Implementation of the Robust MRAC Adaptive Control for a DC Motor: A Method Based on the Lyapunov’s Quadratic Functional

Abstract

This paper proposes a robust model reference adaptive controller (MRAC) based on the approach of Lyapunov’s quadratic functional to control the speed of a DC motor and to guarantee the desired performances of the motor under critical operating conditions such as varying working conditions due to changing environment, temperature increase or process aging. In this paper, the parameters armature’s resistance Ra, armature’s inductance La and moment of inertia J of the machine were varied from 10% to 100% of their nominal values. To achieve this, the methodological approach is as follows. The first step is to present the system, DC motor; then make its modeling. Then, a Lyapunov’s quadratic functional is chosen. It allows to guarantee the global stability of the system and in which (this function) derives the mechanism of adaptation of the motor parameters. The implemented parameter adjustment mechanism into the Matlab / Simulink environment gives the following results: a settling time of 13.6 milliseconds, a system rise time of 10.25 milliseconds, a final value of 0.165 for a step input. For the reference model this final value is 0.166, hence a tracking error of 0.001004. The overall relative uncertainty of the system is 0.6%; and the measurement uncertainty on the speed at the motor output is 0.000234% by applying a variation of 10% to 100% on its nominal parameters. Compared to the results obtained by implementing an MRAC controller based on the MIT rule and those of the classical PID controller, the results obtained from the analysis of the Lyapunov’s quadratic candidate function meet the objectives set and correct the shortcomings and inadequacy of the PID and the MIT controller in the face of the problems of varying working conditions, hence a robust MRAC-Lyapunov controller. These simulation results also show and prove the effectiveness of this controller based on the Lyapunov’s quadratic function, as it ensures the global stability of the controlled system and a judicious choice of the adaptation gain coefficients of this controller improves the output performances of the controlled system by cancelling the deviation between the two processes, DC motor and reference model, in presence or not of the parametric variations.