Parametrization of multivariable systems using output injections: Alpha canonical forms

Abstract In this paper the parametrization of multi-output systems is considered. The method developed is based on the notion of output injections. A similar method has also been used in connection with the identification of a special class of systems. There, the resulting canonical form is called the α-canonical form. Here, using the same terminology, we address the following issues on the α-canonical form. First, we show that the α-canonical forms can be used to parametrize a more general class of systems, namely all minimal (reachable and observable) systems. This is achieved by the use of isomorphisms of the output space. Then, it is proven that the output injections used in constructing the α-canonical forms need to have a special structure in order to guarantee the uniqueness of the parametrization. This result also reveals the key role played by dead-beat observers in constructing the α-canonical forms. Finally, the connections between the α-canonical forms and input-output relations are investigated.