Dynamic operability of mimo systems with time delays and transmission zeroes—I. Assessment

Abstract

Right half-plane transmission (RHPT) zeroes close to the origin of the complex plane impose severe limitations in the closed-loop performance of multivariable systems. Accurate information on their location is essential for the assessment of dynamic operability. A rational system has finite transmission zeroes, and there exist robust and reliable tools (e.g. the QZ algorithm) for their computation. In multivariable systems with time delays, infinite transmission zeroes may be present which makes their numerical determination and therefore the assessment of dynamic operability very difficult. Treatment of the time delays by Padé approximations can lead to erroneous results. This work presents a novel method for the characterization of the transmission zeroes of multivariable time delay systems. The theory of the distribution of roots of quasi-polynomials is explored to develop a test that identifies systems with infinite RHPT zeroes. The test also extracts information that allows fast computation of the infinite transmission zeroes by asymptotic formulas. The traps involved in the use of numerical root finders are avoided, since only the system's transfer function matrix is needed. The advantages of this approach over the QZ algorithm are discussed. This method provides a valuable tool for the fast assessment of dynamic operability of MIMO delay systems. It can identify processes with potential control difficulties due to RHPT zeroes close to the origin and evaluate alternative designs. Several examples illustrate its application and effectiveness.