Gap metric computation for time-varying linear systems on finite horizons

Abstract

It is shown how to compute the aperture, or gap, between the finite-horizon graphs of linear time-varying dynamical systems, given state-space models of the continuous-time input-output maps. The approach involves four quadratic matrix first-order differential equations, three with the symplectic structure of Riccati equations. All four are subject to single-ended boundary conditions. So standard numerical methods can be used to compute the solutions as required. Three of the differential equations, two Riccati and the other not, need to be solved once to construct normalized graph representations, and to test an invertibility condition, respectively. When this invertibility condition is verified, the directed gaps are equal, and the remaining Riccati differential equation is solved repeatedly in a bisection search to determine one of these. Computation of both directed gaps to find the maximum would, by contrast, involve repeatedly solving two such Riccati differential equations, each in a bisection search.