2. 2
Any bounded input yields bounded output, i.e.
For linear systems:
BIBO Stability โAll the poles of the transfer function lie in the LHP.
Characteristic Equation
Solve for poles of the transfer function T(s)
Bounded-Input Bounded-Output (BIBO) stablility
Definition: For any constant N, M >0
3. 3
Asymptotically stable โ All the eigenvalues of the A matrix
have negative real parts
(i.e. in the LHP)
Solve for the eigenvalues for A matrix
Asymptotic stablility
Note: Asy. Stability is indepedent of B and C Matrix
For linear systems:
4. 4
Asy. Stability from Model Decomposition
Suppose that all the eigenvalues of A are distinct.
Coordinate Matrix
Hence, system Asy. Stable โ all the eigenvales of A at lie in the LHP
Let the eigenvector of matrix A with respect to eigenvalue
6. 6
In the absence of pole-zero cancellations, transfer function poles are
identical to the system eigenvalues. Hence BIBO stability is
equivalent to asymptotical stability.
Conclusion: If the system is both controllable and observable, then
BIBO Stability โ Asymptotical Stability
Asymptotic Stablility versus BIBO Stability
โข Asymptotically stable
โข All the eigenvalues of A lie in the LHP
โข BIBO stable
โข Routh-Hurwitz criterion
โข Root locus method
โข Nyquist criterion
โข ....etc.
Methods for Testing Stability
7. 7
Set ๐ข(๐ก) = 0,
we get
0 1
โ2 โ3
๐ฅ1๐
๐ฅ2๐
=
0
0
โ
๐ฅ1๐
๐ฅ2๐
=
0
0
Example:
Equilibrium point
Lyapunov Stablility
In other words, consider the system
Find the equilibrium point ?
8. 8
Definition: An equilibrium state of an autonomous system is
stable in the sense of Lyapunov if for every , exist a
such that for
9. 9
Definition: An equilibrium state of an autonomous system is
asymptotically stable if
(i) it is stable
(ii) there exist a such that
10. 10
Lyapunov Theorem
Consider the system (1)
A function V(x) is called a Lapunov fuction V(x) if
Then eq. state of the system (1) is stable.
Then eq. state of the system (1) is asy. stable.
Moreover, if the Lyapunov function satisfies
and
11. 11
Explanation of the Lyapunov Stability Theorem
1. The derivative of the Lyapunov function along the trajectory is negative.
2. The Lyapunov function may be consider as an energy function of the system.
12. 12
Lyapunovโs method for Linear system:
Proof: Choose
The eq. state is asymptotically stable.
โ
For any p.d. matrix Q , there exists a p.d. solution of the
Lyapunov equation
Hence, the eq. state x=0 is asy. stable by Lapunov theorem.
13. 13
Asymptotically stable in the large
( globally asymptotically stable)
(1) The system is asymptotically stable for all the initial states .
(2) The system has only one equilibrium state.
(3) For an LTI system, asymptotically stable and globally
asymptotically stable are equivalent.
Lyapunov Theorem (Asy. Stability in the large)
If the Lyapunov function V(x) further satisfies
(i)
(ii)
Then, the (asy.) stability is global.
14. 14
Sylvesterโs criterion
A symmetric matrix Q is p.d. if and only if all its n
leading principle minors are positive.
Definition
The i-th leading principle minor of an
matrix Q is the determinant of the matrix extracted from
the upper left-hand corner of Q.
15. 15
Remark:
(1) are all negative Q is n.d.
(2) All leading principle minors of โQ are positive Q is n.d.
๐(๐ฅ)
= 2๐ฅ1
2
+ 4๐ฅ1๐ฅ3 + 3๐ฅ1
2
+ 6๐ฅ2๐ฅ3 + ๐ฅ3
2
Q is not p.d.
Example:
16. 16
P is p.d.
System is asymptotically stable
Example: Test the stability of the system
17. Q.1 Check the stability of the equilibrium state of the system
described by
X1ห = X2
17
1
๐ห2 = โ๐1 โ ๐2๐2
2
Solution:
Select Lyapunov function
V X = ๐2+๐2
1
Which is positive definite function
Its derivative is
V Xฬ =
๐V dX1
+
๐V dX2
๐X1 dt ๐X2 dt
18. 1
V X
ห = 2X1X2 + 2X2 โX1 โ X2X2
18
1 2
V X
ห = 2X1X2 โ 2X2๐1 โ 2๐2๐2
1 2
V X
ห = โ2X2X2
V X
ห is always negative definite.
V X โ โ as X โ โ Then system is asymptotically stable in-the-large.
19. Q.2 Consider the nonlinear system described by the equations
19
X1ห = X2
ห =
X2 1 โ X1 X2 โ X1
Find the region in the state plane for which the equilibrium state of
the systems asymptotically stable.
Solution:
Select Lyapunov function
1 2
V X = ๐2+๐2
Which is positive definite function
Its derivative is
V Xฬ =
๐V dX1
+
๐V dX2
๐X1 dt ๐X2 dt
20. V X
ห = 2X1X2 + 2X2 โ 1 โ X1
20
2
V X
ห = 2X1X2 โ 2๐2 1 โ X1
X2 โ X1
โ 2๐1๐2
2
V X
ห = โ2๐2 1 โ X1
Case1:
V X
ห to be negative definite if 1 โ X1 > 0 , X2 โ 0
or ๐1 < 1
If these conditions are fulfil then system remain asymptotically
stable.
Case2:
V X
ห = 0 if ๐1 = 1 at this point rate of change of energy will
become zero. At this point there is no trajectory. But energy is not
zero at this point.
21. Case3:
V X
ห is not negative definite if ๐1 > 1
In this region system will become unstable.
21
Possibly unstable
Stable Stable
Possibly unstable
X1
X2