Stabilizing control of symmetric affine systems by Direct Gradient Descent Control

This paper is concerned with control of nonholonomic systems. As is well known, symmetric affine system is uncontrollable with continuous time-invariant differentiable state feedback. In this paper we apply Direct Gradient Descent Control (DGDC) for the symmetric affine system. The 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, but also uncontrollable part can be converged to the origin by choosing the initial condition appropriately. Applying the DGDC, we can control the symmetric affine system without transforming it into the "chained form". Simulation results for a four wheeled vehicle and a flying robot demonstrate the effectiveness of the proposed method.