非线性分段光滑动力系统的最优控制及稳定性
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摘要
根据三维水平井轨道设计的实际背景,研究了一类非线性分段光滑动力系统最优控制问题。在此基础上又对固定时刻脉冲微分系统的稳定性进行了定性分析,并讨论了非线性动力系统关于初始时刻偏差的稳定性。该项研究,一方面可以丰富非线性动力系统最优控制理论和稳定性理论,另一方面可以为水平井轨道优化设计提供理论指导,因此具有一定的理论意义和应用价值。所取得的主要结果概括如下:
1.根据三维水平井轨道控制的特征,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,曲率半径、工具面角和弧长为控制变量的非线性分段光滑动力系统。以入靶精度和总弧长的加权和为性能指标,建立一个三维水平井轨道最优控制系统。本文利用极大值原理得到了非线性分段光滑动力系统最优控制的必要条件。由于性能指标的非线性程度高,这就限制了传统的依赖梯度的的优化算法的应用;遗传算法和模拟退火算法这类算法在求解这种最优控制问题时计算量很大。故我们构造了一个基于均匀设计的改进的Hooke-Jeeves算法。首先利用均匀设计选择初始迭代点,将可行域分解成多个子域。在每个子域上用改进的Hooke-Jeeves算法求解。数值结果表明改进的Hooke-Jeeves算法能够很好的解决该问题。我们进一步考虑了带有扰动的水平井轨道最优控制问题,建立了以脉冲微分方程为动态约束的水平井轨道最优控制模型,证明了最优控制的存在性。为了获得脉冲最优控制系统的最优解,将原最优控制问题转化成一个参数规划问题。现有的参数规划算法理论上都得计算目标函数和约束的梯度,而我们的目标函数的梯度不易计算。所以我们首先证明了参数规划问题的最优解关于参数的稳定性的性质。利用这个性质,以不含扰动的最优控制模型的最优解作为初始点,仍用改进的Hooke-Jeeves算法求解参数规划问题。数值模拟结果与结论相一致,这表明我们的算法是有效的。
2.利用扰动Lyapunov函数方法讨论了固定时刻脉冲微分系统的稳定性。在具体问题中,要找到满足所有条件的Lyapunov函数比较困难,因此,要能够减弱对Lyapunov函数的要求将是非常有意义的。本文利用扰动Lyapunov函数得到了在弱假设条件下的脉冲微分系统的稳定性、渐近稳定性、实用稳定性、有界性和最终有界性的充分条件。并将扰动Lyapunov函数推广到脉冲微分系统的两测度稳定性中,得到了脉冲微分系统的(h_0,h)-稳定性、(h_0,h)-渐近稳定性、(h_0,h)-实用稳定性以及(h_0,h)-有界性。本文所得结论都需要利用两个Lyapunov函数,但对每一个Lyapunov函数的限制较少,更易于在实际问题中应用。
3.研究了非线性动力系统关于初始时刻偏差的稳定性。传统的稳定性概念都假设初始值有扰动而初始时刻不变化,但是在实际应用中,由于各种干扰因素存在,初始时刻也会出现误差。所以研究动力系统关于初始时刻偏差的稳定性在实际应用中是很有意义的。现有的关于这种稳定性的判据不易验证,所以也没有例子来检验。本文证明了一个新的比较引理,利用向量Lyapunov函数得到了非线性动力系统关于初始时刻偏差的等稳定性、实用等稳定性、等有界性的充分条件。为了得到更弱条件下的等稳定性判定准则,我们利用扰动Lyapunov函数证明了非线性动力系统关于初始时刻偏差的等稳定性和实用等稳定性。所得结论条件简洁,且易于检验。我们构造了三个例子来说明所得的结论。
This dissertation, based on the engineering background of design of 3D-trajeetory of horizontal wells, studies the optimal control problem of nonlinear piecewise smooth dynamical systems. Furthermore, the qualitative analysis on the stability of piecewise smooth systems with impulses at fixed times are discussed, and the stability analysis of nonlinear dynamical systems with initial time difference are made. These results can not only develop the theory and application of nonlinear dynamical systems and optimal control, but also provide some guidance for the design of the trajectory of horizontal wells. Therefore, this research is very interesting in both theory and practice. The main results, obtained in this dissertation, may be summarized as follows:
1. According to the features of 3D-trajectory formed in horizontal wells, we construct a nonlinear piecewise smooth dynamical system in which the state variables are inclination, azimuth, north coordinate, east coordinate and vertical depth coordinate and the controls are radius, tool-face angle and arc length. Taking the weight sum of precision of hitting target and the total length of the trajectory as a performance criterion, we construct an optimal control model of the trajectory of horizontal wells. The necessary conditions for optimality of nonlinear piecewise smooth dynamical system are proved via maximum principle. Because the performance criterion is highly nonlinear and computationally expensive to evaluate, it limits the efficient use of classical gradient based optimization methods. So we construct a new algorithm in which the uniform design technique has been incorporated into the revised Hooke-Jeeves algorithm to handle the multimodal function. Firstly, we use uniform design method to generate many initial points in control domain, and decompose the control domain into many subdomains. Then we get the locally optimal solutions in each subdomains by the revised Hooke-Jeeves algorithm. It is shown from the real example that the revised Hooke-Jeeves method is efficient. Furthermore, we take fully into account the effect of unknown disturbances in drilling. We present an impulsive optimal control model to solve the optimal designing problem of the trajectory of horizontal wells with disturbances, and we prove that the optimal control exists. To obtain the optimal solutions, the optimal control problem can be converted into a nonlinear parametric optimization by integrating the state equation. The general algorithms for nonlinear parametric optimization problems are all need to calculate the gradient of objective function. Since it is difficult to achieve the gradient of our objective function, these algorithms are not appropriate for our nonlinear parametric optimization problem. We discuss here that the locally optimal solution depends in a continuous way on the parameters (disturbances) and utilize this property to propose a revised Hooke-Jeeves algorithm. The numerical simulation is in accordance with theoretical results. The numerical results illustrate the validity of the model and efficiency of the algorithm.
2. The stability, practical stability and boundedness of impulsive differential systems are proved by using the method of perturbing Lyapunov functions. It is hard to find a Lyapunov function satisfying all the desired conditions. We obtain the stability, asymptotical stability, practical stability and boundedness properties of nonlinear impulsive differential systems using perturbing Lyapunov functions in a manner that provide weaker assumptions. We generalize the concepts of perturbing Lyapunov functions to impulsive differential systems and prove several stability results of nonlinear impulsive systems in terms of two measures by perturbing Lyapunov functions. We find that the method of perturbing Lyapunov functions yields better results since each function can satisfy less rigid requirement.
3. This dissertation investigates the stability and boundedness of nonlinear differential dynamical systems relative to initial time difference. The traditional stability concept assumes that only state variable perturbs and the initial time keeps unchanged. However, it is impossible not to make errors in the starting time in practical applications. Therefore, it is important to study the deviation in initial time. We establishes several criteria on stability and boundedness for nonlinear differential dynamical systems relative to initial time difference by employing a new comparison principle. The conditions for our results are more easy to test in applications than the existing results. We initiate three examples to test our results.
1.根据三维水平井轨道控制的特征,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,曲率半径、工具面角和弧长为控制变量的非线性分段光滑动力系统。以入靶精度和总弧长的加权和为性能指标,建立一个三维水平井轨道最优控制系统。本文利用极大值原理得到了非线性分段光滑动力系统最优控制的必要条件。由于性能指标的非线性程度高,这就限制了传统的依赖梯度的的优化算法的应用;遗传算法和模拟退火算法这类算法在求解这种最优控制问题时计算量很大。故我们构造了一个基于均匀设计的改进的Hooke-Jeeves算法。首先利用均匀设计选择初始迭代点,将可行域分解成多个子域。在每个子域上用改进的Hooke-Jeeves算法求解。数值结果表明改进的Hooke-Jeeves算法能够很好的解决该问题。我们进一步考虑了带有扰动的水平井轨道最优控制问题,建立了以脉冲微分方程为动态约束的水平井轨道最优控制模型,证明了最优控制的存在性。为了获得脉冲最优控制系统的最优解,将原最优控制问题转化成一个参数规划问题。现有的参数规划算法理论上都得计算目标函数和约束的梯度,而我们的目标函数的梯度不易计算。所以我们首先证明了参数规划问题的最优解关于参数的稳定性的性质。利用这个性质,以不含扰动的最优控制模型的最优解作为初始点,仍用改进的Hooke-Jeeves算法求解参数规划问题。数值模拟结果与结论相一致,这表明我们的算法是有效的。
2.利用扰动Lyapunov函数方法讨论了固定时刻脉冲微分系统的稳定性。在具体问题中,要找到满足所有条件的Lyapunov函数比较困难,因此,要能够减弱对Lyapunov函数的要求将是非常有意义的。本文利用扰动Lyapunov函数得到了在弱假设条件下的脉冲微分系统的稳定性、渐近稳定性、实用稳定性、有界性和最终有界性的充分条件。并将扰动Lyapunov函数推广到脉冲微分系统的两测度稳定性中,得到了脉冲微分系统的(h_0,h)-稳定性、(h_0,h)-渐近稳定性、(h_0,h)-实用稳定性以及(h_0,h)-有界性。本文所得结论都需要利用两个Lyapunov函数,但对每一个Lyapunov函数的限制较少,更易于在实际问题中应用。
3.研究了非线性动力系统关于初始时刻偏差的稳定性。传统的稳定性概念都假设初始值有扰动而初始时刻不变化,但是在实际应用中,由于各种干扰因素存在,初始时刻也会出现误差。所以研究动力系统关于初始时刻偏差的稳定性在实际应用中是很有意义的。现有的关于这种稳定性的判据不易验证,所以也没有例子来检验。本文证明了一个新的比较引理,利用向量Lyapunov函数得到了非线性动力系统关于初始时刻偏差的等稳定性、实用等稳定性、等有界性的充分条件。为了得到更弱条件下的等稳定性判定准则,我们利用扰动Lyapunov函数证明了非线性动力系统关于初始时刻偏差的等稳定性和实用等稳定性。所得结论条件简洁,且易于检验。我们构造了三个例子来说明所得的结论。
This dissertation, based on the engineering background of design of 3D-trajeetory of horizontal wells, studies the optimal control problem of nonlinear piecewise smooth dynamical systems. Furthermore, the qualitative analysis on the stability of piecewise smooth systems with impulses at fixed times are discussed, and the stability analysis of nonlinear dynamical systems with initial time difference are made. These results can not only develop the theory and application of nonlinear dynamical systems and optimal control, but also provide some guidance for the design of the trajectory of horizontal wells. Therefore, this research is very interesting in both theory and practice. The main results, obtained in this dissertation, may be summarized as follows:
1. According to the features of 3D-trajectory formed in horizontal wells, we construct a nonlinear piecewise smooth dynamical system in which the state variables are inclination, azimuth, north coordinate, east coordinate and vertical depth coordinate and the controls are radius, tool-face angle and arc length. Taking the weight sum of precision of hitting target and the total length of the trajectory as a performance criterion, we construct an optimal control model of the trajectory of horizontal wells. The necessary conditions for optimality of nonlinear piecewise smooth dynamical system are proved via maximum principle. Because the performance criterion is highly nonlinear and computationally expensive to evaluate, it limits the efficient use of classical gradient based optimization methods. So we construct a new algorithm in which the uniform design technique has been incorporated into the revised Hooke-Jeeves algorithm to handle the multimodal function. Firstly, we use uniform design method to generate many initial points in control domain, and decompose the control domain into many subdomains. Then we get the locally optimal solutions in each subdomains by the revised Hooke-Jeeves algorithm. It is shown from the real example that the revised Hooke-Jeeves method is efficient. Furthermore, we take fully into account the effect of unknown disturbances in drilling. We present an impulsive optimal control model to solve the optimal designing problem of the trajectory of horizontal wells with disturbances, and we prove that the optimal control exists. To obtain the optimal solutions, the optimal control problem can be converted into a nonlinear parametric optimization by integrating the state equation. The general algorithms for nonlinear parametric optimization problems are all need to calculate the gradient of objective function. Since it is difficult to achieve the gradient of our objective function, these algorithms are not appropriate for our nonlinear parametric optimization problem. We discuss here that the locally optimal solution depends in a continuous way on the parameters (disturbances) and utilize this property to propose a revised Hooke-Jeeves algorithm. The numerical simulation is in accordance with theoretical results. The numerical results illustrate the validity of the model and efficiency of the algorithm.
2. The stability, practical stability and boundedness of impulsive differential systems are proved by using the method of perturbing Lyapunov functions. It is hard to find a Lyapunov function satisfying all the desired conditions. We obtain the stability, asymptotical stability, practical stability and boundedness properties of nonlinear impulsive differential systems using perturbing Lyapunov functions in a manner that provide weaker assumptions. We generalize the concepts of perturbing Lyapunov functions to impulsive differential systems and prove several stability results of nonlinear impulsive systems in terms of two measures by perturbing Lyapunov functions. We find that the method of perturbing Lyapunov functions yields better results since each function can satisfy less rigid requirement.
3. This dissertation investigates the stability and boundedness of nonlinear differential dynamical systems relative to initial time difference. The traditional stability concept assumes that only state variable perturbs and the initial time keeps unchanged. However, it is impossible not to make errors in the starting time in practical applications. Therefore, it is important to study the deviation in initial time. We establishes several criteria on stability and boundedness for nonlinear differential dynamical systems relative to initial time difference by employing a new comparison principle. The conditions for our results are more easy to test in applications than the existing results. We initiate three examples to test our results.
引文
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