Criterion of approximation for designing 2×2 feedback systems with inputs satisfying bounding conditions

A common practice in designing a feedback system with a non-rational transfer matrix is to replace the non-rational matrix with an appropriate rational approximant during the design process so that reliable and efficient computational tools for rational systems can be fully utilized. Consequently, a criterion of approximation is required to ensure that the controller obtained from the approximant still provides satisfactory results for the original system. This paper derives such a criterion for the case of two-input two-output feedback systems in which the design objective is to ensure that the errors and the controller outputs always stay within prescribed bounds whenever the inputs satisfy certain bounding conditions. For a given rational approximant matrix, the criterion is expressed as a set of inequalities that can be solved in practice. It will be seen that the criterion generalizes a known result for scalar systems. Finally, a controller for a binary distillation column is designed by using the criterion in conjunction with the method of inequalities.