Multivariable gain margin

Abstract

Abstract The gain margin for a multivariable system is defined by examining the positive definite hermitian (p.d.h.) portion of the polar decomposition of a non-singular perturbation, A, in the feedback path. The dual result defining the phase margin for a multivariable system has been derived by Bar-on and Jonckheere (1990) by examining the unitary portion of the polar decomposition for non-singular perturbations. This paper focuses on the multivariable gain margin, The main result is an extension of the classical SISO concept of gain margin to MIMO systems. The definition for gain margin presented encompasses both gain reduction and gain amplification tolerance because the measurement is relative to the identity matrix as opposed to zero. Closed loop stability is guaranteed for all p.d.h. matrices in the feedback path whose gain, relative to the identity matrix, is less than the gain margin of the system. Calculation of the gain margin requires solving a constrained optimization problem which is almost a complete dual of the constrained optimization problem solved when calculating the phase margin (see Bar-on and Jonckheere 1990).