A new geometric algorithm with order reduction for robust strictly positive real synthesis

A new geometric algorithm with order reduction for robust strictly positive real (SPR) synthesis is presented. By searching from the boundary of the region of the weak strict positive realness (WSPR) of a polynomial, we can find the intersection of the WSPR regions of the polynomial family. Then the synthesis problem can be transformed to finding a feasible solution in ellipses with two variables, thus the problem becomes simpler and easy to solve, and the computational burden has been significantly reduced. Moreover, the derived conditions are necessary and sufficient for robust SPR synthesis of low-order polynomial segments (n≤5) or interval polynomials (n≤4). The algorithm is computationally efficient for some types of polynomial sets, such as segments, intervals and polytopes with arbitrary order. Illustrative examples are provided.