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摘要:
针对舰载机等复杂动力学系统的高阶时滞模型拟配难题，提出了一种基于通用化等效模型的系统频域辨识算法。建立系统辨识通用化等效模型，引入自适应学习率梯度下降法对模型进行基于样本的迭代学习训练，使模型的频域参数自动收敛，辨识得到复杂系统的高阶时滞模型。以舰载机横航向荷兰滚等效拟配模型进行验证，与传统算法对比，并分析了系统的频域与时域响应，结果表明：所提算法具有较好的辨识效果，解决了高阶时滞模型的直接拟配难题，并通用于广泛的高阶时滞模型拟配。
Abstract:This study provides a frequency domain identification approach aimed at the high-order time-delay model matching problem for complex systems like carrier aircraft. A high-order time-delay model of the complex system is ultimately identified after the generalized frequency-domain identification model is established and the model is iteratively trained by system frequency domain response data using the gradient descent method of adaptive learning rate. This allows the model’s frequency domain parameters to automatically converge.Verification is carried out with the Dutch roll equivalent model of carrier aircraft. Through the identification and comparison of high-level and low-level models, the algorithm in this paper is compared with the traditional algorithm, and the frequency domain and time domain analysis show that the algorithm in this paper has a good identification effect. It solves the problem of direct fitting of high-order time-delay models, and is suitable for universal high-order time-delay models.
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图 3 等效高阶模型迭代训练的频域结果
Figure 3. Frequency domain results of equivalent high-order model’s iterative training
图 4 训练模型与等效高阶模型对比
Figure 4. Comparison between training model and equivalent high-order model
图 5 等效低阶模型迭代训练的频域结果
Figure 5. Frequency domain results of equivalent low-order model’s iterative training
表 1 算例频域辨识模型参数
Table 1. Parameters of arithmetic frequency domain identification model
模型 $ T $ $ Z $ $ N $ $ P $ $ Q $ 等效高阶模型 1 0 1 2 1 等效低阶模型 0 1 0 0 1 
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表 2 超参数
Table 2. Hyperparameters
$ \eta $ $ \rho $ $ \varepsilon $ $ \lambda $ 0.001 0.999 0.1 0.0175
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表 3 等效高阶模型的频域特性参数
Table 3. Frequency domain characteristic parameters of equivalent high-order model
等效高阶模型 $ k $ $ b_1^{(1)} $ $ b_2^{(1)} $ $ a_0^{(1)} $ $ a_0^{(2)} $ $ a_1^{(1)} $ $ a_2^{(1)} $ $ \tau $ $ J $ 训练样本 3 0.2 0.1 0.5 0.5 0.6 0.3 0.01 初始值 1 1 1 1 1 1 1 1 402.8 $ 10 $次训练 1.1331 1.1353 1.1211 0.8680 0.8680 0.8601 0.8741 0.8561 271.3 $ {10^2} $次训练 1.3978 1.4102 1.1029 0.7241 0.7241 0.5794 0.9231 0.4888 89.8 $ {10^3} $次训练 2.0211 1.6973 0.7273 0.7071 0.7071 0.7864 1.0552 −0.0120 3.6 $ 5 \times {10^3} $次训练 2.9523 1.2700 0.6509 1.0063 1.0063 0.6912 0.7027 −0.0298 1.1 $ 8 \times {10^3} $次训练 3.0018 0.1999 0.1006 0.5026 0.5026 0.5955 0.2976 0.0098 6.0×10−4 $ 9 \times {10^3} $次训练 3.0003 0.2005 0.1005 0.4998 0.4998 0.5992 0.2995 0.0099 1.1×10−3 $ {10^4} $次训练 2.9995 0.1996 0.0955 0.5005 0.5005 0.6006 0.3007 0.0103 1.5×10−3
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表 4 梯度下降法的等效低阶模型频域特性参数
Table 4. Frequency domain characteristic parameters of equivalent low-order model based on gradient descent method
梯度下降法的等效低阶模型 $ k $ $ a_1^{(1)} $ $ a_2^{(1)} $ $ b_0^{(1)} $ $ \tau $ $ J $ 初始值 1 1 1 1 1 446.7 $ 10 $次训练 1.1084 0.8665 0.8608 1.1304 0.8554 328.6 $ {10^2} $次训练 0.8880 0.5720 0.5645 1.3620 0.4832 118.7 $ {10^3} $次训练 0.6338 0.4694 0.5553 2.0367 0 13.6 $ {10^4} $次训练 0.3146 0.8629 0.8538 7.2477 0 4.6 $ 5 \times {10^4} $次训练 0.1667 1.0386 0.9570 15.8224 0 2.5 $ {10^5} $次训练 0.1603 1.0447 0.9603 16.5260 0 2.4
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表 5 最小二乘法的等效低阶模型频域特性参数
Table 5. Frequency domain characteristic parameters of equivalent low-order model based on least sqaure method
最小二乘法的等效低阶模型 $ K $ $ {\omega _0} $ $ \xi $ $ T $ $ \tau $ $ J $ 初始值 1 1 0.5 1 1 446.7 终值 0.0274 0.9996 0.5435 101.7832 0 1.7 梯度下降法$ {10^5} $次训练等效值 0.1669 1.0205 0.5119 16.5260 0 2.4
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表 6 模型阶跃响应稳态值
Table 6. Steady state value of model step response
模型 单位阶跃稳态值 样本模型 0 等效高阶模型 0 等效低阶模型(梯度下降法) 0.1603 等效低阶模型(最小二乘法) 0.0274
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表 7 模型阶跃响应性能指标
Table 7. Performance index of model step response
模型 峰值 峰值时间/s 调节时间/s 样本模型 1.52 1.3 4.9 等效高阶模型 1.53 1.3 4.9 等效低阶模型(梯度下降法) 1.50 1.3 6.9 等效低阶模型(最小二乘法) 1.48 1.2 7.2
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