Gain margins for multivariable control systems

The phase margin for a multivariable system is defined by examining the unitary portion of the polar decomposition of a perturbation, Δ, in the feedback path. A dual result defining the gain margin for a multivariable system is derived by examining the positive definite hermitian (PDH) portion of the polar decompositions for nonsingular perturbations. This study focuses on the multivariable gain margin. The main result is an extension of the classical SISO (single input single output) concept for all PDH matrices in the feedback path whose gain 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