Stability of Nonlinear Systems
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Transcript of Stability of Nonlinear Systems
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Stability of Nonlinear Systems
By
Lyapunov Stability
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Introduction to Stability
1. The concept of stability2. Critical points
3. Linear stability analysis
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The Concept of Stability
Imprecise definition
Consider a nonlinear system with the origin as a steady-state point:
Does the system return to the origin if perturbed away from theorigin? If so, the system is stable. Otherwise, the system is unstable.
0yfyfy
!! )()(dt
d
0yyyfy
!!!gp
)(lim)0()( tdt
d
t
H
I
y(0)y(t) 0
y1
y2
Precise definition
Stability: produce a bound I on y(0)
such that y(t) remains within a givenbound H
Asymptotic stability: stable & y(t)
converges to the origin
Commonly known as Lyapunov
stability3
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Critical Points of a Linear System
Two-dimensional system
Divide equations
Critical point
Point where dy2/dy1 becomes undetermined Only the origin for a homogeneous linear system
Five types of critical points depending on the geometric shape of
trajectories near the origin and eigenvalues ofA matrix
222121
2
212111
1
yayadt
dy
yayadt
dy
dt
d
!
!! Ax
y
212111
222121
1
2
1
2
yaya
yaya
dtdy
dtdy
dy
dy
!!
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Types of Critical Points
Proper node
Two identical realeigenvalues
Improper node Two different real
eigenvalues
Saddle point Two real eigenvalues
with different signs
Center
Two imaginaryeigenvalues
Spiral point Two complex
eigenvalues
Degeneratenode
No eigenvectorbasis exists
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Linear Stability Analysis
General solution formfor distinct eigenvalues
Procedure
Compute theeigenvalues ofA
The system isasymptotically stable ifand only if Re(Pi) < 0 fori= 1, 2, , n
The origin is unstable ifRe(Pi) > 0 for any i
Stability allows zeroeigenvalues
Imaginary
RealStable
Region
Unstable
Region
Left-Half Plane Right-Half Plane
0
tn
n
tt nececectPPP )()2(
2
)1(
1
21)( xxxy ! .
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Nonlinear Systems
Steady-state points
Nonlinear models can have multiple steady states
Stability must be determined for each steady state
Consider origin as a generic steady-state point
Nonlinear model linearization about origin
00gygy
ygyyfyy
yyy
0yfyf
y
!d!d
d|d!!d
!
!!
)()(
)()('
)()(
dt
ddt
d
dt
ddt
d
yAy
yfy
d!d
!dt
d
dt
d)(
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Lyapunov Stability
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x0
p1(0)
t - time flow
p2(t)
p1(t)
p2(0)
x(t)
Definition of Lyapunov Exponents
Given a continuous dynamical system in an n-dimensional
phase space, we monitor the long-term evolution of an
infinitesimal n-sphere of initial conditions.
The sphere will become an n-ellipsoid due to the locally
deforming nature of the flow.
The i-th one-dimensional Lyapunov exponent is then defined
as following:
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On more formal level
The Multiplicative Ergodic Theorem of Oseledec states thatThe Multiplicative Ergodic Theorem of Oseledec states that
this limit exists for almost all points xthis limit exists for almost all points x00 and almost alland almost all
directions of infinitesimal displacementdirections of infinitesimal displacement in the same basin ofin the same basin of
attraction.attraction. 10
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Order: 1> 2 >> n
The linear extent of the ellipsoid grows as 2
1t
The area defined by the first 2 principle axes grows
as 2(1+2)t
The volume defined by the first 3 principle axes
grows as 2(1+2+3)t and so on The sum of the firstjexponents is defined by the
long-term exponential growth rate of a j-volumeelement.
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Signs of the Lyapunov exponents
Any continuous time-dependent DS without afixed point will have u1 zero exponents.
The sum of the Lyapunov exponents must be
negative in dissipative DS at least onenegative Lyapunov exponent.
A positive Lyapunov exponent reflects adirection ofstretching andfolding and
therefore determines chaos in the system.
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The signs of the Lyapunov exponents provide a
qualitative picture of a systems dynamics
1D maps: ! 1=:
=0 a marginally stable orbit;
0 chaos.
3D continuous dissipative DS: (1,2,3)
(+,0,-) a strange attractor;
(0,0,-) a two-torus; (0,-,-) a limit cycle;
(-,-,-) a fixed point.
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The sign of the Lyapunov Exponent
P0- the system is chaotic andunstable. Nearby points willdiverge irrespective of how close
they are. 14
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Computation of Lyapunov Exponents
Obtaining the Lyapunov exponents from a system
with known differential equations is no real problem
and was dealt with by Wolf. In most real world situations we do not know the
differential equations and so we must calculate the
exponents from a time series of experimental data.
Extracting exponents from a time series is a complexproblem and requires care in its application and the
interpretation of its results.
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Calculation of Lyapunov spectra from
ODE (Wolf et al.) A fiducial trajectory (the center of the sphere) is
defined by the action of nonlinear equations ofmotions on some initial condition.
Trajectories of points on the surface of the sphereare defined by the action of linearized equations onpoints infinitesimally separated from the fiducialtrajectory.
Thus the principle axis are defined by the evolutionvia linearized equations of an initially orthonormalvector frame {e1,e2,,en} attached to the fiducialtrajectory
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Problems in implementing:
Principal axis diverge in magnitude.
In a chaotic system each vector tends to fall
along the local direction of most rapid growth.
(Due to the finite precision of computer calculations, thecollapse toward a common direction causes the tangent space
orientation of all axis vectors to become indistinguishable.)
Solution:Gram-Schmidt reorthonormalization (GSR)
procedure!
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GSR never affects the direction of the first vector, sov
1
tends to seek out the direction in tangent space
which is most rapidly growing, |v1|~ 21t;
v2 has its component along v1 removed and then isnormalized, so v2 is not free to seek for direction,however
{v1,v2} span the same 2D subspace as {v1,v2}, thusthis space continually seeks out the 2D subspace that
is most rapidly growing|S(v1,v2)|~2(1+2)t
|S(v1,v2,vk)|~2(1+2++k)t
k-volume So monitoring k-volume growth we can find first k
Lyapunov exponents.
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Lyapunov spectrum for experimental data
(Wolf et al.)
Experimental data usually consist of discrete
measurements of a single observable.
Need to reconstruct phase space with delay
coordinates and to obtain from such a time series an
attractor whose Lyapunov spectrum is identical tothat of the original one.
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Procedure for P1
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Procedure for P1+P2
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Lyapunov Simple Example
A 2D mapf:R2 R2.
(from Mathworld)
Define a Lyapunov function.
The derivative is negative sothe origin is stable.
212
21
2)(
)(
xxxf
xxf
!
!
)()()(* xfxVxV !
2/)()( 22
2
1xxxV !
2
2
*
21221
*
2)(
)2()(
xxV
xxxxxxV
!
!
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Matlab Tools for Stability Analysis
Matlab provides several functions forlinear stability analysis of nonlinearsystems
fsolve finds steady-state point fornonlinear ODE system
linmod linearizes nonlinear ODE systemabout given steady state to generate
linear ODE system
Eigenvalue computes eigenvalues oflinear ODE system
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Controllability
Observability
)()(
stateseduncontroll%
),(
corankAlengthunco
BActrbco
!
!
)()(statesunobserved
),(
obranklengthunob
Cobsvob
!
!
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Example
? A
-
!
-
-
-
!
-
2
1
1
2
1
2
1
04761.09558.1
0
1
00415.50
0415.501847.7
x
xy
ux
x
x
x
Model equations
Controllability
2)(;50.04150
7.1847-1.0),(
];0;1[
0];50.0415;0415.501847.7[
!
-
!!
!
!
corankctrbco
2)(;97.8712-16.4343-
0.0476-1.9558),(
0.04761];-1.9558[
0];50.0415;0415.501847.7[
!
-
!!
!
!
obrankCobsvob
CObservability
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Example
? A
-
!
-
-
-
!
-
2
1
1
2
1
2
1
04761.09558.1
0
1
00415.50
0415.501847.7
x
xy
ux
x
x
x
Model equations
Analysis
i
i
Aeig
A
49.91243.5924-49
.912
43.59
24-
)(
0];50.0415;0415.501847.7[
!
asymptotically stable because all the eigenvalues
of the state matrix A have negative real parts
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Example
Continuous Biochemical Reactor
Fresh Media Feed
(substrates)
Exit Gas Flow
Agitator
Exit Liquid Flow
(cells & products)
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Cell Growth Modeling
Specific growth rate
Yield coefficients
Biomass/substrate: YX/S = -(X/(S
Product/substrate: YP/S = -(P/(S
Product/biomass: YP/X =(P/(X
Assumed to be constant
Substrate limited growth
S = concentration of rate limiting substrate
Ks = saturation constant
Qm = maximum specific growth rate (achieved when S >> Ks)
(g/L)ionconcentratbiomass1
!! Xdt
dX
XQ
SK
S
SS
m
!
Q
Q )(
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Continuous Bioreactor ModelAssumptions
Sterile feed
Constant volume
Perfect mixing
Constant temperature &pH
Single rate limitingnutrient
Constant yields
Negligible cell death
Product formation rates Empirically related to specific growth rate
Growth associated products: q =YP/XQ
Nongrowth associated products: q =F
Mixed growth associated products: q =YP/XQF29
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Mass Balance Equations
Cell mass
VR = reactor volume
F= volumetric flow rate
D =F/VR = dilution rate Product
Substrate
S0 = feed concentration of rate limiting substrate
XDXdt
dXXVFX
dt
dXV
RRQQ !!
qXDPdt
dPqXVFP
dt
dPV RR !!
XY
SSDdt
dSXV
YFSFS
dt
dSV
SX
R
SX
R QQ/
0
/
0
1)(
1!!
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Steady-State Solutions Simplified model equations
Steady-state equations
Two steady-state points
),()(1
)(
)(),()(
2
/
0
1
SXfXSY
SSDdt
dS
SK
SSSXfXSDX
dt
dX
SX
S
m
!!
!!!
Q
QQQ
0)(1
)(
)(0)(
/
0 !
!!
XSY
SSD
S
SSXSXD
SX
S
Q
QQQ
0:Washout
)()(:Trivial-Non
0
0/
!!
!
!!
XSS
SSYXD
DSDS
SX
S
QQ
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Model Linearization Biomass concentration equation
Substrate concentration equation
Linear model structure:
? A
S
SK
SX
SK
XXDS
SSS
fXX
X
fSXf
dt
Xd
S
m
S
m
SXSX
d
-
d!
-
x
x
-
x
x$
d
2
,
1
,
1
1
zero
),(
QQQ
SD
S
SX
S
X
YX
S
S
Y
SSS
fXX
X
fSXf
dt
Sd
S
S
SXS
SX
SXSX
d
-
d
!
-
x
x
-
x
x$
d
2
//
,
2
,
2
2
11
zero
),(
QQQ
SaXadt
Sd
SaXadt
Xd
dd!d
dd!d
2221
1211
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Non-Trivial Steady State Parameter values
KS = 1.2 g/L, Qm=0.48 h-1, Y
X/S =0.4 g/g
D =0.15 h-1, S0 = 20 g/L
Steady-state concentrations
Linear model coefficients (units h-1)
529.3
1375.0
1
472.10
2
/
22
/
21
21211
!
!!
!
!
!!
D
S
SX
S
X
Ya
S
S
Ya
S
SX
S
Xaa
S
S
SXS
SX
S
S
QQQ
QQ
g/78.7)(g/545.0 0/ !!!! SSYXD
DK
S SXm
S
Q
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Stability Analysis Matrix representation
Eigenvalues (units h-1)
Conclusion The system is asymptotically stable
Axxdt
dx
S
Xx !
-
!
-
dd
!529.3375.0
472.10
365.3164.0529.3375.0
472.111
!!
! PP
P
PPIA
34