Novel Model Reference Adaptive Control Designed by a Lyapunov Function That is Kept at Low Value by Fixed Point Iteration

Abstract

Even in our days the generally prevailing methodology used for designing adaptive controllers is based on Lyapunov’s “direct method”. This statement holds for the special “Model Reference Adaptive Controllers (MRAC)”, too, that have to satisfy two typical requirements. Besides providing precise trajectory tracking they have to generate an illusion to an external control loop working on purely kinematic/kinetic basis that it controls a system the dynamic properties of which are identical to that of a “Reference System”. Based on the structure of the more or less plausible Lyapunov functions the Reference System used to be stable, linear, time-invariant. To evade the mathematical difficulties of the Lyapunov function-based design, as an alternative approach, “Fixed Point Iteration (FPI)”-based design technique was introduced that had a special MRAC variant, too. In this approach the usual restrictions for the dynamics of the Reference System were naturally released. Later it was shown that this adaptive approach can be interpreted from the point of view of the Lyapunov functions in which non-conventional feedback terms are used for driving the Lyapunov function near zero and keeping it in its vicinity. This approach naturally gave rise to a novel MRAC design idea that is akin to that of the “Backstepping Controllers”. In this paper this idea is presented and illustrated via simulation investigations. Simulation results are provided regarding a completely driven 2 degree of freedom physical system, that consists of two nonlinearly coupled and nonlinearly damped mass-points.