Iterative Learning Identification for a Class of Wiener Nonlinear Time- Varying Systems
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摘要: 针对Wiener非线性时变系统的参数辨识问题，该文提出一种基于重复轴的迭代学习算法来实现对时变甚至突变参数的估计。文中将维纳系统输出非线性部分的反函数进行多项式展开，进而构造了回归模型，未知参数及中间变量用其估计替代，分别给出了采用迭代学习梯度算法和迭代学习最小二乘算法实现时变参数辨识的方法。仿真结果表明，与带遗忘因子的递推算法和迭代学习梯度算法相比，迭代学习最小二乘算法更具有参数估计收敛速度快，辨识精度高，系统输出误差小等优势，验证了所提学习算法的有效性。Abstract: For the parameters identification of Wiener nonlinear time-varying systems, iterative learning algorithms based on repeated axes are proposed to estimate the time-varying or even abrupt parameters. At first, the output nonlinear part of the Wiener system undertaken is tackled based on polynomial expansion, and then the regression model is constructed, the unknown parameters and intermediate variables are replaced by their estimates. Both iterative learning gradient and iterative learning least square algorithms are used to conduct the identification of the time-varying systems. Compared with the recursive algorithm with forgetting factor and iterative learning gradient algorithm, the simulation results demonstrate that the iterative learning least squares algorithm can perform high identification accuracy and efficiency, being of fast convergence speed and less resultant system output error, which verifies the effectiveness of the proposed algorithm.
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Key words:
- Time-varying parameters /
- Wiener systems /
- Gradient algorithms /
- Least squares /
- Iterative learning
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表 1 采用迭代学习梯度算法(W-ILG)计算
${\hat {\boldsymbol{\theta}} _k}(t)$ 的流程图输入：重复激励的一组数列 输出：堆积的输出向量${{\boldsymbol{Y}}_k}(t)$ 1. 对于所有的$t = 0,1, ··· ,N$，给定参数估计初始值${\hat \theta _{ - 1}}(t) = 0$，迭代所需${x_0}(t)$和${v_0}(t)$的值，并置$k = 0$； 2. 在第$ k $次重复运行时，采集输入数据${u_k}(t)$，计算输出数据${y_k}(t)$ 3. While $k < {K_{{\rm{max}}} }$(其中${K_{{\rm{max}}} }$为最大迭代次数) 4. for each $t \in [0,N]$ 5. 通过式(16)和式(17)构造${\hat {\boldsymbol{\phi} } _{1,k} }(t)$和${\hat {\boldsymbol{\phi}} _k}(t)$ 6. 通过式(18)选取合适的${\mu _k}$ 7. 通过式(15)计算得出${\hat {\boldsymbol{\theta}} _k}(t)$ 8. 通过式(12)刷新${\hat {\boldsymbol{x}}_k}(t)$和式(13)刷新${\hat {\boldsymbol{v}}_k}(t)$ 9. end 10. end 
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表 2 采用迭代学习最小二乘算法(W-ILLS)计算
${\hat {\boldsymbol{\theta}} _k}(t)$ 的流程图输入：重复激励的一组数列 输出：堆积的输出向量${{\boldsymbol{Y}}_k}(t)$ 1. 对于所有的$t = 0,1, \cdots ,N$，给定参数估计初始值${\hat \theta _{ - 1}}(t) = 0$，迭代所需${x_0}(t)$和${v_0}(t)$的值，并置$k = 0$； 2. 在第$ k $次重复运行时，采集输入数据 ${u_k}(t)$，计算输出数据${y_k}(t)$ 3. While $k < {K_{{\rm{max}}} }$(其中${K_{{\rm{max}}} }$为最大迭代次数) 4. for each $t \in [0,N]$ 5. 通过式(26)和式(27)构造${\hat {\boldsymbol{\phi}} _{1,k} }(t)$和${\hat {\boldsymbol{\phi}} _k}(t)$ 6. 通过式(21)计算得出${\hat {\boldsymbol{\theta}} _k}(t)$，式(22)刷新${{\boldsymbol{P}}_k}(t)$ 7. 通过式(23)刷新${\hat {\boldsymbol{x}}_k}(t)$，式(24)刷新${\hat {\boldsymbol{v}}_k}(t)$，式(25)刷新${e_k}(t)$ 8. End 9. End
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