SE 517 Lecture_03_131

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L E C T U R E 2

D Y N A M I C B E H A V I O R

37

SE517 Nonlinear Systems

Dr. Sami El Ferik, KFUPM, Term 131.

Introduction (Summary of Previous Lecture)38

State space Models are systems of ODE.

State space Model can be either Linear or non-linear.

Time variant or Time invariant.

Autonomous or Non-autonomous

Deterministic or Stochastic

The model can contain lags (delays at the level of the

input or output)

The model can contain delays at the level of thestates.


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Objectives39

Study the dynamic behavior of nonlinear systems.

Introduce the concept of equilibrium point, stability,and limit cycles.

Learn how to solve differential equations.

Learn how to construct the phase portrait.

Learn how to evaluate the stability of the solution.

Understand the difference between global behaviorand local behavior.


Expected Outcomes40

Use tools to study the dynamic behavior of nonlinearsystems.

Grasp the concept of equilibrium point, stability, andlimit cycles.

Solve differential equations.

Construct the phase portrait.

Evaluate the stability of the solution of a differential

equation. Differentiate between global behavior and local

behavior.


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Structure41

Recall/use some of the motivating examples . Solution of differential equations. Use of modern software tools to solve such equations. Examples.

Qualitative Analysis Phase portrait, equilibrium points, and limit cycles. Examples

Stability Definition.

Stability of linear systems and stability analysis via linearapproximation.

Lyapunov stability analysis.

Parametric and non-local behavior. Examples


General State-Space Model42

linearynecessarilnotsxoffunctionanyisuuxxxxf

Where

uuxxxxf

dt

dx

uuxxxxfdt

dx

uuxxxxfdt

dx

ipni

pnn

n

pn

pn

'),...,,,...,,,(

),...,,,...,,,(

),...,,,...,,,(

),...,,,...,,,(

1321

1321

132122

132111

=

=

=

In General if the system is nonlinear


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General State-Space Model43

Define

then


Comparison: State-Space Representation44

Linear System Non-linear System

ml

tFx

ml

Bx

l

gx

xx

)()sin( 2212

21

+=

=

Xy

tr

M

X

M

B

M

Kdt

dX

]01[

)(1010

=

+

=

( )

( )

dXAX Br t

dt

y CX Dr t

= +

= +


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1-45

1. Phenomena of Nonlinear Dynamics

Linear vs. Nonlinear

System

Input Outputu y

state,

(1)x Ax Bu

y Cx

= +

=

( , )(2)

( )

x f x u

y x

=

=

:

:

n m n

n p

f R R R

R R

Definitions : Linear : when the superposition holds

Nonlinear : otherwise

x


Linearity46

A system is said to be linear if it satisfies thesuperposition theorem (addition) and alsohomogeneity

If y1 is the output due to u1

If y2 is the output of u2

Let u=u1+u2 the new output y=y1+y2.

Let u= u1 then the new output y= y1


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Superposition

47

* Superposition

Sys. Sys.1u 2u1y 2y

= Sys.1 2u u+ 1 2y y+

Is the system linear ?

( )1 0 10

( )2 0 20

( ) ( )

( ) ( )

tAt A t

tAt A t

y t Ce x C e Bu d

y t Ce x C e Bu d

= +

= +

( )

1 2 0 1 202 { ( ) ( )}

tAt A ty y Ce x C e B u u d

+ = + +

+

So is it linear? No, under zero initial conditions only.


Linearity48

What is the linearity when ? ( ) 0u t =

11 0

22 0

( )

( )

A t

A t

y t C e x

y t C e x

=

=

1 21 2 0 0( )

A ty y C e x x+ = +

+

A mnemonic rule for linear system :All functions in RHS of a differential equation

are linear. System is linear at least at zero input or zero initial condition

Ex:2

1 2

2 1

n o n l i n e a rs in

x x

x x u

=

= +


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49

Time invariant vs. Time varying

System (1) is time invariant parameters are constant

Time invariant vs. Time varying

( ) ( )( 3 )

( )

x A t x B t u

y C t x

= +

=

( , , )

( 4 )( , )

x f x u t

y x t

=

=

System (2) is time invariant no function has t as its argument.

- Linear time varying system

- Nonlinear time varying system


1-50

Time invariant system are called autonomous and time varying are

called non - autonomous. In our book, autonomous is reserved for

systems with no external input, i.e.,

Thus autonomous are time invariant systems with no external input.

Autonomous & Non - Autonomous

Ex: ,

( ) , ( )

x A x y C x

x f x y x

= =

= =


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Stability & Output of systems

Stability depends on the systems parameter (linear)

Stability depends on the initial conditions, input signals as well

as the system parameters (nonlinear).

Output of a linear system has the same frequency as the input

although its amplitude and phase may differ.

Output of a nonlinear system usually contains additional frequency

components and may, in fact, not contain the input frequency.

51


Equilibrium Point

Equilibrium Point We start with an autonomous system.

or ( ),n

x Ax x f x x R= =

Definition: is an equilibrium point (or a steady state,

or a singular point)

nsx R

0, ( ) 0, i.e., 0s sA x f x x= = =

0s

x =If det(A)0, the autonomous system has a unique equilibrium

point, (Linear System).

52


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Time variant systems53

x* is said to be an equilibrium point for

If

0 0

( , )

( )

x f x t

x t x

=

=

( *, ) 0f x t


Comparison: Equilibrium relations

Example 1: Mass-Damper54

Xy

tr

M

X

M

B

M

Kdt

dX

]01[

)(1010

=

+

=


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Example 2: Pendulum55

ml

tFx

ml

Bx

l

gx

xx

)()sin( 2212

21

+=

=

0 2 4 6 8 10 12 14-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1


Example3: Predator Prey M156

0

/

0

/

x

or y a b

y

or x d c

=

=

=

=

0

0

x ax bxy

y cxy dy

= =

= =


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Predator Prey M257

( )

( )

x a by x x

y cx d y y

=

=


Comparison:58


ml

tFx

ml

Bx

l

gx

xx

)()sin( 2212

21

+=

=

Xy

tr

M

X

M

B

M

Kdt

dX

]01[

)(1010

=

+

=

Equilibrium Linear SystemEquilibrium nonlinear Sys

gm

tFx

ml

tFx

l

gx

xx

)()sin(

)()sin(0

0

1

12

21

=

+==

==

)(

0

)(1

0

0

0

1

2

21

2

trKx

x

trM

XxMBx

MK

x

dt

dX

=

=

+

=

=

0&00)( 21 === xxtrif0&0)( 21 === xkxtFif


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Simulation

59 mass_spring_sim.

clc

close all

clear all

global m k g r B

m=1; k=2; g=9.8;r=0; B=0.5;

X0=[2,8];

[t,Y]=ode45(@mass_spring,[050], X0);

figure(1)

plot(t,Y(:,1))

grid

figure(2)

plot(Y(:,1),Y(:,2))

grid

Pendulum_sim.m

clc

close all

clear all

global l m g F B

l=2; m=1; B=2; g=9.8; F=2;

X0=[pi/4,0];

[t,Y]=ode45(@pendulum,[050], X0);

figure(1)

plot(t,Y(:,1))

grid

figure(2)

plot(Y(:,1),Y(:,2))

grid


ODE45 Code60

function[dXdt]=mass_spring(t,X)

global m g r k B

dXdt(1,1)=X(2);

dXdt(2,1)=-k/m*X(1)-B/m*X(2)+r/m;

function[dXdt]=pendulum(t,X)

global l m g F B

dXdt(1,1)=X(2);

if abs(sin(X(1)))

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Additional real Example: Tunnel Diode61

[ ]

[ ]

1

1 2

21 2

1( )

1

dxh x x

dt c

dxx Rx E

dt L

= +

= +

Applications for tunnel diodes includedlocal oscillators for UHF television tuners,trigger circuits in Oscilloscopes,


62


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63

h(x1)

Step 1: compute the equilibrium point

[ ]

[ ]R

ux

RxuRxx

L

R

ux

Rxhxxh

c

+=+=

+=+=

1221

1121

110

1)()(

10


Lets Summarize64

Unique equilibriumPoint

Stable Linear systemunder harmonic inputproduces an outputwith the samefrequency.

Single mode ofbehavior

Multiple equilibriumpoints.

Nonlinear systemunder harmonic inputproduces an outputcontaining harmonicsand sub-harmonics

Multiple modes ofbehavior.



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Summary65

Unique equilibriumPoint

Stable Linear systemunder harmonic inputproduces an outputwith the samefrequency.

Single mode of

behavior Infinite escape time.

Multiple equilibriumpoints.

Nonlinear systemunder harmonic inputproduces an outputcontaining harmonicsand sub-harmonics

Multiple steady State

modes of behavior. Finite escape time.



66

Solution of the DynamicEquations


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1-67

Linear Autonomous Systems

linear autonomous system

01

( ) in

tAti i

i

x Ax x t e x a e P

=

= = =

where , 1, , : eigenvectors of A

, 1, , : eigenvalues of A

i

i

P i n

i n

=

=

For 1-dim sys.

For 2-dim sys.

0x0x

0x0 = 0 > 0 Re 0, Im 0 <


1-68

Linear Autonomous Systems (Contd.)

Having , i.e., , we can generate a rich set of patterns,but this would not be the eigenbehavior but the forced behavior.

u x Ax Bu= +

Note :

i tjAll other motions are, basically superpositions of these (along with t e ,

where is the multiplicity of ). Thus linear automonous system can

exhibit only exponential behavior (possible labeled by

ij

harmonic

function). Thus the set of possible patterns relatively poor.


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1-69

Solution of Linear systems

Solution always exists locally.

Solution always exists globally.

Solution is unique each initial condition produces a differenttrajectory.

Solution is continuously dependent on initial conditions forevery finite t,

Equilibrium point is unique (when det A0).

0 0

0 0

, ,

( , ) ( , ) ,

T

x x t x x t t T

x x

< <

SE 517 Lecture_03_131

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