Evolving Takagi–Sugeno model based on online Gustafson-Kessel algorithm and kernel recursive least square method

Abstract

In this paper, we introduce an evolving system utilizing sparse weighted kernel least square as local models and online Gustafson-Kessel clustering algorithm for structure identification. Our proposed online clustering algorithm forms elliptical clusters with any orientation which leads to creating less but more complex shape clusters than spherical ones. Moreover, the clustering algorithm is able to determine number of required clusters by adding new clusters over time and to reduce the redundancy of model by merging similar clusters. Additionally, we propose weighted kernel recursive least square method with a new sparsification procedure based on instant prediction error. Also, we introduce an adaptive gradient-based rule for tuning kernel size. The sparsification procedure and adaptive kernel size improve the performance of kernel recursive least square, significantly. To illustrate our methodology, we apply the introduced model to online identification of a time varying and nonlinear system. Finally, to show the superiority of our approach in comparison to some known online approaches, two different time series are considered: Mackey–Glass as a benchmark and electrical load as a real-world time series.

Appendix A

According to recursive Gauss–Newton algorithm, ${\mathbf{J}}_{j}^{k + 1} \, = \, - \frac{{\partial e^{k + 1} }}{{\partial \sigma_{j}^{k} }}$ can be written as following:

$${\mathbf{J}}_{j}^{k + 1} \, = \,\frac{{\partial \hat{y}_{j}^{k + 1} }}{{\partial \sigma_{j}^{k} }}\varPsi_{j} \left( {{\mathbf{x}}^{k + 1} } \right)\, = \,{\mathbf{a}}_{j}^{T} \left( k \right)\frac{{\partial {\mathbf{g}}_{j} \left( {k + 1} \right)}}{{\partial \sigma_{j}^{k} }}\varPsi_{j} \left( {{\mathbf{x}}^{k + 1} } \right)$$

(A.1)

where $\frac{{\partial {\mathbf{g}}_{j} \left( {k + 1} \right)}}{{\partial \sigma_{j}^{k} }}$ equals to:

$$\frac{{\partial {\mathbf{g}}_{j} \left( {k + 1} \right)}}{{\partial \sigma_{j}^{k} }}\, = \,\left[ {\begin{array}{*{20}c} {\frac{{\left| {\left| {{\mathbf{x}}^{k + 1} - {\mathbf{c}}_{j}^{1} } \right|} \right|^{2} }}{{\left( {\sigma_{j}^{k} } \right)^{3} }}\kappa \left( {{\mathbf{x}}^{k + 1} ,{\mathbf{c}}_{j}^{1} } \right)} \\ \vdots \\ {\frac{{\left| {\left| {{\mathbf{x}}^{k + 1} - {\mathbf{c}}_{j}^{N} } \right|} \right|^{2} }}{{\left( {\sigma_{j}^{k} } \right)^{3} }}\kappa \left( {{\mathbf{x}}^{k + 1} ,{\mathbf{c}}_{j}^{N} } \right)} \\ \end{array} } \right]$$

(A.2)

So by considering all these, we can conclude (47).