StateSpaceTransform

StateSpaceTransform[sys,{p,q}]

transforms the state-space model sys using the matrices p and q.

StateSpaceTransform[sys,{{x1p1[z],},{z1q1[x],}}]

transforms using the variable transformations {x1p1[z],} and {z1q1[x],}.

Details and Options

  • StateSpaceTransform returns a transformed model where the state variables have been transformed. The transformation can be a similarity, equivalence, or restricted equivalence transformation.
  • The system sys can be a standard or descriptor StateSpaceModel, AffineStateSpaceModel, or NonlinearStateSpaceModel.
  • For a standard StateSpaceModel[{a,b,c,d}], the original and transformed systems are related by the transformation and the corresponding equations are given by:
  • Typically p and q are inverses, in which case the transformation is a similarity transformation. The following defaults for p and q are used for standard StateSpaceModel transformations:
  • p or {p,Automatic}{p,Inverse[p]}
    {Automatic,q}{Inverse[q],q}
  • For a descriptor StateSpaceModel[{a,b,c,d,e}], the original and transformed systems related by the transformation and the corresponding equations are given by:
  • Typically p and q are invertible matrices but not inverses, in which case the transformation is an equivalence transformation. The following defaults are used for descriptor StateSpaceModel transformations:
  • p or {p,Automatic}{p,IdentityMatrix[n]}
    {Automatic,q}{IdentityMatrix[n],q}
  • For an AffineStateSpaceModel[{a,b,c,d},x] and NonlinearStateSpaceModel[{f,g},x,u] with j the Jacobian matrix D[p[z],{z}], the original and transformed systems are related by the transformation , and the corresponding equations are given by:
  • Typically p[z] and q[x] are inverses, in which case the transformation is an invertible mapping.
  • {{x1->p1[z],},{z1,}}q[x] is computed if needed
    {Automatic,{z1->q[x],}}p[z] is computed
  • When variable transformation matrices {p,q} are given, the resulting system is of the same type as the input. In the case of nonlinear state-space models, these are taken to represent the transformation rules {{x1->p1.z,},{z1->q1.x,}}.
  • When variable transformation rules {{x1->p1[z],},} are given, the resulting system is always AffineStateSpaceModel or NonlinearStateSpaceModel.
  • StateSpaceTransform accepts the option DescriptorStateSpace.

Examples

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Basic Examples  (1)

A similarity transformation :

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Use the matrix pair:

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Scope  (15)

Generalizations & Extensions  (2)

Options  (1)

Applications  (6)

Properties & Relations  (8)

Possible Issues  (2)

See Also

StateSpaceModel  AffineStateSpaceModel  NonlinearStateSpaceModel  ControllableDecomposition  ObservableDecomposition  JordanModelDecomposition  InternallyBalancedDecomposition  FeedbackLinearize  StateTransformationLinearize

Introduced in 2010
(8.0)
| Updated in 2014
(10.0)