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极大-加线性系统控制与优化理论及其应用研究
中文摘要

 极大-加线性系统是在极大-加代数下呈现出线性形式的一类非线性离散系统,存在于计算机、制造、通信、交通等领域,现已成为控制科学与工程学科里极富理论性与应用性的一个研究分支.本文研究极大-加线性系统的控制与优化理论及其应用,主要涉及系统的维数灾难、全局性和不确定性方面的问题,具体包括极大-加线性系统降维、反馈镇定、全局优化、全局鲁棒和不确定极大-加线性系统能达性、最优输入设计、最优控制、可靠性分析,以及超大规模集成电路阵列处理器全局同步、基因调控网络全局鲁棒、云控制系统任务分配、分布式系统负载调度、数据传输系统时序分析、通信协议正确性验证、轨道交通系统线路规划、康复机器人系统自主控制. 研究极大-加线性系统的降维.引入极大-加线性系统伪等价和极大-加矩阵弱相似概念,并用状态矩阵的弱相似来刻画系统的伪等价.指出极大-加线性系统可以降到的维数取决于原系统状态矩阵行向量组所含弱线性无关向量的个数.证明极大-加代数上向量组所含弱线性无关向量个数的唯一性,由此引入极大-加矩阵行秩概念.给出构造与原系统伪等价且维数等于原系统状态矩阵行秩的简约系统的一个多项式算法,该算法所构造的简约系统保持了原系统的周期性和稳定性.运用简约系统设计镇定原系统的状态反馈控制器.将极大-加线性系统的降维与反馈镇定用于超大规模集成电路阵列处理器的优化设计与同步控制,把一个mn维阵列的维数降至m+n维.提出极大-加线性系统Petri网模型的同时可达性分析方法,以缓解状态空间组合爆炸.给出同时可达图生成算法,并运用同时可达图来验证Petri网活性.同时可达图保留了传统可达图中与Petri网性质密切相关的状态结点,把结点数随可达树层数的指数增长减少为线性增长.将同时可达性分析方法用于通信协议正确性验证. 研究极大-加线性系统的全局性.建立极大-加线性系统全局优化模型,通过求解具有仿射等式约束的极大-加线性方程组,分别给出全局最优解存在和唯一的充分必要条件,以及寻求全局最优解的一个多项式算法.将极大-加线性系统的全局优化用于分布式系统负载调度,通过分配初始就绪任务的执行时间,使总体任务的完成时间最早.引入极大-加线性系统全局鲁棒概念,把极大-加线性系统的全局鲁棒性归结为极大-加线性方程组的可解性.通过构造与系统方程组具有相同可解元的同解方程组,证明任意非可解元都是使系统状态和输出保持不变的可摄动元,并确定相应的摄动域.将极大-加线性系统的全局鲁棒性用于基因调控网络的稳定性分析,以确保蛋白质合成时间免受内部摄动和外部干扰的影响. 研究极大-加线性系统的不确定性.发展区间分析方法,分别给出区间极大-加线性方程组强解区间存在和唯一的充分必要条件,以及强解区间的计算公式;把强解区间唯一性对不可数个子方程组唯一可解性的检验转化为对有限个子方程组唯一可解性的检验,从而给出检验区间极大-加线性方程组唯一可解性的一个多项式算法.将极大-加线性系统能达性概念拓展到不确定极大-加线性系统,并将不确定极大-加线性系统的能达性归结为区间极大-加线性方程组的强可解性,从而建立不确定极大-加线性系统能达的充分必要条件,并提出区间有限检验方法,用有限个子系统来验证不确定系统的能达性.运用极大-加代数上区间向量的偏序关系,将不确定极大-加线性系统的最优输入设计问题转化为最优控制问题.引入不确定极大-加线性系统的区间输入和近似区间输入概念,指出区间极大-加线性方程组的强解区间即为相应不确定极大-加线性系统的区间输入,并给出计算最优区间输入和最优近似区间输入的一个多项式算法.将不确定极大-加线性系统最优输入设计用于多发射机-多接收机不确定数据传输系统的时序最优控制与可靠性分析,在数据传输时间发生有界摄动时,确定发射机发射数据的时间,以使接收机在规定时间范围内接收到全部数据. 本文还运用和发展极大-加线性系统理论来研究若干其他应用问题.设计云控制系统并行任务分配优化方案,以保证任务执行时间最短,并给出计算最短执行时间的一个多项式算法;将通信协议的正确性归结为Petri网模型的活性,分别运用同时可达性和关联矩阵幂零性,给出验证由事件图描述的一类通信协议正确性的两个有效算法;分析轨道交通系统列车的周期运行规律,设计缩短系统周期的列车运行方案;制定康复机器人上肢传递系统的自主控制策略,分析周期稳态相对于参数摄动的鲁棒性. 关键词:非线性系统,不确定性,全局性,模型降维,反馈控制,最优控制,算法实现.

英文摘要

 Max-plus linear systems are a class of nonlinear discrete systems that can be described by a linear model in max-plus algebra, which can be found in computer networks, manufacturing systems, communication systems, traffic systems, etc. Max-plus linear system has become a branch of research with highly theoretical value and practical significance in the field of control science and engineering. This dissertation studies control and optimization of max-plus linear systems and their applications, mainly involving some difficulties in high dimensionality, global behaviors and uncertainty, and specifically including the dimension reduction, feedback stabilization, global optimization and global robustness of max-plus linear systems, and reachability, optimal input design, optimal control and reliability analysis of uncertain max-plus linear systems, and their applications in global synchronization of very large scale integration array processors, global robustness of gene regulatory networks, task assignment of cloud control systems, load scheduling of distributed systems, timing analysis of data transmission systems, correctness verification of communication protocols, route planning of railway transport systems and autonomous control of rehabilitation robots. This dissertation investigates the dimension reduction of max-plus linear systems. The pseudo-equivalence relationship between max-plus linear systems and the weakly similar relationship between max-plus matrices are introduced. The pseudo-equivalence of max-plus linear systems are characterized by the weakly similar of their system matrices. It is pointed out that the dimension of the reduced system is determined by the number of weakly linearly independent vectors in the row space of the original state matrix. It is proven that the number of weakly linearly independent vectors in a max-plus vector space is unique, and then the concept of the row rank of a max-plus matrix is proposed. A polynomial algorithm is developed to construct a reduced system which is pseudo-equivalent to the original system, whose dimension is equal to the row rank of the original state matrix. The reduced system constructed by using such an algorithm has the same periodicity and stability as the original system. The dimension reduction method is then used to design a reduced state feedback controller to stabilize a max-plus linear system. The dimension reduction and feedback stabilization techniques of max-plus linear systems are applied in optimal design and synchronous control of very large scale integration array processors, by which the dimension of an mn dimensional array is reduced to m+n. This dissertation also investigates the dimension reduction of the state space of the Petri net model of a max-plus linear system. The simultaneous reachability analysis of Petri nets is proposed, and the generating algorithm of the simultaneous reachability graph is provided, by which the liveness of Petri nets is verified. The simultaneous reachability graph preserves the nodes that are closely related to the properties of a Petri net, by which the exponential growth of the number of nodes with the number of layers in the traditional reachable tree is decreased to linear growth. The simultaneous reachability analysis is then used to verify the correctness of communication protocols, which helps to alleviate the state space explosion problem. This dissertation investigates the global characterizations of max-plus linear systems. The global optimization model of max-plus linear systems is established. The necessary and sufficient conditions for the existence and uniqueness of the global optimal solution are given, respectively, by solving systems of max-plus linear equations with affine equality constraints. A polynomial algorithm is developed to find the globally optimal solution. The global optimization of max-plus linear systems is applied in load scheduling of distributed systems. By allocating the execution time of the initially ready tasks, the completion time of the overall task becomes to the earliest. The global robustness of max-plus linear systems is introduced, which is reduced to the solvability of systems of max-plus linear equations. By constructing the systems of max-plus linear equations that have the same solutions with the original system, the global robustness condition, which requires all states and outputs are entirely unaffected by the bounded parameter perturbations, is established. The global robustness of max-plus linear systems is applied in stability analysis of gene regulation networks to ensure that the protein synthesis time can be freed from internal perturbations and external disturbances. This dissertation studies the uncertainty of max-plus linear systems. The interval analysis approach is developed. The necessary and sufficient conditions for the existence and uniqueness of interval strong solutions of an interval system of max-plus linear equations are established, respectively, and the formula of interval strong solutions is given. The verification of the uniqueness of interval strong solutions of an interval system, which requires to check the uniqueness of solutions of uncountable subsystems, is reduced to the verification of the uniqueness of solutions of finite subsystems. On this basis, a polynomial algorithm is developed to verify the uniqueness of interval strong solutions of an interval system of max-plus linear systems. This dissertation extends the concept of reachability from max-plus linear systems to uncertain max-plus linear systems. The reachability of an uncertain max-plus linear system is transformed into the strong solvability of an interval system of max-plus linear equations, by which a necessary and sufficient condition of the reachability of an uncertain max-plus linear system is derived. In particular, this dissertation proposes the finite verification method of interval systems, by which the reachability condition that needs to check the reachability of an infinite number of subsystems is reduced to check that of a finite number of subsystems. Based on the partial order of interval vectors over max-plus algebra, the optimal input design for uncertain max-plus linear systems is described as some optimal control problems. The concepts of interval input and approximate interval input of uncertain max-plus linear systems are introduced. It is pointed out that the interval strong solution of the interval system of max-plus linear equations is the interval input of the corresponding uncertain max-plus linear system. A polynomial algorithm is developed to find the optimal interval input, as well as the optimal approximate interval input, of an uncertain max-plus linear system. The optimal input design of uncertain max-plus linear systems is used in the timing optimal control and timing reliability of data transmission systems with multiple transmitters and multiple receivers. The range of sending time of an uncertain data transmission system is determined to ensure that the data can bo received within the required time range, without any influence of transport delays. This dissertation also applies and develops the theory of max-plus linear systems to investigates some other application problems. The optimization scheme for the parallel task assignment in a cloud control system is designed to minimize the execution time, and a polynomial algorithm is developed to compute such a minimum time. The correctness of a communication protocol is converted to the liveness of the Petri net model. Based on the simultaneous reachability analysis and the nilpotency of the incidence matrix, two effective algorithms are proposed to verify the correctness of a class of communication protocols that can be modeled by event graphs. The periodic operation regulations of railway transport systems are analyzed and the train operation scheme are planned to reduce the period of traffic systems. The strategy of autonomous control of upper limb transfer systems of rehabilitation robots is presented, and the robust stability with respect to the parameter perturbation is analyzed. Key words: nonlinear system, uncertainty, global characterization, dimension reduction, feedback control, optimal control, algorithm realization.

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