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.[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.

Gilbert's 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]

References

1. Williams, Robert L. II; Lawrence, Douglas A. (2007), Linear State-Space Control Systems, John Wiley & Sons, p. 185, ISBN 9780471735557.
2. Tangirala, Arun K. (2015), Principles of System Identification: Theory and Practice, CRC Press, p. 96, ISBN 9781439896020.
3. Tangirala (2015), p. 91.
4. Mackenroth, Uwe. (17 April 2013). Robust control systems: theory and case studies. Berlin. pp. 114–116. ISBN 978-3-662-09775-5. OCLC 861706617.