se 517 lecture_02_131
Post on 14-Apr-2018
214 Views
Preview:
TRANSCRIPT
-
7/29/2019 SE 517 Lecture_02_131
1/18
9/8/20
LECTURE 02:DYNAMIC BEHAVIOR
TERM 131
1
SE 517 Nonlinear Systems
Dr. Sami El Ferik, KFUPM, Term 131.
Part I: Modeling
2
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
2/18
9/8/20
Objectives :3
Define key modeling concepts.
Introduce simulation packages for model analysisand simulation.
Describe the modeling methodology.
Introduce examples to illustrate modeling.
Dr. Sami El Ferik, KFUPM, Term 131.
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.
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
3/18
9/8/20
Structure5
Modeling Concepts
Why modeling? and How to obtain a model?
State space model. Examples
A Systematic Approach for Developing DynamicModels Examples
Dr. Sami El Ferik, KFUPM, Term 131.
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.
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
4/18
9/8/20
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.
Dr. Sami El Ferik, KFUPM, Term 131.
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
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
5/18
9/8/20
9
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.
Dr. Sami El Ferik, KFUPM, Term 131.
Example of Multiplicity ofModels: Valve Models
10
STEM
AIR
PRESSURE IN
DIAPHRAGM
SPRING
AIR FORCE
2
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
6/18
9/8/20
Valve models11
C. Garcia / Control Engineering Practice, (2008)
Dr. Sami El Ferik, KFUPM, Term 131.
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
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
7/18
9/8/20
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
Dr. Sami El Ferik, KFUPM, Term 131.
Example II: Mass-Spring14
massofpositionisy
dt
tdyv
dt
tyda
whereyKtvBtrMa
)(
)(
)())(()(
2
2
=
=
=
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
8/18
9/8/20
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
Dr. Sami El Ferik, KFUPM, Term 131.
Example 3: HIV Drug Administration16
F. Doyle et al. / Journal of ProcessControl 17 (2007) 571594
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
9/18
9/8/20
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
Dr. Sami El Ferik, KFUPM, Term 131.
Example 5: Steering Problem Bicycle Model18
Chih-Lyang et al. 2009
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
10/18
9/8/20
Example 6:Vectored thrust aircraft19
Dr. Sami El Ferik, KFUPM, Term 131.
Example 7: Internet Congestion Control20
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
11/18
9/8/20
The System Viewpoint
21
Dr. Sami El Ferik, KFUPM, Term 131.
Actuator Nonlinearities22
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
12/18
9/8/20
Example 8 : Valve nonlinearities23
BacklashValve with Stitcion
Valve with Stiction and Slip Jump
Dr. Sami El Ferik, KFUPM, Term 131.
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
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
13/18
9/8/20
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
==
=
=
Dr. Sami El Ferik, KFUPM, Term 131.
State Space Models: ODE26
General Form
The .number of states n is called the order of thesystem
State Space Model
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
14/18
9/8/20
General State-Space Model27
linearynecessarilnotsxoffunctionanyisuuxxxxf
Where
uuxxxxfdt
dx
uuxxxxfdt
dx
uuxxxxfdt
dx
ipni
pnn
n
pn
pn
'),...,,,...,,,(
),...,,,...,,,(
),...,,,...,,,(
),...,,,...,,,(
1321
1321
13212
2
13211
1
=
=
=
Dr. Sami El Ferik, KFUPM, Term 131.
General State-Space Model28
Define
then
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
15/18
9/8/20
Introduction: State-Space Model of LinearSystems
29
Dr. Sami El Ferik, KFUPM, Term 131.
Example 9: Two-Carts System30
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
16/18
9/8/20
31
Continued in Class to generate SS model
Dr. Sami El Ferik, KFUPM, Term 131.
Example 10: Mass-Spring32
massofpositionisy
dt
tdyv
dt
tyda
whereyKtvBtrMa
)(
)(
)())(()(
2
2
=
=
=
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
17/18
9/8/20
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
=
+
=
Dr. Sami El Ferik, KFUPM, Term 131.
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.
Dr. Sami El Ferik, KFUPM, Term 131.
-
7/29/2019 SE 517 Lecture_02_131
18/18
9/8/20
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
Dr. Sami El Ferik, KFUPM, Term 131.
Linear vs Nonlinear
36
Dr. Sami El Ferik, KFUPM, Term 131.
top related