网络化群体系统编队与一致性控制
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摘要
网络化群体系统的协调控制是近年来研究的热点，其在无人飞行器编队、自动高速公路、传感器网络等工程领域具有重要的意义。这些系统都有一个共同的特点，那就是群体之间的交互作用是局部的，而整体却呈现出较为复杂的协调行为。本文主要研究了三个方面的问题：时滞不确定性网络化群体编队系统的鲁棒稳定性；网络化群体系统一致性的收敛性能和时滞鲁棒性能分析；网络化群体一致性的鲁棒控制。
网络化群体编队系统的控制涉及到各个个体局部控制器之间的协调，网络互联拓扑结构分析以及群体系统的群体编队任务。通常，由于网络带宽的限制以及网络拥塞等因素，通信网络总是伴随着时滞。此外，由于环境以及系统本身的不确定性，以数学模型为基础的理论分析也一定要考虑到模型的不确定性对系统的影响。本文考虑了静态无向和有向通信拓扑结构、个体的模型为更一般的线性系统、控制规律为线性的情形，采用了两种方法来分析了带时滞网络化群体系统的编队稳定性。一种方法是基于线性矩阵不等式的方法，而另外一种是采用类似于Nyquist稳定判据的频域法。第一种方法中，针对网络化群体编队系统，设计了PD控制器，首先给出了通信时滞为时不变的情形下群体编队系统的时滞独立和时滞依赖的稳定性条件，接着给出了系统的模型带有不确定情形的时滞群体编队系统的时滞独立和时滞依赖的鲁棒稳定性条件。在此基础上，为了进一步降低结果的保守性以及将结果扩展到时变时滞的群体系统，采用了自由权矩阵方法结合线性矩阵不等式工具分别给出了时变时滞群体编队系统的稳定性条件，以及带模型不确定性时变时滞群体编队系统的鲁棒稳定性条件。相关的仿真和计算表明了结果的可行性和有效性。在第二种方法中，所设计的控制器为一般的线性控制器，结论同时适用于有向和无向图两种情形。论文分别给出了此时群体编队系统稳定性的等价条件，以及设计控制器的准则。
网络化群体系统编队与一致性控制的问题中，系统的收敛速度和其对时滞鲁棒性的分析也非常重要。在许多工程群体系统中都要求系统拥有一定的收敛速度的同时还需要能容忍一定的通信时滞。因此，如何提高系统的收敛性能和时滞鲁棒性显得十分必要。本文针对如何提高一阶群体系统和高阶群体系统的收敛性和时滞鲁棒性问题分别给出了解决方法。针对一阶群体系统，提出了网络最优权的设计方法，使一阶群体系统在满足一定的收敛速度的前提下能容忍最大的通信时滞。在这种方法中，通过设计网络化群体系统构成的无向通信的拓扑结构图的边权来使系统的在满足较快的收敛速度同时时滞鲁棒性达到最强。因为这个时候群体系统的收敛速度与图Laplacian矩阵的第二小特征值的大小成正比，而系统能容忍的最大时滞则与Laplacian的最大特征值成反比，本文通过将Laplacian进行适当变形后，可以将提高收敛性和时滞鲁棒性的问题等价转换为一个正定矩阵的条件数的优化问题，而这个问题是凸优化问题，可以很方便的进行求解。针对二阶双积分系统，本文采用仿真的方法分析了控制器参数对时滞群体编队系统的收敛性和时滞鲁棒性的影响，得出了控制器的参数对性能影响的规律，在此基础上确定了最优的控制器参数。为了更进一步提高系统的时滞鲁棒性，论文还首次采用了分数阶PD~μ控制器来控制群体编队系统，分析了微分阶次μ与时滞鲁棒性的关系并确定了最优的微分阶次μ。仿真结果表明所设计的分数阶PD~μ控制器能大大提高系统的收敛性能和时滞鲁棒性能。
对于群体系统来说，尤其是实际的工程系统来说，总是存在着这样和那样的干扰与噪声。在存在着扰动和噪声的前提下，原本能达到一致的群体系统未必会再达到一致的状态。所以对群体系统设计合适的控制器使系统达到一致并且对噪声与干扰是性能鲁棒的是非常必要和必需的。本文分别给出了静态全连通、具有某种特殊拓扑结构两种情形下的网络化群体系统一致性的H_2、H_∞、H_2／H_∞控制器的设计方法。论文给出了连续时间群体系统在有限能量噪声、白噪声有限能量噪声同时存在情形下的鲁棒控制器设计方法。接着将结果继续扩展到离散时间群体系统，分别给出了离散时间群体系统在白噪声、有限能量噪声、白噪声有限能量噪声同时存在的情形下的一致性鲁棒控制器条件。仿真结果表明根据论文给出的理论可以设计出状态反馈控制器使系统能达到一致并且满足相应的鲁棒性能指标。
最后，对全文的研究工作做了总结，并对今后要进一步开展的研究工作进行了展望。
Recent years have seen the emergence of formations of swarm agents as a topic ofsignificant interest to the control community. Swarm agents systems have appeared widely inmany applications including formation flight of Unmanned Air Vehicles (UAVs), automatedhighway systems, and sensor networks. One common feature for these systems is thatcomplicated coordinated behaviors are exhibited by interactions among agents whereinformation exchange is local. In this work, three relevant issues are investigated in detail:robust stability of networked swarm agents systems with time delays and model uncertainties,analysis on convergence and robusmess for consensus of networked swarm agents, robustcontroller for consensus of swarm systems with extemal noises or disturbances.
Formation control of multi-agent systems involves harmony among local controller design,interaction topology analysis, and common tasks among networked agents. Usually, due tolimited bandwidth and congestion, time delays always exist in communication networkcomposed of agents. Otherwise, since most engineering systems designs are based onmathematical model, the models and realities they represent are always different. To be practical,model uncertainties must be considered. This study provides a theoretical analysis for stabilityof continuous-time formation system with time delays in fixed, undirected and directed graphbased on two methods. One method is in terms of linear matrix inequlity, and the other is basedon frequency domain theory like Nyquist Criterion. Firstly, time delay is time invariant, and thePD controller is used in formation, Two cases of delay-independent and delay-dependentvehicles formations are considered. Then delay-independent and delay-dependent robuststability conditions with model and feedback uncertainties are presented. The sufficientconditions are given to guarantee that with time delays and uncertainties the vehicles formationcan asymptotically converge to predeflned formation. Furthermore, a way based on free weightsmatrix and linear matrix inequlity is used to reduce conservativeness and extend these results oncase of time invariant delays to the case of time varying delays. Then the stability and robuststability conditions are given for swarm formation system with time varying delays on cases ofwithout model uncertainty and with model uncertainty, respectively. Simulations are provided toverify the feasibility and efficiency of these results. Secondly, a more general linear controller isdesigned. The method based on frequency domain theory is suitable to both cases of undirected and directed graph. It is proved that stability of N Vehicles formations is equivalentto the stability of N-1 subsystems which are related to eigenvalues of graph Laplacian.
Convergence speed and robustnee to delays are very important in coordination of swarmagents. In most engineering systems, a fast convergence speed and robustness are necessary.This dissertation proposes solutions to consensus of one order swarm system and high ordermulti-agent system. Aimed at one order system, a way to design the optimal weights associatedwith edges of undirected graph composed of multi-agent systems is presented. The optimalweights are designed to make the states of multi-agent systems converge to consensus with afast speed as well as the maximum communication time-delay can be tolerated. Theconvergence speed which is determined by the second-smallest eigenvalue of graph Laplacianmatrix is assumed to be a given value, at the same time the maximum communication timedelaywhich is decided by the maximum eigenvalue of Laplacian can be got. In order to getrequired second-smallest eigenvalue and optimal maximum eigenvalues, the order of Laplacianis reduced by variable decomposition. Moreover, designing the optimal weights is equivalent tominimizing condition number of a positive-definite matrix. This is a convex optimizationproblem. Simulation results are coincidental with theoretical analysis. Aimed at second andhigher order system, simulations are provided to show the effects of controller paremeters onstability of formation control. The optimal parameters of controller are decided. In order toimprove the robusmess, fractional PD~μcontroller is presented to control formation system.How optimal parameterμaffects robusmess to delays also is analized by simulations. Thenthe optimal value is got. Simulations show that the fractional PD~μcontroller can improverobustness significantly.
It is very obvious that there are always disturbances and noises in swarm systems,especially in practical engineering systems. With disturbances the system will be not stable anymore. So the controller for the system in case of no noise will do not work again. It is veryimportant and necessary to design a robust controller to make the states of swarm agents reachconsensus with some performance under various of external and internal disturbances. Thisdissertation proposes a way to design robust controller for consensus of the system so as tomake the closed-loop system reach consensus with non-consensus part be Lyapunov stablemeeting certain performance for disturbances attenuation. Two cases are considered. The firstone is that the agents in the swarm can interact with every other agent when each agent can receive information from every other. The other one is that the topology structure composedby the agents is fixed when each agent in the swarm can only exchange information with someagents but not all other agents. The methods to design H_∞H_2/H_∞controllers for consensusof networked continuous-time swarm system are given with white noises and persistant noises.Then how to design H_2、H_∞、H_2/H_∞controller for consensus of discrete-time systems arealso stated. Simulation results demonstrate that designed controller for the system can make theclosed-loop system reach consensus with non-consensus part be Lyapunov stable meetingcorossponding performance for disturbances attenuation.
Finally, a summary has been done for all discussions in the dissertation. The researchworks in further study are presented.
网络化群体编队系统的控制涉及到各个个体局部控制器之间的协调，网络互联拓扑结构分析以及群体系统的群体编队任务。通常，由于网络带宽的限制以及网络拥塞等因素，通信网络总是伴随着时滞。此外，由于环境以及系统本身的不确定性，以数学模型为基础的理论分析也一定要考虑到模型的不确定性对系统的影响。本文考虑了静态无向和有向通信拓扑结构、个体的模型为更一般的线性系统、控制规律为线性的情形，采用了两种方法来分析了带时滞网络化群体系统的编队稳定性。一种方法是基于线性矩阵不等式的方法，而另外一种是采用类似于Nyquist稳定判据的频域法。第一种方法中，针对网络化群体编队系统，设计了PD控制器，首先给出了通信时滞为时不变的情形下群体编队系统的时滞独立和时滞依赖的稳定性条件，接着给出了系统的模型带有不确定情形的时滞群体编队系统的时滞独立和时滞依赖的鲁棒稳定性条件。在此基础上，为了进一步降低结果的保守性以及将结果扩展到时变时滞的群体系统，采用了自由权矩阵方法结合线性矩阵不等式工具分别给出了时变时滞群体编队系统的稳定性条件，以及带模型不确定性时变时滞群体编队系统的鲁棒稳定性条件。相关的仿真和计算表明了结果的可行性和有效性。在第二种方法中，所设计的控制器为一般的线性控制器，结论同时适用于有向和无向图两种情形。论文分别给出了此时群体编队系统稳定性的等价条件，以及设计控制器的准则。
网络化群体系统编队与一致性控制的问题中，系统的收敛速度和其对时滞鲁棒性的分析也非常重要。在许多工程群体系统中都要求系统拥有一定的收敛速度的同时还需要能容忍一定的通信时滞。因此，如何提高系统的收敛性能和时滞鲁棒性显得十分必要。本文针对如何提高一阶群体系统和高阶群体系统的收敛性和时滞鲁棒性问题分别给出了解决方法。针对一阶群体系统，提出了网络最优权的设计方法，使一阶群体系统在满足一定的收敛速度的前提下能容忍最大的通信时滞。在这种方法中，通过设计网络化群体系统构成的无向通信的拓扑结构图的边权来使系统的在满足较快的收敛速度同时时滞鲁棒性达到最强。因为这个时候群体系统的收敛速度与图Laplacian矩阵的第二小特征值的大小成正比，而系统能容忍的最大时滞则与Laplacian的最大特征值成反比，本文通过将Laplacian进行适当变形后，可以将提高收敛性和时滞鲁棒性的问题等价转换为一个正定矩阵的条件数的优化问题，而这个问题是凸优化问题，可以很方便的进行求解。针对二阶双积分系统，本文采用仿真的方法分析了控制器参数对时滞群体编队系统的收敛性和时滞鲁棒性的影响，得出了控制器的参数对性能影响的规律，在此基础上确定了最优的控制器参数。为了更进一步提高系统的时滞鲁棒性，论文还首次采用了分数阶PD~μ控制器来控制群体编队系统，分析了微分阶次μ与时滞鲁棒性的关系并确定了最优的微分阶次μ。仿真结果表明所设计的分数阶PD~μ控制器能大大提高系统的收敛性能和时滞鲁棒性能。
对于群体系统来说，尤其是实际的工程系统来说，总是存在着这样和那样的干扰与噪声。在存在着扰动和噪声的前提下，原本能达到一致的群体系统未必会再达到一致的状态。所以对群体系统设计合适的控制器使系统达到一致并且对噪声与干扰是性能鲁棒的是非常必要和必需的。本文分别给出了静态全连通、具有某种特殊拓扑结构两种情形下的网络化群体系统一致性的H_2、H_∞、H_2／H_∞控制器的设计方法。论文给出了连续时间群体系统在有限能量噪声、白噪声有限能量噪声同时存在情形下的鲁棒控制器设计方法。接着将结果继续扩展到离散时间群体系统，分别给出了离散时间群体系统在白噪声、有限能量噪声、白噪声有限能量噪声同时存在的情形下的一致性鲁棒控制器条件。仿真结果表明根据论文给出的理论可以设计出状态反馈控制器使系统能达到一致并且满足相应的鲁棒性能指标。
最后，对全文的研究工作做了总结，并对今后要进一步开展的研究工作进行了展望。
Recent years have seen the emergence of formations of swarm agents as a topic ofsignificant interest to the control community. Swarm agents systems have appeared widely inmany applications including formation flight of Unmanned Air Vehicles (UAVs), automatedhighway systems, and sensor networks. One common feature for these systems is thatcomplicated coordinated behaviors are exhibited by interactions among agents whereinformation exchange is local. In this work, three relevant issues are investigated in detail:robust stability of networked swarm agents systems with time delays and model uncertainties,analysis on convergence and robusmess for consensus of networked swarm agents, robustcontroller for consensus of swarm systems with extemal noises or disturbances.
Formation control of multi-agent systems involves harmony among local controller design,interaction topology analysis, and common tasks among networked agents. Usually, due tolimited bandwidth and congestion, time delays always exist in communication networkcomposed of agents. Otherwise, since most engineering systems designs are based onmathematical model, the models and realities they represent are always different. To be practical,model uncertainties must be considered. This study provides a theoretical analysis for stabilityof continuous-time formation system with time delays in fixed, undirected and directed graphbased on two methods. One method is in terms of linear matrix inequlity, and the other is basedon frequency domain theory like Nyquist Criterion. Firstly, time delay is time invariant, and thePD controller is used in formation, Two cases of delay-independent and delay-dependentvehicles formations are considered. Then delay-independent and delay-dependent robuststability conditions with model and feedback uncertainties are presented. The sufficientconditions are given to guarantee that with time delays and uncertainties the vehicles formationcan asymptotically converge to predeflned formation. Furthermore, a way based on free weightsmatrix and linear matrix inequlity is used to reduce conservativeness and extend these results oncase of time invariant delays to the case of time varying delays. Then the stability and robuststability conditions are given for swarm formation system with time varying delays on cases ofwithout model uncertainty and with model uncertainty, respectively. Simulations are provided toverify the feasibility and efficiency of these results. Secondly, a more general linear controller isdesigned. The method based on frequency domain theory is suitable to both cases of undirected and directed graph. It is proved that stability of N Vehicles formations is equivalentto the stability of N-1 subsystems which are related to eigenvalues of graph Laplacian.
Convergence speed and robustnee to delays are very important in coordination of swarmagents. In most engineering systems, a fast convergence speed and robustness are necessary.This dissertation proposes solutions to consensus of one order swarm system and high ordermulti-agent system. Aimed at one order system, a way to design the optimal weights associatedwith edges of undirected graph composed of multi-agent systems is presented. The optimalweights are designed to make the states of multi-agent systems converge to consensus with afast speed as well as the maximum communication time-delay can be tolerated. Theconvergence speed which is determined by the second-smallest eigenvalue of graph Laplacianmatrix is assumed to be a given value, at the same time the maximum communication timedelaywhich is decided by the maximum eigenvalue of Laplacian can be got. In order to getrequired second-smallest eigenvalue and optimal maximum eigenvalues, the order of Laplacianis reduced by variable decomposition. Moreover, designing the optimal weights is equivalent tominimizing condition number of a positive-definite matrix. This is a convex optimizationproblem. Simulation results are coincidental with theoretical analysis. Aimed at second andhigher order system, simulations are provided to show the effects of controller paremeters onstability of formation control. The optimal parameters of controller are decided. In order toimprove the robusmess, fractional PD~μcontroller is presented to control formation system.How optimal parameterμaffects robusmess to delays also is analized by simulations. Thenthe optimal value is got. Simulations show that the fractional PD~μcontroller can improverobustness significantly.
It is very obvious that there are always disturbances and noises in swarm systems,especially in practical engineering systems. With disturbances the system will be not stable anymore. So the controller for the system in case of no noise will do not work again. It is veryimportant and necessary to design a robust controller to make the states of swarm agents reachconsensus with some performance under various of external and internal disturbances. Thisdissertation proposes a way to design robust controller for consensus of the system so as tomake the closed-loop system reach consensus with non-consensus part be Lyapunov stablemeeting certain performance for disturbances attenuation. Two cases are considered. The firstone is that the agents in the swarm can interact with every other agent when each agent can receive information from every other. The other one is that the topology structure composedby the agents is fixed when each agent in the swarm can only exchange information with someagents but not all other agents. The methods to design H_∞H_2/H_∞controllers for consensusof networked continuous-time swarm system are given with white noises and persistant noises.Then how to design H_2、H_∞、H_2/H_∞controller for consensus of discrete-time systems arealso stated. Simulation results demonstrate that designed controller for the system can make theclosed-loop system reach consensus with non-consensus part be Lyapunov stable meetingcorossponding performance for disturbances attenuation.
Finally, a summary has been done for all discussions in the dissertation. The researchworks in further study are presented.
引文
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