Stability Criterion for Continuous-Time System Polynomials with Uncertain Complex Coefficients

Let a polynomial Pn(s) = p0 + p1s + … + pnsⁿ have coefficients Pi = ai + jbi that may vary, or only known to be, in intervals. Kharitonov's criterion asserts that Pn(s) has all its zeros in the left half of the complex plane (is Hurwitz) for all admissible values of the coefficients, if, and only if, some well-defined 8 complex fixed coefficient polynomials are Hurwitz. When the uncertain polynomial is real the criterion involves only four fixed real polynomials. We restate and give a simple proof for Kharitonov's criterion for both real and complex polynomials. Our derivation is based on evaluation of complex rational lossless positive real functions and their relation to Hurwitz polynomials.