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Time-convexity and time-gain-scheduling in finite-horizon robust H∞-control

A simple efficient method for addressing finite-horizon control problems is suggested for a class of linear time-varying systems by solving ‘standard’ (algebraic) linear matrix inequalities (LMIs). Considering the Lyapunov function xTP(t)x, a class of time-varying solutions for P(t) is derived without using differential inequalities. The core idea is to seek a P(t) which is linear in time, and to exploit convexity in normalized time over the scenario duration (or over each time sub-interval, in the piecewise version) in order to reduce the differential LMIs, which result from the bounded real lemma, to algebraic LMIs at the scenario (or sub-intervals) end-points. The resulting state-feedback gain is explicitly time-scheduled. The method can simultaneously cover polytopic parametric uncertainties, and can be applied together with the ‘best-mean’ approach.