The majorant Lyapunov equation: A nonnegative matrix equation for robust stability and performance of large scale systems

A new robust stability and performance analysis technique is developed. The approach involves replacing the state covariance by its block-norm matrix, i.e., the nonnegative matrix whose elements are the norms of subblocks of the covariance matrix partitioned according to subsystem dynamics. A bound (i.e., majorant) for the block-norm matrix is given by the majorant Lyapunov equation, a Lyapunov-type nonnegative matrix equation. Existence, uniqueness, and computational tractability of solutions to the majorant Lyapunov equation are shown to be completely characterized in terms of
M
matrices. Two examples are considered. For a damped simple harmonic oscillator with uncertain but constant natural frequency, the majorant Lyapunov equation predicts unconditional stability. And, for a pair of nominally uncoupled oscillators with uncertain coupling, the majorant Lyapunov equation shows that the range of nondestabilizing couplings is proportional to the frequency separation between the oscillators, a result not predictable from quadratic or vector Lyapunov functions.