A Closed-Form Approach to Tracking Control of Nonlinear Uncertain Systems Using the Fundamental Equation

In this paper a new approach to tracking control problems for nonlinear uncertain systems is proposed. Real-life complex multi-body systems are usually uncertain, the uncertainties arising from the lack of precise knowledge in modeling the systems. These uncertainties, which we assume to be time varying and unknown yet bounded, are considered in this paper. The uncertainties being unknown, what is available in hand is therefore just the so-called `nominal system,' which is simply our best assessment and description of the actual real-life situation. The aim in this paper is to develop a general control methodology that when applied to an actual real-life uncertain system, causes this system to track a desired reference trajectory that is pre-specified for the nominal system to follow. Our control methodology is developed in two steps. First, we utilize the fundamental equation that provides a nonlinear optimal controller for the nonlinear nominal system so that the reference trajectory is exactly tracked. No approximations/linearizations are done related to the nonlinear nature of the system. In the next step, we ensure that the unknown, uncertain system tracks this reference trajectory by augmenting an additional controller—based on the idea of sliding mode control—to the fundamental equation, providing a general approach for obtaining the equations of motion for a nonlinear uncertain constrained mechanical system. While based on the idea of a sliding surface, the approach permits the use of a large class of control laws that can be adapted to the specific real-life practical limitations of a given, particular controller being used. Thus, in sum, a methodology is developed to obtain a set of closed-form controllers for nonlinear uncertain multi-body systems that can track a desired reference trajectory that the nominal system is required to follow. The set of controllers shows good transient behavior and is robust with respect to the uncertainties involved in the description of the real-life system.