Non-Convex Projection Adaptive Hammerstein Filtering
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摘要: 该文研究了Hammerstein系统参数辨识和非线性系统预测问题，提出一种基于非凸投影的自适应滤波算法。论文将问题归结为具有非凸可行域的约束优化问题，并建立了基于交替方向乘子法(ADMM)和递归最小二乘相结合的算法框架。在该算法框架下，非凸约束优化问题的全局最优解可通过岭回归和欧几里得(Euclid)投影循环计算得到。将提出的算法分别应用于Hammerstein系统的参数辨识、非线性未知系统预测以及非线性声学回声消除，并进行仿真实验，结果显示所提算法具有较好的收敛性和稳定性，能够得到较准确的辨识和预测效果。
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关键词:
- 系统辨识 /
- 非线性预测 /
- 自适应滤波 /
- 参数估计 /
- Hammerstein系统
Abstract: This paper focuses on adaptive filtering techniques for parameter identification of Hammerstein systems and output prediction of nonlinear systems. By formulating the underlying filtering problem as a recursive bilinear least-squares optimization with the non-convex feasible region constraint, an algorithmic framework is developed based on recursive least-squares and Alternating Direction Method of Multipliers (ADMM). Under this framework, the solution to nonconvex constraint optimization problem can be obtained by implementing ridge regression and Euclidean projection. Simulation results in the context of system identification, nonlinear predication, and acoustic echo cancellation, reveal that the proposed algorithm has good convergence and stability, and can obtain more accurate identification and prediction results. -
算法1 提出的RncPLS算法 输入：d(n), x(n) 输出：${\hat {\boldsymbol{a}}}(n),{\hat {\boldsymbol{b}}}(n),{\boldsymbol{\theta}} (n) = {\bar {\boldsymbol{\theta}} _L}(n)$ 初始值：${ {\bar {\boldsymbol{\theta} } }_L}(0),\bar {\boldsymbol{R} }_L^{ - 1}(0),{ {\boldsymbol{\varphi} } }(0),{{\boldsymbol{\eta}} }(0)$ 算法迭代：在每个时刻n，由输入按照以下步骤得到估计值输出，迭代直至收敛。
1. ${ {\boldsymbol{k} } }(n) = \dfrac{ { {\bar {\boldsymbol{R} } }_L^{ - 1}(n - 1){\boldsymbol{ {h} } }(n)} }{ {\gamma + { {h}^{\rm{T} } }(n){\bar {\boldsymbol{R} } }_L^{ - 1}(n - 1){ {\boldsymbol{h} } }(n)} }$2. ${\bar {\boldsymbol{\theta}} _0}(n) = ({{\boldsymbol{I}}} - {{\boldsymbol{k}}}(n){ {{\boldsymbol{h}}}^{\rm{T}}}(n)){\bar {\boldsymbol{\theta}} _L}(n - 1) + {{\boldsymbol{k}}}(n)d(n)$，${\bar {\boldsymbol{R} } }_0^{ - 1}(n) = {\gamma ^{ - 1} }{\bar {\boldsymbol{R} } }_L^{ - 1}(n - 1) - {\gamma ^{ - 1} }{{\boldsymbol{k}}}(n){ { {\boldsymbol{h} } }^{\rm{T} } }(n){\bar {\boldsymbol{R} } }_L^{ - 1}(n - 1)$ 3. for l = 1, 2, ···, L 4. ${\tau _l}(n - 1) = \dfrac{ {\sqrt {\boldsymbol{\xi}} ({\chi _l}(n - 1) - \gamma {\chi _l}(n - 2))} }{ {1 - \gamma } }$，${ {{\boldsymbol{g}}}_l} = \dfrac{ {\sqrt {\boldsymbol{\xi}} {\bar {\boldsymbol{R}}}_{l - 1}^{ - 1}(n){ {{\boldsymbol{e}}}_l} } }{ {1 + {\boldsymbol{\xi}} {{\boldsymbol{e}}}_l^{\rm{T}}{\bar {\boldsymbol{R}}}_{l - 1}^{ - 1}(n){ {{\boldsymbol{e}}}_l} } }$ 5. ${\bar {\boldsymbol{\theta} } _l}(n) = ({\boldsymbol{I} } - \sqrt {\boldsymbol{\xi} } { {\boldsymbol{g} }_l}{\boldsymbol{e} }_l^{\rm{T} }){\bar {\boldsymbol{\theta} } _{l - 1} }(n) + {{\boldsymbol{g}}_l}{\tau _l}(n - 1)$，${\bar {\boldsymbol{R}}}_l^{ - 1}(n) = {\bar {\boldsymbol{R}}}_{l - 1}^{ - 1}(n) - \sqrt {\boldsymbol{\xi}} { {{\boldsymbol{g}}}_l}{{\boldsymbol{e}}}_l^{\rm{T}}{\bar {\boldsymbol{R}}}_{l - 1}^{ - 1}(n)$ 6. end for 7. $\omega (n) = {\boldsymbol{\theta} } (n) + {\boldsymbol{\eta}} (n - 1)$
8. ${\boldsymbol{\hat a} }(n) = {\boldsymbol{u} }(n){ {\boldsymbol{u} }^{\rm{T}}}(n){\omega _1}(n)$，${\hat {\boldsymbol{b} } }(n) = \dfrac{ { {\boldsymbol{\varTheta} } _2^{\rm{T} }(n){u}(n)} }{ {\omega _1^{\rm{T}}(n){u}(n)} }$9. $\varphi (n) = ({ {\boldsymbol{\varTheta} } ^{\rm{T} } }(n){\boldsymbol{u} }(n)) \otimes {\boldsymbol{u} }(n)$，${{\boldsymbol{\eta}} }(n) = {{\boldsymbol{\eta}} }(n - 1) + {{\boldsymbol{\theta}} }(n) - {\varphi }(n)$ 10. ${\boldsymbol{\chi} } (n) = {[{\chi _1}(n),{\chi _2}(n), \cdots ,{\chi _L}(n)]^{\rm{T}}}{\text{ = } }\varphi (n) - {\boldsymbol{\eta} } (n)$ 11. n = n+1，返回步骤1，直至达到收敛条件 
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