Input and Output Normal Forms for Model Order Reduction of Linear Systems

This paper considers the problem of model order reduction by transforming the system into input and output normal forms. The reachability gramian in the input normal form is the identity matrix and the observability gramian is a diagonal matrix. Conversely, the observability gramian in the output normal form is the identity matrix and the reachability gramian is a diagonal matrix. The elements of the non-identity diagonal gramians in both normal forms are the squares of the system Hankel singular values. This fact determines the equivalent role which both normal forms play in model order reduction. The implemented projection is nearly orthogonal up to a scaling with the elements of a diagonal matrix. In the paper are shown the relations between the transformed system descriptions and the reachability and observability operators. Major influence for the output energy distribution in the input normal form has the observability operator, while the input energy is uniformly distributed. Alternatively, the input energy distribution in the output normal form is due to the reachability operator action, while the output energy is uniformly distributed. Several experiments are performed confirming the equivalent role of the input and output normal forms in the procedures of system approximation.