In control theory, given any transfer function, any state-space model that is both controllable and observable and has the same input-output behaviour as the transfer function is said to be a minimal realization of the transfer function.[1][2] The realization is called "minimal" because it describes the system with the minimum number of states.[2]
The minimum number of state variables required to describe a system equals the order of the differential equation;[3] more state variables than the minimum can be defined. For example, a second order system can be defined by two or more state variables, with two being the minimal realization.
Given a matrix transfer function, it is possible to directly construct a minimal state-space realization by using Gilbert's method (also known as Gilbert's realization).[4]
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