Professional training of Engineer on
system Dynamics
Yang BingZheng
wu Xu Guang Yang Futau
Marine College
Northwestern Polytechnical Vihecle Establishment
University Xi’! an, P. R. CHINA Xi! an, P. Repulic. CHINA
710072. Mar. 28, 1992 710705
ABSTRACT
In this paper a professional training schedule on system dynamics for engineers is discussed on the basis
of ten years teaching and training practice.
‘Three main parts are considered in detail. These are theory, practice, and final performance test. All
examples included are to meet the specific necessity of engineers, and are coming from practical
problems.
An overall block diagram of Training is presented. Practice shows this schedule is efficient and attrac-
tive.
1. INTRODUCTION
High technique development calls for high potential managers and engineers, not only to meet pro-
duction reqirements such as model of quality control, sales price prediction, decision of upgrading
products, etc, but also to bridge the gap between theoretical achievements and new technology cre~
ation. In this case for most engineering areas, system dynamics plays an important role, e.g. robotics,
signal processing, CNC technique, power plant control, traffic management etc.
In order to fit different requirements, five main courses are selected for general, these are: Linear sys-
tem, Nonlinear system, Optimization and Optimal control, signal processing, and System Identifica-
tion. Besides, special topics such as information acquisition, estimation theory, philosophical method,
fussy control etc. are inserted optionally as the implemental tutorial.
2. THEORY TEACHING
Engineers in the duration of service are mainly working on practical problems, and have accumulated
a lot of questions relating to system dynamics. This is because in the real world every part must belong
to some “running” system, and a lot of system factor, state or parameters have to be evaluated or con-
trolled in dynamic situation. According to the report from Enterprise Community in China most engi-
neers need the help of related theory. But the schedule of theory teaching hinges on a central problem;
how to modify the conventional plan for students and create an optimal new routine for engineers.
Many educators are finally coming to realize that “concepts, tools, and practice” is the best answer.
Following this the kernel portion of theory teaching is divided into four parts, these are;
a. Fundamental theory and its academic concept;
b. Engineering practical example for demonstration 5
c. Applied computer programs learning;
d. Problem discussion and solving.
- 79 -
The key points of this arrangement can be summarized as follows:
a. Engineers have to master theories and concepts, even in mathematics. The specific condition here is
that they also have to collect a lot of intuitive examples to enhance the relation between theory and
practice;
b. The advent of computer aids the realization of recent advances in theory. A large number of CAD/
CAM/CAE applied software package links concept and engineer application. Engineers are infinitely
better equipped to learn them when they have mastered a few theories or concepts.
c. For engineers, "To touch over with to think, observation more than books, experience rather than
persons” , are the prime principles of training. If the applied theories and concepts tie in properly with
engineering application. In the case practice always induces their creative idea and reinforces their the-
oretical understanding in a mode of "positive feedback”.
d. The four parts are internally connected and mutually affecting each other. But how to match them
in a proper ratio is still a problem left.
In order to fit the special need for engineers along with explanation of theories aditional effective
methods of teaching are incoperated. They are listed in the following paragraphs. The criteria is "easy
to understand, more to get, flexible to manipulate”.
‘A. Physical understanding
One strong point of engineer is to form idea with physical comprehension, so if theory teaching com-
bines with physical explanation they will easily grasp the spirit. In this case a better way is to rewrite
the conventional text, with more qualitative interpretation. An example is the physical explanation of
Kalman filter equation set for linear time invariant discrete system. The system equations are;
Xi = RX +GiWy a.)
Ze=WXetVe (1. 2)
where
X.—state vector
Z,—output vector
‘Wi, Vs,— independent white noise of system and of measure with zero means
ELV» VFJ=Rida
E[W,,WI]=Qdu
Su-—kronecker delta, which is 1 for k=1 and 0 otherwise.
‘The Kalman filter equation comprises the system are as follows:
* main equation
X= Fini Ki Zi Hein 1X1) (1.3)
* gain equation
Ky =Pin—iHi (H,Pya—HE +R) a.4)
* one step prediction equation
Ren =Fin- ier (1.5)
* covariance equation
Pani =FiniPaiFla1 + G1 Q-1GE-1 1.6)
%* update covariance equation
P= (KH Pini 1-H) + RK? a7
where ‘X——estimatied state
P—covariance matrix
In the following paragraphs a number of physical properties of kalman filter are listed both in general
description and in individual terms, ‘
A-1. General description
(a) Kalman filter is a state estimator the results of estimation should contain two terms,
‘X—state estimated;
P—covariance showing the error range.
it is the same as the nominal dimension and the tolerance in measurement.
(b) Kalman filter is a step by step process, the occasion of step change is due to the appearance of Z,,
as shown in Fig. 2-1.
Za Z, 2a So Z, is the major information. In equation (1.
YA Wh 3)HFixiX.—1is the computed output, Z; is the
fw Oe 4 teal output, the difference is called “innovation”
— Kan use used for modification of X,-1. Also before
Sy { > ‘\ attedance of Z, estimation belongs to prior condi-
NI CNT Ne LS tional probability problem otherwise posterior.
The reason of using recursive step by step
Fig. 2-1 algorithm is due to the on-line operation and
memory saving.
(c) Kalman filter exits mainly because measure noise V, exits.
Suppose [V]=[0], [H1] full rank, [P] positive definite, then equation (1,4) changes to
K,=Hi! 1.8)
and X= Fan iMe-1 FHHE Zi HiFi ¢ Xi) =H Zi =X a.9)
P= (KH, Pra Ks)" +KRK7=0 (4.10)
It implies if the measurement is exact there is no need using Kalman filter.
(@) The main function set of Kalman filter are essentially deduced from the output equation.
Suppose Z=Hx+V (orm)
X~NQ,p), V~N(O,R)
they are mutually independent, then it is easy to prove
E(X|Z) =p-+-P,H"(HP,H"+R)"!(Z—Hy) (1, 12)
Var (X |Z) =P,—P,H" (HP,H™-+R)~"HP; (1. 13)
they have probably the similar structures as the Kalman filter equations.
A-2, Remarks about some individual terms
(a) The main equation (1. 3) is a kind of typical Ttructure of linear recursive Algorithm. From (1.
3) easy to see X, is a linear combination of former estimated state X,_, and observed value Z,. If com-
paring it with the equation of recursive averaging, a probably similar structure emerges.
take X= [1 xe500+ 9X0]
the average value
K=12x, 1)
a
a> 1
and Rena RR apy Mor Re) (2.2)
here X,41is the new data same as Z,. The difference is that the Kalman filter is a dynamic system, but
the later is a stationary.
(b) Gain matrix Ky is a measure of the covariance ratio.
From equation (1. 4)
Ki=Pra Hf (Pra iHE+Ri)
or
K.=PHIR (2.3)
without loss of generality
take = H= [1]
-8l-
(2.4)
(2.5)
then
u/oh Pr2/o}
—|Pa/ot Pr/o}
Pa/o} Pa2/03 Paa/ Oz
It shows some definite and relative behaviors between these two uncertainties. More intuitively, when
Q, grows, Ky also goes up, but when R, grows, K, goes down, it means the information from Z, is less
believable.
(c) Covariance matrix P, is a measure of the optimality of state estimation. P, is an overall compre-
hensive term, consisting three main error sources Py, Rx,» Q, coming through different transfer routine
F,, Gx, Hy. Norm of P; means the width of error band of state estimation. The lower the norm value
the better the estimated Value X. No matter what the initial Pp selected P, will reach a definite P at
last.
One interesting question is; why P, does not directly deal with Z,? The statistical reason is that any one
measured value can not change the overall comprehensive property of the system.
(da) R, Q are prior statistical values, it can be obtained either from test or directly from the company
catalog. Anyway matrix R is demanded to be positive definite. In order to force positive definite of
(H,Pxn4:H?-+R,) for inverse operation. In practice the upper limit of R should be chosen because of
mathematical model with error or unpredictable system disturbance. Usually the dimension of R is less
or equal to the dimension of state X, it implies some state output value can not be measured. In this
case Kalman filter is also valid if the system noise is less heavy.
Poo/oa
Ky (2.6)
B. Example Demonstration
T. Jefferson [ * 1] said; “+++ more honorable and more profitable, too, to set 2 good example than to
follow a bad one”. Examples exist in most books. The emphasis here is focused on how to select
property examples concerning both theory and engineering practice.
For demonstration of model formulation some engineering real problems are used to show the whole
procedure. An example is an assembly process as depicted in Fig. 2-2
where 1—steel bush
2——massive aluminum body
is i ie Before assembly part 1 is soaked in liquid hydrogen to —
is : Te 150'C s and part 2 is put in constant temperature oven to +
é 150°C.
yy -@ ‘After assembly surface A and B are closely contacted together
o t in the beginning, but in the last with no exception clearance
t A happens. For revealing the mechanism of this phenomena
! hg mathematical model formulation on principles of system dy-
L | namics is necessary. Suppose part 1 and part 2 are pulled out
T Figa-2 from their original temperature situation to room temperature
(20°C) separately at time to. Then the time history of these processes is shown in Fig. 2—3 as line 1
and line 2. The line 1—2 means part 1 is put into part 2 in assembly form.
For curve 2 the mathematical model is
- 82-
iis
Te
+150
TH+ Ky=20, — y2(0)=150 (2.7) y al
T,——time constant for part 2,
Suppose the temperature change of part 2 is not af-
fected by part 1 due to its large heat capacity. 2.
Then the curve 1-2 can be written as o
7, Btkin=ve),y(O=—150 (2.8) 7
It
20 , (150K,—20)T2 _%,
10 =e (gh—KT ke
20, (150K,—20)T, 7_¥
Dirge tok
(2.9) Fig 2-3
T,,T; can either measured or calculated. Another related prob-
Jem is about the shape deformation of part 1 due to the tempera-
$ ture gradient field. In order to determine the specific point c of
iq contact during the heat transfer dynamical process, for simplifi-
cation take a thin slice of part 1 as shown in Fig. 2-4. The
boundary condition are TH and TL, which means the high and
low temperature. This field problem can be approximated by a
series of difference equations involving the magnitudes
Fig. 2-4 of required variable at the mesh points. The difference equation
corresponding to Laplace’ s equation is based on the heat balance principle i. e. temperature at any point
is equal to the average value of the temperatures at the four neighbouring points.
T1=1/4(T+T:+Tu th]
T2=1/4(T14-T34+T.+-TL]
/4(Te+Te+T +h]
(2.10)
/ALT AT ATLAT
rewrite it in matrix from, as
3 =-1 T, TatTy
“1 4 -f +i 7: T
ms, = t
4-1 -1 vlel ot
1 A zm :
a oT
:
:
(11)
solve it, the temperature gradient gives that the bush part 1 deformation is probably along the dash line
as shown in Fig. 2-2 and the contact point ¢ would probably at L/2. Suppose the exact contact time is
at (Fig. 2-3), then the clearance due to free expansion and shrinkage hereafter can be estimated by
A=L[(T.+20)e+ (To 20)e2]/2 (2.12)
where €; ,€ are the corresponding coefficients of heat expansion of material 1 and 2.
C. Computer aided Education
= §3 =
CAE has been widely used at some universities and research centers. A lot of video tapes for self-study
are also well spreaded. For teaching system dynamics such as time discrete system, switching theory ,
nonlinear system, control theory ete. IEEE Transaction of Education recommends the following pack-
ages as the major category, these are;
Simon, PC-Matlab, Ms-Kermit, Phaser, Mi-
crologic, Ventura, Volkswriter, Wordperfect,
Msword, Microtex +++, This especially benefits
‘engineers to feel and to look at.
An example is the demonstration of chaos phe-
nomena, which is very hard to explain. With
the aid of software package ” phaser” a phase
plot displays all the ” strange” characters as
shown in Fig. 2-5.
D. Mapping
Mapping here means to transfer the idea from one area to another. This is a general way of analog”
for teaching engineers. e. g. It is well known that telecommunication with lunar satellite is using a spe-
cial frequency radio signal which can easily pass through the ion zone without too much decay. A simi-
lar question is that, Is there the possibility by using a special wave length laser to pass though the liquid
media without too much decay? A ”mapping” answer may be helpful to evaluate the possibility of us-
ing laser. This is; According to the theory of system dynamics, natural modes are the signals for
which the loop transmission is unity, hence the answer is theoretically positive.
E. Extended questions
For inducing the creative idea of engineers on system dynamics, extended questions are helpful. For
example
»* Relation between harmonic balance and least square method.
» Hardware of Vector transformation in real world.
* Check the following transition matrix, is this system linear?
_fexp(—2t) 1
Fro=[ 0 eet]
* What is negative frequency?
many universities have fixed their own problem based package with different level. Practice shows its
effectiveness and attractiveness. All questions are as compact as possible but still show the intended
points.
F. Advanced technique introduction
System dynamics is well developed in both directions, theory and application. New idea and new sys-
tem gradually become indispensable technologies in industry, such as expert system, neural networks,
variable structure control, fuzzy control, etc. Training program can not cover all these topics but a
primary introduction is valuable, It is not only benefitial to engineers but also to directors and decision
makers.
G. Summary mode
For engineers it is helpful first to show the global situation of new technologies or ideas, and then the
specific theory. In this case teaching material is better composed in summary mode. An example is the
introduction of white noise in signal analysis as shown in the following table.
DISTRIBUTION OF WHITE NOISE FORMULA
NORMAL NQO,1) #1
UNIFORM uC0,1)+U
EXPONENTIAL —LOG(1—U)
LOGISTIC LOG(U/(1—U))
CAUCHY TAN(PI(U—1/2))
EXTREME VALUE LOG(—LOG(1—U))
LOGNORMAL EXP(Z)
DOUBLE EXPONENTIAL LOG(2U), IF U<0.5
—LOG(2(1—U)), IF U>0.5
Like the design sheet used for engineering this method gives quick reference and a distinct picture.
3. PRACTICE
” AM experience is an rich, to build upon” [ * 1]. Practice is an intuitive source of knowledge, either
for social science or for natural science. It helps both enhancing the understanding of related theory
and serving as a criteria of theory. The practice frame work suggested consists of four parts, i.e,
(a) Analog computer practice
This is mainly for continuous system. The whole system can work independently or combine with
other real facilities. For example, XY-plot of limit cycle explains why clearance in mechanical con-
nections in harmful? Correlation method in system identification shows how to reduce the noise distur-
bance in state estimation event.
(b) Digital computer practice
The softwares recommended are as follows,
% Cubic spline and least square;
% Difference and Differential equation ;
% System analysis and simulation;
* Optimal control system design;
* Operational Research;
* Signal analysis;
% System Identification.
Others are only used for demonstration in class.
(c) Typical control system manipulation
According to the modified training schedule the following control systems are selected,
* Temperature control system;
* 3-Phase asynchronous motor speed control system;
% CNC system;
* Paper production control system.
Data in operation are gathered, analyzed, and compared with the computed values on the basis of for-
mulated mathematical model.
(d) Plant existing problem solving
‘This is an important portion for engineers, because they have to treat practical or theoretical problems
existed in company or research establishment during study system dynamics. Company will act as a
sponsor if some engineering problems can be solved or some methods of treatment are suggested. This
: — >
se
-
reflects the specific tie of training engineers and social requirement. An real example of this work is as
follows,
The problem from machine shop is about the undercut effect of milling helix slot, as shown in Fig.
3-1. In order to get accurate surface a modified blade curve is needed. Take XOY as the coordinate,
then in C-C section plane the equations of machine part and cutter are as follows,
sin'Bxttyt= lps G1
where (x—e)?-+ (y—f)?=de? (3. 2)
e= Seu, t= 4 O+4) @.3)
Machine part
Cer pT t\ oo
3
x —+—}
: Undereut area
Sega e e
Fig. 3-1
The optimal cutter diameter de * in different section can be determined by lagrangian function.
F(x, 5d) = (x—e)?-+ (a Ceingte +y*—4 D4) G4)
—lagrangian multiplier.
Take (x* ,y*) as the tangent point, then (x* ,y*) satisfies the following equations.
e. .
FTL" +2 + sin?B + x* =O
Foyt vyt=
gy 720" +2 +y"=0 @.5)
Faain'pe ¥+G"—LD=0
: . 2
“te. ey’ a syed
ae -radtramly SE a) Povey
@.6) |
de" = Ge ey at @.7)
After succesively evaluating dc" in different section. An optimal blade curve can be obtained. Practice
shows the final accuracy is satisfactory.
4. PERFORMANCE ITEMS
“In order to check the training quality, five items have to get through, these are ;
(a) Analytical ability
It means the ability of modelling, analysis, simulation, design optimization, and engineering realiza-
tion. The key point is that problem solving or decision making should be based on the theory and appli-
cation of system dynamics. One interesting example is the population growth control model in social
science.
(tb) Computing ability
Training program provides two branches to manipulate digital computers, application program practice
and utilization of AD/DA card for information acquisition, system analysis, and control. It would be
checked on computer operation.
(c) Physical understanding
It means to explain some events with practical idea on the basis of their experience and concept mas-
tered. It represents their analytical and investigative ability.
(d) Operation skill
This will be done on real operation of laboratory facilities to check their carefulness, correctness and
skillfulness.
(e) Creative idea and Orgainzation
In homework and discussion seminar, their creative idea will pour out. This is a very important per-
formance of training.
5. TRAINING DIAGRAM
After ten years training practice a summarized block diagram is formed. It covers all the activities
scheduled, including theory teaching and practical operation. It makes a feature of flexibility and effi-
ciency. All training procedures are closely connected and aided by digital computers.
Master of theory, practice of problem, and creation of idea will be the main aim of the whole trining.
Figure. 5-1. shows the skeleton diagram of engineer training.
[Theoretical] [MATLAB,] [Problem
[Practical ILS-N,OR Library
nT [Program| [Problem] [Analytical] A
Ta8 THEORY [EXAMPLE [Learning| [Solving [Ability
Practical Ss
ADVISING Demo} [Lab. work} {Problem i
|Treatment|
Making [Test o
v ¥
[Schedule expansion |Analytical ability
jor shrink \Computing ability
\Seminar \Phys. Understanding
|Video tape help |Operation skill
|Others |Creative idea
Fig. 5-1
. - 87 -
6. CONCLUSION
1, System dynamics covers a lot of areas not only in the sense of technology but also of methodology
for decision.
2. For engineers and managers system dynamics is one important factor of their academic potential.
3. Ten years training practice shows the schedule is efficient and attractive. The kernel portion is
"Theory, Demonstration, program, and exercise”. The main problem is the time duration for
training.
4. This schedule (with some changes) also can be used for graduate students.
7. REFERENCES
Brain D. O. Anderson and John B. Moore. 1979, Optimal filtering. Prentice-Hall. , Inc.
Alan Jennings. 1977. Matrix Computation for Engineers and Scientists. Johe wiley & Sons.
JR Acton and P T Squire. 1985. Solving Equations with physical understanding. Adam Hilger Ltd.
Huseyin Kocak. 1986. Phaser. Spinger-verlag New York Inc.
John C. Nash and Mary Walker-Smith. 1987. Nonlinear Parameter Estimation methods.
Nash Information Service, Inc.
MP PER 1976. FSR Bai. AMARA).
He = 1982, BARES RRB, REE BAL).
BHL1991, LAAKRESE. ALM ASHB).
John L. Casti, 1977. Dynamical systems and their Applications; Vol 135. Academic Press, Inc.
Bruce Bohle. 1986. American Quotations; Gramercy Publishing Company. [ + 1]
