基于时变采样周期的网络控制系统稳定性分析与综合
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摘要
传统的控制系统都假定受控对象按等周期采样,但是在网络控制系统中,等采样周期的假设并不成立,在采样器的采样周期为时变时,研究网络控制系统的建模、分析和控制器设计等问题具有较大的理论意义与现实意义。
对一个控制系统来说,采样器的采样周期越小,系统的性能越好。网络控制系统由于信息传递通道被多个用户共享,倘若采样周期变小,网络负载会增加。随着信息传递通道上传递信息的增加,时滞和信息丢失不可避免,甚至发生网络拥塞,这些可能破坏系统的稳定与性能。如果采用定长的采样周期采样,采样周期应该在可能的范围内尽量大,以避免网络由于拥塞而带来的诸多不良后果,这也导致了在网络闲置时不能充分地利用网络信息传递通道的带宽。为了保证网络的正常运行以及控制系统获得良好的性能,基于网络的繁忙或闲置可以采用不同的采样周期进行采样,这种方法称之为主动时变采样周期技术。另一种情况是,在采样控制系统中,故障等会导致采样周期发生时变,从而影响系统运行的性能,甚至影响系统的稳定性,我们称这种情况为被动时变采样周期。
本文主要从主动时变采样和被动时变采样两个方面对网络控制系统的建模、分析和控制器设计等问题进行了研究,主要工作如下。
第一项工作是研究了具有主动时变采样周期的网络控制系统。假定传感器是时钟驱动和事件驱动相结合的,控制器和执行器都是事件驱动的,应用主动时变采样周期技术研究了具有网络诱导时滞和数据包丢失的网络控制系统的H∞控制器设计问题。假定采样周期在一个已知有限集内随机切换,将具有主动时变采样周期的网络控制系统模型化为参数不确定系统,给出了保持系统渐近稳定的充分条件,将H∞控制器设计问题转化为求解多目标优化下的线性矩阵不等式问题。主动时变采样周期方法能够保证网络带宽得到充分的利用,所设计的控制器也适用于定常采样周期情形。
第二项工作是研究了被动时变采样周期按Markov链随机切换的网络控制系统,包含三个方面的内容。
不考虑时滞的影响,将连续时间线性时不变系统的被动时变采样周期视为一个分布已知的、在有限个值之间随机切换的Markov链,建立起采样控制系统的数学模型；然后,给出了闭环系统随机稳定的充分必要条件,将相应的控制器设计问题转化为矩阵不等式组的求解问题,尽管矩阵不等式组不是线性矩阵不等式组,但是锥补线性化等方法可解矩阵不等式组,相应的算法附在定理之后；接着,在考虑外部扰动的情况下,给出了闭环系统具有H∞性能的随机稳定的充分条件,相应的控制器设计问题被转化为线性矩阵不等式组的求解问题。
将连续时间线性时不变系统的被动时变采样周期视为一个分布已知的、在有限个值之间随机切换的Markov链,又将网络诱导时滞视为一个与被动时变采样周期独立的、分布已知的、在有限个值之间随机切换的Markov链,建立起采样控制系统的数学模型；然后,给出了闭环系统随机稳定的充分必要条件,相应的控制器设计问题被转化为矩阵不等式组的求解问题,同样可以应用锥补线性化等方法求解。
将前两个模型推广到更为一般的Markov链随机跳变系统,给出了闭环系统随机稳定的充分必要条件。在不存在公共控制器的情况下,所设计的模型依赖型控制器可以保证闭环系统随机稳定,从仿真结果可以看到这种模型依赖的控制器设计方法的优越性。
第三项工作是研究了具有确定性模型的被动时变采样周期网络控制系统。考虑一个具有网络诱导时滞、数据包丢失和被动时变采样周期的连续时间线性时不变网络控制系统,并允许执行器在一个采样周期内收到两个或两个以上的控制输入,给出了采样控制系统的数学模型。然后分别应用Jensen不等式和自由加权矩阵两种方法,给出了闭环系统渐近稳定,且具有相应的H∞范数界的充分条件。相应的H∞控制器设计问题被转化为在一组线性矩阵不等式条件约束下的多目标优化问题,应用一些方法可以求解这类多目标优化问题。所提出的控制器设计方法也适用于执行器在一个采样周期内收到零个或一个控制输入的情形。所建立的数学模型比已有成果更具有一般性,所设计的控制器在H∞性能指标和可允许的最大连续数据包丢失数两方面比已有成果更好。
The sampling period is often assumed to be constant in traditional control systems, but the assumption will not stand in networked control systems (NCSs). When the sampling period is time-varying, it is significant to study the modeling, analysis and controller design for Networked control systems.
For Networked control systems, the shorter the sampling period, the better the system performance. Since the communication channels in Networked control systems are shared by multiple users, if the sampling period is short, the network load will increase inevitably. With the increase of information transmitted through communication channels, time delay and packet dropout is unavoidable, and network congestion will also occur, which may destroy the stability and performance of systems. If the sampling period is constant, it should be large enough to avoid network congestion, however, the network bandwidth can not be sufficiently used when the network is idle. To ensure the normal functioning and good system performance, the sampling period may be different when the network is busy or idle, which is called active varying sampling period methodology. On the other hand, the failure in control systems will also lead to time-varying sampling period, which will deteriorate system performance and stability, and it is called passive time-varying sampling period.
This thesis study the modeling, analysis and controller design for Networked control systems with active time-varying sampling period and passive time-varying sampling period, and the main works are given as follows.
First, the thesis studies on a class of NCSs with active time-varying sampling period. Suppose the sensor is both time driven and event driven, controller and actuator are event driven, the H∞controller design for Networked control systems with active time-varying sampling period, time delay and packet dropout is studied. Suppose the sampling period switch stochastically in a finite set, the Networked control systems with active time-varying sampling period are modeled as parameter-uncertain system, the sufficient conditions ensuring the asymptotic stability of systems are presented, and the problem of H∞controller design is converted into a LMI problem with multiple object optimization. The active time-varying sampling period method can ensure the sufficient use of network bandwidth, and the designed controller is also applicable for Networked control systems with constant sampling period.
Second, the thesis studies on NCSs with passive time-varying sampling period driven by Markov chain, it consistes of three parts.
Without considering the influence of time delay, passive time-varying sampling periods of continuous time linear time-invariant systems are modeled as Markov chain that switch in finite values, and the mathematics model for sampling control systems is presented; then the stochastic stability conditions for closed-loop systems are proposed, and the controller design is converted into a set of matrix inequalities, although the matrix inequalities are not linear, they can be transferred into linear matrix inequalities by using the cone complementarity approach, and the corresponding algorithm is given behind the theorem; then with the consideration of external disturbances, the sufficient conditions ensuring the H∞performance of systems are presented, and the controller design is transferred into an LMI problem.
By considering the passive time-varying sampling period of continuous time linear time-invariant systems as distribution known and stochastic switching Markov chain, and considering the network-induced time delay which is independent on passive time-varying sampling period as distribution known, stochastic switching in finite values, the mathematics model is presented for sampling control system; then the sufficient and necessary conditions ensuring the stochastic stability of closed-loop system are proposed, and the problem of controller design is converted into the solution of matrix inequalities, which can also be solved by cone complementarity approach.
The above two models are extended to more general Markov chain stochastic jumping system, and the sufficient and necessary conditions ensuring the stochastic stability of closed-loop system are presented. When there does not exist common controller, the designed model-dependent controller can ensure the stochastic stability of closed-loop system, the merits of the proposed methods are shown by numerical examples.
Third, the thesis studies on a class of NCSs with passive time-varying sampling period and certain model. Considering continuous time linear time-invariant systems with network-induced time delay, packet dropout and passive time-varying sampling period, and the actuator may receive two or more than two control inputs during a sampling period, the model for sampling control system is presented. By using Jensen inequality and free weight matrix method, the sufficient conditions ensuring the asymptotic stability and H∞norm bounds are presented. The problem of H∞controller design is converted into a multiple object optimization problem with the constraint of a set of LMIs, which can be solved effectively. The proposed controller design is also applicable when the actuator receives zero or one control input during a sampling period. The given model is more general than the existing ones, and the designed controller can ensure better H∞performance and admissible maximum number of consecutive packet dropout than the existing results.
对一个控制系统来说,采样器的采样周期越小,系统的性能越好。网络控制系统由于信息传递通道被多个用户共享,倘若采样周期变小,网络负载会增加。随着信息传递通道上传递信息的增加,时滞和信息丢失不可避免,甚至发生网络拥塞,这些可能破坏系统的稳定与性能。如果采用定长的采样周期采样,采样周期应该在可能的范围内尽量大,以避免网络由于拥塞而带来的诸多不良后果,这也导致了在网络闲置时不能充分地利用网络信息传递通道的带宽。为了保证网络的正常运行以及控制系统获得良好的性能,基于网络的繁忙或闲置可以采用不同的采样周期进行采样,这种方法称之为主动时变采样周期技术。另一种情况是,在采样控制系统中,故障等会导致采样周期发生时变,从而影响系统运行的性能,甚至影响系统的稳定性,我们称这种情况为被动时变采样周期。
本文主要从主动时变采样和被动时变采样两个方面对网络控制系统的建模、分析和控制器设计等问题进行了研究,主要工作如下。
第一项工作是研究了具有主动时变采样周期的网络控制系统。假定传感器是时钟驱动和事件驱动相结合的,控制器和执行器都是事件驱动的,应用主动时变采样周期技术研究了具有网络诱导时滞和数据包丢失的网络控制系统的H∞控制器设计问题。假定采样周期在一个已知有限集内随机切换,将具有主动时变采样周期的网络控制系统模型化为参数不确定系统,给出了保持系统渐近稳定的充分条件,将H∞控制器设计问题转化为求解多目标优化下的线性矩阵不等式问题。主动时变采样周期方法能够保证网络带宽得到充分的利用,所设计的控制器也适用于定常采样周期情形。
第二项工作是研究了被动时变采样周期按Markov链随机切换的网络控制系统,包含三个方面的内容。
不考虑时滞的影响,将连续时间线性时不变系统的被动时变采样周期视为一个分布已知的、在有限个值之间随机切换的Markov链,建立起采样控制系统的数学模型；然后,给出了闭环系统随机稳定的充分必要条件,将相应的控制器设计问题转化为矩阵不等式组的求解问题,尽管矩阵不等式组不是线性矩阵不等式组,但是锥补线性化等方法可解矩阵不等式组,相应的算法附在定理之后；接着,在考虑外部扰动的情况下,给出了闭环系统具有H∞性能的随机稳定的充分条件,相应的控制器设计问题被转化为线性矩阵不等式组的求解问题。
将连续时间线性时不变系统的被动时变采样周期视为一个分布已知的、在有限个值之间随机切换的Markov链,又将网络诱导时滞视为一个与被动时变采样周期独立的、分布已知的、在有限个值之间随机切换的Markov链,建立起采样控制系统的数学模型；然后,给出了闭环系统随机稳定的充分必要条件,相应的控制器设计问题被转化为矩阵不等式组的求解问题,同样可以应用锥补线性化等方法求解。
将前两个模型推广到更为一般的Markov链随机跳变系统,给出了闭环系统随机稳定的充分必要条件。在不存在公共控制器的情况下,所设计的模型依赖型控制器可以保证闭环系统随机稳定,从仿真结果可以看到这种模型依赖的控制器设计方法的优越性。
第三项工作是研究了具有确定性模型的被动时变采样周期网络控制系统。考虑一个具有网络诱导时滞、数据包丢失和被动时变采样周期的连续时间线性时不变网络控制系统,并允许执行器在一个采样周期内收到两个或两个以上的控制输入,给出了采样控制系统的数学模型。然后分别应用Jensen不等式和自由加权矩阵两种方法,给出了闭环系统渐近稳定,且具有相应的H∞范数界的充分条件。相应的H∞控制器设计问题被转化为在一组线性矩阵不等式条件约束下的多目标优化问题,应用一些方法可以求解这类多目标优化问题。所提出的控制器设计方法也适用于执行器在一个采样周期内收到零个或一个控制输入的情形。所建立的数学模型比已有成果更具有一般性,所设计的控制器在H∞性能指标和可允许的最大连续数据包丢失数两方面比已有成果更好。
The sampling period is often assumed to be constant in traditional control systems, but the assumption will not stand in networked control systems (NCSs). When the sampling period is time-varying, it is significant to study the modeling, analysis and controller design for Networked control systems.
For Networked control systems, the shorter the sampling period, the better the system performance. Since the communication channels in Networked control systems are shared by multiple users, if the sampling period is short, the network load will increase inevitably. With the increase of information transmitted through communication channels, time delay and packet dropout is unavoidable, and network congestion will also occur, which may destroy the stability and performance of systems. If the sampling period is constant, it should be large enough to avoid network congestion, however, the network bandwidth can not be sufficiently used when the network is idle. To ensure the normal functioning and good system performance, the sampling period may be different when the network is busy or idle, which is called active varying sampling period methodology. On the other hand, the failure in control systems will also lead to time-varying sampling period, which will deteriorate system performance and stability, and it is called passive time-varying sampling period.
This thesis study the modeling, analysis and controller design for Networked control systems with active time-varying sampling period and passive time-varying sampling period, and the main works are given as follows.
First, the thesis studies on a class of NCSs with active time-varying sampling period. Suppose the sensor is both time driven and event driven, controller and actuator are event driven, the H∞controller design for Networked control systems with active time-varying sampling period, time delay and packet dropout is studied. Suppose the sampling period switch stochastically in a finite set, the Networked control systems with active time-varying sampling period are modeled as parameter-uncertain system, the sufficient conditions ensuring the asymptotic stability of systems are presented, and the problem of H∞controller design is converted into a LMI problem with multiple object optimization. The active time-varying sampling period method can ensure the sufficient use of network bandwidth, and the designed controller is also applicable for Networked control systems with constant sampling period.
Second, the thesis studies on NCSs with passive time-varying sampling period driven by Markov chain, it consistes of three parts.
Without considering the influence of time delay, passive time-varying sampling periods of continuous time linear time-invariant systems are modeled as Markov chain that switch in finite values, and the mathematics model for sampling control systems is presented; then the stochastic stability conditions for closed-loop systems are proposed, and the controller design is converted into a set of matrix inequalities, although the matrix inequalities are not linear, they can be transferred into linear matrix inequalities by using the cone complementarity approach, and the corresponding algorithm is given behind the theorem; then with the consideration of external disturbances, the sufficient conditions ensuring the H∞performance of systems are presented, and the controller design is transferred into an LMI problem.
By considering the passive time-varying sampling period of continuous time linear time-invariant systems as distribution known and stochastic switching Markov chain, and considering the network-induced time delay which is independent on passive time-varying sampling period as distribution known, stochastic switching in finite values, the mathematics model is presented for sampling control system; then the sufficient and necessary conditions ensuring the stochastic stability of closed-loop system are proposed, and the problem of controller design is converted into the solution of matrix inequalities, which can also be solved by cone complementarity approach.
The above two models are extended to more general Markov chain stochastic jumping system, and the sufficient and necessary conditions ensuring the stochastic stability of closed-loop system are presented. When there does not exist common controller, the designed model-dependent controller can ensure the stochastic stability of closed-loop system, the merits of the proposed methods are shown by numerical examples.
Third, the thesis studies on a class of NCSs with passive time-varying sampling period and certain model. Considering continuous time linear time-invariant systems with network-induced time delay, packet dropout and passive time-varying sampling period, and the actuator may receive two or more than two control inputs during a sampling period, the model for sampling control system is presented. By using Jensen inequality and free weight matrix method, the sufficient conditions ensuring the asymptotic stability and H∞norm bounds are presented. The problem of H∞controller design is converted into a multiple object optimization problem with the constraint of a set of LMIs, which can be solved effectively. The proposed controller design is also applicable when the actuator receives zero or one control input during a sampling period. The given model is more general than the existing ones, and the designed controller can ensure better H∞performance and admissible maximum number of consecutive packet dropout than the existing results.
引文
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