Semi-Discretization for Fractional-Order Feedback Systems with Input Delay

Abstract

In this paper, we apply the semi-discretization method to controllable single-input linear systems with input delay and fractional-order feedback. This requires the fractional derivative to be discretized in a way that the resulting discrete map is linear and time-invariant. To this end, three different techniques are investigated, namely, the short-memory principle, the application of adaptive time steps, and the exponential approximation of the derivative weights. For all three approaches, we construct a linear map that describes the dynamics of the approximate semi-discrete system. The matrices of these linear maps can be used to investigate the stability of the system. As an example, stability charts are determined for the fractional-order proportional-derivative delayed control of the inverted pendulum, and the numerically obtained stability charts are compared with the exact stability boundaries.