stability of nonlinear systems

8/7/2019 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

stability of nonlinear systems

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