For a 🇺🇸 State Space Model, the choice of state variables may be arbitrary (eg, which units are used), and the same model can be represented with new states via a simple

For the 🕹️ Controllability and 👁️ Observability Grammians, this means the transformed Grammians are respectively

But reveals the responsiveness of controllability and reveals the sensitivity of observability, and transforming the state components affects these respective properties. For instance, one could artificially increase controllability by scaling, but it also decreases observability.

Instead of choosing an arbitrary scaling, balanced realizations find the coordinate system that equates the controllability and observability Grammians, thereby allowing them to reflect intrinsics properties of the system. This way, we can observe which states (those relatively larger values) are both easier to control and observe.

Using and of a system, we can compute and use 🤝 Schur Decomposition,

where is orthogonal and is diagonal with positive entries (since Grammians are PD). Then, the balanced realization has where

where we order .

With this definition, we can use the transformation

to build the corresponding system,

Hankel Singular Values#

The values in are called Hankel singular values, which are square roots of the eigenvalues of . Note that these eigenvalues don't change for any arbitrary state transformation , so these singular values don't depend on the specific realization of the system; for the special case , we can see that these values are also the square roots of eigenvalues of .

Balanced Truncation#

Balanced realizations can be used to simplify a model; that is, to reduce the dimensionality of the states. For the ordered Hankel singular values , we can take the first largest values, so that the balanced realization can be decomposed into

with the reduced system being

Intuitively, this keeps the most controllable and observable states, while discarding the rest.

Note that balanced truncation should only be performed on the stable portion of a system's dynamics; the reduction above assumes that the balanced realization is controllable and observable. The other unstable portions must be preserved, as changing them will alter closed-loop stability properties.