Worst-case inter frequency grid behavior of transfer functions identified via finite frequency response data

Experimental data is crucial during the design and analysis of feedback control systems. Especially for LTI systems, the frequency domain has appeared to be a favorable domain to describe the dynamics of the system at hand. However, due to practical constraints, frequency response data can only be measured on a finite number of frequency points. On the other hand, system properties such as stability and performance can only be obtained by considering the frequency response behavior on a continuum of frequencies. An estimate for the behavior between the measured data points could be estimated via interpolation. However, the results of such an interpolation procedure heavily depends on the prior assumptions made. Different assumptions result in different interpolants. In order to minimize the number of prior assumptions, this paper focusses on a set description of all interpolants. A method is proposed to compute the envelope that contains the frequency response behavior of all possible underlying data-generating systems that corresponds to the measured plant behavior at the frequency grid points and prior assumption made upon the relative stability of the underlying system. It appears that the ratio between amount of relative stability and the chosen frequency grid-size is the key parameter that determines size of the uncertainty set.