# Switched system stability khudaydus present ```Stability of Switched Systems in Sense of Lyapunov’s
Theory
M.Khudaydus
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Department of Mathematics,
a fulfillment requirement for the degree of
MSc
in
Control Theory
April 8, 2018
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
1 /Contr
38
What is Stability ?
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
2 /Contr
38
What is Stability ?
Motions of a ball in a bowl
Figure: The trajectory of the ball is stable when it goes to steady state (origin)
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
2 /Contr
38
What is Stability ?
Motions of a ball in a bowl
Figure: The trajectory of the ball is stable when it goes to steady state (origin)
We say a system is stable when it comes back to its original state after it
was affected with an external disturbance.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
2 /Contr
38
Equilibria
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
3 /Contr
38
Equilibria
Equilibrium Points
Figure: The pendulum motion has only
two equilibrium positions
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
3 /Contr
38
Equilibria
Equilibrium Points
Equation of motion
Pendulum 2nd order ODE Equation
θ̈ = − gl sin θ −
k
m θ̇
Figure: The pendulum motion has only
two equilibrium positions
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
3 /Contr
38
Stability Theory of a Dynamical System
Whose the foundation of the stability theorems in dynamical systems?
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
4 /Contr
38
Stability Theory of a Dynamical System
Whose the foundation of the stability theorems in dynamical systems?
Figure: a Russian mathematician
and engineer who laid the
foundation of the stability theory
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
4 /Contr
38
Stability Theory of a Dynamical System
Whose the foundation of the stability theorems in dynamical systems?
Stability of equilibrium points are
characterized in the sense of Lyapunov
Figure: a Russian mathematician
and engineer who laid the
foundation of the stability theory
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
4 /Contr
38
Stability Theory of a Dynamical System
Whose the foundation of the stability theorems in dynamical systems?
Stability of equilibrium points are
characterized in the sense of Lyapunov
Stability of equilibrium Pt.
Theorem 1
An equilibrium point is stable if all solutions
of the dynamical system starting at nearby
points stay nearby; otherwise, it is unstable
Figure: a Russian mathematician
and engineer who laid the
foundation of the stability theory
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
4 /Contr
38
State-Space Models
Pendulum Equation
2nd Order ODE Equation
M.Khudaydus
g
k
θ̈ = − sin θ − θ̇
l
m
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
5 /Contr
38
State-Space Models
Pendulum Equation
2nd Order ODE Equation
g
k
θ̈ = − sin θ − θ̇
l
m
2nd ODE Dynamical system can be transformed into a finite number
of 1st ODE equations by carefully choosing the state variables:
State-Space Model
(
ẋ1 = x2
ẋ2 = − gl sin x1 −
k
m x2
where we set the state variables as x1 = θ and x2 = θ̇.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
5 /Contr
38
Autonomous and Non-autonomous Dynamical systems
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
6 /Contr
38
Autonomous and Non-autonomous Dynamical systems
Dynamical system is represented in a Vector-notation as
ẋ = f (t, x)
M.Khudaydus
(1)
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
6 /Contr
38
Autonomous and Non-autonomous Dynamical systems
Dynamical system is represented in a Vector-notation as
ẋ = f (t, x)
(1)
Autonomous system (LTI dynamical systems):
Definition 0.1
The system (1) is said to be autonomous when the function f does not
depend on t; that is,
ẋ = f (x)
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
6 /Contr
38
Autonomous and Non-autonomous Dynamical systems
Dynamical system is represented in a Vector-notation as
ẋ = f (t, x)
(1)
Autonomous system (LTI dynamical systems):
Definition 0.1
The system (1) is said to be autonomous when the function f does not
depend on t; that is,
ẋ = f (x)
Non-autonomous system:
Definition 0.2
The system (1) is said to be non-autonomous when the function f depend
on both t and x ; that is,
ẋ = f (t, x)
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
6 /Contr
38
Stability and Asymptotic Stability of Autonomous Sys.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
7 /Contr
38
Stability and Asymptotic Stability of Autonomous Sys.
Theorem 2
Let x = 0 be an equilibrium point of the nonlinear system
ẋ = f (x)
where f : D 7→ R n is continuously differentiable and D is a neighborhood
of the origin.Let
∂f (x)
A=
∂x x=0
Then
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
7 /Contr
38
Stability and Asymptotic Stability of Autonomous Sys.
Theorem 2
Let x = 0 be an equilibrium point of the nonlinear system
ẋ = f (x)
where f : D 7→ R n is continuously differentiable and D is a neighborhood
of the origin.Let
∂f (x)
A=
∂x x=0
Then
The origin is asymptotically stable if Re(λi ) &lt; 0 for all eigenvalues
of A.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
7 /Contr
38
Stability and Asymptotic Stability of Autonomous Sys.
Theorem 2
Let x = 0 be an equilibrium point of the nonlinear system
ẋ = f (x)
where f : D 7→ R n is continuously differentiable and D is a neighborhood
of the origin.Let
∂f (x)
A=
∂x x=0
Then
The origin is asymptotically stable if Re(λi ) &lt; 0 for all eigenvalues
of A.
The origin is unstable if ∃ λi such that Re(λi ) &gt; 0.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
7 /Contr
38
Example:
Consider the pendulum equation
M.Khudaydus
ẋ1
= x2
ẋ2
= −
g
k
sin x1 − x2
l
m
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
8 /Contr
38
Example:
Consider the pendulum equation
ẋ1
= x2
g
k
sin x1 − x2
l
m
has two equilibrium points at (x1 , x2 ) = (0, 0) and (x1 , x2 ) = (π, 0).
Using linearization,the Jacobian matrix is given by:
&quot;
# ∂f1
∂f1
∂f
0
1
∂x1
∂x2
= ∂f2 ∂f2 =
k
− gl cos x1 − m
∂x
∂x1
∂x2
ẋ2
M.Khudaydus
= −
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
8 /Contr
38
Example:
Consider the pendulum equation
ẋ1
= x2
g
k
sin x1 − x2
l
m
has two equilibrium points at (x1 , x2 ) = (0, 0) and (x1 , x2 ) = (π, 0).
Using linearization,the Jacobian matrix is given by:
&quot;
# ∂f1
∂f1
∂f
0
1
∂x1
∂x2
= ∂f2 ∂f2 =
k
− gl cos x1 − m
∂x
∂x1
∂x2
ẋ2
= −
Evaluating the Jacobian at x = 0
∂f
A=
∂x
M.Khudaydus
x=0
0
=
− gl
1
k
−m
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
8 /Contr
38
Example:
Consider the pendulum equation
ẋ1
= x2
g
k
sin x1 − x2
l
m
has two equilibrium points at (x1 , x2 ) = (0, 0) and (x1 , x2 ) = (π, 0).
Using linearization,the Jacobian matrix is given by:
&quot;
# ∂f1
∂f1
∂f
0
1
∂x1
∂x2
= ∂f2 ∂f2 =
k
− gl cos x1 − m
∂x
∂x1
∂x2
ẋ2
= −
Evaluating the Jacobian at x = 0
∂f
A=
∂x
x=0
0
=
− gl
1
k
−m
The eigenvalues of A are :
s
λ1,2
k
1
=−
&plusmn;
2m 2
k 2 4g
−
m
l
If the eigenvalues satisfy Re(λi ) &lt; 0, the equilibrium point at the origin is
asymptotically stable.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
8 /Contr
38
Stability based on the Position of Eigenvalues on
complex-plane
Consider the LTI autonomous system
ẋ = A x
where the matrix A is n &times; n and assume that the eigenvalues of A take the form
M.Khudaydus
λ = a &plusmn; jb
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
9 /Contr
38
Stability based on the Position of Eigenvalues on
complex-plane
Consider the LTI autonomous system
ẋ = A x
where the matrix A is n &times; n and assume that the eigenvalues of A take the form
λ = a &plusmn; jb
(1) if eigenvalues are real and
Re(λ) &lt; 0 regardless of the
imaginary part then, the equilibrium
point has a stable-node.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
9 /Contr
38
Stability based on the Position of Eigenvalues on
complex-plane
Consider the LTI autonomous system
ẋ = A x
where the matrix A is n &times; n and assume that the eigenvalues of A take the form
λ = a &plusmn; jb
(1) if eigenvalues are real and
Re(λ) &lt; 0 regardless of the
imaginary part then, the equilibrium
point has a stable-node.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory for April
the degree
8, 2018ofMScin
9 /Contr
38
Stability based on the Position of Eigenvalues on
complex-plane
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
10 /Contr
38
Stability based on the Position of Eigenvalues on
complex-plane
(2) if eigenvalues are real
and Re(λ) ≥ 0 then, the
equilibrium point has a
unstable-node.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
10 /Contr
38
Stability based on the Position of Eigenvalues on
complex-plane
(2) if eigenvalues are real
and Re(λ) ≥ 0 then, the
equilibrium point has a
unstable-node.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
10 /Contr
38
Stability based on the Position of Eigenvalues on
complex-plane
(3) if eigenvalues are complex then,
(2) if eigenvalues are real
and Re(λ) ≥ 0 then, the
equilibrium point has a
unstable-node.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
10 /Contr
38
Stability based on the Position of Eigenvalues on
complex-plane
(3) if eigenvalues are complex then,
(2) if eigenvalues are real
and Re(λ) ≥ 0 then, the
equilibrium point has a
unstable-node.
M.Khudaydus
(I) when Re(λ) &lt; 0, the equilibrium
point has a stable-focus and the
trajectory takes a spiral shape
converges to the origin
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
10 /Contr
38
Stability based on the Position of Eigenvalues on
complex-plane
(3) if eigenvalues are complex then,
(2) if eigenvalues are real
and Re(λ) ≥ 0 then, the
equilibrium point has a
unstable-node.
M.Khudaydus
(I) when Re(λ) &lt; 0, the equilibrium
point has a stable-focus and the
trajectory takes a spiral shape
converges to the origin
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
10 /Contr
38
Stability based on the Position of Eigenvalues on a complex-plane
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
11 /Contr
38
Stability based on the Position of Eigenvalues on a complex-plane
(3) if eigenvalues are complex then,
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
11 /Contr
38
Stability based on the Position of Eigenvalues on a complex-plane
(3) if eigenvalues are complex then,
(II) when Re(λ) &gt; 0, the
equilibrium point has a
unstable-focus and the
trajectory takes a spiral shape
diverges away from the origin
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
11 /Contr
38
Stability based on the Position of Eigenvalues on a complex-plane
(3) if eigenvalues are complex then,
(II) when Re(λ) &gt; 0, the
equilibrium point has a
unstable-focus and the
trajectory takes a spiral shape
diverges away from the origin
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
11 /Contr
38
Stability based on the Position of Eigenvalues on a complex-plane
(3) if eigenvalues are complex then,
(III) when Re(λ) = 0, the
(II) when Re(λ) &gt; 0, the
equilibrium point has a center
equilibrium point has a
unstable-focus and the
and the trajectory takes a circle
trajectory takes a spiral shape
shape (closed orbits)
diverges away from the origin
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
11 /Contr
38
Stability based on the Position of Eigenvalues on a complex-plane
(3) if eigenvalues are complex then,
(III) when Re(λ) = 0, the
(II) when Re(λ) &gt; 0, the
equilibrium point has a center
equilibrium point has a
unstable-focus and the
and the trajectory takes a circle
trajectory takes a spiral shape
shape (closed orbits)
diverges away from the origin
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
11 /Contr
38
Lyapunov Exponential Stability
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
12 /Contr
38
Lyapunov Exponential Stability
Theorem 3
The equilibrium point x̄ = 0 of the autonomous system
ẋ = f (x(t))
(2)
where f : D 7→ R n is a locally Lipschitz and x(t) is the system state victor,
is exponentially stable, if there exist positive constants α, β and δ such that if
||x(0)|| &lt; δ, then
||x(t)|| ≤ αe −βt ||x(0)|| ∀t ≥ 0
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
12 /Contr
38
Lyapunov Exponential Stability
Theorem 3
The equilibrium point x̄ = 0 of the autonomous system
ẋ = f (x(t))
(2)
where f : D 7→ R n is a locally Lipschitz and x(t) is the system state victor,
is exponentially stable, if there exist positive constants α, β and δ such that if
||x(0)|| &lt; δ, then
||x(t)|| ≤ αe −βt ||x(0)|| ∀t ≥ 0
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
12 /Contr
38
Lyapunov Exponential Stability
Theorem 3
The equilibrium point x̄ = 0 of the autonomous system
ẋ = f (x(t))
(2)
where f : D 7→ R n is a locally Lipschitz and x(t) is the system state victor,
is exponentially stable, if there exist positive constants α, β and δ such that if
||x(0)|| &lt; δ, then
||x(t)|| ≤ αe −βt ||x(0)|| ∀t ≥ 0
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
12 /Contr
38
Pros and Cons of the Exponential Stability theorem
Pros.
(1) The exponential function
converges to zero very fast.
(2) Provide a robust measurement tool box to prove many
modern theorems in stability.
M.Khudaydus
Cons.
(I) Its need to give a solution of
the ODEs of the state models
(II) Satisfying the inequality is
too tedious
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
13 /Contr
38
Lyapunov Functions
In 1892, Lyapunov showed that other functions could be used to determine
stability of an equilibrium points.He said, that if Φ(t, x) is a solution of
the autonomous
ẋ = f (x(t))
(3)
then, the derivative of the Lyapunov function V (x) along the trajectories
of (3) which is denoted by
V̇ (x) =
∂V
f (x)
∂ xi
should be dependent on the system’s equation .In addition, if the solution
Φ(t, x) starts at initial state x at time t = 0, then it should be
V̇ (x) =
d
V (Φ(t, x))
dt
&lt;0
t=0
Therefore, V (x) will decrease along the solution of (3).
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
14 /Contr
38
Lyapunov Functions
Theorem 4
Let x = 0 be an equilibrium point for (3) and D ⊂ R n be a domain
containing x = 0.Let V : D → R n be a continuously differentiable
function, such that
V (0) = 0 ; V (x) &gt; 0 ∀x ∈ D, x 6= 0
(4)
V̇ (x) ≤ 0 ∀ x ∈ D
(5)
Then,x(t) = 0 is stable.Moreover, if
V̇ (x) &lt; 0 x ∈ in D , x 6= 0
(6)
Then x = 0 is asymptotically stable.For proof, see Khalil book,p100.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
15 /Contr
38
Example on Lyapunov Function Candidate
Consider the pendulum equation without friction
ẋ1 = x2
ẋ2 = −
M.Khudaydus
g l
sin x1
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
16 /Contr
38
Example on Lyapunov Function Candidate
Consider the pendulum equation without friction
ẋ1 = x2
ẋ2 = −
g l
sin x1
A Lyapunov function candidate is:
g 1
(1 − cos x1 ) + x22
V (x) =
l
2
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
16 /Contr
38
Example on Lyapunov Function Candidate
Consider the pendulum equation without friction
ẋ1 = x2
ẋ2 = −
g l
sin x1
A Lyapunov function candidate is:
g 1
(1 − cos x1 ) + x22
V (x) =
l
2
So,V (0) = 0 and V (x) is positive definite over the domain
−2π &lt; x1 &lt; 2π.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
16 /Contr
38
Example on Lyapunov Function Candidate
Consider the pendulum equation without friction
ẋ1 = x2
ẋ2 = −
g l
sin x1
A Lyapunov function candidate is:
g 1
(1 − cos x1 ) + x22
V (x) =
l
2
So,V (0) = 0 and V (x) is positive definite over the domain
−2π &lt; x1 &lt; 2π.
The derivative of V (x) along the trajectories of the system is given by
g g g x1 sin x1 + x2 x2 =
x2 sin x1 −
x2 sin x1 = 0
V̇ (x) =
l
l
l
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
16 /Contr
38
Example on Lyapunov Function Candidate
Consider the pendulum equation without friction
ẋ1 = x2
ẋ2 = −
g l
sin x1
A Lyapunov function candidate is:
g 1
(1 − cos x1 ) + x22
V (x) =
l
2
So,V (0) = 0 and V (x) is positive definite over the domain
−2π &lt; x1 &lt; 2π.
The derivative of V (x) along the trajectories of the system is given by
g g g x1 sin x1 + x2 x2 =
x2 sin x1 −
x2 sin x1 = 0
V̇ (x) =
l
l
l
Thus, conditions (4) and (5) are satisfied and we conclude that the origin
is stable.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
16 /Contr
38
Example on Lyapunov Function Candidate
Consider the pendulum equation without friction
ẋ1 = x2
ẋ2 = −
g l
sin x1
A Lyapunov function candidate is:
g 1
(1 − cos x1 ) + x22
V (x) =
l
2
So,V (0) = 0 and V (x) is positive definite over the domain
−2π &lt; x1 &lt; 2π.
The derivative of V (x) along the trajectories of the system is given by
g g g x1 sin x1 + x2 x2 =
x2 sin x1 −
x2 sin x1 = 0
V̇ (x) =
l
l
l
Thus, conditions (4) and (5) are satisfied and we conclude that the origin
is stable.
Since V̇ (x) = 0 , we can conclude that the origin is not asymptotically
stable.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
16 /Contr
38
Lyapunov’s Equation based on Lyapunov Quadratic Form
Consider the autonomous system
M.Khudaydus
ẋ = A x
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
17 /Contr
38
Lyapunov’s Equation based on Lyapunov Quadratic Form
Consider the autonomous system
ẋ = A x
Assume that we have a quadratic Lyapunov function candidate
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
17 /Contr
38
Lyapunov’s Equation based on Lyapunov Quadratic Form
Consider the autonomous system
ẋ = A x
Assume that we have a quadratic Lyapunov function candidate
1
V (x) = x T Px
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
17 /Contr
38
Lyapunov’s Equation based on Lyapunov Quadratic Form
Consider the autonomous system
ẋ = A x
Assume that we have a quadratic Lyapunov function candidate
1
V (x) = x T Px
2
P is a real symmetric positive definite matrix
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
17 /Contr
38
Lyapunov’s Equation based on Lyapunov Quadratic Form
Consider the autonomous system
ẋ = A x
Assume that we have a quadratic Lyapunov function candidate
1
V (x) = x T Px
2
P is a real symmetric positive definite matrix
3
V̇ (x) = x T P ẋ + ẋ T Px = x T (PA + AT P)x = −x T Qx
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
17 /Contr
38
Lyapunov’s Equation based on Lyapunov Quadratic Form
Consider the autonomous system
ẋ = A x
Assume that we have a quadratic Lyapunov function candidate
1
V (x) = x T Px
2
P is a real symmetric positive definite matrix
3
V̇ (x) = x T P ẋ + ẋ T Px = x T (PA + AT P)x = −x T Qx
4
V 0 (x)f (x) = −x T Qx &lt; 0
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
17 /Contr
38
Lyapunov’s Equation based on Lyapunov Quadratic Form
Consider the autonomous system
ẋ = A x
Assume that we have a quadratic Lyapunov function candidate
1
V (x) = x T Px
2
P is a real symmetric positive definite matrix
3
V̇ (x) = x T P ẋ + ẋ T Px = x T (PA + AT P)x = −x T Qx
4
V 0 (x)f (x) = −x T Qx &lt; 0
5
which is so-called Lyapunov-Equation
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
17 /Contr
38
Lyapunov’s Equation based on Lyapunov Quadratic Form
Consider the autonomous system
ẋ = A x
Assume that we have a quadratic Lyapunov function candidate
1
V (x) = x T Px
2
P is a real symmetric positive definite matrix
3
V̇ (x) = x T P ẋ + ẋ T Px = x T (PA + AT P)x = −x T Qx
4
V 0 (x)f (x) = −x T Qx &lt; 0
5
which is so-called Lyapunov-Equation
6
Solving PA + AT P &lt; −Q for P
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
17 /Contr
38
Lyapunov’s Equation based on Lyapunov Quadratic Form
Consider the autonomous system
ẋ = A x
Assume that we have a quadratic Lyapunov function candidate
1
V (x) = x T Px
2
P is a real symmetric positive definite matrix
3
V̇ (x) = x T P ẋ + ẋ T Px = x T (PA + AT P)x = −x T Qx
4
V 0 (x)f (x) = −x T Qx &lt; 0
5
which is so-called Lyapunov-Equation
6
Solving PA + AT P &lt; −Q for P
7
we conclude that the origin is asymptotically stable; which is similar to say
that, Reλi &lt; 0 for all eigenvalues of A.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
17 /Contr
38
Example on Lyapunov Candidate Function
Consider the autonomous system : ẋ = A x where
−1 0
A=
0 −1
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
18 /Contr
38
Example on Lyapunov Candidate Function
Consider the autonomous system : ẋ = A x where
−1 0
A=
0 −1
Solving the LMI AT P + PA &lt; −Q where Q = [I ]
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
18 /Contr
38
Example on Lyapunov Candidate Function
Consider the autonomous system : ẋ = A x where
−1 0
A=
0 −1
Solving the LMI AT P + PA &lt; −Q where Q = [I ]
Based on MATLAB Calculations!
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
18 /Contr
38
Example on Lyapunov Candidate Function
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
19 /Contr
38
Example on Lyapunov Candidate Function
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
19 /Contr
38
Example on Lyapunov Candidate Function
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
19 /Contr
38
Example on Lyapunov Candidate Function
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
19 /Contr
38
Example on Lyapunov Candidate Function
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
19 /Contr
38
Example on Lyapunov Candidate Function
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
19 /Contr
38
Example on Lyapunov Candidate Function
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
19 /Contr
38
Example on Lyapunov Candidate Function
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
19 /Contr
38
Example on Lyapunov Candidate Function
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
19 /Contr
38
Example on Lyapunov Candidate Function
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
19 /Contr
38
Stability under Switching Signals
M.Khudaydus
Switched Systems
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
20 /Contr
38
Hybrid Systems
Switched System
Switched systems are combination of dynamical systems whose governed
by a constructed Switching-Signals (switching events).
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
21 /Contr
38
Hybrid Systems
Switched System
Switched systems are combination of dynamical systems whose governed
by a constructed Switching-Signals (switching events).
dynamical system may be represented by a linear system
ẋ = Ax + Bu
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
21 /Contr
38
Hybrid Systems
Switched System
Switched systems are combination of dynamical systems whose governed
by a constructed Switching-Signals (switching events).
dynamical system may be represented by a linear system
ẋ = Ax + Bu
x ∈ R n is state vector and u ∈ R n is control input
A x denotes a Continuous-Dynamic System.
B u denotes a Discrete-dynamic System.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
21 /Contr
38
Hybrid Systems
Switched System
Switched systems are combination of dynamical systems whose governed
by a constructed Switching-Signals (switching events).
dynamical system may be represented by a linear system
ẋ = Ax + Bu
x ∈ R n is state vector and u ∈ R n is control input
A x denotes a Continuous-Dynamic System.
B u denotes a Discrete-dynamic System.
Hybrid Systems
Hybrid systems are an interaction between the continuous and discrete
dynamic systems that are form a complete dynamical systems.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
21 /Contr
38
Hybrid Systems in our Life
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
22 /Contr
38
Hybrid Systems in our Life
Continuous systems
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
22 /Contr
38
Hybrid Systems in our Life
Continuous systems
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
22 /Contr
38
Hybrid Systems in our Life
Continuous systems
1
M.Khudaydus
The engine that provide the
change in position x and
velocity are the continuous
state variables
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
22 /Contr
38
Hybrid Systems in our Life
Continuous systems
1
The engine that provide the
change in position x and
velocity are the continuous
state variables
2
The continuous dynamical
systems described with ODE
Motion Equations
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
22 /Contr
38
Hybrid Systems in our Life
Continuous systems
Discrete Systems
1
The engine that provide the
change in position x and
velocity are the continuous
state variables
2
The continuous dynamical
systems described with ODE
Motion Equations
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
22 /Contr
38
Hybrid Systems in our Life
Continuous systems
Discrete Systems
1
The engine that provide the
change in position x and
velocity are the continuous
state variables
2
The continuous dynamical
systems described with ODE
Motion Equations
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
22 /Contr
38
Hybrid Systems in our Life
Continuous systems
Discrete Systems
1
2
M.Khudaydus
The engine that provide the
change in position x and
velocity are the continuous
state variables
1
The Transmission Box
(GearBox) is the discrete state
The continuous dynamical
systems described with ODE
Motion Equations
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
22 /Contr
38
Hybrid Systems in our Life
Continuous systems
Discrete Systems
1
2
M.Khudaydus
The engine that provide the
change in position x and
velocity are the continuous
state variables
The continuous dynamical
systems described with ODE
Motion Equations
1
The Transmission Box
(GearBox) is the discrete state
2
The discrete state affects the
continuous trajectory of the
continuous dynamical system
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
22 /Contr
38
Recasting Hybrid Systems into Switched Systems
Equations
The motion of an automobile might takes the form
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
23 /Contr
38
Recasting Hybrid Systems into Switched Systems
Equations
The motion of(an automobile might takes the form
ẋ1 = x2
ẋ2 = f (a, i)
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
23 /Contr
38
Recasting Hybrid Systems into Switched Systems
Equations
The motion of(an automobile might takes the form
ẋ1 = x2
ẋ2 = f (a, i)
x1 is the position , x2 is the velocity and a is the acceleration input wheres
i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
23 /Contr
38
Recasting Hybrid Systems into Switched Systems
Equations
The motion of(an automobile might takes the form
ẋ1 = x2
ẋ2 = f (a, i)
x1 is the position , x2 is the velocity and a is the acceleration input wheres
i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position.
x1 , x2 are the continuous states
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
23 /Contr
38
Recasting Hybrid Systems into Switched Systems
Equations
The motion of(an automobile might takes the form
ẋ1 = x2
ẋ2 = f (a, i)
x1 is the position , x2 is the velocity and a is the acceleration input wheres
i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position.
x1 , x2 are the continuous states
i is the discrete state
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
23 /Contr
38
Recasting Hybrid Systems into Switched Systems
Equations
The motion of(an automobile might takes the form
ẋ1 = x2
ẋ2 = f (a, i)
x1 is the position , x2 is the velocity and a is the acceleration input wheres
i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position.
x1 , x2 are the continuous states
i is the discrete state
The switching events of the discrete states are called switching signal
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
23 /Contr
38
Recasting Hybrid Systems into Switched Systems
Equations
The motion of(an automobile might takes the form
ẋ1 = x2
ẋ2 = f (a, i)
x1 is the position , x2 is the velocity and a is the acceleration input wheres
i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position.
x1 , x2 are the continuous states
i is the discrete state
The switching events of the discrete states are called switching signal
The Complete Form of Hybrid Switched system
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
23 /Contr
38
Recasting Hybrid Systems into Switched Systems
Equations
The motion of(an automobile might takes the form
ẋ1 = x2
ẋ2 = f (a, i)
x1 is the position , x2 is the velocity and a is the acceleration input wheres
i = {1, 2, 3, 4, 5, R ≡ −1} is the gear shift position.
x1 , x2 are the continuous states
i is the discrete state
The switching events of the discrete states are called switching signal
The Complete Form of Hybrid Switched system
The switched systems can be described by
ẋ(t) = fσ(t) (x(t)) , σ(t) ∈ P = {1, 2, &middot; &middot; &middot; }
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
23 /Contr
38
The Switching Signal
The switching events can be classified into
State Dependent switching
Time Dependent switching
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
24 /Contr
38
The Switching Signal
The switching events can be classified into
State Dependent switching
Time Dependent switching
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
24 /Contr
38
The Switching Signal
The switching events can be classified into
State Dependent switching
Time Dependent switching
The thick curves denotes
the switching surfaces
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
24 /Contr
38
The Switching Signal
The switching events can be classified into
State Dependent switching
Time Dependent switching
The thick curves denotes
the switching surfaces
The thin curves denotes
the continuous dynamical
system trajectories
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
24 /Contr
38
The Switching Signal
The switching events can be classified into
State Dependent switching
Time Dependent switching
The thick curves denotes
the switching surfaces
The thin curves denotes
the continuous dynamical
system trajectories
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
24 /Contr
38
The Switching Signal
The switching events can be classified into
State Dependent switching
Time Dependent switching
The thick curves denotes
the switching surfaces
The thin curves denotes
the continuous dynamical
system trajectories
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
24 /Contr
38
The Switching Signal
The switching events can be classified into
State Dependent switching
Time Dependent switching
The thick curves denotes
the switching surfaces
The thin curves denotes
the continuous dynamical
system trajectories
M.Khudaydus
The switching signal
σ(t) generates a finite
number to specify which
subsystem should be
activate till time reach
specified time
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
24 /Contr
38
The Switching Signal
The switching events can be classified into
State Dependent switching
Time Dependent switching
The thick curves denotes
the switching surfaces
The thin curves denotes
the continuous dynamical
system trajectories
M.Khudaydus
The switching signal
σ(t) generates a finite
number to specify which
subsystem should be
activate till time reach
specified time
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
24 /Contr
38
Stable to Unstable Switched Systems and Vice-Versa
Consider the following systems
ẋ = fσ(t) x(t) ,
σ(t) = q, ∀ q ∈ P = {1, 2}
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
25 /Contr
38
Stable to Unstable Switched Systems and Vice-Versa
Consider the following systems
ẋ = fσ(t) x(t) ,
σ(t) = q, ∀ q ∈ P = {1, 2}
Assume that the two systems are asymptotically stable
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
25 /Contr
38
Stable to Unstable Switched Systems and Vice-Versa
Consider the following systems
ẋ = fσ(t) x(t) ,
σ(t) = q, ∀ q ∈ P = {1, 2}
Assume that the two systems are asymptotically stable
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
25 /Contr
38
Stable to Unstable Switched Systems and Vice-Versa
Consider the following systems
ẋ = fσ(t) x(t) ,
σ(t) = q, ∀ q ∈ P = {1, 2}
Assume that the two systems are asymptotically stable
Applying switching signal to activate each swiched systems might give us
stable or unstable switched systems
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
25 /Contr
38
Stable to Unstable Switched Systems and Vice-Versa
Consider the following systems
ẋ = fσ(t) x(t) ,
σ(t) = q, ∀ q ∈ P = {1, 2}
Assume that the two systems are asymptotically stable
Applying switching signal to activate each swiched systems might give us
stable or unstable switched systems
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
25 /Contr
38
Switching Between two Unstable Switched Systems
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
26 /Contr
38
Switching Between two Unstable Switched Systems
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
26 /Contr
38
Switching Between two Unstable Switched Systems
Both switched systems are unstable
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
26 /Contr
38
Switching Between two Unstable Switched Systems
Both switched systems are unstable
Applying the strategy of switching between them, give a
complete stable switched systems
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
26 /Contr
38
A Way to Guarantee Own a Stable Switched Systems
Theorem 5
Consider the autonomous switched systems
ẋ = Ai x
with Ai = {1, 2}, assume that,
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
27 /Contr
38
A Way to Guarantee Own a Stable Switched Systems
Theorem 5
Consider the autonomous switched systems
ẋ = Ai x
with Ai = {1, 2}, assume that,
1
A1 , A2 are asymptotically stable matrices.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
27 /Contr
38
A Way to Guarantee Own a Stable Switched Systems
Theorem 5
Consider the autonomous switched systems
ẋ = Ai x
with Ai = {1, 2}, assume that,
1
A1 , A2 are asymptotically stable matrices.
2
A1 , A2 are commute.
Then,
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
27 /Contr
38
A Way to Guarantee Own a Stable Switched Systems
Theorem 5
Consider the autonomous switched systems
ẋ = Ai x
with Ai = {1, 2}, assume that,
1
A1 , A2 are asymptotically stable matrices.
2
A1 , A2 are commute.
Then,
The complete switched systems are stable under any arbitrary
switching.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
27 /Contr
38
A Way to Guarantee Own a Stable Switched Systems
Theorem 5
Consider the autonomous switched systems
ẋ = Ai x
with Ai = {1, 2}, assume that,
1
A1 , A2 are asymptotically stable matrices.
2
A1 , A2 are commute.
Then,
The complete switched systems are stable under any arbitrary
switching.
There is a positive symmetric matrix P solution to the Lyapunov
Equation: Ai T P + PA &lt; −Q
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
27 /Contr
38
A Way to Guarantee Own a Stable Switched Systems
Theorem 5
Consider the autonomous switched systems
ẋ = Ai x
with Ai = {1, 2}, assume that,
1
A1 , A2 are asymptotically stable matrices.
2
A1 , A2 are commute.
Then,
The complete switched systems are stable under any arbitrary
switching.
There is a positive symmetric matrix P solution to the Lyapunov
Equation: Ai T P + PA &lt; −Q
There is a quadratic Lyapunov function V (x) = x T Px working as a
common function for both individual switched systems.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
27 /Contr
38
Example
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
28 /Contr
38
Stability Under State-Dependent Switching
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
29 /Contr
38
Stability Under State-Dependent Switching
1
M.Khudaydus
Standing requires
balancing between
the lift switched
system and the right
switched system.
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
29 /Contr
38
Stability Under State-Dependent Switching
1
Standing requires
balancing between
the lift switched
system and the right
switched system.
2
Both switched
system are
unstable(individual).
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
29 /Contr
38
Stability Under State-Dependent Switching
1
Standing requires
balancing between
the lift switched
system and the right
switched system.
2
Both switched
system are
unstable(individual).
3
The
ACCELEROMETER
representing the
switching surfaces.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
29 /Contr
38
State-Dependent Example
Consider the autonomous switched system ẋ = Aσ(t) x(t)
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
30 /Contr
38
State-Dependent Example
Consider the autonomous switched system ẋ = Aσ(t) x(t)
0 10
1.5
2
A1 =
A2 =
0 0
−2 −0.5
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
30 /Contr
38
State-Dependent Example
Consider the autonomous switched system ẋ = Aσ(t) x(t)
0 10
1.5
2
A1 =
A2 =
0 0
−2 −0.5
√
the eigenvalues are, λ(A1 ) = 0 and λ1,2 (A2 ) = 0.5 &plusmn; i 3
Thus, both systems are unstable.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
30 /Contr
38
State-Dependent Example
Consider the autonomous switched system ẋ = Aσ(t) x(t)
0 10
1.5
2
A1 =
A2 =
0 0
−2 −0.5
√
the eigenvalues are, λ(A1 ) = 0 and λ1,2 (A2 ) = 0.5 &plusmn; i 3
Thus, both systems are unstable.
The phase portrait for two unstable systems as follow:
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
30 /Contr
38
State-Dependent Example
Consider the autonomous switched system ẋ = Aσ(t) x(t)
0 10
1.5
2
A1 =
A2 =
0 0
−2 −0.5
√
the eigenvalues are, λ(A1 ) = 0 and λ1,2 (A2 ) = 0.5 &plusmn; i 3
Thus, both systems are unstable.
The phase portrait for two unstable systems as follow:
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
30 /Contr
38
Cont.. State-Dependent Example
Applying the state-dependent switching
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
31 /Contr
38
Cont.. State-Dependent Example
Applying the state-dependent switching

and x2 (t) = −0.25x1 (t)
1, if σ(t − ) = 2
σ(t) =

2, if σ(t − ) = 1
and x2 (t) = +0.50x1 (t)
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
31 /Contr
38
Cont.. State-Dependent Example
Applying the state-dependent switching

and x2 (t) = −0.25x1 (t)
1, if σ(t − ) = 2
σ(t) =

2, if σ(t − ) = 1
and x2 (t) = +0.50x1 (t)
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
31 /Contr
38
Stability Under Time-Dependent Switching
The concept of Dwell Time τ [DT].
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
32 /Contr
38
Stability Under Time-Dependent Switching
The concept of Dwell Time τ [DT].
Theorem 6
Consider the switched system
ẋ = Ai x ∀i ∈ P
(7)
If we assume that all the individual subsystems are asymptotically stable
(i.e when all subsystem matrices Ai are Hurwitz stable) then the
switched system is exponentially stable if and only if the dwell − time
is sufficiently large to allow each subsystem reaching the steady-state.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
32 /Contr
38
Stability Under Time-Dependent Switching
The concept of Dwell Time τ [DT].
Theorem 6
Consider the switched system
ẋ = Ai x ∀i ∈ P
(7)
If we assume that all the individual subsystems are asymptotically stable
(i.e when all subsystem matrices Ai are Hurwitz stable) then the
switched system is exponentially stable if and only if the dwell − time
is sufficiently large to allow each subsystem reaching the steady-state.
The ADT (average dwell time) [Enhancement DT].
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
32 /Contr
38
Stability Under Time-Dependent Switching
The concept of Dwell Time τ [DT].
Theorem 6
Consider the switched system
ẋ = Ai x ∀i ∈ P
(7)
If we assume that all the individual subsystems are asymptotically stable
(i.e when all subsystem matrices Ai are Hurwitz stable) then the
switched system is exponentially stable if and only if the dwell − time
is sufficiently large to allow each subsystem reaching the steady-state.
The ADT (average dwell time) [Enhancement DT].
ADT might be applied to stabilize between stable matrices and
unstable matrices.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
32 /Contr
38
Determining the Average Dwell Time
A1 , A2 are unstable and stable Hurwitz matrices,respectively.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
33 /Contr
38
Determining the Average Dwell Time
A1 , A2 are unstable and stable Hurwitz matrices,respectively.
Determine λ1 , λ2 , .. such that the eigenvalues of (A1 − λ1 I ) , (A2 + λ2 I )
are Hurwitz stable matrices.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
33 /Contr
38
Determining the Average Dwell Time
A1 , A2 are unstable and stable Hurwitz matrices,respectively.
Determine λ1 , λ2 , .. such that the eigenvalues of (A1 − λ1 I ) , (A2 + λ2 I )
are Hurwitz stable matrices.
Solve for Pi such that
(A1 − λ1 I )T P1 + P1 (A1 − λ1 I ) &lt; 0
(A2 + λ2 I )T P2 + P2 (A2 + λ2 I ) &lt; 0
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
33 /Contr
38
Determining the Average Dwell Time
A1 , A2 are unstable and stable Hurwitz matrices,respectively.
Determine λ1 , λ2 , .. such that the eigenvalues of (A1 − λ1 I ) , (A2 + λ2 I )
are Hurwitz stable matrices.
Solve for Pi such that
(A1 − λ1 I )T P1 + P1 (A1 − λ1 I ) &lt; 0
(A2 + λ2 I )T P2 + P2 (A2 + λ2 I ) &lt; 0
Calculate Dwell-Time:
M.Khudaydus
ln &micro;
2(λ∗ −λ)
,
&micro;=
λM (P1,P2)
λm (P1,P2)
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
33 /Contr
38
Determining the Average Dwell Time
A1 , A2 are unstable and stable Hurwitz matrices,respectively.
Determine λ1 , λ2 , .. such that the eigenvalues of (A1 − λ1 I ) , (A2 + λ2 I )
are Hurwitz stable matrices.
Solve for Pi such that
(A1 − λ1 I )T P1 + P1 (A1 − λ1 I ) &lt; 0
(A2 + λ2 I )T P2 + P2 (A2 + λ2 I ) &lt; 0
Calculate Dwell-Time:
The Switching Law:
Ts
Tu
≥
ln &micro;
2(λ∗ −λ)
λu −λ∗
λs −λ∗
,
&micro;=
, λ∗ ∈ (λ, λs )
λM (P1,P2)
λm (P1,P2)
, λ ∈ (0, λs )
where T s , T u denote the total activate time of Hurwitz stable subsystems to hurwitz unstable subsystems.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
33 /Contr
38
Example
A1 =
M.Khudaydus
−9 10
−20 10
and A2 =
−10 11
10 −20
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
34 /Contr
38
Example
A1 =
−9 10
−20 10
and A2 =
−10 11
10 −20
The λ(A1 ) = 1, 1 (Unstable) system and λ(A2 ) = −10, −30
(Stable) systems
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
34 /Contr
38
Example
A1 =
−9 10
−20 10
and A2 =
−10 11
10 −20
The λ(A1 ) = 1, 1 (Unstable) system and λ(A2 ) = −10, −30
(Stable) systems
The trajectories of both systems
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
34 /Contr
38
1
The result of V (x), V̇ (x) according to time t:
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
35 /Contr
38
1
The result of V (x), V̇ (x) according to time t:
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
35 /Contr
38
1
The result of V (x), V̇ (x) according to time t:
2
V̇ (x) &lt; 0 not satisfied
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
35 /Contr
38
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
36 /Contr
38
1
Evaluate &micro; =
M.Khudaydus
0.6
0.4
= 1.5
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
36 /Contr
38
1
2
Evaluate &micro; =
0.6
0.4
= 1.5
Evaluate
= 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are
Hurwitz, respectively
M.Khudaydus
λ+
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
36 /Contr
38
1
2
3
Evaluate &micro; =
0.6
0.4
= 1.5
Evaluate
= 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are
Hurwitz, respectively
λ+
let λ ∈ (0, λ− , we choose λ = 1 ⇒ λ∗ ∈ (λ, λ− ) ⇒ λ∗ = 6.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
36 /Contr
38
1
2
Evaluate &micro; =
0.6
0.4
= 1.5
Evaluate
= 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are
Hurwitz, respectively
λ+
3
let λ ∈ (0, λ− , we choose λ = 1 ⇒ λ∗ ∈ (λ, λ− ) ⇒ λ∗ = 6.
4
τd =
M.Khudaydus
ln 1.5
2(6−1)
= 0.04s.
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
36 /Contr
38
1
2
Evaluate &micro; =
0.6
0.4
= 1.5
Evaluate
= 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are
Hurwitz, respectively
λ+
3
let λ ∈ (0, λ− , we choose λ = 1 ⇒ λ∗ ∈ (λ, λ− ) ⇒ λ∗ = 6.
4
τd =
5
So, we need to activate A2 for more than 0.04.Let say 0.08.
M.Khudaydus
ln 1.5
2(6−1)
= 0.04s.
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
36 /Contr
38
1
2
Evaluate &micro; =
0.6
0.4
= 1.5
Evaluate
= 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are
Hurwitz, respectively
λ+
3
let λ ∈ (0, λ− , we choose λ = 1 ⇒ λ∗ ∈ (λ, λ− ) ⇒ λ∗ = 6.
4
τd =
5
So, we need to activate A2 for more than 0.04.Let say 0.08.
6
According to
M.Khudaydus
ln 1.5
2(6−1)
= 0.04s.
T−
T+
≥
λ+ −λ∗
λ− −λ∗
=
16
3
= 5.3 .
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
36 /Contr
38
1
2
Evaluate &micro; =
0.6
0.4
= 1.5
Evaluate
= 10, λ− = 9 such that (A1 − λI ), (A2 + λI ) are
Hurwitz, respectively
λ+
3
let λ ∈ (0, λ− , we choose λ = 1 ⇒ λ∗ ∈ (λ, λ− ) ⇒ λ∗ = 6.
4
τd =
5
So, we need to activate A2 for more than 0.04.Let say 0.08.
6
According to
7
ln 1.5
2(6−1)
T+ ≤
0.015
M.Khudaydus
T−
5.3 ,
= 0.04s.
T−
T+
≥
λ+ −λ∗
λ− −λ∗
=
16
3
= 5.3 .
need to activate the unstable system for equal or less than
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
36 /Contr
38
Successful Results based on applying ADT
The result of V (x), V̇ (x) according to time t :
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
37 /Contr
38
Successful Results based on applying ADT
The result of V (x), V̇ (x) according to time t :
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
37 /Contr
38
Successful Results based on applying ADT
The result of V (x), V̇ (x) according to time t :
V̇ (x) &lt; 0 Satisfied.
M.Khudaydus
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
37 /Contr
38
M.Khudaydus
Thank You! ( ˆ . ˆ )
Supervisor: Dr.Wajdi Kallel
Stability
Department
of SwitchedofSystems
Mathematics,
in Sense
a fulfillment
of Lyapunov’s
requirement
Theory forApril
the 8,
degree
2018 ofMScin
38 /Contr
38
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