Universal Singular Optimal Control: Affine Systems

Abstract

In this study, the problem of finding an optimal controller for nonlinear systems with one input and a reference tracking signal is approached. With the problem's formulation, any desired signal can be tracked instantly with a closed-loop controller without the need for integral terms. Presentation lies at the heart of optimal control. This study, however, does not consider the integral term, allowing tracking and stability to occur naturally. It has a broad scope with a wide range of applications, namely when dealing with affine nonlinear systems, which provide geometric control unification with asymptotic stability in some cases. A common scenario that comes from optimal control involves the minimization of integral cost functionals. Issues like asymptotic stability or even tracking to the desired reference signal have always been the main limitations. In this study, the main theorem allows the solution of optimal control problems with no-integral terms, in other words tracking problems with input/state constraints, providing closed-loop controllers. A DC motor with a pendulum in upright position is an example of an application for which singular optimal control is tested in this study. The results confirm both asymptotic stability and optimal tracking with an accuracy of 95%. The main contributions of this study include an optimal closed-loop controller with no mixed initial/final conditions, input/state constraints, asymptotic stability guarantee, a strong connection with geometric tools and finally the possibility to generalize to systems with multiple inputs. As a conclusion, general nonlinear control systems can be included in the optimal control methodology presented in this study including input/state constraints. Due to the lack of integral terms, the problem can be solved in closed form by using an optimal closed-loop controller.