Continuous System Modeling - Arizona Center of Integrative

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Continuous System Modeling
Need various types models
• Advances in system development ultimately rely on well-constructed
predictive models
• Applications:
– traditional fields such as electrical and mechanical engineering
– newer domains such as information and bio-technologies
• Using appropriate simulation software, we can derive solutions to
difficult problems using such models
• Success often depends on having a variety of modeling approaches
available to formulate the right model for the particular issue at hand
• Therefore, a broad familiarity with different types of models is
desirable
Continuous System Models
• Continuous system models were the first widely
employed models and are traditionally described by
ordinary and partial differential equations.
• Such models originated in such areas as physics and
chemistry, electrical circuits, mechanics, and aeronautics.
• They have been extended to many new areas such as
bio-informatics, homeland security, and social systems.
• Continuous differential equation models remain an
essential component in multi-formalism compositions.
Multi-formalism Compositions
• A host of formalisms have emerged in the last few decades that
greatly increase our ability to express features of the real world and
employ them in engineering systems.
• Such formalisms include Neural Networks, Fuzzy Logic Systems,
Cellular Automata, Evolutionary and Genetic Algorithms, among
others.
• Hybrid models combine two or more formalisms, e.g., fuzzy logic
control of continuous manufacturing process.
• Most often, applications will require such hybrids to address the
problem domain of interest.
Fundamental Systems Problems
Systems Problem
Does source of the data exist?
What are we trying to learn
about it?
Which level transition is
involved?
systems analysis
The system being analyzed may
exist or may be planned. In either
case we are trying to understand its
behavioral characteristics.
moving from higher to lower
levels, e.g., using generative
information to generate the data in
a data system
systems inference
The system exists. We are trying to moving from lower to higher
infer how it works from
levels, e.g., having data, finding a
observations of its behavior.
means to generate it
systems design
The system being designed does
not yet exist in the form that is
being contemplated. We are trying
to come up with a good design for
it.
moving from lower to higher
levels, e.g. having a means to
generate observed data,
synthesizing it with components
taken off the shelf.
M&S Entities and Relations
Device for
executing model
Real World
Simulator
Data: Input/output
relation pairs
modeling
relation
Each entity is represented
as a dynamic system
Each relation is represented
by a homomorphism or other
equivalence
simulation
relation
Model
structure for generating behavior
claimed to represent real world
M&S Framework: Continuous case
Real World
Simulator
modeling
relation
Validity:
•Accuracy of
-retro-diction
-prediction
simulation
relation
Model
d q(t) / dt = x(t)
Numerical Integration:
•Accuracy
•Error effects
Specification Levels for Differential Equation Systems
Level
0
1
Specification Name What we know at this level
Observation Frame how to stimulate the system with inputs;
what variables to measure and how to
observe them over a time base;
I/O Behavior
time-indexed data collected from a source
system; consists of input/output pairs
Differential Equation System Specification
Input and output ports with continuous variables
Input/output pairs
described by relational equations using first and higher order
derivatives, usually linear and some non-linear
f(y(t), d y(t)/dt, ..., d yn(t)/dtn , x1(t), x2(t),..., xm(t))=0
e.g.
d y2(t)/dt2 - (1 – y2) * d y(t)/dt - x1(t) =0
2
I/O Function
knowledge of initial state; given an initial
state, every input stimulus produces a
unique output.
3
State Transition
how states are affected by inputs; given a
Canonical Ordinary Differential Equation Model
state and an input what is the state after the
input stimulus is over; what output event is d q1(t)/dt = f1(q1(t), q2(t), ..., qn(t), x1(t), x2(t),..., xm(t))
generated by a state.
State Operator description
y[0,t] = L(y(0),x(0,t))
...
d qn(t)/dt = fn(q1(t), q2(t), ..., qn(t), x1(t), x2(t),..., xm(t))
<y1(t), y2(t),..., yn(t) > = g(<q1(t), q2(t),..., qm(t)>)
x,y = input and output vectors
q = state vector
Model, usually linear, can be induced from level 2 by realization methods
4
Coupled
Component
components and how they are coupled
together. The components can be specified
at lower levels or can even be structure
systems themselves – leading to hierarchical
structure.
Components can be atomic DESS systems e.g. Integrators, or couplings of them, in
hierarchical structure
Canonical Ordinary Differential
Equation Model
d q1(t)/dt = f1(q1(t), q2(t), ..., qn(t), x1(t), x2(t),..., xm(t))
d q2(t)/dt = f2(q1(t), q2(t), ..., qn(t), x1(t), x2(t),..., xm(t))
...
d qn(t)/dt = fn(q1(t), q2(t), ..., qn(t), x1(t), x2(t),..., xm(t))
x
q
x
f1
q
x
f2
d q1/dt

q1
d q2/dt

q2
d qn/dt

qn
...
q
x
fn
Numerical Integration
x
= f(q(t),x(t))
x(ti)
q
dq (t )
q (t  h)  q(t )
 lim
h0
dt
h
f(q(ti),x(ti))
q(ti)
1
Euler or rectangular method.
q (t  h)  q (t )  h 
q(h)
q(0)
dq (t )
dt
0
= f(q(t),x(t))
h
q(2h)
q(3h)
2h
3h
q((n+1)h)=q(nh)+h*f(q(nh),x(nh))
Feedback Coupling
qn
f1

q1
f2
dq1
 f1 (qn )
dt
dq2
 f 2 (q1 )
dt
...
dqn
 f n (qn 1 )
dt

q2
f3

q3
dqi
 f i (qi 1 )
dt
Direct influence is negative if
sign( f i (qi 1 ))  sign(qi 1 ) (-)
and positive if
sign( f i (qi 1 ))  sign(qi 1 ) (+)
There is no connection if
dq
f i (qi 1 )  0, i.e., i  0
dt
...

qn
Feedback Qualitative Analysis
qn
f1

q1
f2

q2
f3

q3
...

qn
For a feedback connection there are no zero influences.
Then the feedback loop is negative (positive) if
Direct influences multiply in sign:
(+)(+)= +
N is odd (even)
where N is the number of negative direct influences
i.e., sign is determined by (-1) N (1) P  (-1) N
(+)(-)= (-)(-)= +
e.g. N  0, P  1  positive feedback
N  1, P  0  negative feedback
N  2, P  0  positive feedback
f
But: N  1, P  1  oscillation
2nd Order Linear System (undamped)
v

v



x
P  2  exponential growth

-
x
v

v


x
N  2  exponential growth
-
x
v

v


-

x
N  1, P  1  oscillation
x
angular frequency = 
2
period = T =

Continuous system simulation languages
and systems
state-space description languages:
• Continuous System Simulation Language (CSSL) standard, e.g., ACSL
• block oriented simulation systems, e.g., Simulink
Van der Pol Oscillator
CSSL PROGRAM Van der Pol
INITIAL
constant
k = -1, x0 = 1, v0 = 0,
tf = 20
END
DYNAMIC
DERIVATIVE
x = integ(v, x0)
v = integ((1 – x**2)*v – k*x, v0)
END
termt (t.ge.tf)
END
END
Simulink building blocks
1/s
+
+
integrator: dq / dt = x, y = q
f(x)
Sum: y = x1 + x2
*
Multiplier: y = x1 * x2
c
Gain: y = c * x
Function: y = f(x)
Sinusgenerator: y = sin (t)
Subsystem: Placeholder for a subnetwork model
Inport: Input from an external model
1
Constant: y = c
c
Outport: Output to an external model
1
Van der Pol Oscillator in Simulink Block Diagram
k
*
1
+
*
+
v
1/s
x
1/s
1
1st Order Linear System –
exponential growth/decay
c
c0
q
dq
 f (q)
dt
f (q )  cq
c0
1st Order– constant input to
exponential decay
-c
x
+
dq
 f ( q, x )
dt

q
f (q)  x  cq
At equilibrium,
dq
0=
 x  cq  qeq  x / c
dt
difference(t )  x  cq(t )
qeq  x / c
Damped Linear oscillator of second order
-d
m=1
F(t)
k
F(t)
m
+

v
d
-k
x
(a)
(b)

x
Van der Pol Oscillator Dynamic Behavior
Lotka Volterra Model and Behavior
Exercise:
• write the ODE for the model
• find the equilibrium point of the Lotka-Volterra
model
•investigate the oscillations around this equilibrium.
•where do the maximum and minimum populations
occur?
•show that small oscillations around the equilibrium
are approximated by the 2nd order linear oscillator.
py = prey population
pd = predator population
b
k
d
pyeq 
c
pd eq 
pd
pdmax
b
pd eq 
k
pymin
  db
pymax
pd min
d
pyeq 
c
py
b
k
d
py  py 
c
pd   pd 
d py
d
dk
 [b  k pd ] py  kpd ( py  )  
pd   o( py, pd )
dt
c
c
d pd 
b
cb
 [d  cpy ]( pd   ) 
py  o( py, pd )
dt
k
k
dk cb

 db
c k
Locating Min/Max using Zero Crossings
dq
dt
q
dq
dt
q
dq
 0 at t  t '
dt

q is maximum or minimum
at t  t '
Limit Cycle and Chaos are Opposites
•
•
•
•
•
limit cycles – initial state eventually winds up in a periodic loop or
cycle
chaos – trajectories are sensitive to initial states – small difference in
initial state results in large difference in trajectory
Note – ODE models are deterministic – if the input is zero, then if a
trajectory returns to an earlier state, it will get into a cycle
If a chaotic model has a trajectory that comes close to an earlier
state than it diverges from that earlier portion due to its sensitivity to
initial states
BUT – a chaotic model can have a “strange attractor” i.e., a subset
to which always returns, though not with a fixed period.
Rössler Model and Chaotic Behavior
fz (x, z) = b + (x – c)*z

+
y
-1

+
x
fz

z
dx / dt  - ( y  z )
dy / dt  x  a y
dz / dt  b  z ( x - c)
-1
-a
a = b = 0, c = 4.7
dx / dt =0  y=-z
dy / dt  0  x= 0
dz / dt  0  z=0 or x=c  z=0
 equilibrium point = (0,0,0)
state plane (v , z) to x
time behavior
interactive applet at: http://www.geom.uiuc.edu/~worfolk/apps/Rossler/
Rössler Behavior
a = b = 0.2, and c = 8.0.
http://mathforum.org/advanced/robertd/rossler.html
http://astronomy.swin.edu.au/~pbourke/fractals/rossler/
Lorenz Attractor– Butterfly Effect
dX / dt  - c( X - Y )
dY / dt  aX - Y - XZ
dZ / dt  b( XY - Z )
b  8 / 3 c  10
For a < 1 the solution rapidly decays to the origin X=Y=Z=0. This corresponds
to no motion in the fluid context.
For a > 1 (e.g. a=5) the orbit approaches one of two fixed points (depending
on the initial values) away from the origin. The fixed points are at X 2 =
Y 2=Z=a-1. In the convection context this corresponds to nonzero but steady
fluid flow (in a circulating "roll" configuration).
At larger values of a, for example a=24.1, the long time dynamics may either
approach one of the fixed points or a strange attractor (depending on the
choice of initial values), which coexist at these values of a. (Choose nearby
initial values to find solutions that converge to the fixed points.)
For a>24.74 the strange attractor collides with the fixed points, which become
unstable so that practically all initial values lead to the familiar butterfly
dynamics.
a=28 gives the usual picture.
Java applet: http://www.cmp.caltech.edu/~mcc/chaos_new/Lor_docs/intro.html
http://astronomy.swin.edu.au/~pbourke/fractals/lorenz/
References/Literature
•
Course Notes from: B. P., H. Praehofer and T. G. Kim (2000). Theory of Modeling
and Simulation: Integrating Discrete Event and Continuous Complex Dynamic
Systems, (2nd Ed.) Academic Press, NY.)
•
On reserve: A First Course in Differential Equations: The Classic Fifth Edition
(Hardcover) by Dennis G. Zill, Brooks Cole; 5 edition (December 8, 2000)
Others:
•
–
–
–
–
–
–
The Nonlinear Workbook: Chaos, Fractals, Celluar Automata, Neural Networks, Genetic
Algorithms, Gene Expression Programming, Support Vector Machine, Wavelets, Hidden
Markov M (Paperback) by Willi-Hans Steeb, 588 pages, Publisher: World Scientific
Publishing Company; 3rd edition (July 15, 2005)
Modeling and Analysis of Post-Conflict Reconstruction, Damon B. Richardson, Richard F.
Deckro, and Victor D. Wiley, JDMS: The Journal of Defense Modeling and
Simulation,October 2004, Volume 1 Number 4
Fernando J. Barros, A Formal Representation of Hybrid Mobile Component, SIMULATION,
May 2005; 81: 381 - 393.
What is signal and what is noise in the brain? A.Knoblauch, G.Palm, Biosystems 79(1-3), pp
83-90, 2005.
Discrete Event Multi-Level Models for Systems Biology, Uhrmacher, A.M. and Degenring, D.
and Zeigler, B.P, LNCS Transactions on Computational Systems Biology, Vol. 1, 3380/2005,
pp. 66-85.
Modifications of the Helbing-Molnár-Farkas-Vicsek Social Force Model for Pedestrian
Evolution, Taras I. Lakoba, D. J. Kaup, and Neal M. Finkelstein, SIMULATION 2005 81: 339352.
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