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Dynamical Systems 2010

Class 1

Department of Electrical EngineeringEindhoven University of Technology

Siep Weiland

Class 1 (TUE) Dynamical Systems 2010 Siep Weiland 1 / 42

Part I

Organization of the course


Outline of Part I

1 Organization of the courseMaterialScheduleExams and gradingsPurpose


Organization Material

website and course material

Website:

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

Material:

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

Steven H. Strogatz,about EURO 50.

Slides and handoutsClass 1 (TUE) Dynamical Systems 2010 Siep Weiland 4 / 42

Organization Schedule

schedule course 2010

1 August 30: Organization; phase flows; stability

2 September 3: Existence and uniqueness; potential functions; Matlab

3 September 6: Bifurcations

4 September 13: Applications of bifurcations

5 September 17: Two dimensional flows

6 September 20: Stability, dissipativity and Lyapunov

7 September 27: Hamiltonian systems

8 October 1: Limit cycles

9 October 4: Poincare-Bendixson and oscillators

10 October 11: Discrete time evolutions

11 October 15: Chaos: strange attractors, sensitivity

12 October 18: Chaos: demonstrations and applications


Organization Exams and gradings

exams and gradings

ExercisesWe will make exercises in and outside class hoursMay need to bring your notebook (will be announced)Solutions posted on website.

ExamsYour choice of two projects (take home style)Details follow

GradingDetermined by grade of take home exam, possibly averaged withsome exercises that need to be handed in.


Organization Purpose

purpose of this course

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 NL evolution laws appreciate this research area

Topics:

existence and uniqueness of solutions of (nonlinear) differentialequations.

periodic, quasi-periodic and chaotic systems. fixed points, limit sets, periodic orbits, Lyapunov functions and

stability, Bendixson theorem, Poincare maps, bifurcations.

Chaotic attractors, Lyapunov exponents, iteration of functions andperiodic points.


Part II

Todays lecture


Outline of Part II

2 Motivating examples

3 Linear systemsLinearization

4 Nonlinear systemsGeneral structureFixed points

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

6 Summary


Motivating examples

Outline





6 Summary


Motivating examples

Example 1: computational fluid dynamics

Control variables:

temperature velocity

Constraints

maximum temp. fuel constraints emission constraints temp. gradients


Motivating examples

Example 2: phase-lock loops (PLLs)

Classical PLL control loop configuration

VCOLFPD

1/N

- - -vovi

6

-

PD: phase detector LF: loop filter VCO: voltage controlled

oscillator

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 2010 Siep Weiland 12 / 42

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 turbulence Tropical cyclone path prediction

Tropical storm Gustav path forecast, Thursday August 28, 2008

Turbulence and cyclone path predictions are difficult for good reasons


Motivating examples

Example 5: Unpredictable circuit behavior

A very simple electronic circuit

1 nonlinear diode characteristic

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

Liberalization of 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


Linear systems

Outline





6 Summary


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 dAAAAAh?

Fgravity

Model of pendulum of length L

+g

Lsin() = u

Linear ?? Nonlinear ??



linear vs. nonlinear systems

Definition

By a dynamical system we mean any collection B of functions w : TWdefined on a time set T R and producing values in a signal space W.A dynamical system is linear (over R) if this collection is a linear space:if w1,w2 B then also 1w1 + 2w2 B for any 1, 2 R.

Example: Pendulum modelLet (1, u1) and (2, u2) be two solutions and set = 1 + 2. Then

1 +gL sin(1) = u1

2 +gL sin(2) = u2

}6 + g

Lsin() = u1 + u2

So the pendulum model is not linear for this reason.



linearization and state representations

With x1 = 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 gL

)

A

(x1x2

)+

(01

)B

u



linearization -definitions from earlier courses

Linear systems often obtained from linearization of 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) a linearization point and (, , ) a perturbation ofstate, input and output.

Taylor expansion of f and g 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.Class 1 (TUE) Dynamical Systems 2010 Siep Weiland 22 / 42


linearization -definitions

Assume that (x0, u0, y0) is a fixed 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 the previous frame isthe linearization of the nonlinear model x = f (x , u), y = g(x , u) aroundthe fixed point (x0, u0, y0).

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

Equivalently represented by its transfer function

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

its frequency response H(i), or its impulse responseh(t) = C exp(At)B + D(t), etc.



linearization -example

Example

Linearize the nonlinear model

x = f (x , u) = x2 + u sin(x) + u2x2 1y = g(x , u) = x2 + sin(u) exp(x)

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

Solution:

Compute partial derivatives of f and g :

f

x= 2x + u cos(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 linearized model

= 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 + 2 sin(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 designBut:

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





6 Summary


Nonlinear systems General structure

general structure of nonlinear systems

We consider the general form

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

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

Important special cases:

homogeneous or 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 xsin x

|

Hence, if x(0) = x0 then C = log |1+cos x0sin x0 | so that

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

|



example (ctd)

With x0 = pi/3 we can determine x(t) for t = 12 by solving

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

|

for x .This is no easy task!!

Questions:

is it solvable at all? if it is, is solution unique? what happens with x(t) as t ?


Nonlinear systems Fixed points

fixed points -definition

Definition

A point x is a fixed point of the homogeneous flow x = f (x) if f (x) = 0.

x is fixed point means that constant x(t) = x, t R is solution ofx = f (x) with initial condition x(0) = x.

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

Example

x = sin(x) has x = kpi with k Z as its fixed points.


Stability of fixed points

Outline





6 Summary


Stability of fixed points Stable fixed points

stability of fixed points

Homogeneous flow x = sin(x) flows

to the right if sin(x) > 0 (velocity is positive) to the left if sin(x) < 0 (velocity is negative)

6 4 2 0 2 4 61.5

1

0.5

0

0.5

1

1.5

Fixed points at x = kpi, k odd: x is stable fixed point. k even: x is unstable fixed point.


Stability of fixed points Stable fixed points

stability of fixed points

Definition

A fixed point x is stable if 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 condition x0 with |x0 x| < wehave that |x(t) x| for all t 0.

Important to note that

stability is a local property of a fixed point. may depend on , that is some initial conditions should be chosen

closer to x than others. The definition does not say that x(t) x as t . x is also called Lyapunov stable


Stability of fixed points Unstable fixed points

instability of fixed points

Definition

A fixed point x is unstable if it is not stable.

Example

Consider x = x2 1. Fixed points are x = 1 and x = 1. Phasediagram 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 1010

8

6

4

2

0

2

4

6

8

10

t

solu

tion

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 flow x = f (x), then x is stable if f (x) < 0. unstable if f (x) > 0.

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


Summary

Outline





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 hasdimension 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 of f (x).

to next class


Organization of the courseOrganization of the courseMaterialScheduleExams and gradingsPurpose

Today's lectureMotivating examplesLinear systemsLinearization

Nonlinear systemsGeneral structureFixed points

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

Summary

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