计算智能在控制、优化和决策中的应用研究
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摘要
经过半个多世纪的发展，最优化理论和相关的优化算法已逐渐成熟。这些在运筹学(Operational Research)和数学规划工具(Mathematical Programming)基础上形成的最优化算法，具有理论完备、算法效率高、稳定性好等优点，因而在许多需要进行优化计算的场合被广泛的使用。但是，这些算法本质上都基于梯度计算，有些甚至对目标函数有连续二阶可导的要求，并且一般不能含有离散的变量(除混合整数规划外)，严重的限制了应用范围。而且，基于梯度的算法对初始点的选取位置严重依赖，不能保证收敛到全局最优解或者近似的全局最优解。随着社会的进步，技术的创新，当需要求解的问题比较复杂时，特别是当需要进行不确定系统研究而带来的多层优化命题产生时，这些算法在求解上就遇到了困难。这时，人们往往就需要求助于计算智能。
本论文的研究目标是探索计算智能在优化领域的应用，不仅研究传统的单层优化命题，而且研究以minimax优化为代表的多层优化命题；不仅考虑算法的有效性，而且兼顾算法的稳定性。针对一般的带有等式约束和不等式约束的非线性优化问题，提出了神经动力学求解模型，对多模的非线性优化问题改进了粒子群算法以获得更好近似全局最优解。随后，同时利用神经网络和进化计算，对minimax双层优化命题进行算法设计，得到了两种不同特点的算法。在能够求解minimax问题的基础上，探索了相关的应用领域。比如，在不确定系统中，通过minimax算法进行后悔度计算，并把区间数优化问题转化为等价的多目标优化问题以求解。另一个应用实例是鲁棒PID的设计，通过minimax计算整定控制器参数，使最恶劣工况下系统性仍然能够保持在一个令人满意的水平上。
本文的主要研究工作与贡献如下：
1．对计算智能在优化方面的应用的历史和现状进行了详尽的分析和综述，并提出将优化计算拓展到不确定系统中的思想。特别详述了参数不确定系统的一个重要分支，即用区间数描述的参数不确定系统的优化命题的研究现状。
2．神经网络动力学原理用于解决单目标的优化命题。带有等式约束和不等式约束的一般非线性优化问题，是单层优化的难点。使用增广Lagrange乘子法求解时，虽然可以避免罚参数无限增大的弊病，但同时也提出了一个难以求解的子命题。本文中运用Lyapunov函数和Lassalle不变性原理，采
中文摘要
用微分动力学系统为增广Lagrange乘子法的子命题设计了一个神经网络模
型，从而使得目标函数和约束条件一阶可微的一般非线性优化问题能够顺
利求解。
3.单层优化的一个难题是全局最优解或者近似全局最优解的获得。进化算法
能够很好的避免陷入局部最优解从而备受青睐，粒子群算法就是近年来新
发展起来的一种仿生算法。但是进化算法中的一些参数设置通常是通过比
较实验获得的，缺乏理论上的依据。本文分析了粒子群中独立粒子的运动
轨迹，以及整个粒子群系统的稳定性，在此基础之上提出新的参数设置方
法。新方法不仅符合理论上的解释，而且通过标准测试函数库的检验表明
新算法是有效的。文中涉及的稳定性研究在粒子群算法研究中是不多见
的。
4.不确定系统的建模求解，离不开minimax问题的最终解决。本文
为minimax优化这个几乎是崭新的优化命题设计了多种求解算法:首
先是神经网络算法。神经网络的计算速度快，利于电路实现，同时算法的
稳定性也得到了证明。其次是基于进化计算的算法。该算法综合了遗传算
法和单纯形法的优点，主要应用于求解目标函数不可微的minimax全局优
化。
5.对不确定系统中的一类问题，即用区间数作为参数进行建模求解的区间数
规划问题，本文受顾基发研究员的“物理一事理一人理(WSR)”〔26]的
系统科学思想的启发，创造性的提出了一个结合目标函数期望，不确定度
和后悔度的三目标鲁棒优化命题，本优化命题可作为原不确定系统优化命
题的替代命题。此外对含区间数参数的不等式约束的处理也提出了新的转
化准则。
6.由于参数不确定性和滞后特性的存在，整定好的控制器在环境产生波动的
情况下有可能失效。针对这种工业过程，本论文基于LQR性能指标(该指
标和衰减系数和自然频率密切相关)，研究了具有滞后的不确定频域系统
的鲁棒PID控制器设计。通过这种方式设计出来的控制器不仅在通常工况
下有和其他控制器相近的良好性能，当工况发生改变乃至其他控制器失控
的情况下，仍然能够对过程进行平稳的控制。
Having developed for half an century, the conventional optimization algorithms which are based on the Operational Research Theory and some Mathematical Programming tools come into mature. Such algorithms have been widely used in many fields due to their high efficiency and robustness. However, they usually require the optimized functions to be continuous even high order differentiable. What is more, they heavily rely on the initial position of search, without guarantee of the global solutions. When coming to difficult problems especially those exist in the Uncertain System Optimization, people sometimes resort to Computational Intelligence.
The object of this thesis is the application of Computational Intelligence in Optimization, not only the single level optimization, but also the multi level optimization. Both the efficiency and the stability of algorithms are considered. For general nonlinear optimization with constraints, a neural network model and a revised PSO algorithm are proposed. For minimax two level optimization problem, neural network and genetic algorithm solutions are provided, respectively. After that, the minimax optimization is applied into the uncertain system analysis. Through minimax algorithm the regret index could be calculated, then the Interval Programming is translated into equivalent Multiobjective Programming. Another application example of minimax optimization is the robust PID controller design. By this design measurement the control quality is guaranteed even in the worst working conditions.
The main contributions of this thesis are listed as following:
1. The history and the current research progress of Computational Intelligence is synthesized, especially those algorithms applied in the optimization with interval number parameters, an important branch of the uncertain system.
2. A neural network solver for the Augmented Lagrange multiplier (ALM) method is provided, which has a wide application in the constrained nonlinear optimization propositions. A dynamic approach for the minimization subproblem in ALM method is discussed, and then a neural network iterative algorithm is proposed for general constrained nonlinear optimization.
3. A PSO with an increasing inertia weight, distinct from a widely used PSO with a decreasing inertia weight, is proposed for the single level optimization problems. Far from drawing conclusions from sole empirical studies or rule of thumb, this algorithm is derived from particle trajectory study and convergence analysis. Four standard test functions with asymmetric initial range settings are used to confirm the validity of the PSO with an increasing inertia weight. From the experiments, it is clear that a PSO with an increasing inertia weight outperforms the one with a decreasing inertia weight, both in convergent speed and solution precision, with no additional computing load compared with the PSO with a decreasing inertia weight.
4. The minimax problem is a significant topic in signal process and process control, which is relevant to robustness, parameters uncertainty, and signal noise etc. However, efficient algorithms are scarce, especially those for general minimax problem with nonlinear equality and inequality constraints. A novel neural network for general minimax problem has been constructed based on a penalty function approach. The only request on objective function and constraint functions is that they should be first-order differentiable. A Lyapunov function is established for the global stability analysis. The SGA (Simplex-Genetic Algorithm), an improved algorithm of Genetic Algorithm for solving Stackelberg-Nash Equilibrium, is also proposed for the minimax optimization.
5. For the uncertainty optimization with interval coefficients in the objective function, a robust optimization framework is proposed, in which the concept of
"regret" is incorporated. This framework is inspired by the methodology of "Wuli-Shili-Renli" [26] raised by J. Gu. Through this method an uncertainty optimization problem may be transferred i
本论文的研究目标是探索计算智能在优化领域的应用，不仅研究传统的单层优化命题，而且研究以minimax优化为代表的多层优化命题；不仅考虑算法的有效性，而且兼顾算法的稳定性。针对一般的带有等式约束和不等式约束的非线性优化问题，提出了神经动力学求解模型，对多模的非线性优化问题改进了粒子群算法以获得更好近似全局最优解。随后，同时利用神经网络和进化计算，对minimax双层优化命题进行算法设计，得到了两种不同特点的算法。在能够求解minimax问题的基础上，探索了相关的应用领域。比如，在不确定系统中，通过minimax算法进行后悔度计算，并把区间数优化问题转化为等价的多目标优化问题以求解。另一个应用实例是鲁棒PID的设计，通过minimax计算整定控制器参数，使最恶劣工况下系统性仍然能够保持在一个令人满意的水平上。
本文的主要研究工作与贡献如下：
1．对计算智能在优化方面的应用的历史和现状进行了详尽的分析和综述，并提出将优化计算拓展到不确定系统中的思想。特别详述了参数不确定系统的一个重要分支，即用区间数描述的参数不确定系统的优化命题的研究现状。
2．神经网络动力学原理用于解决单目标的优化命题。带有等式约束和不等式约束的一般非线性优化问题，是单层优化的难点。使用增广Lagrange乘子法求解时，虽然可以避免罚参数无限增大的弊病，但同时也提出了一个难以求解的子命题。本文中运用Lyapunov函数和Lassalle不变性原理，采
中文摘要
用微分动力学系统为增广Lagrange乘子法的子命题设计了一个神经网络模
型，从而使得目标函数和约束条件一阶可微的一般非线性优化问题能够顺
利求解。
3.单层优化的一个难题是全局最优解或者近似全局最优解的获得。进化算法
能够很好的避免陷入局部最优解从而备受青睐，粒子群算法就是近年来新
发展起来的一种仿生算法。但是进化算法中的一些参数设置通常是通过比
较实验获得的，缺乏理论上的依据。本文分析了粒子群中独立粒子的运动
轨迹，以及整个粒子群系统的稳定性，在此基础之上提出新的参数设置方
法。新方法不仅符合理论上的解释，而且通过标准测试函数库的检验表明
新算法是有效的。文中涉及的稳定性研究在粒子群算法研究中是不多见
的。
4.不确定系统的建模求解，离不开minimax问题的最终解决。本文
为minimax优化这个几乎是崭新的优化命题设计了多种求解算法:首
先是神经网络算法。神经网络的计算速度快，利于电路实现，同时算法的
稳定性也得到了证明。其次是基于进化计算的算法。该算法综合了遗传算
法和单纯形法的优点，主要应用于求解目标函数不可微的minimax全局优
化。
5.对不确定系统中的一类问题，即用区间数作为参数进行建模求解的区间数
规划问题，本文受顾基发研究员的“物理一事理一人理(WSR)”〔26]的
系统科学思想的启发，创造性的提出了一个结合目标函数期望，不确定度
和后悔度的三目标鲁棒优化命题，本优化命题可作为原不确定系统优化命
题的替代命题。此外对含区间数参数的不等式约束的处理也提出了新的转
化准则。
6.由于参数不确定性和滞后特性的存在，整定好的控制器在环境产生波动的
情况下有可能失效。针对这种工业过程，本论文基于LQR性能指标(该指
标和衰减系数和自然频率密切相关)，研究了具有滞后的不确定频域系统
的鲁棒PID控制器设计。通过这种方式设计出来的控制器不仅在通常工况
下有和其他控制器相近的良好性能，当工况发生改变乃至其他控制器失控
的情况下，仍然能够对过程进行平稳的控制。
Having developed for half an century, the conventional optimization algorithms which are based on the Operational Research Theory and some Mathematical Programming tools come into mature. Such algorithms have been widely used in many fields due to their high efficiency and robustness. However, they usually require the optimized functions to be continuous even high order differentiable. What is more, they heavily rely on the initial position of search, without guarantee of the global solutions. When coming to difficult problems especially those exist in the Uncertain System Optimization, people sometimes resort to Computational Intelligence.
The object of this thesis is the application of Computational Intelligence in Optimization, not only the single level optimization, but also the multi level optimization. Both the efficiency and the stability of algorithms are considered. For general nonlinear optimization with constraints, a neural network model and a revised PSO algorithm are proposed. For minimax two level optimization problem, neural network and genetic algorithm solutions are provided, respectively. After that, the minimax optimization is applied into the uncertain system analysis. Through minimax algorithm the regret index could be calculated, then the Interval Programming is translated into equivalent Multiobjective Programming. Another application example of minimax optimization is the robust PID controller design. By this design measurement the control quality is guaranteed even in the worst working conditions.
The main contributions of this thesis are listed as following:
1. The history and the current research progress of Computational Intelligence is synthesized, especially those algorithms applied in the optimization with interval number parameters, an important branch of the uncertain system.
2. A neural network solver for the Augmented Lagrange multiplier (ALM) method is provided, which has a wide application in the constrained nonlinear optimization propositions. A dynamic approach for the minimization subproblem in ALM method is discussed, and then a neural network iterative algorithm is proposed for general constrained nonlinear optimization.
3. A PSO with an increasing inertia weight, distinct from a widely used PSO with a decreasing inertia weight, is proposed for the single level optimization problems. Far from drawing conclusions from sole empirical studies or rule of thumb, this algorithm is derived from particle trajectory study and convergence analysis. Four standard test functions with asymmetric initial range settings are used to confirm the validity of the PSO with an increasing inertia weight. From the experiments, it is clear that a PSO with an increasing inertia weight outperforms the one with a decreasing inertia weight, both in convergent speed and solution precision, with no additional computing load compared with the PSO with a decreasing inertia weight.
4. The minimax problem is a significant topic in signal process and process control, which is relevant to robustness, parameters uncertainty, and signal noise etc. However, efficient algorithms are scarce, especially those for general minimax problem with nonlinear equality and inequality constraints. A novel neural network for general minimax problem has been constructed based on a penalty function approach. The only request on objective function and constraint functions is that they should be first-order differentiable. A Lyapunov function is established for the global stability analysis. The SGA (Simplex-Genetic Algorithm), an improved algorithm of Genetic Algorithm for solving Stackelberg-Nash Equilibrium, is also proposed for the minimax optimization.
5. For the uncertainty optimization with interval coefficients in the objective function, a robust optimization framework is proposed, in which the concept of
"regret" is incorporated. This framework is inspired by the methodology of "Wuli-Shili-Renli" [26] raised by J. Gu. Through this method an uncertainty optimization problem may be transferred i
引文
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