Powers of SPR functions and preservation properties

Basic properties of a new class of strictly positive real (SPR) functions are stated. Four problems are studied. The first deals with SPR preservation of transfer functions, obtained under the composition of polynomials with SPR0 functions. The second deals with Hurwitz stability preservation of the numerator of transfer functions, obtained under the composition of polynomials with SPR0 functions. The third deals with making a Hurwitz closed-loop plant, an SPR0 function by substituting s by SPR0 functions. The four deals with the synthesis of simultaneous SPR feedback plants. For the new class of SPR0 functions, a characterization is presented. For the first and second problems, sufficient conditions are presented using the new class of SPR0 functions. For the third and four problems, two examples are presented, the first being for simultaneous SPR closed-loop systems via constant controllers. The second is for simultaneous stabilization via universal feedback adaptive control.