Closed form L2/H2-optimising of zeros for model reduction of linear continuous time systems

Closed form expressions of transfer function responses are applied in this article to model reduction of nth order continuous time systems with respect to the location of zeros for a given location of poles. Explicit closed form formulae are derived for the numerator coefficients in the transfer function of the reduced system that minimise the integrated square deviation from the original system with respect to the impulse response or higher order responses, i.e. effective L2 or H2-optimisation. The relative degree of the reduced model can be selected freely, e.g. as the original model's one, by selecting the number of numerator coefficients. Constraints are dealt with by introducing a vector of Lagrange multipliers corresponding to the response order. The formulae are also related to solutions of Lyapunov and Sylvester equations based on the companion matrices of the original and reduced systems. The formulae derived can be used to enhance the results obtained from other reduction techniques such as those based on balanced Grammian reduction and singular value decomposition for mid-sized systems. Two examples, demonstrating this, are presented. The formulae can also be used as a basis for a more general optimisation approach, where the optimisation with respect to the numerator coefficients or the zeros, resulting in the solution of a linear system of equations, is combined with non-linear optimisation with respect to the denominator coefficients or the poles.