### Abstract

Based on the observation that it is very difficult to combine the mathematical frameworks of the prevailing Lyapunov function-based adaptive controllers and the traditional, optimal control-based “Model Predictive Controllers (MPC)”, a novel adaptive solution was introduced to improve the operation of the “Receding Horizon Controllers (RHC)”. Because at the local optima the gradient of the auxiliary function is zero, Lagrange’s original “Reduced Gradient Method (RGM)” was replaced by a “Fixed Point Iteration (FPI)”-based algorithm that directly drove this gradient toward zero. According to “Banach’s Fixed Point Theorem” the convergence of the method was guaranteed by a contractive function that generated the iterative sequence. The greatest modeling burden in this approach was the need for the calculation of the Jacobian of the problem, i.e. the gradient of the gradient of the auxiliary function. In the first simulations only a single “Degree of Freedom (DoF)” 2^{nd} order nonlinear system, a van der Pol oscillator was investigated. The attempts that were made to evade the calculation of the Jacobian were finished with the conclusion that at least a rough numerical approximation of this Jacobian generally must be utilized. Though the MPC approach allows the use a great variety of cost functions and dynamical models, mathematically well established results are available only for quadratic cost functions and “Linear Time-invariant (LTI)” models. For other cost functions and models careful numerical analysis is needed. In this paper the use of non-quadratic cost functions is numerically investigated in the FPI-based adaptive RHC control of 2 DoF 2^{nd} order nonlinear system that consists of two, nonlinearly coupled van der Pol oscillators, is considered. To guarantee lucid calculations simple functions are introduced that map the active parts of the horizon under consideration to the elements of the gradient of the auxiliary function that are calculated analytically. For the calculation of the Jacobian only a rough numerical estimation is applied. The simulation results reveal certain limitations of the suggested method.

Original language | English |
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Pages (from-to) | 135-154 |

Number of pages | 20 |

Journal | Acta Polytechnica Hungarica |

Volume | 16 |

Issue number | 9 |

DOIs | |

Publication status | Published - Jan 1 2019 |

### Keywords

- Adaptive Control
- Banach Space
- Fixed Point Iteration-based Adaptive Control
- Model Predictive Control
- Optimal Control
- Receding Horizon Control
- Van der Pol Oscillator

### ASJC Scopus subject areas

- Engineering(all)