Online near optimal control of unknown nonaffine systems with application to HCCI engines

Multi-input and multi-output (MIMO) optimal control of unknown nonaffine nonlinear systems is a challenging problem due to the presence of control inputs inside the unknown nonlinearity. In this paper, the optimal control of MIMO nonlinear nonaffine discrete-time systems in input-output form is considered when the internal dynamics are unknown. First, the nonaffine nonlinear system is converted into an affine-like equivalent nonlinear system under the assumption that the higher-order terms are bounded. Next, a forward-in-time Hamilton-Jaccobi-Bellman (HJB) equation-based optimal approach is developed to control the affine-like nonlinear system using neural network (NN). To overcome the need to know the control gain matrix of the affine-like system for the optimal controller, an online identifier is introduced. Lyapunov stability of the overall system including the online identifier shows that the approximate control input approaches the optimal control with a bounded error. Finally, the optimal control approach is applied to the cycle-by-cycle discrete-time representation of the experimentally validated HCCI engine which is represented as a nonaffine nonlinear system. Simulation results are included to demonstrate the efficacy of the approach in presence of actuator disturbances.