8. Realizations

A realization is any pair of equations

or the corresponding quadruple of the matrices , or, for the purposes of this guide, the corresponding state-space object StateSpace[a, b, c, d] that leads to the required input-output relations for the system (usually specified as a transfer function matrix ). A realization can also refer to a discrete-time state-space realization (with the obvious change from the differential to difference in the equations). Because there can be innumerable ways to satisfy the input-output relations, a physical system can have an infinite number of realizations. In this chapter, several means of converting between different realizations are described, including means of obtaining realizations of a smaller order. All the functions operate on continuous- and discrete-time objects.

The first group of functions represents the means to convert between different types of realizations. By convention, their names end with Form. For example, to convert a system to Kalman controllable form, the function KalmanControllableForm would be applied to the system. The list of all forms available in Control System Professional can be obtained with ?ControlSystems`*`*Form.

The other group comprises functions that select a subsystem (typically a subspace) possessing certain properties. These functions end with Subsystem (e.g., ControllableSubsystem or DominantSubsystem). The exception to this convention is MinimalRealization, used for computing a minimal realization, which represents the intersection of the controllable and observable subspaces; therefore, this function does effectively select a subsystem. Also introduced is SimilarityTransform, a function that transforms between equivalent realizations of the same system.