A Novel Expression for Computing Time Response of LTI Systems of Arbitrary Order with Applications to Fractional and Stochastic Control

A novel approach is employed to obtain closed-form and exact expressions for the time response of LTI systems of arbitrary order in terms of the univariate Fox H-Function. This presents a convenient and efficient method to study the system output to a variety of test functions, including the step, impulse, sinusoidal, exponential and ramp response, simply by altering the parameters of the H-Function. The resulting expressions are verified for accuracy by comparing the system response with simulations of different models. The representation of the time response as an H-Function allows us to obtain the time response of fractional order systems by approximating fractional operators with higher order rational transfer functions. Furthermore, it also leads to a novel expression for the expected value of the output of a fractional order system when driven by a wide-sense stationary (WSS) random process, which is verified against Monte-Carlo simulations.