New Continuous Control Methodology for Nonlinear Dynamical Systems with Uncertain Parameters

This paper presents a new reference-tracking control methodology for nonlinear dynamical systems in the presence of unknown, but bounded, uncertainties in the system. To this end, two controllers are combined. A nonlinear controller is first developed to exactly track the desired reference trajectory assuming no uncertainties in the nonlinear nominal system. The entire nonlinear dynamics of the nominal system is included and no approximations / linearizations are made. Next, an additional continuous controller is developed in closed form to compensate for uncertainties in the physical model by generalizing the concept of sliding surfaces. Unlike conventional sliding mode control, this Lyapunov-based approach eliminates the chattering problem by replacing a signum function with a set of continuous
functions that may have different forms depending on practical considerations related to actuator implementation. Among these possible forms, special attention is paid to a controller with a PID form. By using Lyapunov stability theory it is shown that this additional controller forces the tracking errors that arise because of the uncertainties in the system to move into a small, user-specified region around the generalized sliding surface. Once these tracking errors enter this small region, if the original nonlinear system is assumed to be linearizable, then linear control theory will ensure that they will further converge to even smaller values. A numerical example is provided, in which a satellite in the presence of air drag is required to maintain a specific, circular orbit around the Earth whose gravity field is imprecisely known. The example demonstrates the accuracy, efficiency, and ease of implementation of the control methodology.