Parameter estimation of unstable, limit cycling systems using adaptive feedback linearization: example of delta wing roll dynamics

The problem of estimating the unknown parameters of a nonlinear (plant) model for an unstable, limit cycling system is considered. A feedback linearizing control is used that aims to cancel the nonlinear terms in the plant model and to alter the linear terms such that the closed-loop plant model matches with a specified linear reference model. The controller parameters are unknown and are evolved, starting from zero, by an adaptive law that aims to drive them towards their ideal values that would provide perfect model matching between the reference model and the closed-loop plant model. The converged controller parameters would then provide good estimates for the unknown plant parameters. An external forcing signal is considered, common to both the reference model and the plant, and an adaptation law in the presence of this forcing function is derived using Lyapunov methods. Significantly, the same stability analysis is used both to derive a controller for the limit cycling system and also to provide a solution to the problem of parameter estimation. Simulations using an exponentially decaying sinusoidal forcing show that good estimates of the linear plant parameters are obtained for a wide range of values of the amplitude, frequency, and decay time of the forcing function, and also in the presence of measurement noise in the plant output. The problem of non-convergence of the nonlinear controller parameters is examined in some detail. The parameter estimation method is demonstrated in our paper on a nonlinear model for a rolling delta wing; however, it should be equally applicable to a wide class of limit cycling systems modeled by a set of nonlinear differential equations.