统跟踪误差的极限值。结果显示，网络环境下，反馈系统的性能极限值与系统的非最小相位零点、被控对象传递函数在频域下的增益等控制系统本身的一些特征密切相关。
研究了部分分散网络系统的最小能量调整问题，综合采用传递函数的互质分解，控制器的Youla参数化，以及Bezout等式等工具，建立了系统最小调整能量与被控对象的幅值及其不稳定极点以及该不稳定极点的方向向量之间的定量关系。
研究了含有交互耦合时变时滞主从节点复杂网络的同步问题，在这样的网络中，主节点有自己独立的动态行为，所有从节点跟随他的动态行为而调整自己的运动轨迹。基于Lyapunov稳定性理论，给出了这种主从节点网络的变结构控制策略。数值实例证明了所得的结论的正确性和所设计的控制器的有效性。
最后对全文进行了归纳总结，并对网络通信环境下，网络控制系统进一步的研究和发展方向进行了展望。
The design constraints of feedback control systems and optimizations of networkedcontrol systems are focal and hot topics of great interest recently in science and engineering.The relationship between the performance limitations and system intrinsic characteristic,such as, nonminimum phase zeros, unstable poles and reference signal, is vitalreference to the controller design. The perfect performance limitations for tracking problemshave been uncovered. But when the steady tracking error is zero, how about therelationship between the the minimum control energy and the characteristics of controlsystems? The networked control system structure will prevail in the future. With theimpact of network circumstance, many issues emerged inevitably, such as network induceddelay, limited bandwidth, data packet dropouts, error caused by the quantization,disturbance signals etc. Under the networked environment condition, how to considerthe performance limitations caused by communications? This thesis, based on previousworks of the others, systematically and deeply investigates the minimum control energyfor the systems with zero steady tracking error. And furthermore, the initial results aboutthe performance limitations for networked control systems are got.
The continuous feedback control system is considered. The problem under considerationis the minimum input energy for tracking control. The reference signal is unity stepsignal. Using a time-domain integral of the square of the input as a performance index,we derived the relations between the limitations and plant characteristics that depend onthe unstable poles, nonminimum phase zeros, the stable part of the plant and the referencesignal direction vector. Two degree of freedom controller is effective to eliminatethe influence of unstable poles for the performance index.
Minimum tracking energy for multi-variable discrete feedback systems were studied.The fundamental limitations imposed by the plant structure are given. The referencesignal is unity step sequences. Using the input usage as a measure of tracking energy.We derived relations that represent fundamental limitations on the achievable infimumof the tracking energy for general MIMO discrete systems. The relations depend on the unstable poles and the stable part of the plant and the reference signal direction vector.Two-degree-of-freedom controller case is considered, too.
A class of impulsive switched control systems are considered. Where the objectiveis to minimize a cost functional defined on the output and input. The optimal controllerscan be obtained through the solutions of a series of Riccati matrix equations. The performancefor optimal control and cheap optimal control are respectively given. Sufficientconditions for passivity of impulsive switched control systems are obtained, and the optimalinput controllers of system are addressed, too. The conclusions showed the impulsesare critical elements to the performance indices. Which are useful references to the controllersdesign.
The optimal tracking problems for networked feedback systems were investigated.The reference signal is step signal. Two degree of freedom controller is adopted. Based onthe Youla parametrization of controller, suppose there is an impulsive disturbance fromthe sensor to the controller, the best performance for tracking limitation under energyconstraints is given. Continuously, we considered the tracking limitation in the case thatthe disturbance is Gaussian white noise. The performance limitation is tightly dependenton the nonminimum phase zeros, and the plant gain in the frequency range etc.
The minimum energy regulation problem for partly networked control systemswas investigated. We adopted the coprime factorization of transfer function, Youlaparametrization of controller and Bezout identity. The explicit expression for the minimumregulation energy is related to the magnitude of plant, unstable poles and the directionsof unstable poles.
Synchronization in leader-follower networks with varying time-delay interconnectionswas investigated. In such a network, the leader had its own dynamics and wasfollowed by all the other nodes. Specifically, a variable structure control strategy was developedfor the synchronization of the leader-follower networks based on the Lyapunovstability theory. An illustrative example was given to demonstrate the effectiveness of theproposed method.
Finally, a summary has been done for all discussions in the dissertation. Theresearch works further study are presented for the networked control systems.
以连续控制系统为研究对象，参考输入为单位阶跃信号，揭示了单位反馈控制系统的最小跟踪能量与被控对象的不稳定极点、控制对象稳定部分的幅值以及参考信号的输入方向向量直接的精确关系表达式。同时，证明了二自由度控制器在消除控制对象不稳定极点对系统性能指标影响的有效性。
研究多变量离散系统的最小跟踪能量的极限值。参考信号为单位阶跃序列，将进入被控对象的输入信号的范数作为性能指标，来衡量跟踪能量的大小。得到了跟踪能量的极值与被控对象单位圆外零极点以及被控对象的幅值之间的精确的关系式。同时，该最优性能指标还与参考输入信号的方向向量有直接关系。给出了二自由度控制器下的最优性能指标，并将两种情况下得到的结果进行了比较，体现了二自由度控制器在控制系统的设计与分析中的优越性。
基于脉冲切换系统模型，研究由系统的输入输出给出的性能指标的最小化问题。通过求解一系列的Riccati方程，可以求得最优控制器。主要目标是求取由系统输入和输出定义的系统的传递函数的范式的最小值。通过推导，给出了最优控制和廉价控制的性能指标式的下确界，并且得到了脉冲切换系统无源性的充分性条件，以及保证整个系统无源性的最优控制器输入。结论显示，脉冲矩阵是性能指标中的重要参量，该性能指标对于控制器的设计有参考意义。
讨论了网络环境下反馈控制系统的跟踪和干扰抑制问题。参考信号是单位阶跃信号，采用二自由度控制器。基于Youla参数化控制器，考虑从传感器到控制器受到脉冲信号的干扰，得到了能量受限下跟踪问题的性能指标的最优值。接着考虑干扰信号为高斯白噪声信号，求解了网络控制系统跟踪误差的极限值。结果显示，网络环境下，反馈系统的性能极限值与系统的非最小相位零点、被控对象传递函数在频域下的增益等控制系统本身的一些特征密切相关。
研究了部分分散网络系统的最小能量调整问题，综合采用传递函数的互质分解，控制器的Youla参数化，以及Bezout等式等工具，建立了系统最小调整能量与被控对象的幅值及其不稳定极点以及该不稳定极点的方向向量之间的定量关系。
研究了含有交互耦合时变时滞主从节点复杂网络的同步问题，在这样的网络中，主节点有自己独立的动态行为，所有从节点跟随他的动态行为而调整自己的运动轨迹。基于Lyapunov稳定性理论，给出了这种主从节点网络的变结构控制策略。数值实例证明了所得的结论的正确性和所设计的控制器的有效性。
最后对全文进行了归纳总结，并对网络通信环境下，网络控制系统进一步的研究和发展方向进行了展望。
The design constraints of feedback control systems and optimizations of networkedcontrol systems are focal and hot topics of great interest recently in science and engineering.The relationship between the performance limitations and system intrinsic characteristic,such as, nonminimum phase zeros, unstable poles and reference signal, is vitalreference to the controller design. The perfect performance limitations for tracking problemshave been uncovered. But when the steady tracking error is zero, how about therelationship between the the minimum control energy and the characteristics of controlsystems? The networked control system structure will prevail in the future. With theimpact of network circumstance, many issues emerged inevitably, such as network induceddelay, limited bandwidth, data packet dropouts, error caused by the quantization,disturbance signals etc. Under the networked environment condition, how to considerthe performance limitations caused by communications? This thesis, based on previousworks of the others, systematically and deeply investigates the minimum control energyfor the systems with zero steady tracking error. And furthermore, the initial results aboutthe performance limitations for networked control systems are got.
The continuous feedback control system is considered. The problem under considerationis the minimum input energy for tracking control. The reference signal is unity stepsignal. Using a time-domain integral of the square of the input as a performance index,we derived the relations between the limitations and plant characteristics that depend onthe unstable poles, nonminimum phase zeros, the stable part of the plant and the referencesignal direction vector. Two degree of freedom controller is effective to eliminatethe influence of unstable poles for the performance index.
Minimum tracking energy for multi-variable discrete feedback systems were studied.The fundamental limitations imposed by the plant structure are given. The referencesignal is unity step sequences. Using the input usage as a measure of tracking energy.We derived relations that represent fundamental limitations on the achievable infimumof the tracking energy for general MIMO discrete systems. The relations depend on the unstable poles and the stable part of the plant and the reference signal direction vector.Two-degree-of-freedom controller case is considered, too.
A class of impulsive switched control systems are considered. Where the objectiveis to minimize a cost functional defined on the output and input. The optimal controllerscan be obtained through the solutions of a series of Riccati matrix equations. The performancefor optimal control and cheap optimal control are respectively given. Sufficientconditions for passivity of impulsive switched control systems are obtained, and the optimalinput controllers of system are addressed, too. The conclusions showed the impulsesare critical elements to the performance indices. Which are useful references to the controllersdesign.
The optimal tracking problems for networked feedback systems were investigated.The reference signal is step signal. Two degree of freedom controller is adopted. Based onthe Youla parametrization of controller, suppose there is an impulsive disturbance fromthe sensor to the controller, the best performance for tracking limitation under energyconstraints is given. Continuously, we considered the tracking limitation in the case thatthe disturbance is Gaussian white noise. The performance limitation is tightly dependenton the nonminimum phase zeros, and the plant gain in the frequency range etc.
The minimum energy regulation problem for partly networked control systemswas investigated. We adopted the coprime factorization of transfer function, Youlaparametrization of controller and Bezout identity. The explicit expression for the minimumregulation energy is related to the magnitude of plant, unstable poles and the directionsof unstable poles.
Synchronization in leader-follower networks with varying time-delay interconnectionswas investigated. In such a network, the leader had its own dynamics and wasfollowed by all the other nodes. Specifically, a variable structure control strategy was developedfor the synchronization of the leader-follower networks based on the Lyapunovstability theory. An illustrative example was given to demonstrate the effectiveness of theproposed method.
Finally, a summary has been done for all discussions in the dissertation. Theresearch works further study are presented for the networked control systems.
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