Carleman approximation based quasi-analytic model predictive control for nonlinear systems

This manuscript aims at developing a nonlinear model predictive controller formulation based on Carleman approximation. It approximates the nonlinear dynamic constraints with polynomial ones through Taylor expansion. Then, it extends the state variables to higher orders following the Kronecker product rule and expresses the nonlinear dynamic constraints with an extended bilinear representation. With little loss of nonlinear information, the formulation enables analytical prediction of future states. It also analytically calculates the sensitivity of the cost function to the manipulated inputs to facilitate the search algorithm by serving as the gradient. We present a brief analysis of error accumulation caused by Carleman approximation and then improve the accuracy of the approach by resetting extended states periodically. The idea of efficient temporal discretization is embedded in control vector parameterization to improve the controller performance. The advantages are illustrated in two applications where we solve a tracking problem and a regulation problem.