Determination of optimal feedback gain matrix for a class of nonlinear systems

The paper studies optimal control of a class of nonlinear systems with quadratic/non-quadratic cost functions and suggests a new methodology to determine the control gain matrix for the nonlinear system. The nonlinear system model is approximated as a sequence of linear time varying approximations in which the classical controller design methods can be utilized. Then, optimal control is designed for the approximated linear time varying system where the optimal control is determined to minimize a given cost function. The results of convergence of the successive linear time varying approximations to the nonlinear system are used and an optimal feedback gain matrix for the nonlinear system is obtained. The convergence of optimal control gain matrix designed for the successive linear time varying systems to a nonlinear gain matrix is proved. Once the gain matrix is determined from the successive approximations, then it is used as an optimal feedback gain matrix for the nonlinear system in a range of implementation domain. The method suggested in this study eliminates the disadvantages of the successive linear time varying approximation approach where the optimal control is obtained from a series of approximations for all implementation cases. The suggested methodology is compared with similar types of approximation based approaches studied in the literature.