Gramian for Control of Fractional Order Multivariate Dynamic System

Abstract

In the paper the concept of Gramian for, dynamic state space multivariate fractional order system is discussed, vis-à-vis integer order systems. Interestingly the basic state transition matrix for fractional order systems with Caputo’s formulation has two types, both participate in the solution to give state trajectory, is elucidated in this paper. These state transition matrices for fractional order systems are from higher transcendental functions in matrix form, called alpha-exponential functions which are shown to be eigenvectors for Caputo and Riemann-Liouvelli derivative based fractional order homogeneous system. These alpha-exponential functions are one-parameter Mittag-Leffler, and Robotnov-Hartley functions. With these state transition matrices, control Gramian, state trajectories, input control vector and control energy are derived, for fractional order system. The differences and modifications for fractional order system vis-à-vis integer order counterparts are elaborated and reasoned out. Several examples are given to compute state transition matrices, Gramian, state trajectory and effort.