**AUTHORS:**Hamza Khan, Jozsef K. Tar, Imre J. Rudas, Gyorgy Eigner

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**ABSTRACT:**
Nonlinear Programming provides a practical, reduced-complexity solution for the realization of Model Predictive Controllers in which a cost function representing contradictory limitations is minimized under the constraints that express the dynamical properties of the system under control. For nonlinear system models and non-quadratic cost functions the solution over a finite time-grid can be obtained by the use of Lagrange’s Reduced Gradient Method that needs complicated numerical calculations. In this paper it is shown that under not too limiting conditions this procedure can be replaced by a simple fixed point seeking iteration based on Banach’s Fixed Point Theorem. The simplicity of the proposed algorithm widens the possibility for the practical applications of the Receding Horizon Control method. The same algorithm is used for adaptively and precisely tracking the “optimized trajectory” that can be constructed by the use of a dynamic model of “overestimated” parameters in order to evade dynamical overloads in the control process. To illustrate the efficiency of the method the Receding Horizon Control of a strongly nonlinear, oscillating system, the van der Pol oscillator is presented. In the simulations three different parameter settings are considered: one of them produces the trajectory to be tracked, the second one is used for the optimization, and the third one serves as the model of the controlled system.

**KEYWORDS:**
Nonlinear Programming, Model Predictive Control, Receding Horizon Controller, Adaptive Control, Fixed Point Transformation

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