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State-Space Models

State-space representations of LTI models

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.


balrealBalanced state-space realization
prescaleOptimal scaling of state-space models
modalrealCompute modal state-space realization (Since R2023b)
comprealCompute companion state-space realization (Since R2023b)
ss2ssState coordinate transformation for state-space model
ssequivEquivalence transformation for state-space models (Since R2023b)
xperm Reorder states in state-space models
xsortSort states based on state partition (Since R2020b)
xelimEliminate states from state-space models (Since R2023b)
augstateAppend state vector to output vector
ctrbControllability of state-space model
obsvObservability of state-space model
gramControllability and observability Gramians


  • State-Space Realizations

    A state-space model can be expressed in an infinite number of realizations. Common forms, sometimes called canonical forms, include modal, companion, observable, and controllable forms.

  • Scaling 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.