Minimal Systems

Abstract

In this chapter the notion of a minimal system is considered. If two systems are similar, then they have the same transfer function. The converse statement is not true. In fact, systems with rather different state spaces may have the same transfer function. For minimal systems this phenomenon does not occur. In Section 7.1 minimal systems are defined as systems that are controllable and observable. The latter two notions are explained in more detail for finite-dimensional systems in Section 7.2. In the finite-dimensional case the connection between a minimal system Θ and its transfer function W_Θ is very close. For example in that case Θ is uniquely determined up to similarity by W_Θ. This result, which is known as the state space similarity theorem, will be proved in Section 7.3. Several examples, presented in Section 7.4, show that a generalization of the finite-dimensional theory to an infinite-dimensional setting is not possible in a straightforward way. An appropriate generalization requires a further refinement of the state space theory. In Section 7.5 the notion of minimality is considered for Brodskii systems, Kreîn systems, unitary systems, monic systems, and polynomial systems.