分段仿射系统的控制器设计及预测控制方法研究
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摘要
在实际工业过程中,大量存在着一类用分段仿射结构(PWA)表示的混杂系统,这类系统常用一组线性子模型与其相应的作用域凸多面体表示。该系统被证明完全等价于其它结构类型的混杂系统。混杂系统是由离散事件动态系统与连续时间动态系统或离散时间动态系统相互混合、相互作用而形成的统一动态系统。混杂系统理论的提出既是社会经济发展的需要,也是计算机科学和控制科学发展的必然结果。分段仿射系统是混杂系统的重要分支之一,它由若干子系统组成,即由切换将逻辑动态和连续动态连接在一起。理论上,分段仿射系统可以逼近任意的非线性系统。
近十年来,国内外学者在分段仿射系统的理论与工程应用方面做了大量的工作,并取得了一系列的研究成果。由于分段仿射系统本质上是属于一类非光滑、非连续的复杂非线性系统,同时由于分段仿射系统具有广泛的工程背景,因此分段仿射系统的理论与应用研究是系统工程与自动控制领域当前研究的难点与热点之一,具有重要的理论价值和实际应用意义。
本文在综述前人研究的基础上,对于分段仿射系统的理论与应用作了进一步研究和探索。本文的研究工作主要集中在以下五个方面:
对分段仿射系统的产生背景,概念,以及研究的必要性和重要性作了综述,着重对分段仿射模型的建模方法作了研究,列举了基于分段仿射系统建模的工程应用实例。
基于简化约束区域计算的目的,采用分段区域划分技术,把分段仿射系统的每个子系统的状态作用空间用椭圆集的形式表示出来,每个子系统在其相应的区域划分内发生作用,并且在不同的区域划分之间发生切换。基于分段Lyapunov函数设计出能使PWA系统在各个区域划分之间任意切换的稳定的控制器;并且研究了当分段仿射系统带有扇区不确定性或者参数摄动不确定性时系统的鲁棒控制器设计问题,给出相应的设计方法。
在实际工程中,具有确定切换序列的分段仿射系统极其普遍,而且控制量带有饱和约束的分析和综合问题也是经常遇到的。针对这种具有确定切换序列的并且带有输入约束的分段仿射系统,为降低一般切换控制器设计的保守性,提出了多模预测控制方法。将多模型准无穷时域预测的指标函数分割成有限时域与无限时域两部分;其中有限时域性能指标优化求解出的自由变量可以使分段仿射系统按照确定的序列切换到指定的子系统模型区域内;无限时域性能指标优化求解出的反馈控制器使系统稳定到期望的平衡点;理论上将系统的稳定问题归结为线性矩阵不等式的凸优化求解可行性问题。在此基础上,通过附加的终端椭圆的渐近收敛约束保证系统更快收敛到平衡点,基于LMI解决了系统的输入受约束的问题。最后通过对一个聚合反应釜(CSTR)模型的仿真证实了上述两种算法的有效性。
而实际系统中系统模型往往是不确定或者缓慢变化的,针对多胞型不确定分段仿射系统,提出了鲁棒预测控制算法。基于预测控制的凸组合理论,给出了多步鲁棒预测控制的性能指标上确界,并将上确界的最小化问题转化为LMI优化问题;该方法不但考虑了模型的不确定性,而且证明了闭环系统的鲁棒稳定性和算法滚动实现的可行性。最后通过仿真验证了算法的有效性。
此外,针对切换顺序已知,并且输入受约束的子系统带有时滞项的分段仿射系统,提出基于椭圆集约束的模型预测控制。该算法基于Lyapunov-Krasovskii函数给出了性能指标上界和系统稳定的充分条件,将性能指标最小化、稳定性约束、输入约束转化为容易求解的LMI约束。通过陆地自主车的仿真验证了该方法的有效性。
In engineering practice, there are many hybrid systems described by piecewise affine systems (PWA) which are composed of linear subsystems and convex polytopic regions. Hybrid systems are composed of discrete event dynamic systems and continuous time dynamic systems or discrete time dynamic systems, which interact on each other. The hybrid system theory, which is proposed for the demand of the economic development, is the result of the development of computer science and control theory. Piecewise affine system is one of the most important branches of hybrid system, and is also straightforward. It consists of some subsystems that integrate the logical and continuous dynamics by switching. Theoretically, any nonlinear system can be approximated as piecewise affine system.
For decades, scientists have conducted much research work in PWA system and its applications in many engineering problems, and make a series of satisfied achievements. But, because of the difficulty and the popularity of the PWA system (i.e. PWA is essentially a non-smooth and even non- continuous nonlinear system and hybrid system has a extensive engineering background), up to today, the topic of PWA system and its applications is one of the most difficult research work and also one of the hot spots of studies in the field of system and control science.
On the basis of summarizing and reviewing previous research work, this dissertation makes some further research work and exploration in PWA and its applications in nonlinear systems. The research work of this thesis consists of following five aspects:
The concept and the background of the development of PWA, the importance and necessity of studies of PWA are explained. Various widely used modeling approaches for hybrid systems, their characteristics and the equivalences of typical hybrid models are summarized. The work focuses mainly on important subclass of hybrid systems: piecewise affine systems (PWA). Typical examples of the applications of PWA in engineering practice are also illustrated.
Using the convex polytopes technique, the corresponding operating region of the PWA systems in state space is described as ellipsoid which can be characterized by a set of vector inequalities. With these ellipsoid partitions, each model is valid in a given region of a certain space, and the PWA systems switches between different partitions. Based on the common Lyapunov and multiple Lyapunov functions, the stable controller is designed for the PWA systems involving mode switching between different operating regions. Furthermore, the design methods of the robust controllers are proposed for the PWA with sector uncertainty or parameter perturbation respectively.
In practice, the PWA systems with certain switching order exist universally. Simultaneously, the analysis and synthesis of control systems with actuator saturation are met frequently. To decrease the conservation of the controller which designed for the PWA systems with constrained control input based on certain switching order, a multi-model predictive control (MPC) algorithm is developed by a sequence of piecewise models along the transition trajectory. The control algorithm is a receding horizon scheme with a quasi-infinite horizon objective function that has finite and infinite horizon cost components. The finite horizon cost consists of free input variables that direct the system towards a terminal region that contains the desired operating point. The infinite horizon cost has an upper bound and takes the system to the final operating point. The control problem is formulated as a convex optimization in terms of Linear Matrix Inequalities (LMIs). Furthermore, on the basis of former algorithm, an additional terminal ellipsoid is introduced to ensure the states converging to the equilibrium faster. The problem of constrained control input is solved by convex optimization involving LMIs. Lastly, a simulation involving Continuous Stirred Tank Reactor (CSTR) model results verify the effectiveness of these proposed methods.
In practice, systems are generally uncertain and time-varying. A robust MPC is proposed for PWA systems with polytopic uncertainties. The problem of minimizing an upper bound on the worst-case objective function is reduced to a convex optimization involving LMIs. The feasible free input variables guarantee the PWA systems switching in certain order, and the receding horizon state-feedback control guarantees closed-loop robust asymptotic stability and input constraints. The approach allows explicit incorporation of the description of plant uncertainty. Furthermore, the closed-loop robust stability and feasibility of receding horizon implementation are proved. The simulation results verify the effectiveness of the proposed method.
A model predictive control (MPC) is proposed for a time-delay PWA systems with constrained input and certain switching order. The corresponding operating region of the considered systems in state space is described as ellipsoid which can be characterized by a set of vector inequalities. The control law is obtained by convex optimization based on MPC involving linear matrix inequalities (LMIs). And the constrained control input of the considered systems is solved in terms of LMIs. The simulation results verify the effectiveness of the proposed method. It is shown that, based on LMI constraints, it is easy to get the MPC for the PWA systems with time-delay and a simulation involving Autonomous Land Vehicle (ALV) model results verify the effectiveness of these proposed methods.
近十年来,国内外学者在分段仿射系统的理论与工程应用方面做了大量的工作,并取得了一系列的研究成果。由于分段仿射系统本质上是属于一类非光滑、非连续的复杂非线性系统,同时由于分段仿射系统具有广泛的工程背景,因此分段仿射系统的理论与应用研究是系统工程与自动控制领域当前研究的难点与热点之一,具有重要的理论价值和实际应用意义。
本文在综述前人研究的基础上,对于分段仿射系统的理论与应用作了进一步研究和探索。本文的研究工作主要集中在以下五个方面:
对分段仿射系统的产生背景,概念,以及研究的必要性和重要性作了综述,着重对分段仿射模型的建模方法作了研究,列举了基于分段仿射系统建模的工程应用实例。
基于简化约束区域计算的目的,采用分段区域划分技术,把分段仿射系统的每个子系统的状态作用空间用椭圆集的形式表示出来,每个子系统在其相应的区域划分内发生作用,并且在不同的区域划分之间发生切换。基于分段Lyapunov函数设计出能使PWA系统在各个区域划分之间任意切换的稳定的控制器;并且研究了当分段仿射系统带有扇区不确定性或者参数摄动不确定性时系统的鲁棒控制器设计问题,给出相应的设计方法。
在实际工程中,具有确定切换序列的分段仿射系统极其普遍,而且控制量带有饱和约束的分析和综合问题也是经常遇到的。针对这种具有确定切换序列的并且带有输入约束的分段仿射系统,为降低一般切换控制器设计的保守性,提出了多模预测控制方法。将多模型准无穷时域预测的指标函数分割成有限时域与无限时域两部分;其中有限时域性能指标优化求解出的自由变量可以使分段仿射系统按照确定的序列切换到指定的子系统模型区域内;无限时域性能指标优化求解出的反馈控制器使系统稳定到期望的平衡点;理论上将系统的稳定问题归结为线性矩阵不等式的凸优化求解可行性问题。在此基础上,通过附加的终端椭圆的渐近收敛约束保证系统更快收敛到平衡点,基于LMI解决了系统的输入受约束的问题。最后通过对一个聚合反应釜(CSTR)模型的仿真证实了上述两种算法的有效性。
而实际系统中系统模型往往是不确定或者缓慢变化的,针对多胞型不确定分段仿射系统,提出了鲁棒预测控制算法。基于预测控制的凸组合理论,给出了多步鲁棒预测控制的性能指标上确界,并将上确界的最小化问题转化为LMI优化问题;该方法不但考虑了模型的不确定性,而且证明了闭环系统的鲁棒稳定性和算法滚动实现的可行性。最后通过仿真验证了算法的有效性。
此外,针对切换顺序已知,并且输入受约束的子系统带有时滞项的分段仿射系统,提出基于椭圆集约束的模型预测控制。该算法基于Lyapunov-Krasovskii函数给出了性能指标上界和系统稳定的充分条件,将性能指标最小化、稳定性约束、输入约束转化为容易求解的LMI约束。通过陆地自主车的仿真验证了该方法的有效性。
In engineering practice, there are many hybrid systems described by piecewise affine systems (PWA) which are composed of linear subsystems and convex polytopic regions. Hybrid systems are composed of discrete event dynamic systems and continuous time dynamic systems or discrete time dynamic systems, which interact on each other. The hybrid system theory, which is proposed for the demand of the economic development, is the result of the development of computer science and control theory. Piecewise affine system is one of the most important branches of hybrid system, and is also straightforward. It consists of some subsystems that integrate the logical and continuous dynamics by switching. Theoretically, any nonlinear system can be approximated as piecewise affine system.
For decades, scientists have conducted much research work in PWA system and its applications in many engineering problems, and make a series of satisfied achievements. But, because of the difficulty and the popularity of the PWA system (i.e. PWA is essentially a non-smooth and even non- continuous nonlinear system and hybrid system has a extensive engineering background), up to today, the topic of PWA system and its applications is one of the most difficult research work and also one of the hot spots of studies in the field of system and control science.
On the basis of summarizing and reviewing previous research work, this dissertation makes some further research work and exploration in PWA and its applications in nonlinear systems. The research work of this thesis consists of following five aspects:
The concept and the background of the development of PWA, the importance and necessity of studies of PWA are explained. Various widely used modeling approaches for hybrid systems, their characteristics and the equivalences of typical hybrid models are summarized. The work focuses mainly on important subclass of hybrid systems: piecewise affine systems (PWA). Typical examples of the applications of PWA in engineering practice are also illustrated.
Using the convex polytopes technique, the corresponding operating region of the PWA systems in state space is described as ellipsoid which can be characterized by a set of vector inequalities. With these ellipsoid partitions, each model is valid in a given region of a certain space, and the PWA systems switches between different partitions. Based on the common Lyapunov and multiple Lyapunov functions, the stable controller is designed for the PWA systems involving mode switching between different operating regions. Furthermore, the design methods of the robust controllers are proposed for the PWA with sector uncertainty or parameter perturbation respectively.
In practice, the PWA systems with certain switching order exist universally. Simultaneously, the analysis and synthesis of control systems with actuator saturation are met frequently. To decrease the conservation of the controller which designed for the PWA systems with constrained control input based on certain switching order, a multi-model predictive control (MPC) algorithm is developed by a sequence of piecewise models along the transition trajectory. The control algorithm is a receding horizon scheme with a quasi-infinite horizon objective function that has finite and infinite horizon cost components. The finite horizon cost consists of free input variables that direct the system towards a terminal region that contains the desired operating point. The infinite horizon cost has an upper bound and takes the system to the final operating point. The control problem is formulated as a convex optimization in terms of Linear Matrix Inequalities (LMIs). Furthermore, on the basis of former algorithm, an additional terminal ellipsoid is introduced to ensure the states converging to the equilibrium faster. The problem of constrained control input is solved by convex optimization involving LMIs. Lastly, a simulation involving Continuous Stirred Tank Reactor (CSTR) model results verify the effectiveness of these proposed methods.
In practice, systems are generally uncertain and time-varying. A robust MPC is proposed for PWA systems with polytopic uncertainties. The problem of minimizing an upper bound on the worst-case objective function is reduced to a convex optimization involving LMIs. The feasible free input variables guarantee the PWA systems switching in certain order, and the receding horizon state-feedback control guarantees closed-loop robust asymptotic stability and input constraints. The approach allows explicit incorporation of the description of plant uncertainty. Furthermore, the closed-loop robust stability and feasibility of receding horizon implementation are proved. The simulation results verify the effectiveness of the proposed method.
A model predictive control (MPC) is proposed for a time-delay PWA systems with constrained input and certain switching order. The corresponding operating region of the considered systems in state space is described as ellipsoid which can be characterized by a set of vector inequalities. The control law is obtained by convex optimization based on MPC involving linear matrix inequalities (LMIs). And the constrained control input of the considered systems is solved in terms of LMIs. The simulation results verify the effectiveness of the proposed method. It is shown that, based on LMI constraints, it is easy to get the MPC for the PWA systems with time-delay and a simulation involving Autonomous Land Vehicle (ALV) model results verify the effectiveness of these proposed methods.
引文
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