Nonlinear Control for a Class of Non-Control Affine Systems

Author(s):Ortner P.
Very many results for optimal control of nonlinear control affine systems are available, while only few results for the non-control affine case exist. This thesis addresses the problem of optimal control for a class of nonlinear non-control affine systems called Extended Hammerstein System (EHS), which is composed of a nonlinear control affine dynamical system and a nonlinear static map. The standard solution to solve the optimal control problem for non-control affine systems is to extend the state of the non-control affine system and adding an integrator to obtain a control affine system. A drawback of this standard method is the need for an integrator which produces poor robustness properties in a control loop. Another drawback is that a partial differential equation, the Hamilton Jacobi Bellmann (HJB) equation, must be solved in the design procedure, which is usually a very difficult task and therefore often solved using inverse techniques, which allow a modification of the original performance index to increase the degrees of freedom in solving the HJB. In this thesis an alternative approach is presented using the EHS class. With this class of systems an approximation of the optimal control problem can be solved. A transformation is presented which optimally maps the original non-control affine problem into an EHS approximation minimizing the quadratic error in the state equation. The transformation is based on either a least squares or an instrumental variables optimization both inspired by system identification. The standard way to solve the HJB for an unknown storage function and for a given system and performance index usually is very difficult. In this thesis the degrees of freedom in the optimal transformation can be used to simplify the solution of this partial differential equation via a so called converse approach. In this converse approach the HJB equation is solved for an unknown part of the system with a given parameterized storage function and a giv
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