不确定线性时滞系统时滞相关非脆弱鲁棒控制
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摘要
传统的最优和鲁棒控制器设计要求控制器必须准确实现,这样有可能出现脆弱的控制器,即控制器参数发生极其微小的偏移,将导致闭环系统的稳定性被破坏或性能下降。另外,时滞现象普遍存在于各种工业过程系统中,时滞的存在往往是系统不稳定和系统性能变差的根源。针对上述问题,本文主要以Lyapunov稳定性理论为基础,利用最近建立的分析时滞系统时滞相关稳定性新方法,分别研究了不确定线性时滞系统、中立型时滞系统、离散时滞系统的非脆弱鲁棒H_∞风控制及保成本控制等问题。
(1)利用时滞二分法分析了线性时滞系统时滞相关鲁棒稳定性
目前分析时滞系统时滞相关鲁棒稳定性时将Lyapunov-Krasovskii泛函中时滞信息作为整体进行考虑,即Lyapunov-Krasovskii泛函中的正定矩阵Q、R在[-h,0]上是固定的。针对由此带来时滞界限的保守性等问题,通过引入时滞区间进行分段处理方法(时滞二分法),针对d(t)为时不变和时变(连续可微)两种情形分类、分段定义了新Lyapunov-Krasovskii泛函,获得了线性时滞系统时滞相关稳定性的新条件,数值实例表明所得到的结论具有更小保守性。
(2)利用时滞二分法设计了线性时滞系统非脆弱鲁棒控制器
在时滞系统鲁棒稳定性分析的基础上,首先研究了具有状态和控制输入不确定,即具有不确定项乘积形式的线性时滞系统时滞相关非脆弱鲁棒H_∞控制问题,针对加性和乘性不确定性非脆弱控制器两种类型,分别建立了使闭环系统不仅鲁棒稳定,而且在零初始条件下具有给定的H_∞扰动抑制水平γ的时滞相关充分条件及非脆弱控制器参数设计方法。然后,将时滞二分法引入到线性时滞系统非脆弱鲁棒H_∞控制研究中,针对d(t)为时不变和时变(连续可微)两种情形分别获得了非脆弱鲁棒稳定的时滞相关充分条件和非脆弱控制控制器参数设计方法。数值实例说明了本方法的有效性。
(3)给出了中立型时滞系统非脆弱鲁棒控制器设计方法
讨论了具有加法和乘法不确定中立型时滞系统的非脆弱H_∞控制问题,首先给出闭环系统内部稳定的有界实(BRL)条件。获得了在非脆弱控制器作用下中立型时滞系统不仅内部渐近稳定而且具有给定的H_∞性能的时滞相关条件,同时给出了非脆弱控制器参数设计方法,这一方法依赖于LMI的解,不需要调节任何参数,数值实例说明本方法的有效性。
(4)给出了离散时滞系统非脆弱鲁棒控制器设计方法
利用有限和不等式方法和Lyapunov-Krasovskii稳定性理论获得了不确定时滞系统在非脆弱控制器作用下不仅内部鲁棒稳定,而且具有给定的H_∞扰动抑制水平γ的时滞相关有界实条件。然后,采用迭代算法分别给出了控制器具有加法不确定性和乘法不确定性两种情况的非脆弱控制器参数的设计方法,借助Matlab的LMI工具箱求解方便,数值实例说明该方法的有效性。
(5)给出了中立型时滞系统非脆弱保成本控制器设计方法
对于不确定中立型时滞系统,通过引入增广Lyapunov泛函来获得最优的非脆弱保成本控制器,讨论了具有状态和控制输入时滞的中立型时滞系统的非脆弱保成本控制器的设计方法。给出了具有加性和乘性增益扰动非脆弱保成本控制器基于LMI的可行解。数值实例表明该方法与现有的结论相比较具有更小保成本性能指标。
As well known,the precondition for conventional designs of optimal and robut controllers,that the controller should be realized exactly,always results in fragile controllers,so that small perturbation of the coefficients of the designed controller destabilize the closed-loop control system.In addition,since time-delay is a normal phenomenon widely existed in many industrial process control systems,it is indeed the critical source of instability and degradation in control performance of control systems.To handle this problem,based on the Lyapunov stability theory and the new method of delay-dependent robust stability developed in recent years,the problems of non-fragile robust H_∞control and guaranteed cost control were investigated for uncertain linear systems,neutral systems,discrete systems with time-delays in this thesis.
(1) Analysis on the delay-dependent robust stability of linear systems with time delays by using bisection of time-delay
So far in the study on the delay-dependent robust stability with delays, the delay information in Lyapunov-Krasovskii functional is always viewed as an integer.That is,the coefficient matrices of positive definite Q and R are constant at the interval[-h,0]in Lyapunov-Krasovskii functional.That induces conservation of the bound of delays.To overcome this shortage,the method decomposing the delay interval into two equidistance subintervals was introduced,on which different new Lyapunov functionals were defined in two intervals and two cases,with respect to the cases of d(t) with constant and time-varying delays.Numerical example was given to show that,the results were less conservative.
(2) The non-fragile robust controller designed for linear systems with time delays using bisection of time-delay
Based on the robust stabilization of delay systems,firstly,considering the uncertainties with state and control input,which are always in the terms of the product of two uncertainties,the non-fragile delay-dependent robust H_∞control of linear delayed systems was investigated.A new delay-dependent condition,which ensured that the closed-loop system was internally robust stable with a given H_∞disturbance attenuation levelγunder zero condition via a non-fragile controller,was obtained with respect to additive and multiplicative uncertainties.As a result,the method of non-fragile H_∞controller design was proposed.Secondly,bisection of time-delay was introduced to the non-fragile robust controller designed for linear systems with time delays using bisection of time-delay,so that,with respect to two cases of d(t) with time-invarying and time-varying delays,the new delay-dependent conditions are obtained,which could ensure that,via the non-fragile controller,the closed-loop system was internally robust stable with a given H_∞disturbance attenuation levelγunder zero condition. Numerical simulation was given to show the validity of this approach.
(3) The non-fragile robust controller for neutral systems with time delays
The problem of delay-dependent non-fragile H_∞control for neutral time-delay systems was concerned with additive and multiplicative controller uncertainties.At first,a sufficient Bounded Real Lemma condition of the closed-loop system was given to show the internally stable,and also the H_∞delay-dependent disturbance performance via a non-fragile controller. The result only depended on the solutions of linear matrix inequalities,so that no parameter needs to be tuned.Numerical simulation was given to show the validity of the proposed approach.
(4) The non-fragile robust controller for discrete systems with time delays
Based on an inequality method for finite sums and Lyapunov stability theory,a new delay-dependent Boundary Real Lemma was obtained,which could ensure that the closed-loop system was internally robust stable with a given H_∞disturbance attenuation level via a non-fragile controller for uncertain discrete-delay systems.Then,the design of non-fragile H_∞controller was proposed by using iterative algorithm with respect to additive and multiplicative uncertainties.It can be easily solved by linear matrix inequalities(LMI) in Matlab Toolbox.A numerical simulation was given to show the validity of this approach.
(5) The non-fragile guaranteed cost controller for neutral systems with time delays
For the neutral time-delay systems,by introducing augmented Lyapunov-Krasovskii functional,the non-fragile guaranteed cost controller was obtained.The design method of non-fragile guaranteed cost control for uncertain neutral systems with state and control input delay was discussed. The feasible solutions were proposed to non-fragile guaranteed cost controller with respect to additive and multiplicative gain disturbance. Numerical examples were given to show the less guaranteed cost performance than the existing ones.
(1)利用时滞二分法分析了线性时滞系统时滞相关鲁棒稳定性
目前分析时滞系统时滞相关鲁棒稳定性时将Lyapunov-Krasovskii泛函中时滞信息作为整体进行考虑,即Lyapunov-Krasovskii泛函中的正定矩阵Q、R在[-h,0]上是固定的。针对由此带来时滞界限的保守性等问题,通过引入时滞区间进行分段处理方法(时滞二分法),针对d(t)为时不变和时变(连续可微)两种情形分类、分段定义了新Lyapunov-Krasovskii泛函,获得了线性时滞系统时滞相关稳定性的新条件,数值实例表明所得到的结论具有更小保守性。
(2)利用时滞二分法设计了线性时滞系统非脆弱鲁棒控制器
在时滞系统鲁棒稳定性分析的基础上,首先研究了具有状态和控制输入不确定,即具有不确定项乘积形式的线性时滞系统时滞相关非脆弱鲁棒H_∞控制问题,针对加性和乘性不确定性非脆弱控制器两种类型,分别建立了使闭环系统不仅鲁棒稳定,而且在零初始条件下具有给定的H_∞扰动抑制水平γ的时滞相关充分条件及非脆弱控制器参数设计方法。然后,将时滞二分法引入到线性时滞系统非脆弱鲁棒H_∞控制研究中,针对d(t)为时不变和时变(连续可微)两种情形分别获得了非脆弱鲁棒稳定的时滞相关充分条件和非脆弱控制控制器参数设计方法。数值实例说明了本方法的有效性。
(3)给出了中立型时滞系统非脆弱鲁棒控制器设计方法
讨论了具有加法和乘法不确定中立型时滞系统的非脆弱H_∞控制问题,首先给出闭环系统内部稳定的有界实(BRL)条件。获得了在非脆弱控制器作用下中立型时滞系统不仅内部渐近稳定而且具有给定的H_∞性能的时滞相关条件,同时给出了非脆弱控制器参数设计方法,这一方法依赖于LMI的解,不需要调节任何参数,数值实例说明本方法的有效性。
(4)给出了离散时滞系统非脆弱鲁棒控制器设计方法
利用有限和不等式方法和Lyapunov-Krasovskii稳定性理论获得了不确定时滞系统在非脆弱控制器作用下不仅内部鲁棒稳定,而且具有给定的H_∞扰动抑制水平γ的时滞相关有界实条件。然后,采用迭代算法分别给出了控制器具有加法不确定性和乘法不确定性两种情况的非脆弱控制器参数的设计方法,借助Matlab的LMI工具箱求解方便,数值实例说明该方法的有效性。
(5)给出了中立型时滞系统非脆弱保成本控制器设计方法
对于不确定中立型时滞系统,通过引入增广Lyapunov泛函来获得最优的非脆弱保成本控制器,讨论了具有状态和控制输入时滞的中立型时滞系统的非脆弱保成本控制器的设计方法。给出了具有加性和乘性增益扰动非脆弱保成本控制器基于LMI的可行解。数值实例表明该方法与现有的结论相比较具有更小保成本性能指标。
As well known,the precondition for conventional designs of optimal and robut controllers,that the controller should be realized exactly,always results in fragile controllers,so that small perturbation of the coefficients of the designed controller destabilize the closed-loop control system.In addition,since time-delay is a normal phenomenon widely existed in many industrial process control systems,it is indeed the critical source of instability and degradation in control performance of control systems.To handle this problem,based on the Lyapunov stability theory and the new method of delay-dependent robust stability developed in recent years,the problems of non-fragile robust H_∞control and guaranteed cost control were investigated for uncertain linear systems,neutral systems,discrete systems with time-delays in this thesis.
(1) Analysis on the delay-dependent robust stability of linear systems with time delays by using bisection of time-delay
So far in the study on the delay-dependent robust stability with delays, the delay information in Lyapunov-Krasovskii functional is always viewed as an integer.That is,the coefficient matrices of positive definite Q and R are constant at the interval[-h,0]in Lyapunov-Krasovskii functional.That induces conservation of the bound of delays.To overcome this shortage,the method decomposing the delay interval into two equidistance subintervals was introduced,on which different new Lyapunov functionals were defined in two intervals and two cases,with respect to the cases of d(t) with constant and time-varying delays.Numerical example was given to show that,the results were less conservative.
(2) The non-fragile robust controller designed for linear systems with time delays using bisection of time-delay
Based on the robust stabilization of delay systems,firstly,considering the uncertainties with state and control input,which are always in the terms of the product of two uncertainties,the non-fragile delay-dependent robust H_∞control of linear delayed systems was investigated.A new delay-dependent condition,which ensured that the closed-loop system was internally robust stable with a given H_∞disturbance attenuation levelγunder zero condition via a non-fragile controller,was obtained with respect to additive and multiplicative uncertainties.As a result,the method of non-fragile H_∞controller design was proposed.Secondly,bisection of time-delay was introduced to the non-fragile robust controller designed for linear systems with time delays using bisection of time-delay,so that,with respect to two cases of d(t) with time-invarying and time-varying delays,the new delay-dependent conditions are obtained,which could ensure that,via the non-fragile controller,the closed-loop system was internally robust stable with a given H_∞disturbance attenuation levelγunder zero condition. Numerical simulation was given to show the validity of this approach.
(3) The non-fragile robust controller for neutral systems with time delays
The problem of delay-dependent non-fragile H_∞control for neutral time-delay systems was concerned with additive and multiplicative controller uncertainties.At first,a sufficient Bounded Real Lemma condition of the closed-loop system was given to show the internally stable,and also the H_∞delay-dependent disturbance performance via a non-fragile controller. The result only depended on the solutions of linear matrix inequalities,so that no parameter needs to be tuned.Numerical simulation was given to show the validity of the proposed approach.
(4) The non-fragile robust controller for discrete systems with time delays
Based on an inequality method for finite sums and Lyapunov stability theory,a new delay-dependent Boundary Real Lemma was obtained,which could ensure that the closed-loop system was internally robust stable with a given H_∞disturbance attenuation level via a non-fragile controller for uncertain discrete-delay systems.Then,the design of non-fragile H_∞controller was proposed by using iterative algorithm with respect to additive and multiplicative uncertainties.It can be easily solved by linear matrix inequalities(LMI) in Matlab Toolbox.A numerical simulation was given to show the validity of this approach.
(5) The non-fragile guaranteed cost controller for neutral systems with time delays
For the neutral time-delay systems,by introducing augmented Lyapunov-Krasovskii functional,the non-fragile guaranteed cost controller was obtained.The design method of non-fragile guaranteed cost control for uncertain neutral systems with state and control input delay was discussed. The feasible solutions were proposed to non-fragile guaranteed cost controller with respect to additive and multiplicative gain disturbance. Numerical examples were given to show the less guaranteed cost performance than the existing ones.
引文
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