8.4 Internally Balanced Realizations
The Kalman minimal realization algorithm may result in structurally unstable models. In such cases another model reduction technique, based on the internally balanced realization, may be applied (Moore (1981)). The controllable and observable realization is said to be internally balanced if its controllability and observability Gramians are represented by the same (positive definite) diagonal matrix. InternallyBalancedForm attempts to construct such a realization.
Finding the internally balanced form.
Consider a SISO system in its controllable canonical form (see Example 1 in the above-cited paper of Moore (1981)). Evidently, the realization is poorly balanced.
This finds the internally balanced realization.
We can verify that the realization has equal and diagonal controllability and observability Gramians. The built-in function Chop rounds small numerical errors down to the exact zeros.
Finally, we can see that the two forms are in fact different realizations of the same transfer function.
Similarly to KalmanControllableForm and related functions, InternallyBalancedForm takes the option DecompositionMethod. The default Automatic value for this option invokes the built-in function Eigensystem for the continuous-time systems and SingularValues for the discrete-time systems, in the most important case of inexact systems. For exact systems, the Eigensystem-based solution will be attempted.