Fixed Point Transformation-based Adaptive Optimal Control Using NLP

To reduce the effects of modeling imprecisions, in the traditional "Receding Horizon Control" that works with finite horizon lengths, in the consecutive horizon-length cycles, the actually measured state variable is used as the starting point in the next cycle. In this design, within a horizon-length cycle, a cost function is minimized under a constraint that mathematically represents the dynamic properties of the system under control. In the "Nonlinear Programming" (NLP) approach the state variables as well as the control signals are considered over a discrete time-resolution grid, and the solution is computed by the use of Lagrange's "Reduced Gradient" (RG) method. It provides the "estimated optimal control signals" and the "estimated optimal state variables" over this grid. The controller exerts the estimated control signals but the state variables develop according to the exact dynamics of the system. In this paper an alternative approach is suggested in which, instead of exerting the estimated control signals, the estimated optimized trajectory is adaptively tracked within the given horizon. Simulation investigations are presented for a simple "Linear Time-Invariant" (LTI) model with strongly non-linear cost and terminal cost functions. It is found that the transients of the adaptive controller that appear at the boundaries of the finite-length horizons reduce the available improvement in the tracking precision. In contrast to the traditional RHC, in which decreasing horizon length improves the tracking precision, in our case some increase in the horizon length improves the precision by giving the controller more time to compensate the effects of these transients.