# 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

open allclose all

## Basic Examples(1)

A similarity transformation :

 In[1]:=
 In[2]:=
 Out[2]=

Use the matrix pair:

 In[3]:=
 Out[3]=