基于线性矩阵不等式的非线性预测控制研究
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摘要
非线性预测控制是控制理论研究的一个重要分支。经过近二十年的发展，非线性预测控制理论研究已经取得一些较为成熟的成果。但由于非线性预测控制的复杂性，一些比较基础的问题，如稳定性、鲁棒性、算法实时性问题还没有很好地解决，这使得非线性预测控制理论和实践之间存在着较大的差距，阻碍了非线性预测控制的发展和工业应用。因此稳定性、鲁棒性、算法实时性问题仍是非线性预测控制需要研究的主要问题。
本文以线性矩阵不等式为设计工具，对非线性预测控制的上述三个问题进行了研究。在非线性预测控制器设计中采用线性矩阵不等式，易造成算法适用范围小等问题，从而导致其实际应用性较差。本文从理论上详细分析了造成该问题的主要原因，并给出了一系列有效的解决办法，为非线性预测控制的研究提供了一种思路。
第一章首先回顾了非线性预测控制的产生背景、研究现状与发展趋势，分析了阻碍非线性预测控制在实际过程工业领域应用的原因。作为预备知识，介绍了线性矩阵不等式的概念、基本问题以及线性矩阵不等式的解法。
接着介绍了用线性矩阵不等式进行非线性预测控制设计的一般性思路及其优缺点，指出造成该类算法适用范围小的主要原因有两个，一是稳定性约束条件过强，二是非线性系统的多面体模型表示保守性过大。
针对第一个问题，在第二章中提出了可调参数的线性矩阵不等式非线性预测控制算法。该算法通过调整参数大小可有效放宽约束条件，从而可扩大算法的适用范围；在第三章中，提出了基于可行解的线性矩阵不等式非线性预测控制算法。该算法针对非线性优化中最优解很难得到的问题，采用可行解对非线性预测控制进行设计，同样可以有效放宽约束条件。
针对第二个问题，在第四章中将多模型控制和非线性预测控制相结合，提出了多模型线性矩阵不等式非线性预测控制算法。该方法通过对局部控制器的巧妙设计，可保证控制器切换后的系统稳定性。
第五章提出了用扩展线性矩阵不等式对非线性预测控制算法进行设计的思路。扩展线性矩阵不等式具有较小的保守性和较大的设计自由度，因此该类算法在减小保守性的同时，还可综合考虑到多个性能指标，具有较好的控制效果。
第六章将预测控制应用到温室控制中，提出了基于切换控制的温室建模和控制方法。
本文详细分析了所提出的算法的有效性，对大部分结果给出了理论证明，并通过仿真试验与传统方法进行了对比，验证了算法的优越性。
Nonlinear model predictive control (NMPC) is an important branch of control theory. After twenty years of development, much progress has been made in NMPC and some theory has reached a relative mature stage. While, comparing with the great success in the applications of the linear model predictive control, few applications of NMPC have been reported. This may attributes to the complexcity of NMPC, and some fundamental problems of NMPC theory have not been well resolved yet. Among them, the problems of stability, robustness and efficiency of the optimization are the main three ones, and make the main obstacles in the applications of NMPC. So the current NMPC researches still focus on the problems of stability, robust, and optimization efficiency.
In this thesis the linear matrix inequality (LMI) is used as the main design tool to study the above problems in NMPC. While by integrating LMI into NMPC, it may make the applicable domain very small. This thesis gives a detailed analysis about this problem and presents a series of resolving methods, which afford a new way in NMPC research.
In this thesis, the background, current research condition and future direction of NMPC is first reviewed. And as the preliminary knowledge, LMI is introduced, including its definition, main problem and its optimal algorithm.
Then the general concept of designing NMPC algorithm with LMI is introduced. As the discussions pointed, two reasons mainly account for the small applicable domain of this class algorithms. One is the stable constraints are too strong, and the other is the polynomial model representation of nonlinear systems is too conservative.
Aiming at the first problem, a parameterized algorithm is proposed in chapter 2, which can enlarge the applicable domain efficiently by tuning the parameters. In chapter 3, a feasible solution based LMI NMPC algorithm is proposed. For the optimal solution in NMPC is usually difficult to achieved, using feasible solution can loosen the
To the second problem, multi-model method is integrated in NMPC in chapter 4, and a multi-model and LMI based NMPC algorithm is proposed. In this method, after delicately design for local controllers, the stability after switching controllers can be guaranteed.
In this thesis, the concept of using extended LMI in NMPC design is also introduced. For the extended LMI has many good properties, such as smaller conservativeness and more agilities in design, two or more optimal index can be achieved simultaneously while the conservativeness is not enlarged.
In this thesis, MPC and LMI is also used in the greenhouse environment system control, and in chapter 6, the new modeling method and MPC method is proposed which are based on the switching system control.
In this thesis, all the proposed algorithms are detailed analyzed and most of the theorems are proved. And some simulated experiments are given to illuminate the effectiveness of algorithms.
本文以线性矩阵不等式为设计工具，对非线性预测控制的上述三个问题进行了研究。在非线性预测控制器设计中采用线性矩阵不等式，易造成算法适用范围小等问题，从而导致其实际应用性较差。本文从理论上详细分析了造成该问题的主要原因，并给出了一系列有效的解决办法，为非线性预测控制的研究提供了一种思路。
第一章首先回顾了非线性预测控制的产生背景、研究现状与发展趋势，分析了阻碍非线性预测控制在实际过程工业领域应用的原因。作为预备知识，介绍了线性矩阵不等式的概念、基本问题以及线性矩阵不等式的解法。
接着介绍了用线性矩阵不等式进行非线性预测控制设计的一般性思路及其优缺点，指出造成该类算法适用范围小的主要原因有两个，一是稳定性约束条件过强，二是非线性系统的多面体模型表示保守性过大。
针对第一个问题，在第二章中提出了可调参数的线性矩阵不等式非线性预测控制算法。该算法通过调整参数大小可有效放宽约束条件，从而可扩大算法的适用范围；在第三章中，提出了基于可行解的线性矩阵不等式非线性预测控制算法。该算法针对非线性优化中最优解很难得到的问题，采用可行解对非线性预测控制进行设计，同样可以有效放宽约束条件。
针对第二个问题，在第四章中将多模型控制和非线性预测控制相结合，提出了多模型线性矩阵不等式非线性预测控制算法。该方法通过对局部控制器的巧妙设计，可保证控制器切换后的系统稳定性。
第五章提出了用扩展线性矩阵不等式对非线性预测控制算法进行设计的思路。扩展线性矩阵不等式具有较小的保守性和较大的设计自由度，因此该类算法在减小保守性的同时，还可综合考虑到多个性能指标，具有较好的控制效果。
第六章将预测控制应用到温室控制中，提出了基于切换控制的温室建模和控制方法。
本文详细分析了所提出的算法的有效性，对大部分结果给出了理论证明，并通过仿真试验与传统方法进行了对比，验证了算法的优越性。
Nonlinear model predictive control (NMPC) is an important branch of control theory. After twenty years of development, much progress has been made in NMPC and some theory has reached a relative mature stage. While, comparing with the great success in the applications of the linear model predictive control, few applications of NMPC have been reported. This may attributes to the complexcity of NMPC, and some fundamental problems of NMPC theory have not been well resolved yet. Among them, the problems of stability, robustness and efficiency of the optimization are the main three ones, and make the main obstacles in the applications of NMPC. So the current NMPC researches still focus on the problems of stability, robust, and optimization efficiency.
In this thesis the linear matrix inequality (LMI) is used as the main design tool to study the above problems in NMPC. While by integrating LMI into NMPC, it may make the applicable domain very small. This thesis gives a detailed analysis about this problem and presents a series of resolving methods, which afford a new way in NMPC research.
In this thesis, the background, current research condition and future direction of NMPC is first reviewed. And as the preliminary knowledge, LMI is introduced, including its definition, main problem and its optimal algorithm.
Then the general concept of designing NMPC algorithm with LMI is introduced. As the discussions pointed, two reasons mainly account for the small applicable domain of this class algorithms. One is the stable constraints are too strong, and the other is the polynomial model representation of nonlinear systems is too conservative.
Aiming at the first problem, a parameterized algorithm is proposed in chapter 2, which can enlarge the applicable domain efficiently by tuning the parameters. In chapter 3, a feasible solution based LMI NMPC algorithm is proposed. For the optimal solution in NMPC is usually difficult to achieved, using feasible solution can loosen the
To the second problem, multi-model method is integrated in NMPC in chapter 4, and a multi-model and LMI based NMPC algorithm is proposed. In this method, after delicately design for local controllers, the stability after switching controllers can be guaranteed.
In this thesis, the concept of using extended LMI in NMPC design is also introduced. For the extended LMI has many good properties, such as smaller conservativeness and more agilities in design, two or more optimal index can be achieved simultaneously while the conservativeness is not enlarged.
In this thesis, MPC and LMI is also used in the greenhouse environment system control, and in chapter 6, the new modeling method and MPC method is proposed which are based on the switching system control.
In this thesis, all the proposed algorithms are detailed analyzed and most of the theorems are proved. And some simulated experiments are given to illuminate the effectiveness of algorithms.
引文
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