Multivariate positive functions under coefficient perturbation

Testing of positive realness of a rational function when its coefficients vary over given intervals is required in practical applications such as design of adaptive control systems. The k-variate positive function is first related to a (k+1)-variate Hurwitz polynomial. Then, using the Kharitonov-type results for Hurwitz polynomials, similar results are derived for multivariate positive functions and multivariate very strict positive functions. This derivation shows that for a k-variate interval positive function, only 8×2^k extremum positive functions need be tested. This is against the previously reported 16×4^k functions for the multivariate case and 32 for the single-variate case. The interval positive real functions require the testing of 4×2^k extremum functions