Carleman linearization
In mathematics, Carleman linearization (or Carleman embedding) is a technique to transform a finite-dimensional nonlinear dynamical system into an infinite-dimensional linear system. It was introduced by the Swedish mathematician Torsten Carleman in 1932. Carleman linearization is related to composition operator and has been widely used in the study of dynamical systems. It also been used in many applied fields, such as in control theory and in quantum computing. Procedure Consider the following autonomous nonlinear system: where denotes the system state vector. Also, and 's are known analytic vector functions, and is the element of an unknown disturbance to the system. At the desired nominal point, the nonlinear functions in the above system can be approximated by Taylor expansion where is the partial derivative of with respect to at and denotes the Kronecker product. Without loss of generality, we assume that is at the origin. Applying Taylor approximation to the system, we obtain where and . Consequently, the following linear system for higher orders of the original states are obtained: where , and similarly . Employing Kronecker product operator, the approximated system is presented in the following form where , and and matrices are defined in (Hashemian and Armaou 2015). See