广义双线性系统的稳定性与镇定
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摘要
广义系统的稳定性与镇定问题是广义系统理论的基本问题之一,正在受到越来越多的学者的关注,并随着控制理论的发展而不断完善。由于在实际系统中,更多的只能用非线性系统来描述,使得对非线性系统的研究格外有意义。广义双线性系统是一类重要的广义非线性系统,它有两方面的优点:一方面,和广义线性系统相比,它和广义非线性系统更接近;另一方面,在许多不能够用广义线性系统模型描述的实际物理过程,用广义双线性系统模型却能够很好地近似。到目前为止,对广义双线性系统的研究并不多见,所以对广义双线性系统进行研究有着重要意义。
本文针对广义双线性系统的稳定性与镇定问题进行研究。基于Lyapunov理论、变结构控制方法等对广义双线性系统的稳定性,状态观测器的设计、基于状态观测器的反馈控制、最优控制、鲁棒镇定以及变结构控制问题进行研究。本文的主要内容作包括以下几个方面:
一、通过引入E-渐近稳定的概念和广义Lyapunov函数,采用Lyapunov方法研究了广义双线性系统的稳定性问题;给出了广义双线性系统在两种不同情形下系统可镇定的充分条件。
二、对输入受约束的广义双线性系统,采用等价分解和极点配置的方法来设计状态观测器,给出了广义双线性系统存在的充分条件和设计方法;同时,对基于状态观测器的反馈控制问题也进行了研究。
三、利用Lyapunov稳定性理论和广义Lyapunov方程的解来研究广义双线性系统的最优控制问题,使得闭环系统全局渐近稳定且使广义二次性能指标最小;给出了广义双线性系统最优控制器的设计方法。
四、针对不确定广义双线性系统的镇定问题,通过引入广义二次稳定的概念、采用广义Lyapunov方法,依次给出了不确定广义双线性系统在两种不同情形下可镇定的充分条件。
五、研究广义双线性系统的变结构控制问题,通过引进滑动模动态补偿器来选取切换面,进而设计变结构控制器以保证了闭环系统的渐近稳定性;同时,用类似的方法给出了不确定广义双线性系统的变结构控制设计。
本文几乎所有结论都通过具体的数值算例验证,说明了本文给出的方法和结论的有效性和合理性。
The asymptotical stability and stabilization problem is one of the fundamental problems in the theory of singular systems.Its results have been attracting more and more attentions,and are becoming more perfect along the development of the control theory.Because more systems can only be described by nonlinear systems in practical systems rather than linear systems,it has great significance extraordinarily to study nonlinear systems.Singular bilinear system(SBS)is a class of important singular nonlinear systems,and there are two advantages of using SBS models.One advantage is that they provide a better approximation to singular nonlinear systems than those of singular linear systems,the other advantage is that many real physical process may be appropriately modeled as SBS when singular linear systems models are inadequate.Up to now,the study of SBS is scarcely reported in literatures,so to investigate SBS has great significance.
In this dissertation,the asymptotical stability and stabilization problems of SBS are considered.Based on Lyapunov theory,variable structure control method,etc,the stability,the design of state observer,the feedback control of SBS based on state observer,optimal control,robust stabilization and variable structure control problems of SBS are studied,respectively,in this dissertation.The main results obtained in this dissertation are as follows:
1.By citing the concept of E-stability and generalized Lyapunov function the stability of SBS is introduced,and the Lyapunov method is adopted at the same time. The sufficient conditions of stabilization of SBS are given in two differential situations.
2.The design of state observer for SBS with bounded inputs are presented via equivalent decomposition and pole-assignment methods,simultaneously,the feedback control based on the state observer of SBS is discussed.
3.The optimal control of SBS is studied by employing Lyapunov stability theory and the solution of generalized Lyapunov equations.At the same time,the result global asymptotically stabilizes the closed-loop system and minimizes the generalized quadratic performance index.In the last,the design method of optimal controller of SBS is presented.
4.To deal with the problem of robust stabilization for uncertain SBS,the sufficient conditions of stabilization for SBS with uncertainties in two different conditions are presented separately via the concept of generated quadratic stability and Lyapunov method in this dissertation.
5.In order to solve the problem of variable structure control of SBS,the sliding surface is chosen by employing sliding mode compensator,and the variable structure controller is designed,simultaneously,the stability of the closed-loop system is guaranteed.At the same time,the designing of variable structure controller of SBS with uncertainty is presented by the similar method.
Further more,in this dissertation,the effectiveness and rationality of almost all of the design methods and results are illustrated by numerical examples.
本文针对广义双线性系统的稳定性与镇定问题进行研究。基于Lyapunov理论、变结构控制方法等对广义双线性系统的稳定性,状态观测器的设计、基于状态观测器的反馈控制、最优控制、鲁棒镇定以及变结构控制问题进行研究。本文的主要内容作包括以下几个方面:
一、通过引入E-渐近稳定的概念和广义Lyapunov函数,采用Lyapunov方法研究了广义双线性系统的稳定性问题;给出了广义双线性系统在两种不同情形下系统可镇定的充分条件。
二、对输入受约束的广义双线性系统,采用等价分解和极点配置的方法来设计状态观测器,给出了广义双线性系统存在的充分条件和设计方法;同时,对基于状态观测器的反馈控制问题也进行了研究。
三、利用Lyapunov稳定性理论和广义Lyapunov方程的解来研究广义双线性系统的最优控制问题,使得闭环系统全局渐近稳定且使广义二次性能指标最小;给出了广义双线性系统最优控制器的设计方法。
四、针对不确定广义双线性系统的镇定问题,通过引入广义二次稳定的概念、采用广义Lyapunov方法,依次给出了不确定广义双线性系统在两种不同情形下可镇定的充分条件。
五、研究广义双线性系统的变结构控制问题,通过引进滑动模动态补偿器来选取切换面,进而设计变结构控制器以保证了闭环系统的渐近稳定性;同时,用类似的方法给出了不确定广义双线性系统的变结构控制设计。
本文几乎所有结论都通过具体的数值算例验证,说明了本文给出的方法和结论的有效性和合理性。
The asymptotical stability and stabilization problem is one of the fundamental problems in the theory of singular systems.Its results have been attracting more and more attentions,and are becoming more perfect along the development of the control theory.Because more systems can only be described by nonlinear systems in practical systems rather than linear systems,it has great significance extraordinarily to study nonlinear systems.Singular bilinear system(SBS)is a class of important singular nonlinear systems,and there are two advantages of using SBS models.One advantage is that they provide a better approximation to singular nonlinear systems than those of singular linear systems,the other advantage is that many real physical process may be appropriately modeled as SBS when singular linear systems models are inadequate.Up to now,the study of SBS is scarcely reported in literatures,so to investigate SBS has great significance.
In this dissertation,the asymptotical stability and stabilization problems of SBS are considered.Based on Lyapunov theory,variable structure control method,etc,the stability,the design of state observer,the feedback control of SBS based on state observer,optimal control,robust stabilization and variable structure control problems of SBS are studied,respectively,in this dissertation.The main results obtained in this dissertation are as follows:
1.By citing the concept of E-stability and generalized Lyapunov function the stability of SBS is introduced,and the Lyapunov method is adopted at the same time. The sufficient conditions of stabilization of SBS are given in two differential situations.
2.The design of state observer for SBS with bounded inputs are presented via equivalent decomposition and pole-assignment methods,simultaneously,the feedback control based on the state observer of SBS is discussed.
3.The optimal control of SBS is studied by employing Lyapunov stability theory and the solution of generalized Lyapunov equations.At the same time,the result global asymptotically stabilizes the closed-loop system and minimizes the generalized quadratic performance index.In the last,the design method of optimal controller of SBS is presented.
4.To deal with the problem of robust stabilization for uncertain SBS,the sufficient conditions of stabilization for SBS with uncertainties in two different conditions are presented separately via the concept of generated quadratic stability and Lyapunov method in this dissertation.
5.In order to solve the problem of variable structure control of SBS,the sliding surface is chosen by employing sliding mode compensator,and the variable structure controller is designed,simultaneously,the stability of the closed-loop system is guaranteed.At the same time,the designing of variable structure controller of SBS with uncertainty is presented by the similar method.
Further more,in this dissertation,the effectiveness and rationality of almost all of the design methods and results are illustrated by numerical examples.
引文
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[16]Rafael Bru~*,Carmen Coll,Nestor Thome.Compensating periodic descriptor systems[J].Systems & Control Letters,2001,43:133-139.
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