基函数下非线性系统的支持向量机辨识

Support Vector Machine Identification of Nonlinear System Under Basis Function

对于非线性系统的辨识问题,将非线性系统在某组基函数下展开成线性回归形式。当基函数已知时,对于线性回归形式中的未知参数矢量,在性能指标中增加未知参数的惩罚项,采用规范化的最小二乘法来辨识未知参数矢量。当基函数未知时,在性能指标函数中增加预测误差的约束条件。对于约束最优化问题,分别从基-对偶角度考虑其最优解,并分析最优解的统计有偏性。为避免基函数的先验信息,利用最小二乘支持向量机中的核函数来替换线性回归矩阵的乘积运算,直接利用核函数来表示此约束最优化问题中的对偶矢量,并使用此对偶矢量和核函数来近似原非线性函数。最后利用仿真算例验证采用支持向量机的核函数来表示非线性系统的有效性。

To the problem of nonlinear system identification, the nonlinear system is expanded in a linear regression form under one basis function set. When the basis function set is known, the problem of how to identify the nonlinear system is transformed to estimate the unknown parameter vector exists in the linear regression form. One penalty term about the unknown parameter vector is added in the cost function and the regularized least square method is applied to estimate the unknown parameter vector. When this basis function set is unknown, the constraint about prediction error is considered in the cost function. As to this constrain optimization problem, the optimal solution is solved from the primal-dual point and its bias is also analyzed. To avoid the priori information about the basis function, the multiply operation coming from the linear regression matrix is replaced by one kernel function defined in the least square support vector machine theory. Then the dual vector in the constrain optimization problem is denoted by this kernel function directly and further the formal nonlinear function can be also approximated by one weighted sum form which is consisted of the kernel function and dual vector. Finally the simulation example results confirm the effectiveness of approximating the nonlinear system based on kernel function from support vector machine theory.