BUILT-IN MATHEMATICA SYMBOL

StateSpaceTransform

transforms the StateSpaceModel object ss using the similarity matrix

.

StateSpaceTransform

computes the transformation using the matrix pair

.

MORE INFORMATION

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
	Automatic		change of variables

For a unitary matrix , can be supplied instead of .

A transformation using a non-square orthogonal matrix effectively selects a subsystem.

EXAMPLES

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Basic Examples (1)

A similarity transformation

:

A similarity transformation

:

In[1]:=

In[2]:=

Out[2]=

Scope (6)

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:

Applications (3)

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:

Properties & Relations (4)

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:

Possible Issues (1)

The transformation matrix must be orthonormal or invertible:

It is neither orthonormal nor invertible: