Minimal realization

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.{{citation|title=Linear State-Space Control Systems|first1=Robert L. II|last1=Williams|first2=Douglas A.|last2=Lawrence|publisher=John Wiley & Sons|year=2007|isbn=9780471735557|page=185|url=https://books.google.com/books?id=UPWAmAXQu1AC&pg=PA185}}.{{citation|title=Principles of System Identification: Theory and Practice|first=Arun K.|last=Tangirala|publisher=CRC Press|year=2015|isbn=9781439896020|page=96|url=https://books.google.com/books?id=aUHOBQAAQBAJ&pg=PA96}}. The realization is called "minimal" because it describes the system with the minimum number of states.

The minimum number of state variables required to describe a system equals the order of the differential equation;{{harvtxt|Tangirala|2015}}, p. 91. 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.