Control of Symmetric Affine Systems via Direct Gradient Descent Control

Abstract

This paper is concerned with control of nonholonomic systems. As is well known, symmetric affine system is uncontrollable with continuously differentiable state feedback control. In this paper we apply Direct Gradient Descent Control (DGDC) for the symmetric affine system. DGDC is such a method that we manipulate control inputs directly so as to decrease a performance function by the steepest descent method. Note that the DGDC is a dynamic controller that we can adjust not only its gain parameter but also its initial condition. Then, not only controllable part of symmetric affine system is asymptotically stabilized by the DGDC but also uncontrollable part can be made to converge to the origin by choosing the initial condition appropriately. Applying the DGDC, we can control the symmetric affine system without transforming into the “chained form”. Simulation results for a four wheeled vehicle and flying robots demonstrate the effectiveness of the proposed method.