Robust disturbance rejection via P-D state feedback Example for lane keeping of highway vehicles

Abstract

The problem of robust disturbance rejection (RDR) using a Proportional plus Derivative (P-D) state feedback controller is studied for the case of single-input single output (SISO) linear systems with nonlinear uncertain structure. The linear systems are considered to be subjected to nonmeasurable disturbances. Sufficient conditions for the problem to have a solution are established. A family of P-D feedback controllers solving the problem is analytically determined. Sufficient conditions for the solution of the RDR problem with simultaneous robust stability are also established. Furthermore, the latter problem together with normality and asymptotic command matching is solved. The problem of lane keeping of highway vehicles was the motive for the derivation of all the above results. In particular, the problem of zeroing the relative lateral position of a vehicle travelling along a highway is formulated here as an RDR problem. The curvature of the road plays the role of the unknown disturbance. The problem of rejecting robustly the influence of the road curvature to the lateral position is proved to be always solvable, using a P-D feedback controller. The controller is analytically determined using a finite step algorithm. Furthermore, the free responses of the vehicle’s relative lateral position and lateral velocity are proved to be expressed only in terms of arbitrary modes. The general set of P-D feedback controllers solving the problem is proved also to guarantee robust stability of the closed-loop system. Using the infinity norm, the good performance of the rest of the vehicle’s variables, namely the yaw angle, the yaw rate and the front steering angle, is proved. Simulation results illustrate the good performance of the RDR over a wide range of system uncertainties and measurement errors.