Finite time response control of affine systems

Abstract

The paper presents an original method for Finite Time Response (FTR) control of the affine systems. The FTR property is specific to linear systems only, known in the literature as dead-beat algorithms. In this work, it is developed as a new procedure for the affine systems FTR synthesis, called the Equivalent Input Method (EIM). For this purpose it calculates an equivalent input which will determine, according to a quadratic criterion, the best approximation of the affine component. This way the system is approximated by an affine system with an input variable equal to the sum of the original input and the equivalent input, but having only a residual affine component. This residual affine component has a smaller norm than the initial affine component. Considering zero the residual affine component, a FTR linear system synthesis procedure is applied. In the real system, controlled by a FTR control law, the residual affine component creates at each step a disturbance that FTR algorithm seeks to cancel. This approach is justified by the fact that the disturbance residual affine component is much smaller in norm than the original affine component. Under certain circumstances, this residual affine component can be zero. The controllability and algorithm convergence is analyzed. The proposed EIM method can be applied also for nonlinear systems approximated by Piecewise Affine Subsystems (PWAS). An experimental platform has been designed in Matlab environment allowing implementation of various affine systems and their control algorithms. Simulation results are included to support the method presented in the paper.