预测控制在线优化策略的研究
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摘要
预测控制作为一类采用在线滚动优化的控制算法,以其良好的控制性能,灵活的处理各类约束的能力,受到工业界和理论界的广泛重视。通过近十几年的理论研究,现代预测控制综合理论逐渐形成,涌现出很多重要的研究成果。稳定性问题作为现代预测控制综合理论的一个重要问题,虽然得到了较好的解决。但控制性能、在线计算量以及可行域的大小等实际控制中所关心的问题都仍有待研究。
对于预测控制而言,其在线优化策略的设计将决定其稳定性及控制性能等问题。本文由此出发,分别针对开环和闭环预测控制策略,在保证系统稳定性的前提下,就其控制性能、在线计算量以及可行域大小等方面进行了研究,得到以下成果:
●针对以往预测控制集结框架的不足,首先提出了一种更为一般的集结框架,从该框架出发可以更为方便地涵盖以往开环策略中的各种减少优化变量的方法,并为进一步的理论分析和设计奠定了基础;在该框架的基础上,针对集结预测控制的稳定性进行了分析,给出了集结预测控制器稳定性分析的一般方法,并以此为基础研究了集结预测控制器的稳定可行域的估算方法。
●针对集结预测控制器的控制性能问题,根据预测控制滚动优化的特点,即只有第一个控制量实际施加到控制对象,提出了等效集结及拟等效集结概念,并分析了四种条件下的等效集结和拟等效集结设计问题,给出了其设计方法。
●针对有界扰动系统的鲁棒预测控制问题,通过改进原ERPC的设计并利用扰动不变集,设计了基于衰减集结的鲁棒预测控制器,其特点是在线计算量低,控制性能较好;另外,为了降低因离线设计控制不变集带来的保守性,基于广义集结框架,设计了一种集结鲁棒预测控制器,利用双模控制方法,通过增加终端集外的控制量,降低控制器的保守性。
●针对多胞不确定系统的鲁棒控制问题,提出了多步控制集的概念,进而设计反馈鲁棒预测控制器,并在此基础上设计了采用单一或参数依赖Lyapunov函数的反馈鲁棒预测控制器;另外,给出了反馈鲁棒预测控制器的离线设计在线综合的设计方法。
●针对变化速率有限的LPV系统,首先利用该系统参数变化的特点给出了两种预测系统模型未来变化情况的算法,该算法采用代数运算并且不增加系统模型的顶点个数,从而对控制器的在线计算量不产生影响;进一步,基于对未来系统模型的变化情况的预测,给出该LPV系统的反馈鲁棒预测控制器设计。
●针对以往鲁棒预测控制大多采用LMI的形式,从而带来了较大的在线计算量问题,采用闭环策略,将多胞不确定系统的鲁棒预测控制问题转化为QP问题,从而大大地降低了预测控制器的在线计算量。同时,提出了扩展广义插补的方法,从而解决了具有闭环策略的鲁棒预测控制器必须具有固定终端集和反馈律K的问题,降低了设计的保守性。
As a category of control algorithm adopting online receding horizon optimization, model predictive control (MPC) attracts much attention of industrial and theoretical researchers due to its good control performance and capability of handling constraints explicitly. Over the past decade, qualitative synthesis of model predictive control rapidly develops and many important results are proposed. As an important issue of qualitative synthesis of model predictive control, the guaranteed stability of MPC was particularly addressed. However, many other issues such as control performance, online computation burden and feasible region of a MPC controller, which are important in practical applications, are still expected to be further studied.
For a MPC controller, the issues such as stability or control performance are dependent on the online optimization strategy adopted. So this dissertation will focus on the online optimization strategy of MPC. With the guaranteed stability as the precondition, for the open-loop and closed-loop MPC respectively, the control performance, feasible region and online computation burden of MPC are studied in this dissertation and the following contributions are obtained.
●A general aggregation framework is proposed to overcome the weakness of the previous framework and can more easily formulate other aggregation strategies . Based on the general framework, a method to analyze the stability of aggregation based MPC controller is developed. And then an algorithm to estimate its feasible region with guaranteed stability is proposed.●In order to guarantee the control performance of aggregation based MPC, two new concepts, the equivalent aggregation and quasi-equivalent aggregation, are proposed. Making use of the characteristic of MPC, i.e. at each time, only the first control input in the optimal solution is acted on the practical plant, the problems of equivalent aggregation and quasi-equivalent aggregation of MPC are studied for four cases and the corresponding algorithms are developed.
●For linear systems with additive disturbance, two design methods of RMPC (robust MPC) are presented. Based on the disturbance invariant set, an improved ERPC (efficient robust predictive control) and the aggregation strategy are adopted in the two design methods respectively to improve the control performance and reduce the online computation burden.
●For uncertain systems with polytopic description, the concept of multi-step control set is proposed. Based on the concept, the feedback robust MPC controller with single or parameter dependent Lyapunov function is developed. Making use of the characteristic of the method, the design of a sequence of feedback control laws can be completed offline and then a feedback robust MPC controller with lower online computation burden can be obtained.
●For LPV systems (linear parameter varying system) with limited rates of parameters varying, two algorithms to predict the models in the future are developed. The future system is predicted to belong to a sequence of polyhedral sets with the same number of models as the original one. To determine these sets, only algebra calculation is concerned. Based on the two algorithms, the feedback robust MPC for this kind of LPV systems are developed. Compared with the design without considering the limited rates of parameter varying, the design here can reduce the conservativeness and lead to better control performance.
●Although LMI (linear matrix inequalities) is widely used in a number of previous literatures on robust MPC, it makes the online computation burden of robust MPC too heavy for practical applications. For uncertain systems with polytopic description, the min-max optimization problem of a closed-loop robust MPC is converted into a QP (Quadratic Programming) problem and then the online computation burden is reduced greatly. Meanwhile the extended general interpolation is proposed. Based on it, the closed-loop robust QP-MPC with varying feedback control law is also developed to reduce the design conservativeness.
对于预测控制而言,其在线优化策略的设计将决定其稳定性及控制性能等问题。本文由此出发,分别针对开环和闭环预测控制策略,在保证系统稳定性的前提下,就其控制性能、在线计算量以及可行域大小等方面进行了研究,得到以下成果:
●针对以往预测控制集结框架的不足,首先提出了一种更为一般的集结框架,从该框架出发可以更为方便地涵盖以往开环策略中的各种减少优化变量的方法,并为进一步的理论分析和设计奠定了基础;在该框架的基础上,针对集结预测控制的稳定性进行了分析,给出了集结预测控制器稳定性分析的一般方法,并以此为基础研究了集结预测控制器的稳定可行域的估算方法。
●针对集结预测控制器的控制性能问题,根据预测控制滚动优化的特点,即只有第一个控制量实际施加到控制对象,提出了等效集结及拟等效集结概念,并分析了四种条件下的等效集结和拟等效集结设计问题,给出了其设计方法。
●针对有界扰动系统的鲁棒预测控制问题,通过改进原ERPC的设计并利用扰动不变集,设计了基于衰减集结的鲁棒预测控制器,其特点是在线计算量低,控制性能较好;另外,为了降低因离线设计控制不变集带来的保守性,基于广义集结框架,设计了一种集结鲁棒预测控制器,利用双模控制方法,通过增加终端集外的控制量,降低控制器的保守性。
●针对多胞不确定系统的鲁棒控制问题,提出了多步控制集的概念,进而设计反馈鲁棒预测控制器,并在此基础上设计了采用单一或参数依赖Lyapunov函数的反馈鲁棒预测控制器;另外,给出了反馈鲁棒预测控制器的离线设计在线综合的设计方法。
●针对变化速率有限的LPV系统,首先利用该系统参数变化的特点给出了两种预测系统模型未来变化情况的算法,该算法采用代数运算并且不增加系统模型的顶点个数,从而对控制器的在线计算量不产生影响;进一步,基于对未来系统模型的变化情况的预测,给出该LPV系统的反馈鲁棒预测控制器设计。
●针对以往鲁棒预测控制大多采用LMI的形式,从而带来了较大的在线计算量问题,采用闭环策略,将多胞不确定系统的鲁棒预测控制问题转化为QP问题,从而大大地降低了预测控制器的在线计算量。同时,提出了扩展广义插补的方法,从而解决了具有闭环策略的鲁棒预测控制器必须具有固定终端集和反馈律K的问题,降低了设计的保守性。
As a category of control algorithm adopting online receding horizon optimization, model predictive control (MPC) attracts much attention of industrial and theoretical researchers due to its good control performance and capability of handling constraints explicitly. Over the past decade, qualitative synthesis of model predictive control rapidly develops and many important results are proposed. As an important issue of qualitative synthesis of model predictive control, the guaranteed stability of MPC was particularly addressed. However, many other issues such as control performance, online computation burden and feasible region of a MPC controller, which are important in practical applications, are still expected to be further studied.
For a MPC controller, the issues such as stability or control performance are dependent on the online optimization strategy adopted. So this dissertation will focus on the online optimization strategy of MPC. With the guaranteed stability as the precondition, for the open-loop and closed-loop MPC respectively, the control performance, feasible region and online computation burden of MPC are studied in this dissertation and the following contributions are obtained.
●A general aggregation framework is proposed to overcome the weakness of the previous framework and can more easily formulate other aggregation strategies . Based on the general framework, a method to analyze the stability of aggregation based MPC controller is developed. And then an algorithm to estimate its feasible region with guaranteed stability is proposed.●In order to guarantee the control performance of aggregation based MPC, two new concepts, the equivalent aggregation and quasi-equivalent aggregation, are proposed. Making use of the characteristic of MPC, i.e. at each time, only the first control input in the optimal solution is acted on the practical plant, the problems of equivalent aggregation and quasi-equivalent aggregation of MPC are studied for four cases and the corresponding algorithms are developed.
●For linear systems with additive disturbance, two design methods of RMPC (robust MPC) are presented. Based on the disturbance invariant set, an improved ERPC (efficient robust predictive control) and the aggregation strategy are adopted in the two design methods respectively to improve the control performance and reduce the online computation burden.
●For uncertain systems with polytopic description, the concept of multi-step control set is proposed. Based on the concept, the feedback robust MPC controller with single or parameter dependent Lyapunov function is developed. Making use of the characteristic of the method, the design of a sequence of feedback control laws can be completed offline and then a feedback robust MPC controller with lower online computation burden can be obtained.
●For LPV systems (linear parameter varying system) with limited rates of parameters varying, two algorithms to predict the models in the future are developed. The future system is predicted to belong to a sequence of polyhedral sets with the same number of models as the original one. To determine these sets, only algebra calculation is concerned. Based on the two algorithms, the feedback robust MPC for this kind of LPV systems are developed. Compared with the design without considering the limited rates of parameter varying, the design here can reduce the conservativeness and lead to better control performance.
●Although LMI (linear matrix inequalities) is widely used in a number of previous literatures on robust MPC, it makes the online computation burden of robust MPC too heavy for practical applications. For uncertain systems with polytopic description, the min-max optimization problem of a closed-loop robust MPC is converted into a QP (Quadratic Programming) problem and then the online computation burden is reduced greatly. Meanwhile the extended general interpolation is proposed. Based on it, the closed-loop robust QP-MPC with varying feedback control law is also developed to reduce the design conservativeness.
引文
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