An Adaptive Pseudospectral Method Combined With Homotopy Method for Solving Optimal Control Problems
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摘要: 针对弱间断最优控制问题和Bang-Bang最优控制问题，提出一种结合同伦法的自适应拟谱方法.Chebyshev拟谱方法转换原问题成为非线性规划问题.基于同伦法思想，同伦参数改变路径约束的界限，得到一系列比较光滑的最优控制问题.通过解这些问题得到原问题的不光滑解.文中证明了弱间断情况下数值解的收敛性.依据这收敛性和同伦参数，误差指示量可以捕捉不光滑点.本文方法与其他方法在数值算例中的对比表明，本文方法在精度和效率上都有明显优势.
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关键词:
- 最优控制问题 /
- 自适应拟谱方法 /
- 同伦法 /
- 弱间断解 /
- Bang-Bang控制
Abstract: Aiming at the weakly discontinuous and Bang-Bang optimal control problems, an adaptive pseudospectral method combined with the homotopy method is proposed. The Chebyshev pseudospectral method transforms the original problems into the resulting nonlinear programming problems. Based on the idea of homotopy method, the homotopic parameters change the bounds on path constraints to obtain a series of smoother optimal control problems. The nonsmooth original problems are solved by starting from the smoother problems. The convergence of numerical solutions is proved for the weakly discontinuous problems. With the help of the convergence and the homotopic parameters, an error indicator is able to capture the nonsmooth points. Several numerical examples are given to compare the proposed method with other methods. The comparison shows that the proposed method has obvious advantages in terms of accuracy and efficiency.1) 本文责任编委 张卫东 -
表 1 算法1解全部算例使用的参数
Table 1 The parameters of Algorithm 1 in all examples
参数 数值 $N_{\min}$ $4$ $N_{\rm{Initial}}$ $8$ ${P^{\rm CGL}_{\rm stop}}$ $33$ $\delta$ $0$ $\theta$ $0.1$ 
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表 2 算法1解例1的结果
Table 2 The results of Example 1 by Algorithm 1
${Tol}$ 目标值相对误差 时间(s) 误差指示量 配置点数 $1\times10^{-1}$ $3.372\times10^{-9}$ $9.~7$ $2.5452\times10^{-2}$ $49$ $5\times10^{-2}$ $2.311\times10^{-10}$ $11.~4$ $1.2418\times10^{-2}$ $49$ $1\times10^{-2}$ $6.802\times10^{-10}$ $15.~8$ $3.2392\times10^{-3}$ $49$ $5\times10^{-3}$ $5.811\times10^{-10}$ $18.~0$ $1.6129\times10^{-3}$ $49$ $1\times10^{-3}$ $2.152\times10^{-10}$ $22.~4$ $4.0097\times10^{-4}$ $49$ $5\times10^{-4}$ $2.460\times10^{-10}$ $24.~5$ $1.9961\times10^{-4}$ $49$ $1\times10^{-4}$ $1.601\times10^{-10}$ $31.~5$ $2.5017\times10^{-5}$ $49$ $5\times10^{-5}$ $1.512\times10^{-10}$ $34.~0$ $1.2464\times10^{-5}$ $49$ $1\times10^{-5}$ $1.476\times10^{-10}$ $38.~5$ $3.1159\times10^{-6}$ $49$
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表 3 Chebyshev拟谱方法解例1的结果
Table 3 The results of Example 1 by the Chebyshev pseudospectral method
目标值相对误差 时间(s) 配置点数 $4.2145\times10^{-4}$ $0.25$ 9 $5.5397\times10^{-6}$ $0.38$ 17 $5.1118\times10^{-7}$ $0.80$ 33 $4.2394\times10^{-7}$ $5.27$ 65 $3.9478\times10^{-9}$ $14.73$ 129 $4.631\times10^{-10}$ $44.59$ 257
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表 5 算法1解例2的结果
Table 5 The results of Example 2 by Algorithm 1
${Tol}$ 目标值相对误差 时间(s) 误差指示量 配置点数 $1\times10^{-1}$ $8.0706\times10^{-9}$ $5.7$ $6.3108\times10^{-3}$ $49$ $5\times10^{-2}$ $4.7381\times10^{-9}$ $6.4$ $9.0744\times10^{-4}$ $49$
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表 6 Chebyshev拟谱方法解例2的结果
Table 6 The results of Example 2 by the Chebyshev pseudospectral method
目标值相对误差 时间(s) 配置点数 $6.0018\times10^{-3}$ $0.13$ 9 $1.5119\times10^{-3}$ $0.23$ 17 $3.7982\times10^{-4}$ $0.39$ 33 $9.7584\times10^{-5}$ $1.08$ 65 $2.7067\times10^{-5}$ $2.62$ 129 $1.8455\times10^{-5}$ $6.86$ 257
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表 8 算法1解例3的结果
Table 8 The results of Example 3 by Algorithm 1
$Tol$ 数值解目标值 时间(s) 误差指示量 配置点数 $1\times10^{-1}$ $-26.704709$ $334.6$ $5.1952\times10^{-3}$ $~49$ $5\times10^{-2}$ $-26.704676$ $423.4$ $4.4434\times10^{-5}$ $~41$
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表 9 Chebyshev拟谱方法解例3的结果
Table 9 The results of Example 3 by the Chebyshev pseudospectral method
数值解目标值 时间(s) 配置点数 $-26.668531$ $1.7$ $9 $-26.689549$ $4.0$ $17 $-26.702575$ $8.0$ $33 $-26.703963$ $87.4$ $65 $-26.704482$ $153.2$ 129 $-26.704704$ $1\, 380.0$ 257
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