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EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
(^1)
Updated 2 October 2016 Prof. Mohamed Zribi Some Concepts of Stability Handout 8: Fall 2016Lumped Systems Theory EE 510:
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
Some Concepts of Stability
1)^ Four types of stability will be studied:
4) Bounded Input Bounded State (BIBS) stability 3) Stability in the sense of Lyapunov (SISL) 2) Bounded Input Bounded Output (BIBO) stability Asymptotic Stability (AS)
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
(^3)
Definition: 1. Asymptotic Stability
A
linear
continuous
time
system
is
said
to be
Criterion: infinity.initial state x(0), all state variables approach zero as t approachesasymptotically stable if with zero input (u(t)=0) and arbitrary
A linear continuous time system is asymptotically
stable if and only if
every eigenvalue of the system matrix has a
negative real part.
(0) At
x (^) Ax x t e (^) x
For asymptotic stability,
0 At
e as (^) t
(^1) (^1) (^1) ( )
At
adj sI (^) A
e sI (^) A
sI (^) A
The terms of
sI (^) A
are such that
(^1) 2
(^1) (^2) ... n n
s (^) s s (^) s s (^) s
which leads to
(^1) 2
(^1) (^2) ... n
s t s t ns t
e e e
Hence the real part of
i^ s must be negative to have an asymptotically
1)^ Consequences: stable system.
With zero input, x(t) approaches the origin as
(^) t (^)
state as2)^ With constant input, x(t) approaches the unique equilibrium
(^) t (^)
t 3)^ With general input , the natural response will die out as
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
Example 1:
x x u (^) Ax (^) Bu
The eigenvalues of A are -1 and -
the system is asymptotically
Note that if u(t)=0,stable.
(^) lim (^) ( ) lim (0) 0 At
t t
x t e (^) x
^
Note that if u(t)=5,
lim (^) ( ) 15
t x t
Example 2:
0 2^ x^ (^0 ) x (^) Ax
The eigenvalues of A are 0 and -
the system is not
Example 3: asymptotically stable.
0 2^ x^ (^5 ) x (^) Ax
The eigenvalues of A are 5 and -
the system is not
asymptotically stable.
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
(^7)
Example 7:
x x u (^) Ax (^) Bu
The eigenvalues of A are 0 and -
the system is not
If asymptotically stable.
[ 5 2]
y (^) Cx x
, The transfer function is
s
H s s s s
If BIBO stable does not imply Asymptotic stable transfer function is located at s = -5).function matrix has a negative real part (the pole of theThus the system is BIBO stable since the pole of the transfer
[0 3]
y (^) Cx x
, The transfer function is
( (^) 5) s
H s s s s
(the pole of the transfer function is located at s = 0).transfer function matrix does not have a negative real partThus the system is not BIBO stable since the pole of the
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
Example 8:
[ 2 1]
x x u (^) Ax (^) Bu y (^) Cx x
The eigenvalues of A are -1 and -
the system is
The transfer function isasymptotically stable.
(( )^
(^) 2)^ s
H s s s^
^
, the system is BIBO
Asymptotic stable stable.
BIBO stable
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
(^9)
Definition: 3. Stability in the Sense of Lyapunov (SISL)
A continuous linear time invariant system is said to be
Criterion: bounded at all the times.SISL if with u(t)=0 and arbitrary x(0), the state vector x(t) remains
A continuous linear time invariant system is SISL if
and only if:
a) every
eigenvalue
of
the
system
matrix
has
a
non-
of that eigenvalue.of independent eigenvectors is equal to the multiplicityb)^ For each eigenvalue with a zero real part, the numberpositive real part.
Example 9:
x x u (^) Ax (^) Bu
The eigenvalues of A are -1 and -
the system is
The eigenvalues of A are -1 and -2asymptotically stable.
the system is SISL.
Asymptotic stable
stable in the Sense of Lyapunov.
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
Example 10:
x x u (^) Ax (^) Bu
The eigenvalues of A are 1 and -
the system is not
The eigenvalues of A are 1 and -1asymptotically stable.
the system is not
SISL does not imply Asymptotic stable SISL.
Example 11:
x x u (^) Ax (^) Bu
The eigenvalues of A are 0 and 0
the system is not
The eigenvalues of A are 0 and 0asymptotically stable.
the eigenvalue 0 has
number of independent eigenvectors is 1 (it iseigenvectors for the mode 0. It is easy to check that themultiplicity 2, we need to find the number of independent
1 ^ ) 0
the
system is not SISL.
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
(^13)
Example 13:
[0 1]
x x u (^) Ax (^) Bu y (^) Cx x
The eigenvalues of A are -3 and -
the system
The eigenvalues of A are -3 and -2asymptotically stable.
the system is SISL.
The
transfer
function
is
(( )^
s
H s s s s
Thus
the
The system is BIBS stable sincesystem is BIBO stable.
3
(^1) (^1) (^1) 1
(0) t
x (^) t x (^) t x (^) t e (^) x
and
(^2) 2
x (^) t x (^) t u
so that
(^1) ( ) x^ t
and
(^2) ( ) x^ t
are bounded input
Asymptotic stable bounded state stable.
stable in the Sense of Lyapunov.
Asymptotic stable
BIBO stable.
Asymptotic stable
BIBS stable.
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
(^14)
Example 14:
[0 1]
x x u (^) Ax (^) Bu y (^) Cx x
The eigenvalues of A are 0 and -
the system is not
The eigenvalues of A are 0 and -2asymptotically stable.
the system is SISL.
The transfer function is
s
H s s s s
. Thus the system
The system is BIBS stable sinceis BIBO stable.
1 (^1) 1
x (^) t x (^) t x
and
(^2) 2
x (^) t x (^) t u
so that
(^1) ( ) x^
t and
(^2) ( ) x^ t
are bounded input
BIBS stable bounded state stable.
stable in the Sense of Lyapunov.
BIBS stable
BIBO stable.
The system is BIBS stable but not Asymptotically stable.
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
(^15)
Example 15:
x x u (^) Ax (^) Bu
The eigenvalues of A are 0 and -
the system is not
The eigenvalues of A are 0 and -5asymptotically stable.
the system is SISL.
If
[ 5 2]
y (^) Cx x
, The transfer function is
s
H s s s s
If transfer function is located at s = -5).function matrix has a negative real part (the pole of theThus the system is BIBO stable since the pole of the transfer
[0 3]
y (^) Cx x
, The transfer function is
( (^) 5) s
H s s s s
half plane (including the^ a)^ All eigenvalues of the system matrix are in the closed left^ ^ The system is not BIBS stable:(the pole of the transfer function is located at s = 0).transfer function matrix does not have a negative real partThus the system is not BIBO stable since the pole of the
(^) j
axis). The eigenvalues are -
and 0. (true)
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
b) For each eigenvalue on the
(^) j
axis (
) , the number of
c) All poles ofthat eigenvalue. (true)independent eigenvectors is equal to the multiplicity of
1
sI (^) A (^) B ^
are in the open half plane (not
including
the
j
axis).
(False
because
2
1 2
(^) 5 )^ s^ s
sI (^) A (^) B s s s
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
(^19)
Example 17:
[0 1]
x x u (^) Ax (^) Bu y (^) Cx x
The eigenvalues of A are 0 and -
the system is not
The eigenvalues of A are 0 and -2asymptotically stable.
the system is stable in
The transfer function isthe sense of Lyapunov.
s
H s s s s
. Thus the system
The system is BIBS stable. Note thatis BIBO stable.
1 (^1) 1
x (^) t x (^) t x
and
(^2) 2
x (^) t x (^) t u
so that
1
( ) x^ t and
(^2) ( ) x^ t
are bounded input
bounded state stable.
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
(^20)
Example 18:
x x u (^) Ax (^) Bu
The eigenvalues of A are 0 and -
the system is not
The eigenvalues of A are 0 and -5asymptotically stable.
the system is SISL.
If
[ 5
2]
y (^) Cx x
, The transfer function is
s
H s s s s
If Thus the system is BIBO stable.
[ 5
2]
y (^) Cx x
, the system is SISL and BIBO stable
If
[0 3]
y (^) Cx x
, The transfer function is
( (^) 5) s
H s s s s
If Thus the system is not BIBO stable.
[0 3]
y (^) Cx x
, the system is SISL but not BIBO stable
The system is not BIBS stable.
EE 510: Lumped Systems Theory
Prof. Mohamed Zribi
1)^ Remark:
Asymptotic stability
BIBO stability, BIBS stability,
2) BIBS stabilityStability in the Sense of Lyapunov
BIBO stability, Stability in the Sense of
4) The system could be BIBO stable but not SISL.3) The system could be SISL but not BIBO stable.Lyapunov
