The optimal projection approach to model reduction and the relationship between the methods of Wilson and Moore

First-order necessary conditions for optimal reduced-order modelling of linear time-invariant systems are derived in the form of a pair of modified Lyapunov equations coupled by an oblique projection that determines the optimal reduced-order model. This form of the necessary conditions considerably simplifies previous results of Wilson ([1]) and clearly reveals the suboptimality of the balancing method of Moore ([2]).