非线性多阶段最优控制系统理论、算法及应用
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摘要
本文根据三维水平井井眼轨道设计的实际背景，研究一类非线性多阶段最优控制系统，包括系统的状态方程解的存在性、最优性条件、求解该系统的全局优化算法以及把算法应用到实际的三维水平井井眼轨道设计中。本文的主要研究内容可概括如下：
1．对一类非线性多阶段最优控制系统，讨论它的状态方程组解的存在性、最优控制系统最优解的存在性及最优性条件。
2．为了获得非线性多阶段最优控制系统的全局最优解，把该最优控制系统转化为一个非线性规划问题求解。由于该非线性规划问题的目标函数和约束函数可能是隐函数或不可微的，因此给出三种基于随机搜索的全局优化算法。
(1)．把均匀设计、聚类思想与遗传算法结合起来，给出了改进的混合遗传算法解决不带终端约束的非线性多阶段最优控制问题，并进行了收敛性分析。
(2)．针对带有终端约束的非线性多阶段最优控制系统，给出了改进的进化规划算法。在这个算法中，对于目标函数和不同的约束函数，分别在每一点根据两种情况定义两种电荷，基于电磁理论求出合力，把合力方向作为变异搜索方向。为了以较大概率抛弃不可行点，定义一个新的适应性函数。并对算法进行了收敛性分析。
(3)．针对含有等式约束的非线性多阶段最优控制系统，给出了一个全局优化算法。在该算法中，利用凝聚函数构造一个可行集替代目标函数值小于当前目标函数值的可行域，基于随机搜索技术和局部搜索技术寻找该可行集内新的可行点，这个过程直到找不到新的可行点为止。并对算法的收敛性进行了分析。根据此算法的思想给出一个求解含有终端约束的三维水平井轨道设计最优控制系统的全局优化算法。
3．根据三维水平井轨道控制的特性，建立了以井斜角、方位角、北坐标、东坐标和垂深坐标为状态变量表示井眼轨道曲线的非线性多阶段动力系统(即状态方程组)。基于建立的动力系统，从不同角度考虑，建立了四个三维水平井井眼轨道设计最优控制系统。三维水平井井眼轨道设计的目的有两个：第一要求动力系统的终端输出与靶点处的对应值越接近越好，第二要求动力系统确定的三维水平井轨道曲线总长最短。基于这两个目的，建立一个不带终端约束的三维水平井井眼轨道最优控制系统；如果在满足给定的入靶精度条件下，只考虑轨道曲线总长最短，那么建立另一个带终端约束的三维水平井井眼轨道设计最优控制系统；由于在三维水平井井眼轨道实际设计中，造斜点处的初值和靶点处的终值的测量是不精确的，给定的一些变量的约束范围也是不精确的，所以在模糊环境下，建立一个三维水平井轨道设计模糊最优控制系统；在实钻过程中，由于客观条件的随机性，三维水平井的实钻轨道往往偏离最优设计的轨道，因此建立一个三维水平井轨道随钻修正设计最优控制系统。并把给出的三种算法应用到三维水平井井眼
大连理工大学博士学位论文
轨道设计的实际应用中。
4.目前，所有的对CRS(Controlled Random SearCh)算法的改进工作都是围绕着如
何产生新的试验点，但是对CRS算法的收敛性工作一直没有考虑。本文基于以一定概率
接收“可变”单纯形的策略，给出一个改进的CRS算法，证明了改进的算法依概率1收
敛。
This dissertation, based on practical background of design of 3D-trajectory of horizontal wells, studies nonlinear multistage optimal control systems, including the existence of solutions of state equations and optimal solutions of the optimal control system?optimality conditions?global optimization algorithms for solving the optimal control system and applying these algorithms to practical design of 3D-trajectory of horizontal wells. The main results, obtained in this dissertation, may be summarized as follows:
1. For a class of nonlinear multistage optimal control systems, the existence of solutions of the state equations and the existence of optimal solutions of the optimal control system and optimality conditions for the optimal control system are discussed.
2. In order to obtain the global optimal solution of the nonlinear multistage optimal control system, the optimal control system is translated into a nonlinear programming problem. Since the objective function and constraint function can be implicit functions or nondifferentiable in this nonlinear programming problem, three global optimization algorithms, based on stochastic search techniques, are proposed.
(1) Based on a combination of the uniform design?clustering idea and genetic algorithms, an improved hybrid genetic algorithm is proposed for solving the nonlinear multiage optimal control system without terminal constraints. The convergence of the algorithm is analyzed.
(2) An improved evolutionary programming algorithm is developed for solving nonlinear multiage optimal control system with terminal constraints. In this algorithm, each individual can be regarded as a charged particle. According to two cases, two kinds of charge are defined on each individual for objective function and each constraint function, respectively. After calculating these charges, like the electromagnetic force a combination force exerted on each individual is calculated. The direction of this force is taken as a search direction of mutation operator. In order to discard infeasible individual with higher probability, a new fitness function is defined. The convergence of this algorithm is proved.
(3) A global optimization algorithm for solving nonlinear multiage optimal control system with equality terminal constraints is proposed. In this algorithm, firstly, based on aggregate function, a new set that can substitute for the feasible region in which the value of objective function is lower than the current value of objective function is defined, then, in this new set, a feasible point is found by stochastic search and local search. The process stops until the new feasible point cannot be found. The convergence proof of this algorithm is given. According to the idea of the proposed algorithm, a global optimization algorithm is constructed for solving the optimal control system of design of 3D trajectory of horizontal wells with terminal constraints.
3. According to the features of 3D-trajectory formed in horizontal wells, we construct a nonlinear multistage dynamical system (i.e. state equations) in which state variables are inclination?azimuth?north coordinate?east coordinate and vertical depth coordinate. The dynamical system describes the 3D-trajectory of horizontal wells. Based on this dynamical system, four optimal control systems of design of 3D-trajectory of horizontal wells are proposed from different points of view. There are two aims for designing 3D-trajectory of horizontal wells. Firstly, the terminal output of this dynamical system will approximate the corresponding values of the target point as close as possible. Secondly, the total length of 3D-trajectory of horizontal wells is as short as possible. Based on the two aims, one optimal control system is developed for designing 3D-trajectory of horizontal wells; if the total length of 3D-trajectory of horizontal wells is only considered under satisfying condition of the given terminal outpu
t of this dynamical system, then the other optimal control system with terminal constraints is propo
1．对一类非线性多阶段最优控制系统，讨论它的状态方程组解的存在性、最优控制系统最优解的存在性及最优性条件。
2．为了获得非线性多阶段最优控制系统的全局最优解，把该最优控制系统转化为一个非线性规划问题求解。由于该非线性规划问题的目标函数和约束函数可能是隐函数或不可微的，因此给出三种基于随机搜索的全局优化算法。
(1)．把均匀设计、聚类思想与遗传算法结合起来，给出了改进的混合遗传算法解决不带终端约束的非线性多阶段最优控制问题，并进行了收敛性分析。
(2)．针对带有终端约束的非线性多阶段最优控制系统，给出了改进的进化规划算法。在这个算法中，对于目标函数和不同的约束函数，分别在每一点根据两种情况定义两种电荷，基于电磁理论求出合力，把合力方向作为变异搜索方向。为了以较大概率抛弃不可行点，定义一个新的适应性函数。并对算法进行了收敛性分析。
(3)．针对含有等式约束的非线性多阶段最优控制系统，给出了一个全局优化算法。在该算法中，利用凝聚函数构造一个可行集替代目标函数值小于当前目标函数值的可行域，基于随机搜索技术和局部搜索技术寻找该可行集内新的可行点，这个过程直到找不到新的可行点为止。并对算法的收敛性进行了分析。根据此算法的思想给出一个求解含有终端约束的三维水平井轨道设计最优控制系统的全局优化算法。
3．根据三维水平井轨道控制的特性，建立了以井斜角、方位角、北坐标、东坐标和垂深坐标为状态变量表示井眼轨道曲线的非线性多阶段动力系统(即状态方程组)。基于建立的动力系统，从不同角度考虑，建立了四个三维水平井井眼轨道设计最优控制系统。三维水平井井眼轨道设计的目的有两个：第一要求动力系统的终端输出与靶点处的对应值越接近越好，第二要求动力系统确定的三维水平井轨道曲线总长最短。基于这两个目的，建立一个不带终端约束的三维水平井井眼轨道最优控制系统；如果在满足给定的入靶精度条件下，只考虑轨道曲线总长最短，那么建立另一个带终端约束的三维水平井井眼轨道设计最优控制系统；由于在三维水平井井眼轨道实际设计中，造斜点处的初值和靶点处的终值的测量是不精确的，给定的一些变量的约束范围也是不精确的，所以在模糊环境下，建立一个三维水平井轨道设计模糊最优控制系统；在实钻过程中，由于客观条件的随机性，三维水平井的实钻轨道往往偏离最优设计的轨道，因此建立一个三维水平井轨道随钻修正设计最优控制系统。并把给出的三种算法应用到三维水平井井眼
大连理工大学博士学位论文
轨道设计的实际应用中。
4.目前，所有的对CRS(Controlled Random SearCh)算法的改进工作都是围绕着如
何产生新的试验点，但是对CRS算法的收敛性工作一直没有考虑。本文基于以一定概率
接收“可变”单纯形的策略，给出一个改进的CRS算法，证明了改进的算法依概率1收
敛。
This dissertation, based on practical background of design of 3D-trajectory of horizontal wells, studies nonlinear multistage optimal control systems, including the existence of solutions of state equations and optimal solutions of the optimal control system?optimality conditions?global optimization algorithms for solving the optimal control system and applying these algorithms to practical design of 3D-trajectory of horizontal wells. The main results, obtained in this dissertation, may be summarized as follows:
1. For a class of nonlinear multistage optimal control systems, the existence of solutions of the state equations and the existence of optimal solutions of the optimal control system and optimality conditions for the optimal control system are discussed.
2. In order to obtain the global optimal solution of the nonlinear multistage optimal control system, the optimal control system is translated into a nonlinear programming problem. Since the objective function and constraint function can be implicit functions or nondifferentiable in this nonlinear programming problem, three global optimization algorithms, based on stochastic search techniques, are proposed.
(1) Based on a combination of the uniform design?clustering idea and genetic algorithms, an improved hybrid genetic algorithm is proposed for solving the nonlinear multiage optimal control system without terminal constraints. The convergence of the algorithm is analyzed.
(2) An improved evolutionary programming algorithm is developed for solving nonlinear multiage optimal control system with terminal constraints. In this algorithm, each individual can be regarded as a charged particle. According to two cases, two kinds of charge are defined on each individual for objective function and each constraint function, respectively. After calculating these charges, like the electromagnetic force a combination force exerted on each individual is calculated. The direction of this force is taken as a search direction of mutation operator. In order to discard infeasible individual with higher probability, a new fitness function is defined. The convergence of this algorithm is proved.
(3) A global optimization algorithm for solving nonlinear multiage optimal control system with equality terminal constraints is proposed. In this algorithm, firstly, based on aggregate function, a new set that can substitute for the feasible region in which the value of objective function is lower than the current value of objective function is defined, then, in this new set, a feasible point is found by stochastic search and local search. The process stops until the new feasible point cannot be found. The convergence proof of this algorithm is given. According to the idea of the proposed algorithm, a global optimization algorithm is constructed for solving the optimal control system of design of 3D trajectory of horizontal wells with terminal constraints.
3. According to the features of 3D-trajectory formed in horizontal wells, we construct a nonlinear multistage dynamical system (i.e. state equations) in which state variables are inclination?azimuth?north coordinate?east coordinate and vertical depth coordinate. The dynamical system describes the 3D-trajectory of horizontal wells. Based on this dynamical system, four optimal control systems of design of 3D-trajectory of horizontal wells are proposed from different points of view. There are two aims for designing 3D-trajectory of horizontal wells. Firstly, the terminal output of this dynamical system will approximate the corresponding values of the target point as close as possible. Secondly, the total length of 3D-trajectory of horizontal wells is as short as possible. Based on the two aims, one optimal control system is developed for designing 3D-trajectory of horizontal wells; if the total length of 3D-trajectory of horizontal wells is only considered under satisfying condition of the given terminal outpu
t of this dynamical system, then the other optimal control system with terminal constraints is propo
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