Neural networks are often linearized to construct a framework of neural predictive control

but a challenging issue remains that lots of dynamics caused by omitting high-order terms are unmodeled.We took ESNs (Echo State Networks) as a paradigm

and proposed a ridge regression method to compensate unmodeled dynamics.The unmodeled dynamics were observed by collecting the difference of ESN internal states before-and-after linearization

and they were represented by a linear function of ESN states estimated with ridge regressions.The compensation terms for the unmodeled dynamics were then internalized as movements and rotations of attractors in ESN reservoirs.The internalization provided a new possibility:The loss of boundary constraint because of linearization of activation functions can be partially remedied by the attraction effect of attractors.Two examples demonstrated that our compensation method could actively improve the control.