中文摘要：

针对回声状态网络（ESN）传统的训练方法无法解决高维矩阵不可逆时的训练,以及无法应用于需要在线训练的建模当中等问题,提出了两种新的递推训练算法。分别将含遗忘因子递推最小二乘算法（FFRLS）和无先导卡尔曼滤波算法（UKF）应用到回声状态网络输出神经元为线性函数和非线性函数的权值训练中,进而直接对网络的输出权值进行递推更新。与传统的训练方法相比,所提新方法不仅具有在线更新、精度高的优点,而且还可以解决传统训练方法中批量数据构成的向量矩阵不可逆及输出神经元为非线性函数且其反函数不可求的问题。通过对连续搅拌釜式反应器（CSTR）浓度和温度的预测仿真,结果证明了所提新方法的有效性。

英文摘要：

Two new recursive algorithms are proposed in the light of the problems of the irreversibility of the high dimensional matrix, and the inability to apply online training, associated with a traditional echo state network （ESN）. We put forward a forgetting factor recursive [east square （FFRLS） algorithm and an unscented Kalman filter （UKF） algorithm for the training of the connecting weights in association with the linear and nonlinear output neuron functions, which can directly and recursively update the output connecting weights. The proposed methods have the advantages of higher precision and can be updated online and, in addition, can solve problems associated with the traditional echo state network training methods, such as the batch data based matrix inversion being diffi- cult to perform, and the inability to solve the inverse of the nonlinear output function. Simulations of the concentra- tion and temperature in a continuous stirred tank reactor （CSTR） demonstrate the viability and effectiveness of our proposed methods.