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StateSpaceTransform

StateSpaceTransform
transforms the StateSpaceModel object ss using the similarity matrix .
StateSpaceTransform
computes the transformation using the matrix pair .
  • The state-space model ss can be given as StateSpaceModel, where a, b, c, and d represent the state, input, output, and transmission matrices in either a continuous-time or a discrete-time system:
continuous-time system
discrete-time system
  • The pair implements the following transformation:
, continuous-time system
, discrete-time system
  • The variable is the new state vector.
  • Common automatic arguments include:
or {s1, Automatic} change of variables
Automaticchange of variables
  • For a unitary matrix , can be supplied instead of .
  • A transformation using a non-square orthogonal matrix effectively selects a subsystem.
A similarity transformation :
A similarity transformation :
In[1]:=
Click for copyable input
In[2]:=
Click for copyable input
Out[2]=
The transformation :
Use the matrix pair:
The reverse transformation :
Use the matrix pair:
The transformation for a discrete-time system:
An orthogonal matrix transformation:
The matrix pair can use either the transpose or the inverse:
An orthonormal matrix transformation:
A reverse transformation using an orthonormal matrix:
A function to obtain the controllable companion form of a single-input system:
A function to obtain the observable companion form of a single-output system:
Rearrange the states:
The eigenvalues (and consequently, stability characteristics) are invariant under a similarity transformation:
Controllability and observability characteristics are invariant under a similarity transformation:
The eigenvalues of the product of the controllability and observability Gramians are invariant (and positive) under a similarity transformation for a completely controllable and observable system:
The system response is invariant under a similarity transformation:
Both are responses of essentially the same model:
The transformation matrix must be orthonormal or invertible:
It is neither orthonormal nor invertible:
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