Lyapunnov

1. Lyapunov Stability Theory: Linear
Systems
M. S. Fadali
Professor of EE
1
Outline
 Lyapunov’s (first, indirect) linearization
method.
 Linear time-invariant case.
 Domain of attraction.
2
Lyapunov’s Linearization Method
 Linearize nonlinear system in vicinity
of equilibrium :
.
 Find the eigenvalues of the linearized system.
The equilibrium of the nonlinear system is:
◦ Exponentially stable if all the eigenvalues are in the
open LHP.
◦ Unstable if one or more of its eigenvalues is in the
open RHP.
◦ Inconclusive for LHP eigenvalues and one or
more eigenvalues on the imaginary axis.
3
Example
 Determine the stability of the equilibrium
of the mechanical system at the origin
 Equilibrium with
4

2. Nonlinear State Equations
 Physical state variables
 State Equations
5
Linearization and Stability
 Equilibrium state
 Linearized model with
 Characteristic polynomial and stability
,
 Stable ,
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Linear Time-invariant Case
The LTI system
is asymptotically stable if and only if for
any positive definite matrix there exists
a positive definite symmetric solution to
the Lyapunov equation
7
Proof: Sufficiency
 Use a quadratic Lyapunov function
globally exp. stable.
8

3. Proof: Necessity
 Let Hurwitz
→
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Symmetric Positive Definite
for some nonzero
iff is not an observable pair.
for observable.
Note: can be positive semidefinite.
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Uniqueness
Subtract
constant if and only if
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Remarks
 Recall that the original Lyapunov theorem
only gives a sufficient condition.
 If we start with (i.e. with ) and
solve for , the condition the test may or
may not work.
 If we start with (i.e. with the derivative
and we find a the condition is necessary
and sufficient.
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4. Example
Determine the stability of the system with
state matrix
using the Lyapunov equation with .
Note: The system is clearly stable by
inspection since is in companion form.
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Solution
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• Multiply
• Equate to obtain three equations in three unknowns.
Equivalent Linear System
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Choose
 not positive definite.
 No conclusion: sufficient condition only.
 Choose and solve for .
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5. MAPLE
Compute:
with(LinearAlgebra):
Transpose(A).P+P.A
Solve the equivalent linear system: M.p=-q
p is a vector whose entries are the entries
of the P matrix, similarly define q
LinearSolve(M,B)
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Equivalent Linear System
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MATLAB
 Solve a different equation.
 Identical to our equation with
replaced by .
 Eigenvalues are the same!
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MATLAB Example
>> A=[0,1;-6,-5];
>> Q=eye(2)
>> P=lyap(A,eye(2))
P =
0.5333 -0.5000
-0.5000 0.7000
>> eig(P)
ans =
0.1098
1.1236
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6. To Get Earlier Answer
>> P=lyap(A',eye(2))
P =
1.1167 0.0833
0.0833 0.1167
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






1167
.
0
08333
.
0
08333
.
0
1167
.
1
P
Domain (Ball, Region) of Attraction
 Region in which the trajectories of the
system converge to an asymptotically
stable equilibrium point.
 Difficult to estimate, in general.
 Can be estimated using the linearized
system in the vicinity of the asymptotically
stable equilibrium.
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Example
Equilibrium
Lyapunov function candidate for
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Calculate
For
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7. Simulation Results
 The ball of attraction can be estimated
to be
 Although for
we have this
region includes divergent trajectories
because is not an invariant set. For
example, the trajectory starting at
crosses then
diverges.
25
Theorem 3.9
 Equilibrium of
I. compact set containing ,
invariant w.r.t. the solutions of
II.
Then the region of attraction of
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Proof
 Under the assumptions
 is the largest invariant set in
 By La Salle’s Theorem, every solution
starting in approaches as , i.e.
approaches as
 is an estimate of the domain of
attraction.
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Example
For
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8. Invariant Set
Minimum value at edge
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Estimate Using Linearized system
Solve
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Example
Equilibrium
Solve
for
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Contours
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-3 -2 -1 0 1 2 3
-1.5
-1
-0.5
0
0.5
1
1.5
2