On the thirty-two virtual polynomials to stabilize an interval plant

This paper studies the conservatism of the 32 virtual polynomials to stabilize an interval plant. It is shown that working with the 32 virtual vertices is generally less conservative than with the Kharitonov polynomials of the smallest interval polynomial containing the characteristic polynomial polytope. By means of the former, it is possible to find all the controllers such that the value set of the polytope of characteristic polynomials is applied in two quadrants as a maximum for each ω; while using the latter, only some of them can be found. The cases in which both methods coincide are also analyzed, and the conditions on the numerator and denominator of the controller are developed. Thus, this coincidence can be known a priori from the characteristics of the coefficients of the numerator and denominator of the controller. It is shown that these conditions are satisfied by the first-order controllers