non-linear autonomous dynamical systems

NOTES
ON STABILITY
FOR NON-LINEAR SYSTEMS
Michele MICCIO
and
Andrea CAMMAROTA
• Dept. of Industrial Engineering (Università di Salerno)
• Prodal Scarl (Fisciano)
Rev. 2.6 of May 26, 2016
Introduction
For clarity and simplicity, we concentrate
on
 stability of steady-state regimes
or equilibria
QUESTIONS
1. How many steady-state regimes
does a dynamical system have?
2. What
are
the
stability
properties of a given steadystate regime?
2
http://www.scholarpedia.org/article/Equilibria
Steady-state Regimes:
Multiplicity of Equilibrium Points
3
Qualitative concept of stability
An equilibrium may be stable or
unstable or indifferent.
For example,
the equilibrium of a pencil standing
on its tip is unstable;
the equilibrium of a picture on the
wall is (usually) stable;
the equilibrium of a ball on a flat
plane is (usually) indifferent.
Adapted from On-Line Lectures ©Alexei Sharov, Department
of Entomology, Virginia Tech, Blacksburg, VA
http://www.ento.vt.edu/~sharov/PopEcol/
4
Qualitative concept of stability
The concept of stability can be
illustrated by a cone placed on a plane
horizontal surface.
5
STABILITY
FOR INPUT-OUTPUT
DYNAMICAL SYSTEMS
6
Stability
for linear input-output systems
A necessary and sufficient condition for a
linear dynamic system to be stable is that all
the poles of the system transfer function
have negative real parts (BIBO Stability
Theorem).
An
input-output
system
is
defined
marginally stable if only certain bounded
inputs will result in a bounded output.
http://en.wikipedia.org/wiki/Marginal_stability
7
Stability:
general properties
Linear Systems
stability is an
intrinsic
property of the
system itself
and not of its
equilibrium
points
Non-Linear Systems
stability is a property of
each asymptotic regime
and not of the system
itself;
 e.g., the same dynamic
system may exhibit stable
and unstable equilibrium
points in parallel
8
STABILITY
FOR AUTONOMOUS
DYNAMICAL SYSTEMS
9
Lyapunov Stability definition
(for a point of equilibrium xs)
Consider an autonomous
dynamical system  d
nonlinear
 x (t )  f ( x (t ))
 dt
 x (t0 )  x0
A point of equilibrium xs of the above system is said to be
Lyapunov stable if, for every region U of radius  >0, there
exists a neighborhood V of radius  > 0 such that, for each
initial condition x0V, the corresponding trajectory x(t)
remains included in U for every t>t0
that is:
x(t)  x s  

x0

x(t)
xs
Conceptually, the meaning is the following:
Trajectories (e.g., perturbations from equilibrium) starting "close
enough" to xs (within a distance δ from it) remain "close enough"
forever (within a distance  from it). Note that this must be true for
any  that one may want to choose, otherwise the system is10
unstable!
Lyapunov Asymptotic Stability definition
(for a point of equilibrium xs)
Consider the same autonomous nonlinear
dynamical system
d
 x (t )  f ( x (t ))
 dt
 x (t0 )  x0
A point of equilibrium xs of the above system is said to be
asymptotically stable if it is Lyapunov stable and if a
neighborhood V of radius  > 0 exists such that, for each
initial condition x0V and for t  
x(t)  x s  0

x0

x(t)
xs
1st example
2nd example
Conceptually, the meaning is the following:
Trajectories (e.g., perturbations from equilibrium) that start "close
enough" to xs (within a distance δ from it) not only remain close
11
enough, but also eventually converge to the equilibrium.
Instability
A point of equilibrium xs is said to be
unstable if it is not Lyapunov stable
http://www.scholarpedia.org/article/Stability
12
Example:
2nd order dynamical system
 The origin is an equilibrium point
phase
portrait

retrieved from Dr. Victor M. Becerra, University of
Reading (UK)
13
Qualitative concept of stability
vs.
Lyapunov Stability
Stabilità
semplice
stable equilibrium
unstable equilibrium
Instabilità
Asymptotic Stability
Lyapunov Instability
Stabilità asintotica
indifferent or neutral
equilibrium
Lyapunov Stability
 Adapted from Economia dei Sistemi Complessi
dei proff.ri Silvio Giove e Christina Mosele
14
LINEAR
AUTONOMOUS
DYNAMICAL SYSTEMS
15
Study of
linear autonomous dynamical systems
… understanding linear systems and their solutions is a
crucial first step to understanding the more complex nonlinear
systems.
n-order, linear, autonomous dynamical system
dx
 Ax ( t )
dt
IC : x t 0   x 0
x is state:
x   x1 ,..., x n 
A is a non singular state matrix [n · n]
Equilibrium points:
dx
0 
dt
Ax  0
If A is non singular , x=0 is the only steady state.
For this part see ch. 27 of:
 D.R. Coughanowr, L.B. Koppel, Process systems analysis
and control
UniSA library 660.281 5 COU
16
Linear autonomous dynamical systems:
definitions
EIGENVALUES
A  I  0
Solutions of the eq.:
HYPERBOLICITY
A linear autonomous dynamical system is said to be
hyperbolic if all eigenvalues have nonzero real part.
A linear autonomous dynamical system is said to be nonhyperbolic if at least one eigenvalue has a real part equal to
zero .
ORBITS
Hyp.:
1) n distinct Eigenvalues and Eigenvectors,
n
x(t)   c k e  k ( t t 0 ) v k
k1
2) at least i is complex
e i (t t0 )  e Re( i )( t t0 ) cosIm( i )(t  t0 )   j sin Im( i )(t  t0 ) 

INVARIANT
A subset A of the state space X is said to be invariant
if the orbits starting at each of its points are totally
included in A.
17
Linear autonomous dynamical systems:
Stability
x(t) is bounded if the terms.
e s ( t t0 )
e Re( s )( t t0 )
do not diverge.
Therefore:
• Asymptotically Stable if Re(i)<0  i
• Unstable if at least one eigenvalue i with positive
real part exists
18
Autonomous dynamical systems:
an example in the planar case
order n = 2
x 1  f1 ( x1 , x 2 )

x 2  f 2 ( x1 , x 2 )
IC : x t 0   x 0
phase
portrait
orbit
or
trajectory
19
Linear autonomous dynamical systems:
the planar case
ì dx1
ïï dt = a11x1 (t) + a12 x 2 (t)
í
ï dx 2 = a x (t) + a x (t)
21 1
22 2
ïî dt
IC : x ( 0 ) = x 0
x   x1 , x 2 
 a11 a12 
A

a
a
 21 22 
Assumptions:
•A is invertible,
•The eigenvalues 1 and 2 are non-zero.
•v1 and v2 are the corresponding eigenvectors
•c1 and c2 are constants
The origin (0,0) is the only steady-state point
The general solution is:
x(t)  c1e 1 t v1  c 2e  2 t v2
20
Planar linear dynamical systems:
the hyperbolic case
Real
Complex and conjugate
1bis)
1)
1  0
2  0
Re(1 )  0
Re( 2 )  0
2bis)
2)
1  0
Re(1 )  0
2  0
Re( 2 )  0
3)
1  0
2  0
21
Planar linear dynamical systems:
the non-hyperbolic case
Real
4)
Complex and conjugate
8)
Re( 1 )  0
1  0
2  0
Re( 2 )  0
5)
1  0
2  0
6)
1  0
2  0
With 2
indipendent
eigenvectors
1  0
2  0
With only 1
indipendent
eigenvector
7)
22
PPlane
MATLAB and JAVA software for
simulation and stability analysis of
2nd order (planar) systems
23
http://math.rice.edu/~dfield/
Phase Portrait:
1) 1 e 2 are real, distinct and negative
 1 0 
A

0

4


Eigenvalues 1= -1 2= -4
1
v1   
0
Eigenvectors
coincident with axis versors:




0
v 2   
1
x(t )  c1e1t v1  c2e2t v2
x '=-x
y '=-4y

2
1.5
1
y
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2
-1.5
-1
-0.5
0
x
0.5
1
1.5
(NODO STABILE)
Knot or Nodal Sink
2
2.5
24
Phase Portrait:
1) 1 e 2 are real, distinct and negative
 2  1 Eigenvalues 1= -1 2= -4
A


2

3


  1
 1 

Eigenvectors (not orthogonal !) v1    v2  
  1
  2
x(t )  c1e1t v1  c2e2t v2




x '=-2x -y
y '=-2x -3y
2
1.5
invariant
1
y
0.5
0
-0.5
-1
-1.5
-2
-2
-1.5
-1
-0.5
0
x
0.5
1
(NODO STABILE)
Knot or Nodal Sink
1.5
2
25
Phase Portrait:
2) 1 e 2 are real, distinct and positive
 2  2
A

 1 3 
Eigenvalues 1=4 2=1
Eigenvectors (not orthogonal !)


 2
v2   
1
  1
v1   
1


x(t )  c1e1t v1  c2e2t v2
x '=2x -2y
y '=-x +3y
2
1.5
1
y
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2
-1.5
-1
-0.5
0
x
0.5
1
1.5
2
(NODO INSTABILE)
Unstable Knot or Nodal Source
2.5
26
Phase Portrait:
3) 1 e 2 are real, distinct and opposite
Eigenvalues 1= -1 2= 3
1 1
A

4
1


 1 
v1   
  2
Eigenvectors (not orthogonal !)



1
v2   
 2

x(t )  c1e1t v1  c2e2t v2
x '=x +y
y '=4x +y
2
1.5
1
y
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2
-1.5
-1
-0.5
0
x
0.5
(SELLA)
Saddle
1
1.5
2
2.5
27
Phase Portrait:
1bis) 1 e 2 complex with a negative real part
Eigenvalues 1= -1+3j 2= -1-3j
 2  3
A

6

4

 Eigenvectors
 1 
 1 
 v2  
v1  


1  j 
1  j 
(complex !)
x '=2x -3y
y '=6x -4y
2
1.5
1
y
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2
-1.5
-1
-0.5
0
x
0.5
1
1.5
2
2.5
(FUOCO STABILE o POZZO A SPIRALE)
Stable Focus or Spiral Sink
28
Phase Portrait:
2bis) 1 e 2 complex with a positive real part
Eigenvalues 1=1-2j 2=1+2j
 1 2
A

 2 1 Eigenvectors
 j
v1   
 1 
(complex !)
 j
v2   
1
x '=x +2y
y '=-2x +y
2
1.5
1
y
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2
-1.5
-1
-0.5
0
x
0.5
1
1.5
(FUOCO INSTABILE)
Spiral Source
2
2.5
29
Phase Portrait:
8) 1 e 2 imaginary (non hyperbolic)
 0 2
A

 2 0
Eigenvalues 1= -2j 2= 2j
 j
v2   
1
 j
v1   
 1 
Eigenvectors
(complex !)
x '=2y
y '=-2x
2
1.5
1
y
0.5
0
-0.5
-1
-1.5
-2
-2.5
-2
-1.5
-1
-0.5
0
x
0.5
1
1.5
2
2.5
(CENTRO)
CENTER
30
 NOT asymptotically stable !
Planar linear dynamical systems:
Eigenvalues
Det A  I  0
a11  
A  I  
 a21
a12 
a 22   
Det A  I   0 
 (a11   )(a22   )  a12 a21  0 
 2  (a11  a22 )  a11a22  a12 a21  0 
  Tr ( A)  Det ( A)  0
2
31
Planar linear dynamical systems:
real and distinct Eigenvalues
2  Tr ( A)  Det ( A)  0
• The eigenvalues are real and distinct when the
discriminant Δ is positive
D = [Tr(A)]2 - 4Det(A) > 0
ß
[Tr(A)]2
Det(A) <
4
• The sign of eigenvalues is from Carthesius rule:
Possible Cases
Unstable
Knot
1,2R
1>0, 2>0
Det(A)<[Tr(A)]2/4
Tr(A)>0
Det(A)>0
Stable
Knot
1,2R
1<0, 2<0
Det(A)<[Tr(A)]2/4
Tr(A)<0
Det(A)>0
Saddle
1,2R
1<0, 2>0
Det(A)<[Tr(A)]2/4
Det(A)<0
Tr(A)− 0 
there are always a variation and a
permanence in sign, whatever
Tr(A)=0 
there are two eigenvalues, real and
opposed in sign
Planar linear dynamical systems:
real and coincident Eigenvalues
  Tr ( A)  Det ( A)  0
2
• The eigenvalues are real and coincident
when the discriminant Δ is null
D = [Tr(A)] - 4Det(A) = 0
2
This can be viewed as the eq. for a parabola
passing through the origin, having its vertical axis
(Det A) coincident with the y-axis
Possible Cases
Unstable
Knot
1,2R
1 = 2 = Tr(A)/2 > 0
Det(A) = [Tr(A)]2/4
Tr(A)>0
Stable
Knot
1,2R
1 = 2 = Tr(A)/2 < 0
Det(A) = [Tr(A)]2/4
Tr(A)<0
33
Planar linear dynamical systems:
complex and coniugate Eigenvalues
  Tr ( A)  Det ( A)  0
2
• The eigenvalues are complex and coniugate when the
discriminant Δ is negative
D = [Tr(A)]2 - 4Det(A) < 0
ß
[Tr(A)]2
Det(A) >
4
Tr(A)
[Tr(A)]2 - 4Det(A)
l1 =
-j
2
2
Tr(A)
[Tr(A)]2 - 4Det(A)
l2 =
+j
2
2
Possible Cases
Unstable
Focus
1,2C-R
Re(1)=Re(2)>0
Det(A)>[Tr(A)]2/4
Tr(A)>0
Stable
Focus
1,2C-R
Re(1)=Re(2)<0
Det(A)>[Tr(A)]2/4
Tr(A)<0
Center
1,2C-R
Re(1)=Re(2)=0
Det(A)>[Tr(A)]2/4
Tr(A) = 0
34
Planar linear dynamical systems:
General Diagram
2  Tr ( A)  Det ( A)  0
det(A)
Tr(A)
[Tr (A)]2  4Det (A)  0
35
Planar linear dynamical systems:
attractors and repellors
n=2
36
NON-LINEAR
AUTONOMOUS
DYNAMICAL SYSTEMS
37
Linearization
around a steady-state point
d
 x (t )  f ( x (t ))
 dt
 x (t0 )  x0
Let xs be a steady-state point, which can be stable or unstable.
Let’s use Taylor expansion:
f(x)= f(xs)+ J(xs) (x- xs)+ O(| x- xs|2)
f(xs) = 0 as it’s a steady-state;
The linear term is the product between the Jacobian matrix J
and the vector (x – xs).
f1
 f1
(
x
)
 x s x ( x s )
2
 1
 f 2 ( x s ) f 2 ( x s )
J ( x s )   x1
x2
 

 f n
f n
(
x
)
( xs )

s

x

x
 1
2
f1

( xs )
xn

f 2

( xs )
xn


 

f n

( xs )
xn


The term O(| x- xs|2) is an infinitesimal of order of | x- xs|2 and
is negligible when x→ xs.
38
Stability
of non-linear systems:
Linearization Method
1. The non-linear system is linearized with a 1st order
Taylor expansion around xs;
2. y(t) = x(t) - xs is set (deviation variables)
3. the new linearized system is analized according to the
theory of stability for linear systems
dy/dt = J(xs) y(t)
Lyapunov Theorem
(Lyapunov's indirect method)
• If the new linearized system has all eigenvalues with a
negative real part (asymptotically stable steady-state)
then the steady-state is asymptotically stable even for
the original non linear system
• If the new linearized system has at least one eigenvalue
with a positive real part (unstable steady-state) then the
steady-state is unstable even for the original non linear
system
• Nothing can be concluded in other cases
39
Diabatic CSTR:
multiple steady-state
Van Herden stability analysis
case C)
QR
•
QE
 H 
k0 exp   E C A ;
QR C A , T    
 c 
 RT 
p 

•
Estende arbitrariamente
il bilancio stazionario
anche a condizioni
transitorie;
È efficace per individuare
i regimi instabili, ma non
quelli stabili;
 F UA 
T  F T f  UA T j
QE T    
 V Vc 
V
Vc p
p 

At steady state:
QR (CA , T) ss  QE (T) ss
40
Diabatic CSTR:
non-dimensional model
Non-dimensional variables:
x1 
c Af  c A
x2 
c Af
T  Tf
Tf

t’=t/tR
Non-dimensional parameters:

B
E
RT f

 H  cA  
 c pT f
Model:
Da   k0e tR 
V
F  tR
 hAt R 

 V cp 
 

 x
dx1
  x1  Da 1  x1  exp  2
dt
 1  x2








 x
dx2
  x2  B  Da 1  x1  exp  2
dt
 1  x2




    x2  x2 c 


41
Diabatic CSTR:
multiple steady-state
(Da=0.04; beta=1.5)
phase portrait
x ' = - x + Da (1 - x) exp(y/(1 + y/gam))
y ' = - y + B Da (1 - x) exp(y/(1 + y/gam)) - be (y - yref)
Da = 0.031
gam = 22.0
yref = 0.01
B = 22.0
be = 1.5
10
9
8
temperatura
adimensionale
7
6
5
4
3
2
1
0
0
0.1
0.2
0.3
0.4
0.5
conversione
0.6
0.7
0.8
0.9
1
42
Diabatic CSTR:
stable steady-states
low conversion
x ' = - x + Da (1 - x) exp(y/(1 + y/gam))
y ' = - y + B Da (1 - x) exp(y/(1 + y/gam)) - be (y - yref)
Da = 0.031
gam = 22.0
yref = 0.01
Stable Knot
B = 22.0
be = 1.5
Coordinates (0.043,0.388)
0.5
temperatura
adimensionale
0.45
Eigenvalues -1.14, -1.48
0.4
0.35
Eigenvectors
0.3
(0.400, -0.916)
0.25
0.035
0.04
0.045
conversione
0.05
0.055
(-0.09, 0.995)
high conversion
x ' = - x + Da (1 - x) exp(y/(1 + y/gam))
y ' = - y + B Da (1 - x) exp(y/(1 + y/gam)) - be (y - yref)
Da = 0.031
gam = 22.0
yref = 0.01
B = 22.0
be = 1.5
Stable Focus
Coordinates (0.921, 8.11)
8.5
Eigenvalues -2.14+4.01j
8.4
temperatura
adimensionale
8.3
-2.14-4.01j
8.2
8.1
8
Eigenvectors
7.9
7.8
(0.041+0.016j, -0.999)
7.7
0.905
0.91
0.915
0.92
0.925
conversione
0.93
0.935
0.94
0.041-0.016j, -0.999)
43
Diabatic CSTR:
unstable steady-state
phase portrait
x ' = - x + Da (1 - x) exp(y/(1 + y/gam))
y ' = - y + B Da (1 - x) exp(y/(1 + y/gam)) - be (y - yref)
Da = 0.031
gam = 22.0
yref = 0.01
B = 22.0
be = 1.5
4.2
4
temperatura
adimensionale
3.8
3.6
3.4
3.2
3
0.36
0.37
0.38
0.39
0.4
0.41
0.42
conversione
0.43
0.44
0.45
0.46
Saddle
Coordinates (0.412,3.64)
Eigenvalues -0.751, 3.23
Corresponding Eigenvectors: (-0.304, -0.952)
44
(-0.061, 0.998)
Diabatic CSTR:
dynamical (periodic) regime
(Da=0.07; beta=2.3)
Stable
limit cycle
phase portrait
x ' = - x + Da (1 - x) exp(y/(1 + y/gam))
y ' = - y + B Da (1 - x) exp(y/(1 + y/gam)) - be (y - yref)
Da = 0.07
gam = 22.0
yref = 0.01
B = 22.0
be = 2.3
14
12
temperatura
adimensionale
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
conversione
0.6
0.7
0.8
0.9
1
unstable
steady-state
45
Diabatic CSTR:
limit cycle
x ' = - x + Da (1 - x) exp(y/(1 + y/gam))
y ' = - y + B Da (1 - x) exp(y/(1 + y/gam)) - be (y - yref)
Da = 0.07
gam = 22.0
yref = 0.01
B = 22.0
be = 2.3
9
8
temperatura
adimensionale
7
6
5
phase portrait
4
0.7
0.75
0.8
0.85
conversione
0.9
0.95
1
time trajectories (corresponding)
10
0.95
8
0.90
6
0.85
4
0.80
2


1.00
0.75
0
20
40
60
0
80
46
Diabatic CSTR
47
Summary
1. Linearizazion Method
(Lyapunov's indirect method)
2. Lyapunov's 2nd method for stability (1892)
• is more general
• makes use of a Lyapunov function V(x) which has an analogy to
the potential function of classical dynamics.
3. Direct Simulation Methods
48
Concept of bifurcation
A nonlinear dynamic system differs from a linear dynamic
system in that its qualitative properties can change under
small perturbations of the model parameters.
These properties include the number of equilibria, stability
of the equilibria, existence of limit cycles, multiple modes of
behavior, and chaos.
from Zhang, Y. and Henson, M. A. (2001), “Bifurcation analysis of
continuous biochemical reactor models”, Biotechnology Progress,
17, p. 647-660
In generale, si dice che un parametro attraversa un valore di
biforcazione quando determina il passaggio fra due situazioni
dinamiche qualitativamente diverse, dovuto ad esempio alla
creazione o scomparsa di punti di equilibrio o altri tipi di
regimi asintotici, oppure cambiamenti di stabilità.
Gian-Italo Bischi, Rosa Carini, Laura Gardini e Paolo Tenti
pubblicato sul n. 47 di Lettera matematica pristem
49