Outline
Lyapunov Stability Theory: Linear
Systems
M. S. Fadali
Professor of EE

Lyapunov’s (first, indirect) linearization
method.

Linear time-invariant case.

Domain of attraction.
1

2
Lyapunov’s Linearization Method
Example
Linearize nonlinear
of equilibrium :

Determine the stability of the equilibrium
of the mechanical system at the origin

Equilibrium with
system in vicinity
.

Find the eigenvalues of the linearized system.
of the nonlinear system is:
The equilibrium
◦ 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.
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4
Nonlinear State Equations

Physical state variables

State Equations
Linearization and Stability


Equilibrium state
Linearized model with

Characteristic polynomial and stability
,

Stable
,
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6
Linear Time-invariant Case
Proof: Sufficiency
The LTI system

Use a quadratic Lyapunov function
is asymptotically stable if and only if for
any positive definite matrix there exists
a positive definite symmetric solution to
the Lyapunov equation
globally exp. stable.
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8
Proof: Necessity

Let
Symmetric Positive Definite
Hurwitz
for some nonzero
iff
is not an observable pair.
for
observable.
Note: can be positive semidefinite.
→
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Uniqueness
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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.

Subtract
constant if and only if
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Solution
Example
Determine the stability of the system with
state matrix
using the Lyapunov equation with
• Multiply
.
Note: The system is clearly stable by
inspection since is in companion form.
• Equate to obtain three equations in three unknowns.
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Equivalent Linear System
14
Choose

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not positive definite.

No conclusion: sufficient condition only.

Choose
and solve for .
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MAPLE
Equivalent Linear System
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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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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To Get Earlier Answer
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.
 1.1167 0.08333
P

0.08333 0.1167 

>> P=lyap(A',eye(2))
P=
1.1167 0.0833
0.0833 0.1167
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Example
22
Calculate
Equilibrium
Lyapunov function candidate for
For
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Simulation Results
Theorem 3.9
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.


Equilibrium
I.
compact set containing
invariant w.r.t. the solutions of
,
II.
Then
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Proof

of
the region of attraction of
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Example
Under the assumptions
is the largest invariant set in
 By La Salle’s Theorem, every solution
, i.e.
starting in approaches as
approaches as

is an estimate of the domain of
attraction.

For
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Invariant Set
Estimate Using Linearized system
Minimum value at edge
Solve
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Example
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Contours
2
Equilibrium
Solve
1.5
1
0.5
0
-0.5
-1
-1.5
-3
for
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-2
-1
0
1
2
3
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