A method for the order reduction of linear switching systems

The paper deals with the problem of reducing the order of an original high-order asymptotically stable linear switching system by independently approximating the (stable) LTI systems corresponding to every fixed value of the switching signal. Precisely, each reduced-order model is obtained by minimising the L2 norm of a weighted equation error by means of an efficient algorithm that ensures model stability as well as the retention of a number of first- and second-order information indices, such as the Markov parameters and the impulse-response energies. Then, the stability of the switching system is guaranteed, irrespective of the switching law, by realising the aforementioned reduced models in such a way that they share a common Lyapunov function. To this purpose, a simple state-coordinate transformation amenable to online implementation is applied to the state models initially derived. To improve the approximation, the state after every switching is reset, with due care for stability, according to a fast inclusion-projection procedure. Two examples taken from the literature show that the suggested reduction technique compares favourably with existing techniques.