nonminimizable

Nonminimizable is a term used in systems theory and control engineering to describe a property of a system's input-output behavior that cannot be captured by a finite-dimensional minimal state-space model. In the standard framework for linear time-invariant systems, a transfer function G(s) has a minimal realization if there exists a finite-dimensional state-space pair (A, B, C, D) that is both controllable and observable and that realizes G(s). Such a realization has dimension equal to the McMillan degree, the degree of G(s). A system (or its transfer function) is called nonminimizable when no finite-dimensional minimal realization exists; equivalently, the transfer function has infinite McMillan degree or is non-rational.

This situation arises with infinite-dimensional dynamics, such as pure time delays G(s) = e^{-τs} or distributed-parameter systems governed by partial differential equations. For these systems, any finite-state realization can only be an approximation of the exact input-output behavior. In practice, engineers often seek finite-dimensional approximations (for example, Padé approximations) to enable standard control design, accepting that the model is an approximation rather than an exact minimal model. The concept of nonminimizable is thus a statement about the fundamental limits of finite-state modeling for the given system.

Nonminimizable is distinct from nonminimal realizations, which refer to particular finite realizations that are not controllable or observable. In those cases, a smaller, minimal realization can often be obtained. Nonminimizable, by contrast, points to the absence of any finite-dimensional minimal realization for the system’s input-output behavior.