关于时滞系统的弹性保成本控制的研究
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摘要
近年来,如何设计鲁棒控制器使不确定系统满足鲁棒稳定性的同时满足一定的性能指标,已经引起了广泛的关注。解决这个问题的方法之一是Chang和Peng提出的保成本控制的方法。研究这一问题的目的是设计一个保成本控制器,使得闭环系统对于所有允许的不确定性渐近稳定,并且闭环性能指标不超过某个确定的上界。众所周知,参数不确定性和时间滞后经常是系统性能退化和系统不稳定的主要原因。因此,对于时滞系统的保成本控制的研究越来越引起人们的研究兴趣,并且在线性系统中已经取得了较多有价值的成果,但由于非线性系统的特殊性和复杂性,在非线性系统中的研究成果还不多见。
本文利用Lyapunov稳定性理论和线性矩阵不等式(LMI)的方法,研究具有范数有界的时滞系统的弹性保成本控制,主要内容包括以下几方面：
首先,利用Lyapunov函数和线性矩阵不等式方法(LMI),给出变时滞的不确定非线性系统在增益摄动为加性和乘性时滞相关时的弹性保成本控制律的存在条件,以及弹性保成本控制器的设计方法,并通过建立和求解凸优化的问题得出最优保成本。
其次,利用线性矩阵不等式(LMI)处理方法,导出了区间变时滞广义系统对所有容许的不确定性均正则、无脉冲且渐近稳定的时滞相关充分条件和区间变时滞广义系统的系统弹性保性能控制器存在的条件,证明了该条件等价于
一个线性矩阵不等式的可行性问题,并用该线性矩阵不等式的可行解给出了弹性保性能控制器的一个参数化表示。
然后,基于Lyapunov稳定性理论,通过构造广义Lyapunov函数和广义二次性能指标函数,以线性矩阵不等式(LMI)的形式给出了基于观测器状态反馈的弹性保成本控制器的设计方法。
最后,总结了论文的主要工作,并且对不确定广义系统保成本控制的研究进行了展望。
Recently, the problem of designing robust controllers to make uncertain systems not only be stable but also guarantees an adequate level of performance has drawn considerable attention. One approach to this problem is the guaranteed cost control approach first introduced by Chang and Peng.This approach is to design a controller such that the closed-loop system is asymptotically stable for all admissible uncertainties and the closed-loop cost function value is not more than a specified upper bound.It is well known that time-delays as well as parameter uncertainties are frequently the main cause of deterioration of systems performance and instability of systems.Therefore, there has been increasing interest in the guaranteed cost control of uncertain time-delay systems,and many significant results have been presented.However, there have been few results for nonlinear systems, owing to their partieularity and compexity.
Based on, the Lyapunov stability theory and linear matrix inequality approach, the problems of guaranteed control for nonlinear time-delay systems with norm-bounded uncertainties are considered.The main contents in this paper are as follows:
Firstly, a condition for the existence of resilient guaranteed cost controller is derived via linear matrix inequalities (LMI) technology and Lyapunov function. Designing methodology and conditions of existence in resilient guaranteed cost control is diseussed for uncertain nonlinear time-delay systems under two classes of controller gain perturbations, namely, additive form and multiplicative form. Solving a convex optimization problem gives the optimal guaranteed cost value of the system.
Secondly, a condition for the existence of resilient guaranteed cost controller is derived via linear matrix inequalities (LMI) technology. To be regular, impulse- free and robustly stable for all admissible uncertainties and conditions of existence in resilient guaranteed cost control for singular systems with time varying interval time-delay is given. Furthermore, it is shown that this condition is equivalent to the feasibility problem of LMIs, and its solutions provide a parameterized representation of resilient guaranteed cost controllers.
Thirdly, In terms of linear matrix inequalities (LMIs), a design method is obtained by constructing generalized Lyapunov function and generalized quadratic regulator.
The last section summarizes the main work in this paper and prospects the research of uncertain guaranteed cost control problems in the future.
本文利用Lyapunov稳定性理论和线性矩阵不等式(LMI)的方法,研究具有范数有界的时滞系统的弹性保成本控制,主要内容包括以下几方面：
首先,利用Lyapunov函数和线性矩阵不等式方法(LMI),给出变时滞的不确定非线性系统在增益摄动为加性和乘性时滞相关时的弹性保成本控制律的存在条件,以及弹性保成本控制器的设计方法,并通过建立和求解凸优化的问题得出最优保成本。
其次,利用线性矩阵不等式(LMI)处理方法,导出了区间变时滞广义系统对所有容许的不确定性均正则、无脉冲且渐近稳定的时滞相关充分条件和区间变时滞广义系统的系统弹性保性能控制器存在的条件,证明了该条件等价于
一个线性矩阵不等式的可行性问题,并用该线性矩阵不等式的可行解给出了弹性保性能控制器的一个参数化表示。
然后,基于Lyapunov稳定性理论,通过构造广义Lyapunov函数和广义二次性能指标函数,以线性矩阵不等式(LMI)的形式给出了基于观测器状态反馈的弹性保成本控制器的设计方法。
最后,总结了论文的主要工作,并且对不确定广义系统保成本控制的研究进行了展望。
Recently, the problem of designing robust controllers to make uncertain systems not only be stable but also guarantees an adequate level of performance has drawn considerable attention. One approach to this problem is the guaranteed cost control approach first introduced by Chang and Peng.This approach is to design a controller such that the closed-loop system is asymptotically stable for all admissible uncertainties and the closed-loop cost function value is not more than a specified upper bound.It is well known that time-delays as well as parameter uncertainties are frequently the main cause of deterioration of systems performance and instability of systems.Therefore, there has been increasing interest in the guaranteed cost control of uncertain time-delay systems,and many significant results have been presented.However, there have been few results for nonlinear systems, owing to their partieularity and compexity.
Based on, the Lyapunov stability theory and linear matrix inequality approach, the problems of guaranteed control for nonlinear time-delay systems with norm-bounded uncertainties are considered.The main contents in this paper are as follows:
Firstly, a condition for the existence of resilient guaranteed cost controller is derived via linear matrix inequalities (LMI) technology and Lyapunov function. Designing methodology and conditions of existence in resilient guaranteed cost control is diseussed for uncertain nonlinear time-delay systems under two classes of controller gain perturbations, namely, additive form and multiplicative form. Solving a convex optimization problem gives the optimal guaranteed cost value of the system.
Secondly, a condition for the existence of resilient guaranteed cost controller is derived via linear matrix inequalities (LMI) technology. To be regular, impulse- free and robustly stable for all admissible uncertainties and conditions of existence in resilient guaranteed cost control for singular systems with time varying interval time-delay is given. Furthermore, it is shown that this condition is equivalent to the feasibility problem of LMIs, and its solutions provide a parameterized representation of resilient guaranteed cost controllers.
Thirdly, In terms of linear matrix inequalities (LMIs), a design method is obtained by constructing generalized Lyapunov function and generalized quadratic regulator.
The last section summarizes the main work in this paper and prospects the research of uncertain guaranteed cost control problems in the future.
引文
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[12]Gu K. Discretization schemes for Lyapunov-Krasovskii functions in time-delay systems[J].Kybernetiea,2001,37:479-50
[13]Gu K, Han Q L, Luo A C J, Niculesen S I. Discretized Lyapunov functional for systems with distributed delay and piecewise constant coefficient[J]. International Journal of Control,2001,74:734-744
[14]Han Q L. On stability of linear neutral systems with mixed time delays:a discretized Lyapunov function approach[J]. Automatica,2005, 41(7):1209-1218
[15]Yu M, Wang L, Chu T. An LMI approach to networked control systems with data packet dropout and transmission delays[M]. Proc.MTNS.Leuven,2004
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[17]Yue D, Han Q L, Lam J. Network-based robust H∞ control of systems with uncertainty[J].Antomatiea,2005,41:999-1007
[18]Yue D, Peng C, Tang G Y. Guaranteed cost control of linear systems over networks with state and input quantization[J]. IEE proc. Control Theory Appl.,2006,153(6):658-664
[19]朱淑倩,张承慧,李振波,程兆林.不确定奇异时滞系统的时滞相关型鲁棒H∞弹性控制[J],控制理论与应用,2007,24(4)：587-593
[20]Zhu S Q, Zhang C H, Cheng Z L, Feng J. Delay-dependent robust stability criteria for two classes of uncertain singular time-delay Systems[J]. IEEE Transactions on Automatic Control,2007,52(5):880-885
[21]朱淑倩,程兆林.线性奇异时滞系统的状态与不确定输入估计[J],白动化学报,2004,35(5)：778-783
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