DescriptorStateSpace
✖
DescriptorStateSpace
is an option to StateSpaceModel and StateSpaceTransform that specifies whether to use descriptor or standard representation.
Details

- In StateSpaceModel[{a,b,c,d,e}], the matrix e is the descriptor matrix.
- The equations governing continuous-time standard and descriptor state-space models are:
-
standard state space descriptor state space - Discrete-time systems have analogous forms.
- The option can take the following values:
-
Automatic automatically determine the representation False use standard state-space representation True use descriptor state-space representation - With the setting DescriptorStateSpace->Automatic, StateSpaceModel uses the descriptor representation only if a standard representation cannot be obtained, and StateSpaceTransform does not alter the representation.
Examples
open allclose allBasic Examples (7)Summary of the most common use cases
Invert the descriptor matrix to create a standard state-space model:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-ogrg6

Add an identity matrix to create a descriptor form:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-fuouth

Obtain the descriptor StateSpaceModel:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-3skmp

By default, the standard form is obtained:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-8nk5dm

For some singular systems, the descriptor form is obtained by default:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-4ytis2

This is because it is not possible to obtain the standard form:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-zbxw56


When possible, the standard form is obtained:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-qppcdk


https://wolfram.com/xid/0hyvavxc2h9ob6ba-zbvui6

In some cases, the standard form can have fewer states:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-2e5jpp


https://wolfram.com/xid/0hyvavxc2h9ob6ba-svn8qx

However, the order of the system is the same in both cases:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-20owak

Obtain the result from StateSpaceTransform in the descriptor form:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-o7uvol

By default, the form is unaltered:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-j88l8t

Scope (6)Survey of the scope of standard use cases
Create a standard state-space model from a descriptor state-space model:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-be9wdo

Create a descriptor state-space model from a standard state-space model:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-o542u

Convert a proper transfer-function model to a descriptor state-space model:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-kf064c

Create a descriptor state-space model from an ODE:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-dhwdzd

Obtain the state transformation of a standard StateSpaceModel in descriptor form:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-7wc841

It gives the same result as an explicit descriptor specification:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-15wnrp

Similar results can be obtained with the inverse transformation as well:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-m7lw4k


https://wolfram.com/xid/0hyvavxc2h9ob6ba-22i700

A descriptor StateSpaceModel with an invertible descriptor matrix:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-xa4c05
Its state transformation can be obtained in standard form:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-8qma0j

Properties & Relations (2)Properties of the function, and connections to other functions
Improper transfer-function models will create descriptor state-space models by default:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-bpshvn

There is no standard state-space realization:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-ddn1f


A descriptor StateSpaceModel with a noninvertible descriptor matrix:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-yr31m3
StateSpaceTransform produces a descriptor model:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-bj7isz

The state transformation cannot be obtained in standard form:

https://wolfram.com/xid/0hyvavxc2h9ob6ba-iojb6d


Wolfram Research (2012), DescriptorStateSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/DescriptorStateSpace.html (updated 2014).
Text
Wolfram Research (2012), DescriptorStateSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/DescriptorStateSpace.html (updated 2014).
Wolfram Research (2012), DescriptorStateSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/DescriptorStateSpace.html (updated 2014).
CMS
Wolfram Language. 2012. "DescriptorStateSpace." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/DescriptorStateSpace.html.
Wolfram Language. 2012. "DescriptorStateSpace." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/DescriptorStateSpace.html.
APA
Wolfram Language. (2012). DescriptorStateSpace. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DescriptorStateSpace.html
Wolfram Language. (2012). DescriptorStateSpace. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DescriptorStateSpace.html
BibTeX
@misc{reference.wolfram_2025_descriptorstatespace, author="Wolfram Research", title="{DescriptorStateSpace}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/DescriptorStateSpace.html}", note=[Accessed: 19-June-2025
]}
BibLaTeX
@online{reference.wolfram_2025_descriptorstatespace, organization={Wolfram Research}, title={DescriptorStateSpace}, year={2014}, url={https://reference.wolfram.com/language/ref/DescriptorStateSpace.html}, note=[Accessed: 19-June-2025
]}