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Linear systems solutions [ edit ]
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:[1] [2]
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 [ edit ]
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 matrix converges uniformly and absolutely to a solution that exists and is unique.[2] The series has a formal sum that can be written as
Φ
(
t
,
τ
)
=
exp
T
∫
τ
t
A
(
σ
)
d
σ
{\displaystyle \mathbf {\Phi } (t,\tau )=\exp {\mathcal {T}}\int _{\tau }^{t}\mathbf {A} (\sigma )\,d\sigma }
where
T
{\displaystyle {\mathcal {T}}}
is the time-ordering operator, used to ensure that the repeated product integral is in proper order. The Magnus expansion provides a means for evaluating this product.
Other properties [ edit ]
The state transition matrix
Φ
{\displaystyle \mathbf {\Phi } }
satisfies the following relationships. These relationships are generic to the product integral .
1. It is continuous and has continuous derivatives.
2, It is never singular; in fact
Φ
−
1
(
t
,
τ
)
=
Φ
(
τ
,
t
)
{\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )=\mathbf {\Phi } (\tau ,t)}
and
Φ
−
1
(
t
,
τ
)
Φ
(
t
,
τ
)
=
I
{\displaystyle \mathbf {\Phi } ^{-1}(t,\tau )\mathbf {\Phi } (t,\tau )=I}
, where
I
{\displaystyle I}
is the identity matrix.
3.
Φ
(
t
,
t
)
=
I
{\displaystyle \mathbf {\Phi } (t,t)=I}
for all
t
{\displaystyle t}
.[3]
4.
Φ
(
t
2
,
t
1
)
Φ
(
t
1
,
t
0
)
=
Φ
(
t
2
,
t
0
)
{\displaystyle \mathbf {\Phi } (t_{2},t_{1})\mathbf {\Phi } (t_{1},t_{0})=\mathbf {\Phi } (t_{2},t_{0})}
for all
t
0
≤
t
1
≤
t
2
{\displaystyle t_{0}\leq t_{1}\leq t_{2}}
.
5. It satisfies the differential equation
∂
Φ
(
t
,
t
0
)
∂
t
=
A
(
t
)
Φ
(
t
,
t
0
)
{\displaystyle {\frac {\partial \mathbf {\Phi } (t,t_{0})}{\partial t}}=\mathbf {A} (t )\mathbf {\Phi } (t,t_{0})}
with initial conditions
Φ
(
t
0
,
t
0
)
=
I
{\displaystyle \mathbf {\Phi } (t_{0},t_{0})=I}
.
6. The state-transition matrix
Φ
(
t
,
τ
)
{\displaystyle \mathbf {\Phi } (t,\tau )}
, given by
Φ
(
t
,
τ
)
≡
U
(
t
)
U
−
1
(
τ
)
{\displaystyle \mathbf {\Phi } (t,\tau )\equiv \mathbf {U} (t )\mathbf {U} ^{-1}(\tau )}
where the
n
×
n
{\displaystyle n\times n}
matrix
U
(
t
)
{\displaystyle \mathbf {U} (t )}
is the fundamental solution matrix that satisfies
U
˙
(
t
)
=
A
(
t
)
U
(
t
)
{\displaystyle {\dot {\mathbf {U} }}(t )=\mathbf {A} (t )\mathbf {U} (t )}
with initial condition
U
(
t
0
)
=
I
{\displaystyle \mathbf {U} (t_{0})=I}
.
7. Given the state
x
(
τ
)
{\displaystyle \mathbf {x} (\tau )}
at any time
τ
{\displaystyle \tau }
, the state at any other time
t
{\displaystyle t}
is given by the mapping
x
(
t
)
=
Φ
(
t
,
τ
)
x
(
τ
)
{\displaystyle \mathbf {x} (t )=\mathbf {\Phi } (t,\tau )\mathbf {x} (\tau )}
Estimation of the state-transition matrix [ edit ]
In the time-invariant case, we can define
Φ
{\displaystyle \mathbf {\Phi } }
, using the matrix exponential , as
Φ
(
t
,
t
0
)
=
e
A
(
t
−
t
0
)
{\displaystyle \mathbf {\Phi } (t,t_{0})=e^{\mathbf {A} (t-t_{0})}}
. [4]
In the time-variant case, the state-transition matrix
Φ
(
t
,
t
0
)
{\displaystyle \mathbf {\Phi } (t,t_{0})}
can be estimated from the solutions of the differential equation
u
˙
(
t
)
=
A
(
t
)
u
(
t
)
{\displaystyle {\dot {\mathbf {u} }}(t )=\mathbf {A} (t )\mathbf {u} (t )}
with initial conditions
u
(
t
0
)
{\displaystyle \mathbf {u} (t_{0})}
given by
[
1
,
0
,
…
,
0
]
T
{\displaystyle [1,\ 0,\ \ldots ,\ 0]^{T}}
,
[
0
,
1
,
…
,
0
]
T
{\displaystyle [0,\ 1,\ \ldots ,\ 0]^{T}}
, ...,
[
0
,
0
,
…
,
1
]
T
{\displaystyle [0,\ 0,\ \ldots ,\ 1]^{T}}
. The corresponding solutions provide the
n
{\displaystyle n}
columns of matrix
Φ
(
t
,
t
0
)
{\displaystyle \mathbf {\Phi } (t,t_{0})}
. Now, from property 4,
Φ
(
t
,
τ
)
=
Φ
(
t
,
t
0
)
Φ
(
τ
,
t
0
)
−
1
{\displaystyle \mathbf {\Phi } (t,\tau )=\mathbf {\Phi } (t,t_{0})\mathbf {\Phi } (\tau ,t_{0})^{-1}}
for all
t
0
≤
τ
≤
t
{\displaystyle t_{0}\leq \tau \leq t}
. The state-transition matrix must be determined before analysis on the time-varying solution can continue.
See also [ edit ]
References [ edit ]
^ a b Rugh, Wilson (1996). Linear System Theory . Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-441205-2 .
^ Brockett, Roger W. (1970). Finite Dimensional Linear Systems . John Wiley & Sons. ISBN 978-0-471-10585-5 .
^ Reyneke, Pieter V. (2012). "Polynomial Filtering: To any degree on irregularly sampled data" . Automatika . 53 (4 ): 382–397. doi :10.7305/automatika.53-4.248 . hdl :2263/21017 . S2CID 40282943 .
Further reading [ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=State-transition_matrix&oldid=1225875814 "
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