8.1 Irreducible (Minimal) Realizations
The internal structure of a system may allow some of the integrators (or delay elements) to be shared by several input-output pairs and still result in the same transfer matrix. The system that realizes the maximum possible degree of sharing (and, consequently, the smallest possible dimension of the associated state space) is called the irreducible (or minimal) realization (see, e.g., Brogan (1991), Section 12.4). The function MinimalRealization tries to find such a realization.
Finding the irreducible (minimal) realization.
The input system for MinimalRealization can be in either state-space or transfer function form; the resultant system is always a state-space one. For SISO transfer function systems, MinimalRealization constructs a state-space realization after an attempt to cancel common pole-zero pairs (the underlying function, PoleZeroCancel, can also be accessed directly, see Section 8.6). Otherwise, MinimalRealization constructs a state-space realization first and then uses the functions ControllableSubsystem and ObservableSubsystem consecutively to select first the controllable and then the observable subspaces. The result is therefore a subsystem that is both completely observable and controllable. See Section 8.2 for more on the definitions of the controllable and observable subspaces. In contrast, DominantSubsystem eliminates weakly controllable and observable modes (see Section 8.5).
Selecting controllable and observable subspaces.
Load the application.
Consider a third-order state-space system with two inputs and two outputs. The first mode is uncontrollable, and the second one is unobservable. These modes, then, have no effect on the input-output relations and so can be dropped without changing the transfer function matrix.
We can verify that this system is not controllable.
Neither is it observable.
This selects the controllable subspace.
This selects the observable subspace.
By selecting the observable subspace of the controllable subspace, we arrive at a minimal realization.
The same result can be obtained directly.
The minimal realization is both controllable and observable.
The method that ControllableSubsystem and ObservableSubsystem use to reduce the dimension of the system can be chosen through the option ReductionMethod. ReductionMethod Kalman specifies that the Kalman decomposition is to be used. With the default option value Automatic, the functions first try to use the structural information about the input system that might be already available. Additionally, ControllableSubsystem takes, and passes along, the options specific to the method it employs. ObservableSubsystem takes both the options for ControllableSubsystem and DualSystem. MinimalRealization, as an interface function, inherits options from its constituents ControllableSubsystem and ObservableSubsystem or PoleZeroCancel, whichever is applicable.
Specifying the reduction method.