nonlinear Dynamic course

8/12/2019 nonlinear Dynamic course

1/12

Dynamical Systems 2013

Class 1

Department of Electrical EngineeringEindhoven University of Technology

Siep Weiland

Class 1 (TUE) Dynamical Systems 2013 Siep W eila nd 1 / 46

Part I

Organization of the course


Outline of Part I

1 Organization of the course

Material and contactsSchedule and topicsExams and gradings


Organization Material and contacts

website and course material

Website:

http://w3.ele.tue.nl/en/cs/education/courses/dynamical systems/

Material:

Book

Nonlinear Dynamics and Chaos:with applications to Physics, Biology, Chemistry andEngineering

Author: Steven H. Strogatz

Publisher Perseus Books

new USD 175, used USD 43

Slides and handouts

http://w3.ele.tue.nl/en/cs/education/courses/dynamical_systems/http://w3.ele.tue.nl/en/cs/education/courses/dynamical_systems/


2/12

Organization Material and contacts

Contact information

Contact information

Siep WeilandDepartment of Electrical EngineeringPotentiaal 4.34;

Phone: +31.40.247.5979Email:[email protected]

StudentassistantHandumant ShekhawatDepartment of Electrical EngineeringPotentiaal 4.34;Phone: +31.40.247.xxxxEmail:[email protected]


Organization Schedule and topics

Purpose of this course

Aim:After this course you can

distinguish among particular classes of NL systems analyze stability, periodicity, chaotic behavior, bifurcations, hysteresis,

attractors, repellers, limit cycles efficiently simulate and analyse NL evolution laws

appreciate this research area Lectures

What ? When ? Where ?

Classes Thursdays 13.45-15.30 Auditorium 10Instructions Mondays 15.45-17.30 Auditorium 13

but subject to changes.. .

Questions:Ask questions !! (Why? How? What? So what? When? . . . )



Schedule and topics

Weekly schedule 2013.

Week Monday(Aud 13) Thursday(Aud 10)

1 - September 52 September 9 September 12

3 - September 194 September 23 September 265 - October 36 October 7 October 107 - October 178 October 21 October 24



Schedule and topics

Week 1 Organization an phase flows

Week 2 Potential functions, Matlab implementation

Week 3 Bifurcations and their applications

Week 4 Two dimensional flows and stability Week 5 Dissipativity and Hamiltonian systems

Week 6 Limit cycles

Week 7 Poincare-Bendixson and oscillators

Week 8 Chaotic systems

http://dyns03.pdf/http://dyns04.pdf/http://dyns06.pdf/http://dyns06.pdf/http://dyns07.pdf/http://dyns08.pdf/http://dyns09.pdf/http://dyns01.pdf/http://dyns01.pdf/http://dyns02.pdf/http://dyns02.pdf/http://dyns03.pdf/http://dyns03.pdf/http://dyns04.pdf/http://dyns04.pdf/http://dyns06.pdf/http://dyns07.pdf/http://dyns08.pdf/http://dyns09.pdf/http://dyns01.pdf/http://dyns02.pdf/http://dyns03.pdf/http://dyns04.pdf/http://dyns06.pdf/http://dyns07.pdf/http://dyns08.pdf/http://dyns09.pdf/http://dyns01.pdf/http://dyns02.pdf/http://dyns03.pdf/http://dyns04.pdf/http://dyns06.pdf/http://dyns07.pdf/http://dyns08.pdf/http://dyns09.pdf/http://dyns09.pdf/http://dyns08.pdf/http://dyns07.pdf/http://dyns06.pdf/http://dyns04.pdf/http://dyns03.pdf/http://dyns02.pdf/http://dyns01.pdf/


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Organization Exams and gradings

exams and gradings

ExercisesWe will makeexercises in and outside class hoursMay need to bring your notebook (will be announced)Solutions are alwaysposted on website.

ExamsRegular exam EProject P (take home style)Details follow

Final grade

G= round (E+ (1 )P) with = 12 .


Part II

Todays lecture


Outline of Part II

2 Motivating examples

3 Linear systemsLinearization

4 Nonlinear systems

General structureFixed points

5 Stability of fixed pointsStable fixed pointsUnstable fixed pointsVerifying stability of fixed points

6 Summary


Motivating examples

Outline



4 Nonlinear systems



6 Summary



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Motivating examples

Example 1: dynamics in air flows

Integrated Roof Wind Energy System (IRWES project)

modeling of wind flow inside andaround funnel

design of control system to regulatelouvers

- prevent turbulence- constant wind speed


Motivating examples

Example 2: phase-lock loops (PLLs)

Classical PLL control loop configuration

VCOLFPD

1/N

vovi

PD: phase detector

LF: loop filter

VCO: voltage controlledoscillator

1/N: Divider

Used in many, many applications for

carrier synchronization

carrier recovery

frequency division and multiplication

demodulation schemes.

Aim: lock frequency of output voltage vo to frequency of input voltage viClass 1 (TUE) Dynamical Systems 2013 Siep W eila nd 14 / 46

Motivating examples

Example 3: Laser beams

Monochromatic, coherent anddirectional light produced viastimulated photon emission(1958)

Application of lasers in

CD/DVD players eye surgery

optical communication

welding, cutting, blasting

concerts

dental drills

. . .


Motivating examples

Example 4: Meteorology

Atmospheric turbulenceHurricane path prediction

Tropical storm TORAJI path forecast, September 5, 2013

Turbulence and cyclone path predictions are difficult for good reasons



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Motivating examples

Example 5: Unpredictable circuit behavior

A very simple electronic circuit

2 nonlinear diode characteristics

Its voltage behavior

2. 5 2 1. 5 1 0. 5 0 0.5 1 1.5 2 2.50.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4


Motivating examples

Example 6: Electrical power networks

Main trends

Liberalizationof power market From monopolistic to competitive

market Increase of complexity

Increase of distributed and renewablepower generation

wind turbines, photovoltaic cells,.. . contribute to power generation but

not to stabilization changes of transmission structures

Aim:Stable operation of power net


Motivating examples

Example 7: computational fluid dynamics

Control variables:

temperature

velocity

Constraints

maximum temp.

fuel constraints emission constraints

temp. gradients


Linear systems

Outline



4 Nonlinear systems



6 Summary



6/12

Linear systems Linearization

linear systems

So far in your E curriculum:

All systems, physical components, models were assumed to haveidealized linear dynamics

You have seen different formats: State space models

x= Ax+Bu, y= Cx+ Du

Transfer function models

Y(s) = H(s)U(s)

Models of differential equations

my+ by+ ky= u

What means linearity precisely and how realistic is this property ??



linear vs. nonlinear systems

Example: Pendulum

Fgravity

Model of pendulum of length L

+g

Lsin() = u

Linear ?? Nonlinear ??




Example: Magnetic levitation

FgravFmag

z

v: voltage actuator

i: current coil

z: vertical position

Model:

Md2z

dt2 =Mg k

i2

z2

Ldi

dt

+ Ri= v

with parameters

Mmass of ball

ggravitation constant

kmagnetisation constant

Lcoil inductance

Rcoil resistance

Linear ?? Nonlinear??Class 1 (TUE)

Dynamical Systems 2013 Siep W eila nd 23 / 46



Definition

By adynamical systemwe mean any collection Bof functions w: T Wdefined on atime set T Rand producing values in asignal space W.

A dynamical system is linear(over R) if this collection is a linear space:ifw1, w2 B then also 1w1+2w2 B for any 1, 2 R.

Example: Pendulum model: Bis solution set of diff. eqn.Take solutions (1, u1) and (2, u2) and set (, u) = (1+2, u1+u2).Then

1+ gL

sin(1) = u12+

gL

sin(2) = u2

+

g

Lsin() = u

So the pendulum model is not linear for this reason.



7/12


linearization and state representations

Withx1 = and x2 = this gives:

nonlinear model in state space form:

x1x2

=

x2

gL

sin(x1) +u

=

f1(x1, x2, u)

f2(x1, x2, u)

linearized model in differential form:

Around = 0 gives sin() so that

+g

L = u

linearized model in state space form:

x1x2

=

0 10 g

L

A

x1x2

+

01

B

u



linearization -definitions from earlier courses

Linear systems often obtained from linearizationof nonlinear system

x= f(x, u), y= g(x, u)

Set

x(t) = x0+(t), u(t) = u0+(t), y(t) = y0+(t)

with (x0, u0, y0) alinearization pointand (, , ) aperturbationofstate, input and output.

Taylor expansionoff andg around (x0, u0, y0) yields:

f(x, u) = f(x0, u0) +f

x(x0, u0)[x x0] +

f

u(x0, u0)[u u0] +. . .

g(x, u) = g(x0, u0) +g

x(x0, u0)[x x0] +

g

u(x0, u0)[u u0] +. . .

where. . . stands for higher order terms [x x0]2, [u u0]

2 etc.



linearization -definitions

Assume that (x0, u0, y0) is afixed point, that is

f(x0, u0) = 0 and y0 = g(x0, u0)

Since x= , = x x0, = u u0 and = y y0, we have

=f

x(x0, u0)+

f

u(x0, u0)+. . .

=g

x(x0, u0)+

g

u(x0, u0)+. . .

Ignoring the higher order terms yields a model of the form

= A+B, = C+D

where

A=f

x(x0, u0), B=

f

u(x0, u0), C =

g

x(x0, u0), D=

g

u(x0, u0)



linearization -definitions

Definition

The model = A+B, = C+D defined on theprevious frameisthe linearization of the nonlinear model x= f(x, u), y= g(x, u) aroundthe fixed point (x0, u0, y0).

It represents an approximationof the dynamic behavior of thenonlinear model for small perturbations around the linearization point(x0, u0, y0). So, it haslocal validity.

Equivalently represented by its transfer function

H(s) = C(Is A)1B+ D

its frequency response H(i), or its impulse response

h(t) = Cexp(At)B+ D(t), etc.



8/12


linearization -example

Example

Linearize the nonlinear model

x= f(x, u) = x2 +usin(x) +u2x2 1

y= g(x, u) = x2 + sin(u) exp(x)

around point (x0, u0, y0) = (1, 0,1).

Solution:

Compute partial derivatives off and g:

f

x = 2x+ ucos(x) + 2xu2;

f

u= sin(x) + 2ux2

g

x = 2x+ sin(u)exp(x);

g

u= cos(u)exp(x)



linearization -example

Evaluate these at fixed point (x0, u0, y0) = (1, 0,1):

f

x(1, 0) = 2;

f

u(1, 0) = sin(1);

g

x(1, 0) = 2;

g

u(1, 0) = exp(1).

Hence,

A=2, B= sin(1), C= 2, D= exp(1)

yields the linearizedmodel

= 2+ sin(1), = 2+ exp(1)

Equivalently, the transfer function

H(s) = 2(s 2)1 sin(1) + exp(1)

with pole in 2 and zero in 2 + 2sin(1)/ exp(1).



linearization -pros and cons

Why are linearizations so important?

Linear models have local validity(only valid for small perturbations)

We can easily analyse linear models(freq. responses, stability, interconnections, robustness)

Allow many equivalent representations

(state space,transfer functions, differential equations, convolutions)

Extremely suitable for control system design

But:

We ignore global dynamics

We ignore phenomena beyond small perturbations(periodicity, attractors, chaos, bifurcations)

Qualitatitive properties of nonlinear dynamics not (always) capturedin linearized models


Nonlinear systems

Outline



4 Nonlinear systems



6 Summary



9/12

Nonlinear systems General structure

general structure of nonlinear systems

We consider the general form

x= f(x, u), y= g(x, u)

where

x(t) Rn, u(t) Rm, y(t) Rp

Important special cases:

homogeneousor autonomous system: no input u.

one-dimensional flows: case n = 1.Hence, state is one dimensional vector.

linear system: both f and g linear in x and u.



some questions

x= f(x, u), y= g(x, u)

What do we mean by a solution ??

Do solutions x(t) exist for any input, init. condition and time ??

Can we compute them ??



example

Determine solutions of the autonomous system

x= sin(x)

Solution by separation of variables:

dt=

dx

sin x

Integrate:

t+ C= log |1 + cos x

sin x |

Hence, ifx(0) =x0 then C = log |1+cosx0

sin x0| so that

t= log |[1 + cos(x0)] sin(x)

sin(x0)[1 + cos(x)]|



example (ctd)

Withx0 = /3 we can determine x(t) for t= 12 by solving

12 = log |[1 + cos(/3)]sin(x)

sin(/3)[1 + cos(x)]|

forx.This is no easy task!!

Questions:

is it solvable at all?

if it is, is solution unique?

what happens with x(t) as t ?



10/12

Nonlinear systems Fixed points

fixed points -definition

Definition

Apointx is afixed pointof the homogeneous flow x= f(x) iff(x) = 0.

x is fixed point means that constant x(t) = x, t Ris solution of

x= f(x) with initial condition x(0) =x.

Fixed pointsare also called equilibrium points,constant solutions,working points,steady solutions,stagnation points.

Example

x= sin(x) has x =k with k Zas its fixed points.


Stability of fixed points

Outline



4 Nonlinear systems



6 Summary


Stability of fixed points Stable fixed points

stability of fixed points

Homogeneous flow x= sin(x) flows

to therightif sin(x)> 0 (velocity ispositive)

to theleftif sin(x)< 0 (velocity isnegative)

6 4 2 0 2 4 61.5

1

0.5

0

0.5

1

1.5

Fixed points at x =k,

k odd: x isstablefixed point.

keven: x isunstablefixed point.


Stability of fixed points Stable fixed points

stability of fixed points

Definition

Afixed point x isstableif any solution x(t) stays near x for all t 0 ifthe initial condition x0 starts near enough to x

. Precisely, if for all >0there exist >0 such that for all initial conditionx0 with |x0 x| < wehavethat|x(t) x| for all t 0.

Important to note that

stability is a local propertyof a fixed point.

may depend on , that is some initial conditions should be chosencloser to x than others.

The definition does not say that x(t) x as t .

x is also calledLyapunov stable



11/12

Stability of fixed points Unstable fixed points

instability of fixed points

Definition

A fixed point x isunstableif it is not stable.

Example

Consider x= x2 1. Fixed points are x = 1 and x = 1. Phase

diagram tells us that x = 1 is unstable, x = 1 is stable.

Stable or unstable??



solutions of x= sin(x)

Flow patterns x(t) for 0 t 10 of x= sin(x) for various initialconditions:

0 1 2 3 4 5 6 7 8 9 1 010

8

6

4

2

0

2

4

6

8

10

t

solution

x

solutions of xdot=sin(x)

Note stable and unstable fixed points on vertical axis!



some refinements on stability definitions

Definition

A fixed point x is said to be

attractive if there exist >0 such that limt |x(t) x| = 0

whenever the initial condition x0 satisfies |x0 x

| . asymptotically stable if it is both stable and attractive.

Remark: there exist examples of stable fixed points that are not attractive,and examples of attractive fixed points that are not stable.


Stability of fixed points Verifying stability

how to verify stability?

Theorem

If f is differentiable at a fixed point x of the flowx= f(x), then x is

stableif f(x)< 0.

unstableif f(x)> 0.

So stability can be inferred from sign off(x). No statement on casewhere f(x) = 0.For example, verify stability of fixed points of x= x3 or x= x3.



12/12

Summary

Outline



4 Nonlinear systemsGeneral structureFixed points


6 Summary


Summary

summary

Linear systems admit representations in state space, transfer function,differential equation and convolution format.

Nonlinear systems only allow representations in differential and statespace format. No transfer functions!!

Focused on autonomous (no inputs) and one-dimensional (state has

dimension 1) nonlinear systems. We defined fixed points of nonlinear dynamical systems

Phase diagrams are helpful to decide about stability of fixed points

Introduced precise definitions of stability

Can verify stability of fixed points through sign off(x).

to next class

http://dynsv2.pdf/http://dynsv2.pdf/

nonlinear Dynamic course

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