The representation of a model in state-space is not unique. Coordinate transformation yields state-space models with different matrices but identical dynamics. State coordinate transformation can be useful for achieving minimal realizations of state-space models, or for converting canonical forms for analysis and control design.
Coordinate transformation can also be useful for scaling poorly-conditioned models. Proper scaling of state-space models is important for accurate computations. An example of a poorly scaled model is a dynamic system with two states in the state vector that have units of light years and millimeters. Such disparate units may introduce both very large and very small entries into the A matrix. Over the course of computations, this mix of small and large entries in the matrix could destroy important characteristics of the model and lead to incorrect results.
Canonical State-Space Realizations
Modal, companion, observable and controllable canonical state-space models.
When working with state-space models, proper scaling is important for accurate computations.
Scaling State-Space Models to Maximize Accuracy
This example shows that proper scaling of state-space models can be critical for accuracy and provides an overview of automatic and manual rescaling tools.