Nonlinear PI control of uncertain systems: an alternative to parameter adaptation

A novel approach to stabilization and trajectory tracking for nonlinear systems with unknown parameters and uncertain disturbances is developed. We take a drastic departure from the classical adaptive control approach consisting of a parameterized feedback law and an identifier, which tries to minimize a tracking (or prediction) error. Instead, we propose a simple nonlinear PI structure that generates a stable error equation with a perturbation function that exhibits at least one root. Trajectories are forced to converge to this root by suitably adjusting the nonlinear PI gains. We consider the two basic problems of: (i) matched uncertainties, when the uncertain terms are in the image of the input matrix, and (ii) unknown control directions, when the control signal is multiplied by a gain of unknown sign. We show that, without knowing the system parameters, and with only basic information on the uncertainties we can achieve global asymptotic stability and global tracking, without injecting high gains into the loop. Interestingly, we prove that we can take as our nonlinear PI structure an activation function reminiscent of that used in neural networks. Although most of the results are derived assuming full state measurement, we also present an observer-based solution for a chain of integrators with unknown control direction. The procedure is shown to provide simple solutions to the classical problems of neural network function approximation, as well as eccentricity control and friction compensation of mechanical systems.