SE 517 Lecture_02_131

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LECTURE 02:DYNAMIC BEHAVIOR

TERM 131

1

SE 517 Nonlinear Systems

Dr. Sami El Ferik, KFUPM, Term 131.

Part I: Modeling

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Objectives :3

Define key modeling concepts.

Introduce simulation packages for model analysisand simulation.

Describe the modeling methodology.

Introduce examples to illustrate modeling.


Expected Outcomes4

Recognize the key concepts of Modeling.

Recognize the fact that a system can have manymodels depending on the question we are trying toanswer.

Master simulation packages for analysis andsimulation.

Grasp the modeling methodology.

Apply the modeling methodology for real systems.


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Structure5

Modeling Concepts

Why modeling? and How to obtain a model?

State space model. Examples

A Systematic Approach for Developing DynamicModels Examples


General Modeling Principles6

The model equations are at best an approximation tothe real process.

Adage: All models are wrong, but some are useful.

Modeling involves a compromise between modelaccuracy and complexity on one hand, and the cost,effort, and time required to develop the model, onthe other hand.

Modeling is both an art and a science. Creativity isrequired to make simplifying assumptions that resultin an appropriate/useful model.


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Why modeling?7

Models can be used to improve our understanding of thesystem.

Models allow to reason about a system and predict systemsbehavior under different conditions which may be difficult orimpossible to do because of safety or financial aspects.

Models are needed in system design or for designing bettercontrollers.

A model can be a part of some controllers like feed forwardcontrollers (model-based controllers)

A model can be very valuable in optimizing the operating

conditions to get the best performance of the system. Models are immerging in monitoring of faults and anomalies,

especially for early warning systems.


How to Obtain Models8

Identification(Black-Box)(Experimental)

Conduct an experiment

Collect data

Fit data to a model

Verify/validate the model

The first-principleapproach (white-boxmodels) (Theoretical) Construct a simplified version

using idealized elements

Write element laws

Write interaction laws Combine element laws and

interaction laws to obtain themodel


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Depending on the level of details and focus, a system canhave multiple models.

The model selected depends on what we want to do withit.

Modeling can be costly and time consuming.

Defining the objective of the modeling exercise can savetime and money.

Models should not be more complicated than what isrequired to achieve these objectives.

For control purposes simple models can be used. Themodel does not have to be exact.


Example of Multiplicity ofModels: Valve Models

10

STEM

AIR

PRESSURE IN

DIAPHRAGM

SPRING

AIR FORCE

2


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Valve models11

C. Garcia / Control Engineering Practice, (2008)


Example of Multiplicity of Models:

Glucose-Insulin System

Model I

() = + ())1 ) ()() + ())) = 2 + () ())3 )

() = () + ()

() =

5

()

= ()

Model II BergmansMinimal

12

() = )) = g 2() 3() 4() + ))5 ))

() = )) = 1() ()1

Model III Introducing TimeDelay in one State


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Example I:13

Gravity Drained Tank

h1LT

wI

h1A

C=h2

A

C+

dt

dh2A

w_in=h1

A

C+

dt

dh1(white-box models)

0 20 40 60 80 100 120 140 160 180 2000.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Time

Measured and simulated model output

h2 LT

w2

wout

(Experimental or black-box mod

Schematic Diagram


Example II: Mass-Spring14

massofpositionisy

dt

tdyv

dt

tyda

whereyKtvBtrMa

)(

)(

)())(()(

2

2

=

=

=


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Mass-Spring Model15

Use Hooks law.

Ideal friction element

( )

massofpositiontheisy

massofspeedtheisdt

tdy

yKdt

tdyBtr

dt

tydM

elementfrictionIdealtvBtvB

lawsHookyKyK

)(

)()(

)(

)())((

)'()(

2

2

=

=

=

How many initial conditions arerequired to solve this equation?

We conclude that knowledge of the speed and position are enough to predictthe future dynamic of the system


Example 3: HIV Drug Administration16

F. Doyle et al. / Journal of ProcessControl 17 (2007) 571594


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Example 4: Predator Prey Model17

Volterra-Lokta Predator Prey Model for Small fish inthe Adriatic

x ax bxy

y cxy dy

=

=

( )

( )

x a by x x

y cx d y y

=

=

M1 M2


Example 5: Steering Problem Bicycle Model18

Chih-Lyang et al. 2009


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Example 6:Vectored thrust aircraft19


Example 7: Internet Congestion Control20


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The System Viewpoint

21


Actuator Nonlinearities22


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Example 8 : Valve nonlinearities23

BacklashValve with Stitcion

Valve with Stiction and Slip Jump


Model Formulation

State-Space Model: ODE24

Definition State of a system is a collection of variables that summarize the past of a

system for the purpose of predicting the future.

Lets define

X is called state vector.

In general, where n is the number of state variables. The control variable (in our case r(t)) is represented by

where p is the number of inputs. The output where q is the number of outputs.

dt

dyy &

=

dt

dy

y

X

n

X

pu

qY


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State Space Models: ODE25

Let assume that

=

=

==

=

=

=

2

2

2

2

2

1

2

1

2

1

dt

yd

dt

dx

x

dt

dx

dt

dx

dt

dXX

dt

dyx

yx

x

x

X

122

2

2

2

)(1)()(1)(

)()(

)(

xMKx

MBtr

My

MK

dttdy

MBtr

Mdttyd

yKdt

tdyBtr

dt

tydM

==

=

=

=

12

2

2

1

)(1

xM

Kx

M

Btr

M

x

dt

dx

dt

dx

dt

dX

),(

),(

1

2

1

rXhxY

rXf

dt

dx

dt

dx

dt

dX

==

=

=


State Space Models: ODE26

General Form

The .number of states n is called the order of thesystem

State Space Model


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General State-Space Model27

linearynecessarilnotsxoffunctionanyisuuxxxxf

Where

uuxxxxfdt

dx

uuxxxxfdt

dx

uuxxxxfdt

dx

ipni

pnn

n

pn

pn

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

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

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

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

1321

1321

13212

2

13211

1

=

=

=


General State-Space Model28

Define

then


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Introduction: State-Space Model of LinearSystems

29


Example 9: Two-Carts System30


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Continued in Class to generate SS model


Example 10: Mass-Spring32

massofpositionisy

dt

tdyv

dt

tyda

whereyKtvBtrMa

)(

)(

)())(()(

2

2

=

=

=


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Mass-Spring Model33

( )

massofpositiontheisy

massofspeedtheisdt

tdy

yKdt

tdyBtr

dt

tydM

elementfrictionIdealtvBtvB

lawsHookyKyK

Assume

)(

)()(

)(

)())((

)'()(

2

2

=

=

=

=

=

12

2

2

1

)(1

xM

Kx

M

Btr

M

x

dt

dx

dt

dx

dt

dX

Xy

tr

M

X

M

B

M

K

dt

dX

]01[

)(1010

=

+

=


LTI Block representation34

U(t)

A

CB ++

++

Dy(t)

dX/dtt X

General Form

where A, B, C and D areconstant matrices.

A is called the dynamicsmatrix,

B is called the controlmatrix,

C is called the sensormatrix

D is called the direct term.


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Example 11: Pendulum35

BmgltlFml

Bmgltml

=

=

)sin()(

)sin()(

2

2

mg

F

B

ml

tFx

ml

Bx

l

gx

xx

xx

)()sin(

;

2212

21

21

+=

=

==

)sin(mg

l


Linear vs Nonlinear

36


SE 517 Lecture_02_131

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