State-transition matrix
Tool in control theory
In control theory and dynamical systems theory, the state-transition matrix is a matrix function that describes how the state of a linear system changes over time. Essentially, if the system's state is known at an initial time
t
0
{\displaystyle t_{0}}
, the state-transition matrix allows for the calculation of the state at any future time
t
{\displaystyle t}
.
The matrix is used to find the general solution to the homogeneous linear differential equation
x
˙
(
t
)
=
A
(
t
)
x
(
t
)
{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)}
and is also a key component in finding the full solution for the non-homogeneous (input-driven) case.
For linear time-invariant (LTI) systems, where the matrix
A
{\displaystyle \mathbf {A} }
is constant, the state-transition matrix is the matrix exponential
exp
(
A
(
t
−
t
0
)
)
{\displaystyle \textstyle \exp {(\mathbf {A} (t-t_{0}))}}
. In the more complex time-variant case, where
A
(
t
)
{\displaystyle \mathbf {A} (t)}
can change over time, there is no simple formula, and the matrix is typically found by calculating the Peano–Baker series.
Linear systems solutions
The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form
x
˙
(
t
)
=
A
(
t
)
x
(
t
)
+
B
(
t
)
u
(
t
)
,
x
(
t
0
)
=
x
0
,
{\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} (t)\mathbf {x} (t)+\mathbf {B} (t)\mathbf {u} (t),\;\mathbf {x} (t_{0})=\mathbf {x} _{0},}
where
x
(
t
)
{\displaystyle \mathbf {x} (t)}
are the states of the system;
u
(
t
)
{\displaystyle \mathbf {u} (t)}
is the input signal;
A
(
t
)
{\displaystyle \mathbf {A} (t)}
and
B
(
t
)
{\displaystyle \mathbf {B} (t)}
are matrix functions; and
x
0
{\displaystyle \mathbf {x} _{0}}
is the initial condition at
t
0
{\displaystyle t_{0}}
. Using the state-transition matrix
Φ
(
t
,
τ
)
{\displaystyle \mathbf {\Phi } (t,\tau )}
, the solution is given by:
x
(
t
)
=
Φ
(
t
,
t
0
)
x
(
t
0
)
+
∫
t
0
t
Φ
(
t
,
τ
)
B
(
τ
)
u
(
τ
)
d
τ
.
{\displaystyle \mathbf {x} (t)=\mathbf {\Phi } (t,t_{0})\mathbf {x} (t_{0})+\int _{t_{0}}^{t}\mathbf {\Phi } (t,\tau )\mathbf {B} (\tau )\mathbf {u} (\tau )\ d\tau .}
The first term is known as the zero-input response and represents how the system's state would evolve in the absence of any input. The second term is known as the zero-state response and defines how the inputs impact the system.
Peano–Baker series
The most general transition matrix is given by a product integral, referred to as the Peano–Baker series:
Φ
(
t
,
τ
)
=
I
+
∫
τ
t
A
(
σ
1
)
d
σ
1
+
∫
τ
t
A
(
σ
1
)
∫
τ
σ
1
A
(
σ
2
)
d
σ
2
d
σ
1
+
∫
τ
t
A
(
σ
1
)
∫
τ
σ
1
A
(
σ
2
)
∫
τ
σ
2
A
(
σ
3
)
d
σ
3
d
σ
2
d
σ
1
+
⋯
,
{\displaystyle {\begin{aligned}\mathbf {\Phi } (t,\tau )=\mathbf {I} &+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\,d\sigma _{1}\\&+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{\sigma _{1}}\mathbf {A} (\sigma _{2})\,d\sigma _{2}\,d\sigma _{1}\\&+\int _{\tau }^{t}\mathbf {A} (\sigma _{1})\int _{\tau }^{\sigma _{1}}\mathbf {A} (\sigma _{2})\int _{\tau }^{\sigma _{2}}\mathbf {A} (\sigma _{3})\,d\sigma _{3}\,d\sigma _{2}\,d\sigma _{1}\\&+\cdots ,\end{aligned}}}
where
I
{\displaystyle \mathbf {I} }
is the identity matrix.
This brief starts where responsible research should: with the source description of “State-transition matrix” as tool in control theory. Everything that follows is an evidence route, not borrowed authority.
Why this record matters
The subject matters to the general reference register because the source frames it as tool in control theory. Its deeper value depends on whether names, dates, institutions and citations support that framing.
The citation trail is more important than the brevity of the summary: it shows where individual claims can be examined in context. The source revision retrieved here is dated Jul 31, 2026. The linked authority identifier is Q7602993. None of the 0 selected statements returned an explicit reference.
Overview language is designed for orientation and should not be treated as a substitute for the evidence cited beneath it. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “State-transition matrix”, its source revision and the description used here.
- Expand the search: follow State-transition matrix primary sources, State-transition matrix archive and State-transition research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “State-transition matrix”?
- What terminology or title could unlock a more precise catalogue search?
- Which cited source is closest to the event, object or claim?
Search terms from this dossier
This entry incorporates text from “State-transition matrix” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.