Bounded-Input Bounded-Output Stability Tests for Two-Dimensional Continuous-Time Systems

This paper presents two efficient algorithms to determine whether a bivariate polynomial, possibly with complex coefficients, does not vanish in the cross product of two closed right-half planes (is "2-C stable"). A 2-C stable polynomial in the denominator of a two-dimensional analog filter has been proved (not long ago) to imply bounded-input bounded-output (BIBO) stability. The two algorithms are entirely different but both rely on a recently proposed fraction-free (FF) Routh test for complex polynomials in this transaction. The first algorithm tests the 2-C stability of a bivariate polynomial of degree (n_{1},n_{2}) in order n^{6} of elementary operations (when n_{1}=n_{2}=n ). It is a "tabular type" two-dimensional stability test that can be regarded as a "Routh table" whose scalar entries were replaced by univariate polynomials. The second 2-C stability test is obtained from the first by its telepolation. It carries out the 2-C stability test by a finite collection of FF Routh tests and requires only order n^{4} elementary operations. Both algorithms possess an integer-preserving property that enhances them with additional merits including numerical error-free decision on 2-C stability.