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首页> 外文期刊>Automatic Control, IEEE Transactions on >Recursive Identification for Nonlinear ARX Systems Based on Stochastic Approximation Algorithm
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Recursive Identification for Nonlinear ARX Systems Based on Stochastic Approximation Algorithm

机译：基于随机近似算法的非线性ARX系统的递归辨识

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摘要

The nonparametric identification for nonlinear autoregressive systems with exogenous inputs (NARX) described by $y_{k+1}=f(y_{k},ldots,y_{k+1-n_{0}},u_{k},ldots,u_{k+1-n_{0}})+varepsilon_{k+1}$ is considered. First, a condition on $f(cdot)$ is introduced to guarantee ergodicity and stationarity of ${y_{k}}$ . Then the kernel function based stochastic approximation algorithm with expanding truncations (SAAWET) is proposed to recursively estimate the value of $f(phi^{ast})$ at any given $phi^{ast} triangleq [y^{(1)},ldots,y^{(n_{0})},u^{(1)},ldots,u^{(n_{0})}]^{tau}in {bf R}^{2n_{0}}$. It is shown that the estimate converges to the true value with probability one. In establishing the strong consistency of the estimate, the properties of the Markov chain associated with the NARX system play an important role. Numerical examples are given, which show that the simulation results are consistent with the theoretical analysis. The intention of the paper is not only to present a concrete solution to the problem under consideration but also to profile a new analysis method for nonlinear systems. The proposed method consisting in combining the Markov chain properties with stochastic approximation algorithms may be of future potential, although a restrictive condition has to be imposed on $f(cdot)$, that is, the growth rate of $f(x)$ should not be faster than linear with coefficient less than 1 as $Vert{x}Vert$ tends to infinity.-n-
机译：具有$ y_ {k + 1} = f（y_ {k}，ldots，y_ {k + 1-n_ {0}}，u_ {k}，ldots描述的具有外源输入（NARX）的非线性自回归系统的非参数辨识，u_ {k + 1-n_ {0}}）+ varepsilon_ {k + 1} $。首先，引入$ f（cdot）$的条件以保证$ {y_ {k}} $的遍历性和平稳性。然后提出了一种基于核函数的具有扩展截断的随机逼近算法（SAAWET），以在任意给定的$ phi ^ {ast} triangleq [y ^ {（1）}上递归估计$ f（phi ^ {ast}）$的值。 ，ldots，y ^ {（n_ {0}）}，u ^ {（1）}，ldots，u ^ {（n_ {0}）}] ^ {tau} in {bf R} ^ {2n_ {0} } $。结果表明，该估计以概率一收敛到真实值。在建立估计的强一致性时，与NARX系统相关的马尔可夫链的属性起着重要作用。数值算例表明仿真结果与理论分析吻合。本文的目的不仅是为所考虑的问题提供具体的解决方案，而且还为非线性系统提供一种新的分析方法。尽管必须对$ f（cdot）$施加限制条件，即$ f（x）$的增长速度应受限制，但将马尔可夫链属性与随机逼近算法结合起来的方法可能具有未来的潜力。 $ Vert {x} Vert $趋于无穷大，因此系数小于1的线性速度不会快于线性。-n-

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