正系统动力学性质研究
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摘要
本论文集中研究了正系统的关键动力学性质，并对正系统中一些热点子领域中的相关问题进行了深入讨论，建立了比较完善、系统而重要的结果。全文包括以下八个部分：
第一部分主要介绍了正系统的研究背景和研究现状，提供了在本论文中常用的基本概念，基础知识和重要引理.它们是后续章节讨论的前提。
第二部分研究了含有多时滞的正系统(包括含有常时滞和时变时滞的连续时间和离散时间等四种情形)的稳定性，同时讨论了它们的控制器设计问题。主要有以下结果：1、建立了四类系统稳定的充分必要条件。2、建立了可以保证闭环系统为正且渐近稳定的控制器存在的充分必要条件；如果这类控制器存在，则提供了计算其增益矩阵的方法。所有的结果均以线性规划和线性矩阵不等式来表达。很有趣的是：无论是连续时间还是离散时间，常时滞系统和时变时滞系统的相应结果具有完全一致的形式。这一现象揭示了正系统的固有属性。3、揭示了常时滞正系统的重要性质：如果系统稳定，在一定初始条件下，其解是“单调下降”的。4、揭示了具有相同系统矩阵的常时滞和时变时滞系统的解在一定条件下的深刻联系：前者的解大于后者。可以认为，上述所有结果都表明了正系统具有一种“优雅”的性质。从方法上讲，本章灵活使用了线性copositive Lyapunov泛函，发展了稳定性研究工具，突破了稳定性研究的一般思路。
第三部分研究了含有常时滞和时变时滞的正系统(包括连续时间系统和离散时间系统系统情形)的约束控制问题：如何确定一组界，使得闭环系统的状态为正，且系统状态和控制输入都可以为这组界所限定。这是正系统领域所关注的基本问题之一，也称为有界控制问题。由于对于一般动力学系统来讲，有界控制研究相对较少，处理方法相对单调，这给正系统的有界控制研究造成了一定困难。针对这些系统，本章建立了使得闭环系统满足上述约束的控制器存在的充分必要条件。从方法上看，本章创新了处理有界控制的一般方法，采用了反证法和微积分技巧，改变了这一领域以往不能建立有界控制的充分必要条件的状况。
第四部分讨论了无界时滞离散时间正系统的稳定性。本部分主要贡献为：建立了这类系统渐近稳定的充分必要条件，揭示了一定条件下具有相同系统矩阵但含有不同无界时滞正系统的解之间的关系。这些结果对解决含有无界时滞的连续时间正系统的稳定性问题具有重要的启示。
第五部分讨论了切换正系统的稳定性分析。过去，人们主要采用公共线性copositive Lyapunov函数方法来研究这类系统。非常明显，采用这种方法所建立的稳定性判据的保守性一般较高。本章主要结果是提出了分析切换正系统的一种新方法，即切换线性copositive Lyapunov函数方法，这种方法可以有效地降低分析切换正系统稳定性的保守性。它对时滞切换正系统的研究具有重要作用。
第六部分研究了由Roesser模型描述的含有多时滞的2-D正系统的稳定性。这类系统是当前正系统和2-D系统领域研究的热点问题。我们给出了这类系统稳定的充分必要条件。这个结果稍加修改即能推广到由其它模型描述的时滞2-D正系统上去。方法上，本章采用构造性的证明方法。这些结果和方法对于n—D时滞系统的研究具有借鉴价值。
第七部分研究了大系统在正性约束下的分散控制问题，给出了对应控制器存在的充分必要条件。这是正系统理论在大系统领域中的应用。从结果上看，它们在一定条件下解决了大系统的分散控制这一难点问题。这一成功应用显示了正系统在应用上的优势。
第八部分对全文进行了总结，并指出了今后的研究方向。
Addressed in this dissertation are the key dynamics of positive systems. Several relevantproblems attracting more and more attention in this field are discussed in detail, anda series of well-established, systematical, and important results are obtained. As a whole,this dissertation is consisted of the following eight parts.
Part 1 introduces the backgrounds and research status of positive systems. It alsoprovides the readers with some preliminaries and important definitions and lemmas thatare frequently used in the remaining chapters.
Part 2 treats the positive systems with multiple bounded delays, including four situations:continuous-and discrete-time systems with constant or time-varying delays. Stabilityanalysis and controller designing are considered. The main results lie in the followingfour aspects. 1. The necessary and sufficient stability criteria are established. 2. Thenecessary and sufficient conditions are also provided, which determines whether or not acontroller exists such that the corresponding closed-loop system is positive and asymptoticallystable. Furthermore, if such a controller exists, the method how to compute it isexplicitly provided. All the results are expressed in terms of linear programming and linearmatrix inequality. It is interesting to see that the results for systems with constant andtime-varying delays are same in form, no matter the systems are continuous- or discretetime.3. It reveals some important properties of positive systems with constant delays: ifa system is stable, then its trajectory should monotonically decrease provided that the initialconditions are properly chosen. 4. It also reveals a relationship between the solutionsto systems with identical systems matrices but with different kinds of delays: solution tosystem with time-varying delays is not greater than that to system with constant delays,providing some conditions are satisfied. One may draw a conclusion that the positivesystem is "elegant" based on the obtained results. TechnologicaUy, this chapter employsthe copositive Lyapunov functional in a flexible manner, creates some new approaches toanalyze the system stability, and makes a breakthrough in the aspect of studying stability.
Part 3 is devoted to investigating the constrained control problem of positive systemswith delays(including constant delays and time-varying delays): determining two boundsand designing a controller such that the corresponding closed-loop system is positive, and its states and control input can be bounded by the two preset bounds. Therefore, in thereferences, this problem is also called bounded control one. For the general dynamicsystems, the constrained control issue is generally not a main concern and therefore themethod to treat this problem is lacking, which make our task difficult. This chapter establishessome necessary and sufficient conditions for the existence of such controllers. Bymeans of contradiction and calculus, this chapter proposes some new methods to treat thebounded control problem, resulting a set of brief and easily applied conditions, changingthe fact that no any necessary and sufficient conditions can be established before for thebounded control problem.
Part 4 is concerned with the stability problem of discrete-time positive systems withunbounded delays. The main contribution lies in the following aspects: the necessary andsufficient stability condition is established, and the relationship among positive systemswith the identical system matrices but different unbounded delays is revealed. It is hopefulthat these results shed light on the stability issue of continuous-time positive systemswith unbounded delays.
Part 5 performs stability analysis for switched positive systems. During the pastyears, the common linear copositive Lyapunov approach was the most popular methodto treat this problem. Obviously, this method is too conservative. In order to overcomethis shortage, the chapter proposes a new approach: switched linear copositive Lyapunovmethod. Compared with the common linear copositive Lyapunov method, ours approachis much less conservative and thus can be probably widely used in the future.
Part 6 studies the stability problem of 2-D positive systems with multiple delays anddescribed by the Roesser model. Much attention is now paid to this class systems. A necessaryand sufficient stability condition is established. It can be, after simple mathematicmanipulation, applied to several delayed 2-D positive systems described by other models.In this chapter, a constructive method is used to prove the main theorem. These results,together with these method, hopefully work in the further study of n-D positive systems.
Part 7 is concerned with decentralized control of large-scale systems under the positivenessconstraint, which means that a decentralized controller should be such that theclosed-loop systems are positive. The necessary and sufficient conditions determine theexistence of such controllers are established. These results solve one of the most difficultproblem in the field of decentralized control, under certain conditions. Also, this part shows that positive systems possess great potential in applications.
Part 8 summarizes the main results obtained in this dissertation, and points out thefuture works that have been the author's concerns.
第一部分主要介绍了正系统的研究背景和研究现状，提供了在本论文中常用的基本概念，基础知识和重要引理.它们是后续章节讨论的前提。
第二部分研究了含有多时滞的正系统(包括含有常时滞和时变时滞的连续时间和离散时间等四种情形)的稳定性，同时讨论了它们的控制器设计问题。主要有以下结果：1、建立了四类系统稳定的充分必要条件。2、建立了可以保证闭环系统为正且渐近稳定的控制器存在的充分必要条件；如果这类控制器存在，则提供了计算其增益矩阵的方法。所有的结果均以线性规划和线性矩阵不等式来表达。很有趣的是：无论是连续时间还是离散时间，常时滞系统和时变时滞系统的相应结果具有完全一致的形式。这一现象揭示了正系统的固有属性。3、揭示了常时滞正系统的重要性质：如果系统稳定，在一定初始条件下，其解是“单调下降”的。4、揭示了具有相同系统矩阵的常时滞和时变时滞系统的解在一定条件下的深刻联系：前者的解大于后者。可以认为，上述所有结果都表明了正系统具有一种“优雅”的性质。从方法上讲，本章灵活使用了线性copositive Lyapunov泛函，发展了稳定性研究工具，突破了稳定性研究的一般思路。
第三部分研究了含有常时滞和时变时滞的正系统(包括连续时间系统和离散时间系统系统情形)的约束控制问题：如何确定一组界，使得闭环系统的状态为正，且系统状态和控制输入都可以为这组界所限定。这是正系统领域所关注的基本问题之一，也称为有界控制问题。由于对于一般动力学系统来讲，有界控制研究相对较少，处理方法相对单调，这给正系统的有界控制研究造成了一定困难。针对这些系统，本章建立了使得闭环系统满足上述约束的控制器存在的充分必要条件。从方法上看，本章创新了处理有界控制的一般方法，采用了反证法和微积分技巧，改变了这一领域以往不能建立有界控制的充分必要条件的状况。
第四部分讨论了无界时滞离散时间正系统的稳定性。本部分主要贡献为：建立了这类系统渐近稳定的充分必要条件，揭示了一定条件下具有相同系统矩阵但含有不同无界时滞正系统的解之间的关系。这些结果对解决含有无界时滞的连续时间正系统的稳定性问题具有重要的启示。
第五部分讨论了切换正系统的稳定性分析。过去，人们主要采用公共线性copositive Lyapunov函数方法来研究这类系统。非常明显，采用这种方法所建立的稳定性判据的保守性一般较高。本章主要结果是提出了分析切换正系统的一种新方法，即切换线性copositive Lyapunov函数方法，这种方法可以有效地降低分析切换正系统稳定性的保守性。它对时滞切换正系统的研究具有重要作用。
第六部分研究了由Roesser模型描述的含有多时滞的2-D正系统的稳定性。这类系统是当前正系统和2-D系统领域研究的热点问题。我们给出了这类系统稳定的充分必要条件。这个结果稍加修改即能推广到由其它模型描述的时滞2-D正系统上去。方法上，本章采用构造性的证明方法。这些结果和方法对于n—D时滞系统的研究具有借鉴价值。
第七部分研究了大系统在正性约束下的分散控制问题，给出了对应控制器存在的充分必要条件。这是正系统理论在大系统领域中的应用。从结果上看，它们在一定条件下解决了大系统的分散控制这一难点问题。这一成功应用显示了正系统在应用上的优势。
第八部分对全文进行了总结，并指出了今后的研究方向。
Addressed in this dissertation are the key dynamics of positive systems. Several relevantproblems attracting more and more attention in this field are discussed in detail, anda series of well-established, systematical, and important results are obtained. As a whole,this dissertation is consisted of the following eight parts.
Part 1 introduces the backgrounds and research status of positive systems. It alsoprovides the readers with some preliminaries and important definitions and lemmas thatare frequently used in the remaining chapters.
Part 2 treats the positive systems with multiple bounded delays, including four situations:continuous-and discrete-time systems with constant or time-varying delays. Stabilityanalysis and controller designing are considered. The main results lie in the followingfour aspects. 1. The necessary and sufficient stability criteria are established. 2. Thenecessary and sufficient conditions are also provided, which determines whether or not acontroller exists such that the corresponding closed-loop system is positive and asymptoticallystable. Furthermore, if such a controller exists, the method how to compute it isexplicitly provided. All the results are expressed in terms of linear programming and linearmatrix inequality. It is interesting to see that the results for systems with constant andtime-varying delays are same in form, no matter the systems are continuous- or discretetime.3. It reveals some important properties of positive systems with constant delays: ifa system is stable, then its trajectory should monotonically decrease provided that the initialconditions are properly chosen. 4. It also reveals a relationship between the solutionsto systems with identical systems matrices but with different kinds of delays: solution tosystem with time-varying delays is not greater than that to system with constant delays,providing some conditions are satisfied. One may draw a conclusion that the positivesystem is "elegant" based on the obtained results. TechnologicaUy, this chapter employsthe copositive Lyapunov functional in a flexible manner, creates some new approaches toanalyze the system stability, and makes a breakthrough in the aspect of studying stability.
Part 3 is devoted to investigating the constrained control problem of positive systemswith delays(including constant delays and time-varying delays): determining two boundsand designing a controller such that the corresponding closed-loop system is positive, and its states and control input can be bounded by the two preset bounds. Therefore, in thereferences, this problem is also called bounded control one. For the general dynamicsystems, the constrained control issue is generally not a main concern and therefore themethod to treat this problem is lacking, which make our task difficult. This chapter establishessome necessary and sufficient conditions for the existence of such controllers. Bymeans of contradiction and calculus, this chapter proposes some new methods to treat thebounded control problem, resulting a set of brief and easily applied conditions, changingthe fact that no any necessary and sufficient conditions can be established before for thebounded control problem.
Part 4 is concerned with the stability problem of discrete-time positive systems withunbounded delays. The main contribution lies in the following aspects: the necessary andsufficient stability condition is established, and the relationship among positive systemswith the identical system matrices but different unbounded delays is revealed. It is hopefulthat these results shed light on the stability issue of continuous-time positive systemswith unbounded delays.
Part 5 performs stability analysis for switched positive systems. During the pastyears, the common linear copositive Lyapunov approach was the most popular methodto treat this problem. Obviously, this method is too conservative. In order to overcomethis shortage, the chapter proposes a new approach: switched linear copositive Lyapunovmethod. Compared with the common linear copositive Lyapunov method, ours approachis much less conservative and thus can be probably widely used in the future.
Part 6 studies the stability problem of 2-D positive systems with multiple delays anddescribed by the Roesser model. Much attention is now paid to this class systems. A necessaryand sufficient stability condition is established. It can be, after simple mathematicmanipulation, applied to several delayed 2-D positive systems described by other models.In this chapter, a constructive method is used to prove the main theorem. These results,together with these method, hopefully work in the further study of n-D positive systems.
Part 7 is concerned with decentralized control of large-scale systems under the positivenessconstraint, which means that a decentralized controller should be such that theclosed-loop systems are positive. The necessary and sufficient conditions determine theexistence of such controllers are established. These results solve one of the most difficultproblem in the field of decentralized control, under certain conditions. Also, this part shows that positive systems possess great potential in applications.
Part 8 summarizes the main results obtained in this dissertation, and points out thefuture works that have been the author's concerns.
引文
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[39] X. Liu, "Constrained control of positive systems with delays," Accepted by IEEE Trans. on Automatic Control, 2009.
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