- Source: Time-invariant system
In control theory, a time-invariant (TI) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function depends only indirectly on the time-domain (via the input function, for example), then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a "time-varying system".
Mathematically speaking, "time-invariance" of a system is the following property:: p. 50
Given a system with a time-dependent output function
y
(
t
)
{\displaystyle y(t)}
, and a time-dependent input function
x
(
t
)
{\displaystyle x(t)}
, the system will be considered time-invariant if a time-delay on the input
x
(
t
+
δ
)
{\displaystyle x(t+\delta )}
directly equates to a time-delay of the output
y
(
t
+
δ
)
{\displaystyle y(t+\delta )}
function. For example, if time
t
{\displaystyle t}
is "elapsed time", then "time-invariance" implies that the relationship between the input function
x
(
t
)
{\displaystyle x(t)}
and the output function
y
(
t
)
{\displaystyle y(t)}
is constant with respect to time
t
:
{\displaystyle t:}
y
(
t
)
=
f
(
x
(
t
)
,
t
)
=
f
(
x
(
t
)
)
.
{\displaystyle y(t)=f(x(t),t)=f(x(t)).}
In the language of signal processing, this property can be satisfied if the transfer function of the system is not a direct function of time except as expressed by the input and output.
In the context of a system schematic, this property can also be stated as follows, as shown in the figure to the right:
If a system is time-invariant then the system block commutes with an arbitrary delay.
If a time-invariant system is also linear, it is the subject of linear time-invariant theory (linear time-invariant) with direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. Nonlinear time-invariant systems lack a comprehensive, governing theory. Discrete time-invariant systems are known as shift-invariant systems. Systems which lack the time-invariant property are studied as time-variant systems.
Simple example
To demonstrate how to determine if a system is time-invariant, consider the two systems:
System A:
y
(
t
)
=
t
x
(
t
)
{\displaystyle y(t)=tx(t)}
System B:
y
(
t
)
=
10
x
(
t
)
{\displaystyle y(t)=10x(t)}
Since the System Function
y
(
t
)
{\displaystyle y(t)}
for system A explicitly depends on t outside of
x
(
t
)
{\displaystyle x(t)}
, it is not time-invariant because the time-dependence is not explicitly a function of the input function.
In contrast, system B's time-dependence is only a function of the time-varying input
x
(
t
)
{\displaystyle x(t)}
. This makes system B time-invariant.
The Formal Example below shows in more detail that while System B is a Shift-Invariant System as a function of time, t, System A is not.
Formal example
A more formal proof of why systems A and B above differ is now presented. To perform this proof, the second definition will be used.
System A: Start with a delay of the input
x
d
(
t
)
=
x
(
t
+
δ
)
{\displaystyle x_{d}(t)=x(t+\delta )}
y
(
t
)
=
t
x
(
t
)
{\displaystyle y(t)=tx(t)}
y
1
(
t
)
=
t
x
d
(
t
)
=
t
x
(
t
+
δ
)
{\displaystyle y_{1}(t)=tx_{d}(t)=tx(t+\delta )}
Now delay the output by
δ
{\displaystyle \delta }
y
(
t
)
=
t
x
(
t
)
{\displaystyle y(t)=tx(t)}
y
2
(
t
)
=
y
(
t
+
δ
)
=
(
t
+
δ
)
x
(
t
+
δ
)
{\displaystyle y_{2}(t)=y(t+\delta )=(t+\delta )x(t+\delta )}
Clearly
y
1
(
t
)
≠
y
2
(
t
)
{\displaystyle y_{1}(t)\neq y_{2}(t)}
, therefore the system is not time-invariant.
System B: Start with a delay of the input
x
d
(
t
)
=
x
(
t
+
δ
)
{\displaystyle x_{d}(t)=x(t+\delta )}
y
(
t
)
=
10
x
(
t
)
{\displaystyle y(t)=10x(t)}
y
1
(
t
)
=
10
x
d
(
t
)
=
10
x
(
t
+
δ
)
{\displaystyle y_{1}(t)=10x_{d}(t)=10x(t+\delta )}
Now delay the output by
δ
{\displaystyle \delta }
y
(
t
)
=
10
x
(
t
)
{\displaystyle y(t)=10x(t)}
y
2
(
t
)
=
y
(
t
+
δ
)
=
10
x
(
t
+
δ
)
{\displaystyle y_{2}(t)=y(t+\delta )=10x(t+\delta )}
Clearly
y
1
(
t
)
=
y
2
(
t
)
{\displaystyle y_{1}(t)=y_{2}(t)}
, therefore the system is time-invariant.
More generally, the relationship between the input and output is
y
(
t
)
=
f
(
x
(
t
)
,
t
)
,
{\displaystyle y(t)=f(x(t),t),}
and its variation with time is
d
y
d
t
=
∂
f
∂
t
+
∂
f
∂
x
d
x
d
t
.
{\displaystyle {\frac {\mathrm {d} y}{\mathrm {d} t}}={\frac {\partial f}{\partial t}}+{\frac {\partial f}{\partial x}}{\frac {\mathrm {d} x}{\mathrm {d} t}}.}
For time-invariant systems, the system properties remain constant with time,
∂
f
∂
t
=
0.
{\displaystyle {\frac {\partial f}{\partial t}}=0.}
Applied to Systems A and B above:
f
A
=
t
x
(
t
)
⟹
∂
f
A
∂
t
=
x
(
t
)
≠
0
{\displaystyle f_{A}=tx(t)\qquad \implies \qquad {\frac {\partial f_{A}}{\partial t}}=x(t)\neq 0}
in general, so it is not time-invariant,
f
B
=
10
x
(
t
)
⟹
∂
f
B
∂
t
=
0
{\displaystyle f_{B}=10x(t)\qquad \implies \qquad {\frac {\partial f_{B}}{\partial t}}=0}
so it is time-invariant.
Abstract example
We can denote the shift operator by
T
r
{\displaystyle \mathbb {T} _{r}}
where
r
{\displaystyle r}
is the amount by which a vector's index set should be shifted. For example, the "advance-by-1" system
x
(
t
+
1
)
=
δ
(
t
+
1
)
∗
x
(
t
)
{\displaystyle x(t+1)=\delta (t+1)*x(t)}
can be represented in this abstract notation by
x
~
1
=
T
1
x
~
{\displaystyle {\tilde {x}}_{1}=\mathbb {T} _{1}{\tilde {x}}}
where
x
~
{\displaystyle {\tilde {x}}}
is a function given by
x
~
=
x
(
t
)
∀
t
∈
R
{\displaystyle {\tilde {x}}=x(t)\forall t\in \mathbb {R} }
with the system yielding the shifted output
x
~
1
=
x
(
t
+
1
)
∀
t
∈
R
{\displaystyle {\tilde {x}}_{1}=x(t+1)\forall t\in \mathbb {R} }
So
T
1
{\displaystyle \mathbb {T} _{1}}
is an operator that advances the input vector by 1.
Suppose we represent a system by an operator
H
{\displaystyle \mathbb {H} }
. This system is time-invariant if it commutes with the shift operator, i.e.,
T
r
H
=
H
T
r
∀
r
{\displaystyle \mathbb {T} _{r}\mathbb {H} =\mathbb {H} \mathbb {T} _{r}\forall r}
If our system equation is given by
y
~
=
H
x
~
{\displaystyle {\tilde {y}}=\mathbb {H} {\tilde {x}}}
then it is time-invariant if we can apply the system operator
H
{\displaystyle \mathbb {H} }
on
x
~
{\displaystyle {\tilde {x}}}
followed by the shift operator
T
r
{\displaystyle \mathbb {T} _{r}}
, or we can apply the shift operator
T
r
{\displaystyle \mathbb {T} _{r}}
followed by the system operator
H
{\displaystyle \mathbb {H} }
, with the two computations yielding equivalent results.
Applying the system operator first gives
T
r
H
x
~
=
T
r
y
~
=
y
~
r
{\displaystyle \mathbb {T} _{r}\mathbb {H} {\tilde {x}}=\mathbb {T} _{r}{\tilde {y}}={\tilde {y}}_{r}}
Applying the shift operator first gives
H
T
r
x
~
=
H
x
~
r
{\displaystyle \mathbb {H} \mathbb {T} _{r}{\tilde {x}}=\mathbb {H} {\tilde {x}}_{r}}
If the system is time-invariant, then
H
x
~
r
=
y
~
r
{\displaystyle \mathbb {H} {\tilde {x}}_{r}={\tilde {y}}_{r}}
See also
Finite impulse response
Sheffer sequence
State space (controls)
Signal-flow graph
LTI system theory
Autonomous system (mathematics)
References
Kata Kunci Pencarian:
- Jaringan saraf konvolusional
- Arkea
- Garis waktu peristiwa jauh di masa depan
- Time-invariant system
- Linear time-invariant system
- Linear system
- Time-variant system
- Shift-invariant system
- Time constant
- Transfer function
- Controllability
- High-pass filter
- Dynamical system
