多个体线性系统的一致性分析与控制
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摘要
多个体系统来源于实际控制系统,对其研究不仅是现代控制理论发展的需要,而且是解决大量实际问题的迫切要求.对多个体系统一致性问题的探索和研究是协调控制问题的一个基本问题,并已成为学术界的研究热点.本文在前人工作的基础上,借助线性系统理论、奇异系统理论、控制理论、代数图论和矩阵理论等理论和方法,研究了多个体线性正常状态空间系统和多个体线性奇异系统的一致性条件和控制设计问题.本文的主要研究工作如下.
研究了加速一致性问题.针对固定拓扑结构下离散多个体正常状态空间系统,基于一个修正的代数Riccati方程,提出了一种不需要非线性协议的解决方法,得出了以指定趋同速度因子实现一致的充分条件,并给出了期望趋同函数和指定趋同速度联合设计的协议算法.针对切换拓扑结构下连续和离散多个体系统,首先将通讯拓扑结构的切换特性转化为系统之间的切换,给出了多个体闭环系统实现一致的充分条件,并证明此时的趋同状态为零.最后推出并分析与趋同状态范数相等的一个向量范数的显式表达式,得出了趋同状态有界的条件,并且给出了提高趋同速度的协议算法.虽然已有很多文献研究切换拓扑结构下多个体系统能实现一致的充分条件,但是却鲜有文献讨论此通讯拓扑下高阶多个体系统趋同状态的形式和趋同速度提高的问题.研究发现趋同速度过快,可能会影响多个体闭环系统的稳定性,即印证了高增益对系统的不良影响,过分追求速度是不可取的.
研究了同质多个体奇异系统的协同控制问题.基于受限等价系统,采用了修正型一致性协议.针对连续系统和离散系统两种情形,基于包含有向生成树的通讯拓扑结构,得出了连续多个体奇异系统能实现容许一致的充要条件和离散多个体奇异系统能实现容许一致的充分条件,并给出了趋同状态的显式表达式以及能实现容许一致的趋同协议的算法.与已有的关于连续多个体奇异系统的结果相比较,本文给出的是充要条件,而非充分条件,本文给出的条件是非常容易进行事先验证的,而非待定的线性矩阵不等式.针对连续多个体奇异系统,探讨了单个首领的跟踪问题,得出了能实现跟踪的充要条件为奇异系统个体强能稳,并给出了协议增益的设计算法；提出了多个首领的牵制控制问题,得出了能实现该问题的充要条件仍是个体强能稳,然后给出了协议算法.与已有的多个体奇异系统跟踪问题的结果相比较,本文给出了充要条件,而非充分条件.与多个体正常状态空间系统的多首领牵制控制问题的结果类似,本文推出奇异系统个体的状态收敛于多个首领的动态状态的凸组合.
研究了异质多个体奇异系统的容许一致性问题.针对连续和离散两种情形,为异质个体设计了结构形式相同、参数不同的修正型一致性协议.借助趋同误差向量必须正则、无脉冲(或因果)和渐近稳定的特性,得出了能实现容许一致的充分条件,并构造了一种求解一致性协议增益的算法.本部分给出的控制协议,本质上实现了单个首领的跟踪问题,与同质多个体奇异系统的首领跟踪问题相比较,这里给出的首领和跟随者的动态系统是不同的.与已有的关于异质多个体奇异系统的结果相比较,有三点优势：一是本文给出的充分条件便于验证,且容易满足；二是对输入矩阵的要求为列满秩,而非非奇异矩阵,大大降低了对个体系统的苛刻要求；三是本文所构造的协议解决了已有文献的两个缺陷:一是奇异系统个体可以含有脉冲,二是消除了趋同误差向量动态特性中的脉冲项.另外,针对异质多个体奇异系统的容许一致性问题,本文所给出的理论结果和控制设计算法具有一般性,既适用于同质多个体奇异系统的跟踪首领问题,也适用于异质多个体正常状态空间系统的跟踪首领问题.
Multi-agent systems come from the actual control system. Research on them not only is the need of the development of modern control theory, and is urged to solve a large number of actual demands. The consensus problem of multi-agent systems is a fundamental problem of cooperative control problem, which is currently research hotspot. In the dissertation, combining the linear system theory, singular system theory, control theory, algebraic graph theory and matrix analysis, the consensus problems for both linear normal and linear singular multi-agent systems are investigated systemat-ically. The innovative work and key points of this dissertation can be summarized as follows:
The accelerated consensus problem of linear normal multi-agent systems is stud-ied. Based on a modified algebraic Riccati equation, an approach without nonlinear protocol is proposed and investigated for discrete-time case under time-invariant topolo-gies. A sufficient condition is derived, which guarantees that the multi-agent system solves consensus with prescribed consensus speed index. The algorithm is given to achieve the joint design of an expected consensus function and a prescribed consen-sus speed. For the consensus problems of continuous-time and discrete-time normal multi-agent systems with directed switching topologies, the switched behavior of com-munication topologies is transmitted to the dynamic systems which are transformed into switched subsystems. Then the consensus problem is solved under the assumption that each agent is stabilizable. Finally, it is shown that the consensus state is zero state. The sufficient condition to get bounded consensus state is derived by analyzing a vector whose norm is equal to consensus state's. The algorithms of improving the consensus speed have been given. Although many references have discussed the consensualization problem of multi-agent systems under switching topologies, few gave the expression of the consensus state and improved the consensus speed. It is found that the overquick consensus speed may destroy the stability of the closed-loop multi-agent system, i.e., high gain has bad effect for dynamic system and it is not advisable to only pursue speed.
Cooperative control problem for homogeneous singular multi-agent systems is discussed. Under the restricted system equivalence, the modified consensus proto-cols are proposed. The admissible consensus problems of both continuous-time and discrete-time multi-agent singular systems are investigated. Based on the communica-tion topologies containing a spanning tree, the sufficient and necessary conditions are presented for continuous-time multi-agent singular systems, and the sufficient condi-tions are presented for discrete-time multi-agent singular systems. The formulas of the consensus states are given and the algorithms of constructing the protocols for achieving admissible consensualization are provided. In contrast with the previous results which gave sufficient conditions that are composed of linear matrix inequalities to be deter-mined, this dissertation gives the sufficient and necessary conditions and the conditions are easy to be checked, verified and satisfied for the continuous-time multi-agent sys-tems. On the other hand, for continuous-time multi-agent singular systems, the tracking problem of single leader is discussed, and it is proved that the strong stabilization of each agent is a sufficient and necessary condition, and the algorithm to construct the protocol is given. The containment control problem of multiple leaders is studied. It is also shown that a sufficient and necessary condition of solving the above problem is strong stabilizable. The algorithms of the protocols are given. Compared with the ex-isting results which gave sufficient conditions for the tracking problem of single leader, the condition here is sufficient and necessary. Compared with the consensus state for multi-agent normal systems in the presence of multiple leader, the states of the follow-ers for multi-agent singular systems asymptotically converge to the convex hull formed by the dynamic states of the leaders.
The admissible consensus problem of heterogeneous singular multi-agent systems is investigated. The modified protocols with identical models and different gains are designed for individual agent. The consensus error vector must be regular, impulse free (causal) and asymptotically stable. The above request is applied to obtain the sufficient conditions and construct the algorithms of consensus protocols. In essence, the consen-sus protocols given here solve the tracking problems of single leaders. Compared with the tracking problem for homogeneous singular multi-agent system, here the leader and the followers are heterogeneous. In contrast with the existing references about hetero-geneous singular multi-agent systems, this dissertation has three advantages:First, the sufficient conditions given are mild and easy to be verified. Second, the input matrix which is nonsingular in previous results is only requested to be column full rank here, and this reduces the rigorous condition greatly. Finally, the consensus protocols con-structed here eliminate two shortcomings:singular agent may have impulse term and the impulse term in the dynamics of consensus error vector is removed. Furthermore, the results given here are general, and they may be applied to leader-following con-sensus of multi-agent homogeneous singular systems, and they also may be applied to leader-following consensus of multi-agent heterogeneous normal systems.
研究了加速一致性问题.针对固定拓扑结构下离散多个体正常状态空间系统,基于一个修正的代数Riccati方程,提出了一种不需要非线性协议的解决方法,得出了以指定趋同速度因子实现一致的充分条件,并给出了期望趋同函数和指定趋同速度联合设计的协议算法.针对切换拓扑结构下连续和离散多个体系统,首先将通讯拓扑结构的切换特性转化为系统之间的切换,给出了多个体闭环系统实现一致的充分条件,并证明此时的趋同状态为零.最后推出并分析与趋同状态范数相等的一个向量范数的显式表达式,得出了趋同状态有界的条件,并且给出了提高趋同速度的协议算法.虽然已有很多文献研究切换拓扑结构下多个体系统能实现一致的充分条件,但是却鲜有文献讨论此通讯拓扑下高阶多个体系统趋同状态的形式和趋同速度提高的问题.研究发现趋同速度过快,可能会影响多个体闭环系统的稳定性,即印证了高增益对系统的不良影响,过分追求速度是不可取的.
研究了同质多个体奇异系统的协同控制问题.基于受限等价系统,采用了修正型一致性协议.针对连续系统和离散系统两种情形,基于包含有向生成树的通讯拓扑结构,得出了连续多个体奇异系统能实现容许一致的充要条件和离散多个体奇异系统能实现容许一致的充分条件,并给出了趋同状态的显式表达式以及能实现容许一致的趋同协议的算法.与已有的关于连续多个体奇异系统的结果相比较,本文给出的是充要条件,而非充分条件,本文给出的条件是非常容易进行事先验证的,而非待定的线性矩阵不等式.针对连续多个体奇异系统,探讨了单个首领的跟踪问题,得出了能实现跟踪的充要条件为奇异系统个体强能稳,并给出了协议增益的设计算法；提出了多个首领的牵制控制问题,得出了能实现该问题的充要条件仍是个体强能稳,然后给出了协议算法.与已有的多个体奇异系统跟踪问题的结果相比较,本文给出了充要条件,而非充分条件.与多个体正常状态空间系统的多首领牵制控制问题的结果类似,本文推出奇异系统个体的状态收敛于多个首领的动态状态的凸组合.
研究了异质多个体奇异系统的容许一致性问题.针对连续和离散两种情形,为异质个体设计了结构形式相同、参数不同的修正型一致性协议.借助趋同误差向量必须正则、无脉冲(或因果)和渐近稳定的特性,得出了能实现容许一致的充分条件,并构造了一种求解一致性协议增益的算法.本部分给出的控制协议,本质上实现了单个首领的跟踪问题,与同质多个体奇异系统的首领跟踪问题相比较,这里给出的首领和跟随者的动态系统是不同的.与已有的关于异质多个体奇异系统的结果相比较,有三点优势：一是本文给出的充分条件便于验证,且容易满足；二是对输入矩阵的要求为列满秩,而非非奇异矩阵,大大降低了对个体系统的苛刻要求；三是本文所构造的协议解决了已有文献的两个缺陷:一是奇异系统个体可以含有脉冲,二是消除了趋同误差向量动态特性中的脉冲项.另外,针对异质多个体奇异系统的容许一致性问题,本文所给出的理论结果和控制设计算法具有一般性,既适用于同质多个体奇异系统的跟踪首领问题,也适用于异质多个体正常状态空间系统的跟踪首领问题.
Multi-agent systems come from the actual control system. Research on them not only is the need of the development of modern control theory, and is urged to solve a large number of actual demands. The consensus problem of multi-agent systems is a fundamental problem of cooperative control problem, which is currently research hotspot. In the dissertation, combining the linear system theory, singular system theory, control theory, algebraic graph theory and matrix analysis, the consensus problems for both linear normal and linear singular multi-agent systems are investigated systemat-ically. The innovative work and key points of this dissertation can be summarized as follows:
The accelerated consensus problem of linear normal multi-agent systems is stud-ied. Based on a modified algebraic Riccati equation, an approach without nonlinear protocol is proposed and investigated for discrete-time case under time-invariant topolo-gies. A sufficient condition is derived, which guarantees that the multi-agent system solves consensus with prescribed consensus speed index. The algorithm is given to achieve the joint design of an expected consensus function and a prescribed consen-sus speed. For the consensus problems of continuous-time and discrete-time normal multi-agent systems with directed switching topologies, the switched behavior of com-munication topologies is transmitted to the dynamic systems which are transformed into switched subsystems. Then the consensus problem is solved under the assumption that each agent is stabilizable. Finally, it is shown that the consensus state is zero state. The sufficient condition to get bounded consensus state is derived by analyzing a vector whose norm is equal to consensus state's. The algorithms of improving the consensus speed have been given. Although many references have discussed the consensualization problem of multi-agent systems under switching topologies, few gave the expression of the consensus state and improved the consensus speed. It is found that the overquick consensus speed may destroy the stability of the closed-loop multi-agent system, i.e., high gain has bad effect for dynamic system and it is not advisable to only pursue speed.
Cooperative control problem for homogeneous singular multi-agent systems is discussed. Under the restricted system equivalence, the modified consensus proto-cols are proposed. The admissible consensus problems of both continuous-time and discrete-time multi-agent singular systems are investigated. Based on the communica-tion topologies containing a spanning tree, the sufficient and necessary conditions are presented for continuous-time multi-agent singular systems, and the sufficient condi-tions are presented for discrete-time multi-agent singular systems. The formulas of the consensus states are given and the algorithms of constructing the protocols for achieving admissible consensualization are provided. In contrast with the previous results which gave sufficient conditions that are composed of linear matrix inequalities to be deter-mined, this dissertation gives the sufficient and necessary conditions and the conditions are easy to be checked, verified and satisfied for the continuous-time multi-agent sys-tems. On the other hand, for continuous-time multi-agent singular systems, the tracking problem of single leader is discussed, and it is proved that the strong stabilization of each agent is a sufficient and necessary condition, and the algorithm to construct the protocol is given. The containment control problem of multiple leaders is studied. It is also shown that a sufficient and necessary condition of solving the above problem is strong stabilizable. The algorithms of the protocols are given. Compared with the ex-isting results which gave sufficient conditions for the tracking problem of single leader, the condition here is sufficient and necessary. Compared with the consensus state for multi-agent normal systems in the presence of multiple leader, the states of the follow-ers for multi-agent singular systems asymptotically converge to the convex hull formed by the dynamic states of the leaders.
The admissible consensus problem of heterogeneous singular multi-agent systems is investigated. The modified protocols with identical models and different gains are designed for individual agent. The consensus error vector must be regular, impulse free (causal) and asymptotically stable. The above request is applied to obtain the sufficient conditions and construct the algorithms of consensus protocols. In essence, the consen-sus protocols given here solve the tracking problems of single leaders. Compared with the tracking problem for homogeneous singular multi-agent system, here the leader and the followers are heterogeneous. In contrast with the existing references about hetero-geneous singular multi-agent systems, this dissertation has three advantages:First, the sufficient conditions given are mild and easy to be verified. Second, the input matrix which is nonsingular in previous results is only requested to be column full rank here, and this reduces the rigorous condition greatly. Finally, the consensus protocols con-structed here eliminate two shortcomings:singular agent may have impulse term and the impulse term in the dynamics of consensus error vector is removed. Furthermore, the results given here are general, and they may be applied to leader-following con-sensus of multi-agent homogeneous singular systems, and they also may be applied to leader-following consensus of multi-agent heterogeneous normal systems.
引文
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