When a system's outputs are bounded for every bounded input
In
signal processing , specifically
control theory , bounded-input, bounded-output (BIBO ) stability is a form of
stability for
signals and
systems that take inputs. If a system is BIBO stable, then the output will be
bounded for every input to the system that is bounded.
A signal is bounded if there is a finite value
B
>
0
{\displaystyle B>0}
such that the signal magnitude never exceeds
B
{\displaystyle B}
, that is
For
discrete-time signals:
∃
B
∀
n
(
|
y
n
|
≤
B
)
n
∈
Z
{\displaystyle \exists B\forall n(\ |y[n]|\leq B)\quad n\in \mathbb {Z} }
For
continuous-time signals:
∃
B
∀
t
(
|
y
(
t
)
|
≤
B
)
t
∈
R
{\displaystyle \exists B\forall t(\ |y(t)|\leq B)\quad t\in \mathbb {R} }
Time-domain condition for linear time-invariant systems
Continuous-time necessary and sufficient condition
For a
continuous time
linear time-invariant (LTI) system, the condition for BIBO stability is that the
impulse response ,
h
(
t
)
{\displaystyle h(t)}
, be
absolutely integrable , i.e., its
L1 norm exists.
∫
−
∞
∞
|
h
(
t
)
|
d
t
=
‖
h
‖
1
∈
R
{\displaystyle \int _{-\infty }^{\infty }\left|h(t)\right|\,{\mathord {\operatorname {d} }}t=\|h\|_{1}\in \mathbb {R} }
Discrete-time sufficient condition
For a
discrete time LTI system, the condition for BIBO stability is that the
impulse response be
absolutely summable , i.e., its
ℓ
1
{\displaystyle \ell ^{1}}
norm exists.
∑
n
=
−
∞
∞
|
h
n
|
=
‖
h
‖
1
∈
R
{\displaystyle \ \sum _{n=-\infty }^{\infty }|h[n]|=\|h\|_{1}\in \mathbb {R} }
Proof of sufficiency
Given a
discrete time LTI system with
impulse response
h
n
{\displaystyle \ h[n]}
the relationship between the input
x
n
{\displaystyle \ x[n]}
and the output
y
n
{\displaystyle \ y[n]}
is
y
n
=
h
n
∗
x
n
{\displaystyle \ y[n]=h[n]*x[n]}
where
∗
{\displaystyle *}
denotes
convolution . Then it follows by the definition of convolution
y
n
=
∑
k
=
−
∞
∞
h
k
x
n
−
k
{\displaystyle \ y[n]=\sum _{k=-\infty }^{\infty }h[k]x[n-k]}
Let
‖
x
‖
∞
{\displaystyle \|x\|_{\infty }}
be the maximum value of
|
x
n
|
{\displaystyle \ |x[n]|}
, i.e., the
L
∞
{\displaystyle L_{\infty }}
-norm .
|
y
n
|
=
|
∑
k
=
−
∞
∞
h
n
−
k
x
k
|
{\displaystyle \left|y[n]\right|=\left|\sum _{k=-\infty }^{\infty }h[n-k]x[k]\right|}
≤
∑
k
=
−
∞
∞
|
h
n
−
k
|
|
x
k
|
{\displaystyle \leq \sum _{k=-\infty }^{\infty }\left|h[n-k]\right|\left|x[k]\right|}
(by the
triangle inequality )
≤
∑
k
=
−
∞
∞
|
h
n
−
k
|
‖
x
‖
∞
=
‖
x
‖
∞
∑
k
=
−
∞
∞
|
h
n
−
k
|
=
‖
x
‖
∞
∑
k
=
−
∞
∞
|
h
k
|
{\displaystyle {\begin{aligned}&\leq \sum _{k=-\infty }^{\infty }\left|h[n-k]\right|\|x\|_{\infty }\\&=\|x\|_{\infty }\sum _{k=-\infty }^{\infty }\left|h[n-k]\right|\\&=\|x\|_{\infty }\sum _{k=-\infty }^{\infty }\left|h[k]\right|\end{aligned}}}
If
h
n
{\displaystyle h[n]}
is absolutely summable, then
∑
k
=
−
∞
∞
|
h
k
|
=
‖
h
‖
1
∈
R
{\displaystyle \sum _{k=-\infty }^{\infty }{\left|h[k]\right|}=\|h\|_{1}\in \mathbb {R} }
and
‖
x
‖
∞
∑
k
=
−
∞
∞
|
h
k
|
=
‖
x
‖
∞
‖
h
‖
1
{\displaystyle \|x\|_{\infty }\sum _{k=-\infty }^{\infty }\left|h[k]\right|=\|x\|_{\infty }\|h\|_{1}}
So if
h
n
{\displaystyle h[n]}
is absolutely summable and
|
x
n
|
{\displaystyle \left|x[n]\right|}
is bounded, then
|
y
n
|
{\displaystyle \left|y[n]\right|}
is bounded as well because
‖
x
‖
∞
‖
h
‖
1
∈
R
{\displaystyle \|x\|_{\infty }\|h\|_{1}\in \mathbb {R} }
.
The proof for continuous-time follows the same arguments.
Frequency-domain condition for linear time-invariant systems
Continuous-time signals
For a
rational and
continuous-time system , the condition for stability is that the
region of convergence (ROC) of the
Laplace transform includes the
imaginary axis . When the system is
causal , the ROC is the
open region to the right of a vertical line whose
abscissa is the
real part of the "largest pole", or the
pole that has the greatest real part of any pole in the system. The real part of the largest pole defining the ROC is called the
abscissa of convergence . Therefore, all poles of the system must be in the strict left half of the
s-plane for BIBO stability.
This stability condition can be derived from the above time-domain condition as follows:
∫
−
∞
∞
|
h
(
t
)
|
d
t
=
∫
−
∞
∞
|
h
(
t
)
|
|
e
−
j
ω
t
|
d
t
=
∫
−
∞
∞
|
h
(
t
)
(
1
⋅
e
)
−
j
ω
t
|
d
t
=
∫
−
∞
∞
|
h
(
t
)
(
e
σ
+
j
ω
)
−
t
|
d
t
=
∫
−
∞
∞
|
h
(
t
)
e
−
s
t
|
d
t
{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }\left|h(t)\right|\,dt&=\int _{-\infty }^{\infty }\left|h(t)\right|\left|e^{-j\omega t}\right|\,dt\\&=\int _{-\infty }^{\infty }\left|h(t)(1\cdot e)^{-j\omega t}\right|\,dt\\&=\int _{-\infty }^{\infty }\left|h(t)(e^{\sigma +j\omega })^{-t}\right|\,dt\\&=\int _{-\infty }^{\infty }\left|h(t)e^{-st}\right|\,dt\end{aligned}}}
where
s
=
σ
+
j
ω
{\displaystyle s=\sigma +j\omega }
and
Re
(
s
)
=
σ
=
0.
{\displaystyle \operatorname {Re} (s)=\sigma =0.}
The
region of convergence must therefore include the
imaginary axis .
Discrete-time signals
For a
rational and
discrete time system , the condition for stability is that the
region of convergence (ROC) of the
z-transform includes the
unit circle . When the system is
causal , the ROC is the
open region outside a circle whose radius is the magnitude of the
pole with largest magnitude. Therefore, all poles of the system must be inside the
unit circle in the
z-plane for BIBO stability.
This stability condition can be derived in a similar fashion to the continuous-time derivation:
∑
n
=
−
∞
∞
|
h
n
|
=
∑
n
=
−
∞
∞
|
h
n
|
|
e
−
j
ω
n
|
=
∑
n
=
−
∞
∞
|
h
n
(
1
⋅
e
)
−
j
ω
n
|
=
∑
n
=
−
∞
∞
|
h
n
(
r
e
j
ω
)
−
n
|
=
∑
n
=
−
∞
∞
|
h
n
z
−
n
|
{\displaystyle {\begin{aligned}\sum _{n=-\infty }^{\infty }\left|h[n]\right|&=\sum _{n=-\infty }^{\infty }\left|h[n]\right|\left|e^{-j\omega n}\right|\\&=\sum _{n=-\infty }^{\infty }\left|h[n](1\cdot e)^{-j\omega n}\right|\\&=\sum _{n=-\infty }^{\infty }\left|h[n](re^{j\omega })^{-n}\right|\\&=\sum _{n=-\infty }^{\infty }\left|h[n]z^{-n}\right|\end{aligned}}}
where
z
=
r
e
j
ω
{\displaystyle z=re^{j\omega }}
and
r
=
|
z
|
=
1
{\displaystyle r=|z|=1}
.
The
region of convergence must therefore include the
unit circle .
See also
Further reading
Gordon E. Carlson Signal and Linear Systems Analysis with Matlab second edition, Wiley, 1998,
ISBN
0-471-12465-6
John G. Proakis and Dimitris G. Manolakis Digital Signal Processing Principals, Algorithms and Applications third edition, Prentice Hall, 1996,
ISBN
0-13-373762-4
D. Ronald Fannin, William H. Tranter, and Rodger E. Ziemer Signals & Systems Continuous and Discrete fourth edition, Prentice Hall, 1998,
ISBN
0-13-496456-X
Proof of the necessary conditions for BIBO stability.
Christophe Basso Designing Control Loops for Linear and Switching Power Supplies: A Tutorial Guide first edition, Artech House, 2012, 978-1608075577
Michael Unser (2020). "A Note on BIBO Stability". IEEE Transactions on Signal Processing . 68 : 5904–5913.
arXiv :
2005.14428 .
doi :
10.1109/TSP.2020.3025029 .
References