Error Bounds for Carleman Linearization of General Nonlinear Systems
Arash Amini, Qiyu Sun, Nader Motee
Abstract
In his 1932 paper, Carleman proposed a linearization method to transform a given finite-dimensional nonlinear system with analytic right-hand into an equivalent infinite-dimensional linear system with (usually) unbounded operators. Finite truncation of the transformed system has been used to study dynamic properties, learning, and control of nonlinear systems. One of the remaining outstanding problems in this context is quantifying the effectiveness of such finitely truncated models. Intuitively, one expects that as the truncation length increases, the truncated system will approximate the original nonlinear more effectively. In this paper, we provide explicit error bounds and prove that the trajectory of the truncated system stays close to that of the original nonlinear system over a quantifiable time interval. This is particularly important in applications, such as Model Predictive Control, to choose proper truncation lengths for a given sampling period and employ the resulting truncated system as a good approximation of the nonlinear system. Several examples are discussed to support our theoretical estimates.