A MODERN CONTROL ENGINEERING APPROACH
TO SYSTEM DYNAMICS
R. K. Appiah
Dept. of Electrical Engineering
University of Zambia
Lusaka, Zambia.
R.G, Coyle
System Dynamics Group
University of Bradford
Management Centre
Bradford, U.K.
ABSTRACT
The paper reviews, briefly, the development of
system dynamics (sD) and presents a modern control
engineering approach. It formulates and solves the sD
policy design problem as a model-following control system
design problem in an adaptive control framework. A
computationally simple policy algorithm based on variable-
structure system theory is used in an illustrative example
of the stabilisation of the dynamic characteristics of a
production/raw materials system. Computer simulation
results are given for the modern control approach as well
as the classical sD techniques. Directions in which the
modern control approach could be developed are indicated.
405
2 Introduction
Over the past two decades, system dynamics (SD),
initiated by Forrester [1], has developed into an
effective technique for mathematically modelling and
analysing the dynamics of diverse socio-economic phenomena.
The flow diagram of system dynamics in Fig.1 shows three
paths of development of the subject. Starting with a
given real-world socio-economic process, Forrester showed,
perhaps for the first time, how the basic interactions
between the variables of such a system can be captured as
a tissue of feedback loops in an influence (or causal-loop)
Giagran, With the notions of flows, accummulations and
delays of variables, he provided a nomenclature for setting
up flow diagrams from the influence diagram. He further
@eveloped a computational procedure, DYNAMO, which is a
numerical integration procedure directly suitable for
digital computer simulation,
In the digital computer simulations, variations of
the classical three-term controller have been exclusively
used as policy functions.
In studying certain socio-economic problems, what is
sometimes required is a simple and systematic method of
thinking a problem through, and documenting it, The emphasis
is on qualitative analysis of a complex problem and a
heurgstic appreciation of the consequences of various
decision actions, A methodology for qualitative SD analysis
has recently been developed [4].
406
The thizd approach is that of modern control
engineering. It consists of firstly translating the
influence diagram into an analogue computer flow diagram
and setting up state-space equations from it.
The second stage is to bring all relevant results in
modern system theory and modern controller design techni-
ques to bear on the policy design problem, Digital
computer simulation is use@ as an aid.
A departure may be made from Forrester's precept
that flows and accummulations in socio-economic systems
are continuous-time processes. or even if they are trully
so we may choose to take a sampled-data view of these
processes, Either’way, the influence diagram may be
translated into a digital filter diagram from which
discrete-time state-space equations may also be easily
developed.
This paper reports some initial results in following
the (continuous-time) modern control engineering path in
system dynamics. It has been chosen because Forrester's
Path has the following short comings:
1) it has lea to little attention being paid to the
very important issue of policy design;
2) it has not benefitted from powerful results in
feedback theory, and in modern system theory based
on the structure of state-space models, e.g.
stability, controllability, etc; and
3/..
407
3) it hag not benefitted from modern controller
design techniques.
The authors view their efforts here as an attempt to place
SD in its proper setting of system and* control engineering.
Managed systems constitute one class of so¢io-economic
problems to which the SD methodology has been widely
applied [2].-..We illustrate the modern control engineering
approach with an example from this field.
In a recent paper [3], the stabilisation of the dynamic
characteristics of a production/raw materials system was used
to elucidate some problems of system dynamics modelling of
managed systems. The central points of that paper may be
re-stated briefly as follows:
a) The problem is the interaction between the
production manager and the raw materials manager
in a "typical" consumer goods manufacturing firm.
Orders for goods are received from distributors
and accummulated into an orders backlog which
the production manager attempts to control to a
reasonable level by varying the production rate.
Production uses up raw materials and the raw
materials manager re-orders them so as to keep
stocks to an adequate level.
408
b)
e)
a)
The pattern of orders is generally cyclical
and the pattern of production was also cyclical
with an amplitude roughly one-half that of
orders. Raw material stocks, however, fluctuated
in an explosive manner causing complaints from
raw materials suppliers and problems in the works
and the raw materials manager's competence was
generally in question, See Fig.3 produced by the
model of Fig.2 representing the original system.
The problem is to bring this behaviour under
control,
In Reference 2, system dynamics is defined to be
the application of the attitude of mind, and some
of the approaches, of a control engineer to the
design of regulatory policies for managed systems.
The purpose of Ref.3 was to demonstrate that
viewpoint.
It is recognised, however, that there are
appreciable differences between the design require-
ments for managed and engineering systems so that
the system dynamicist must, perforce, proceed by
a mixture of common sense, experience, and rules
of thumb derived from control engineering practice
in the search for improved policies. This approach
will usually be applied via a simulation model.
Sfevwwe
409
In Ref.3, that approach was applied to generate
three options for alternative control strategies
and it was shown not only that they enabled
stabilisation of the system to be achieved but
that the performance of the system differed
significantly between the options.
This paper addresses the central problem of managed
system analysis, viz policy design, from a modern control
engineering viewpoint, Specifically, it formulates and
solves the SD policy design problem as a model-following
control problem in an adaptive control framework.
The analysis starts in section 2 with the influence
diagram of the production/raw materials system in Fig.2
translated into a continuous-time state-space model. The
policy design problem is formulated in section 3. Linear
model-following control interpretations are discussed in
section 4, together with a policy algorithm derived from
variable-structure system theory.
In the course of the exposition, some observations
are made to highlight some differences of emphasis between
system dynamics and control engineering. The authors view
the differences as posing a potentially rewarding challenge
to the control engineer who wishes to contribute to the
analysis of managed systems.
6/o-
410
Analogue model
The steps followed in forming a system dynamics
model are clearly illustrated in Ref.4. In this exposition
we extend the procedure somewhat to facilitate the develop-
ment of the control engineering viewpoint. Thus starting
with the influence diagram of Fig.2, we proceed to draw an
analogue computer flow diagram for the model using
deviational system variables about some nominal operating
values, Then we develop state-space equations from the
flow diagram.
to develop dynamical equations in terms of deviational
variables, let
wo = NO+w
BLOG = BLOG + x, 2
DBLoG = DBLOG + x,, RMAR = RWAR + u'>
RMS = RNS + x.) 80 = 50 + e,
DRMS = DRMS + x), srM = SRM + e,
aon = FOR + yy apr = APR + yy
where the bar ‘~) denotes a nominal steady-state operating
value. As Ref.3 shows, it is easy to choose nominal values
for the system variables to obtain equilibrium, characterised
by SO = SRM = 0. About equilibrium, the dynamics are
described in terms of the (deviational) state vector, x7
desired state vector, %,; error vector, e; policy (or control)
vector, u; and the exogenous input vector, w.
Teves
ai
It is a basic tenet of system dynamics that the
resource and information flows and accummulations in a
socio-economic system are continuous in time so that
the deviational variables defined above are all analogue.
A procedure is now given for translating an
influence diagram into an analogue computer flow diagram,
using only standard symbols. Level: A level (or state
variable) is the output of an integrator whose inputs are
flows, signed + if they increase the level or - if they
deplete it. See the Table.
Smoothed level: From the influence diagram, Fig.2, a
typical smoothed level is Average Order Rate (AOR) having
the pysmap [5S] equation:
L AOR.K = AOR.J + (DT/AOR)(NO.JK - AOR.J).
The corresponding integral equation is
AOR = fa(NO - AOR) dt , a = 1 /paoR
with analogue computer circuit as shown in the Table.
Delay: Delay in resource or information flow is often
modelled in SD by an exponential delay, typically by a
third-order delay. The DYSMAP equations for a first-order
exponential delay with time delay, DDEL, are:
L LEV.K = LEV.g + DT * (INPUT.JK - OUTPUT.JK)
R OUTPUT.KL =. LEV.K/DDEL
B/...ee
412
The corresponding integro-algebraic equations are
LEV = f(INPUT - OUTPUT) dt
oureUT = 7, LEV
y = lyppen.
An nth-order exponential délay is a cascade of n first-
order delays each with
y = "/ppet,
DDEL being the overall time delay.
The Table shows analogue computer circuits for first- and
third-order exponential delays.
As delay in information flow is a time displacement,
it would seem that a Pade approximant would be a more
appropriate model in such cases. Circuits of Pade
approximants are, however, not given here.
For a linear sD model the units described above, together
with standard analogue computer symbols for potentiometer
and summing amplifier, are sufficient to translate an
influence diagram into an analogue computer flow diagram.
The Table summarises these "building blocks" for ease of
reference. Nonlinearities are easily accommodated in
analogue computer diagrams.
413
In view of the above, then, the influence diagram
of Fig.2 easily translates into the analogue computer
flow diagram of Fig.4, starting with the integrations in
the physical flow modules and adding on the "building
blocks".
Plant and reference model
Controller design techniques in control engineering
require knowledge, in the form of a mathematical model, of
the open-loop system to be controlled, here called the plant.
In a state-space continuous-time formulation a linear plant
is typically described by
E(t) = a z(t) + B u(t) + H y(t) qa)
ult) = c z(t) (2)
where the matrices A, B, H and C may be constant ox time-
varying; 2 is the state vector; u the control vector; va
vector of disturbances; and y the output vector.
To bring control theory to bear on a managed system,
then it is necessary to identify the plant. In managed
systems this identification is not as obvious as in
engineering, We consider as plant the aggregate of parts
of the system, apart from parts generating the desired
state variables, that generate information necessary for
policy design. This will subsume the resource flow modules
and parts of the information modules.
LO/. s+
4t4
Fig.3 specifies the plant with a third-order pipeline
delay, and a reference model (in adaptive control
terminology) corresponding to the production/raw materials
system in open loop.
It must be noted that the system dynamicist has a
flexibility that the control engineer rarely has in being
able to re-structure the system, in addition to designing
policies to obtain a satisfactory overall model. Given the
great complexity, and sometimes severe non-linearity, of
managed systems, it is probably only this freedom to change
the structure of the system which makes policy design a
practical proposition,
Dynamical equations for the units of the system shown
in Fig,3 may be written as follows. Plant equations:
a. YU
6, = -y8, + Y8, a)
fees.
4t5
Reference model equations:
ar 7 7% a + 1 By
(4)
nz * 9 82 Y-
Note that desired values are specified only for
the levels x,, and x4, but not for the levels in the
delay, 0), 8, and @,, Thus the ideal plant (without
delay) is described by:
ete
. (5)
po = Bg 7 By
These sets of first-order differential equations may
be written more elegantly in the vector-matrix notation of
state-space description as follows.
Plant equations:
EO AR+ BUD y (6)
where
416
Without the delay, plants equations are:
+Bou+D ow (7)
yp Be Bw
where oi
aA,= 0,3 = , - sue :
P = eB i =
Reference model equations are:
x =A x +B utD w (8)
Xo 7 An %m * Bh Bt Pm
where =
o 6 a8)
a 1B 1 Dy :
= °
The state generalised error is
ee ee (9)
Delays necessarily increase the dimension of the
plant necessitating some sort of model reduction. The
control engineering literature is replete with linear
model reduction techniques [6,7]. As is generally well-
known socio-economic phenomena exhibit multiple-time-scale
properties and the method of singular perturbation [8,9]
is attractive for dealing with such problems.
As the simulation results show later on, the reduced
model, eqn. 7, is adequate in this case for designing
policies,
1B/aeeee
2.2 Controllability and stabilisability
Policy design using modern control engineering
techniques demands that the plant satisfies controllability
and/or stabilisability conditions.
The system
Zs Az+Bu (Qo)
is said to be completely controllable if there exists a
piecewise continuous control vector, u(t), which steers
the system from an arbitrary initial state, z(t,), to any
terminal state, 2(t,), in finite time, t,~ +t, . Complete
controllability of a system is characterised by the algebraic
condition that the controllability matrix
ae Te, ap, ey a
uc ql)
must be of full rank, i.e.
rank (£) =n (12)
where n is the order of the system.
The system is said to be stabilisable if there exists
a matrix, G, such that all eigenvalues of the modified
system matrix (A + BG) are in the left half of the complex
plane. A system is stabilisable if it is completely
controllable; but the converse is not necessarily true.
It is easy to show that the plant with or without
delay is completely controllable.
BY nes
418
I. Policy in system dynamics
So far in system dynamics, the structure of policy
functions has been no more than variations of the classical
three-term controller
£o=l, eee ym (13)
But the state variables of system dynamics models are all
accessible, except apparently for some state variables of
pipeline delays. Even here, however, there is no limit to
the detailed information which management can collect, if
they wish to do so. It is, therefore, always possible to
re-write the equations simulating the pipeline, to give
access to information about all, or any part of, the
contents. An example of this was OptionIx in [3] in which
a large measure of stability can be induced by allowing the
Raw Material Manager to have access to the Raw Material on
Order, which is the content of a delay.
Policies can therefore be functions of as many
relevant state variables as required to achieve a robust
performance. Robustness in this context means that the
system always behaves as well as possible for any member
of a specified set of input functions - the input space, 2.
And the system is said to be vulnerable if it behaves
unsatisfactorily for at least one member of this input space.
1B/ 605
419
It must be observed here that in managed systems
(as opposed to engineering systems) exogenous inputs are
not necessarily disturbances to be rejected; neither are
they necessarily desired states or outputs to be followed.
A central requirement of a social system is that it should
be able to adapt itself to benefit from favourable changes
in exogenous inputs, and to reject or cope with unfavourable
changes. Such adaptivity must derive from the policies of
the system, or from changes to the structure,
For the production/raw materials system, the policy
design problem may be stated as follows:
Policy design problem: Choose u(t) such that
lim e(t) = 0 ww(t)e Qs a4)
tre - ~
Conventional system dynamics has used linear policy
functions of the form;
ye=GY¥, (15)
where G is a constant policy matrix and ¥ is the information
vector of system errors and smoothed variables defined as
yr = [e" | ; (6)
16/s..e-
420
The structure of G for the original system, Fig. 2, is
° 1.0 0
(a7)
“Voarus 0 1.0
and for option III (Ref 3)
> =
Vensn “Mearns 2°? |
ee] | (18)
-1/ t
° TARMS 0 1.0)
Another possible structure, Option rv, is
Vorasn “Mrarms 1.0
c=
as)
VWrasn “varus 0
Figs. 3, 5 and 6 show the simulation results using eqns.
17, 18 and 19 respectively.
In the following section, policy design from a
modern control engineering viewpoint is presented.
421
Linear model-following control
We recognise the SD policy design problem stated
in section 3 as a linear model-following control (LMFC)
problem assuming only slight system parameter uncertainty.
With poor knowledge of parameter values and/or severe
parameter variations it becomes an adaptive model-following
control (AMFC) problem.
The adaptive control approach is appropriate in such
systems because: 1) parameter estimates in SD models are
often only engineering estimates, ie parameters are not
identified to a high degree of accuracy; 2) parameters of
the model may vary, reflecting changes in the real-world
situation that the model is trying to capture; and
3) compared with the more obvious approach of optimal
control theory, the adaptive control approach addresses the
sensitivity problem directly and also yields computationally
simpler algorithms.
A brief introduction to LFMC is given in this section,
together with a computationally simple algorithm. A
simulation example using the algorithm is also given.
4.1 Linear modei-following control system
In the linear model-following control, the control
system design specifications are embodied in a reference
model whose dynamical behaviour the plant is forced to
18/.
422
reproduce. The LMFC scheme [10] is shown in Fig. 7, in
which the plant is described by
oa; Bou, 20
Bp Bp tp * Ppt a9)
the reference model by
eo =A, e+ Bue (21)
En * An Sm * Pa Sn
The required control is given by
pe Kix +K x + Ky, (22)
e pe Sa Sn u Sm
or equivalently by
pe-K x +K e+ Ky, (23)
5 1%, and x are of dimension n, and
P mo =p
u, and y are respectively of dimensions r and m. rt is
where & “=x
P
a combination of feedback and feedforward controls.
The reference model must be stable, and both plant
and reference model must be stabilisable.
A fundamental problem of LMFC is:
"Perfect model following
Given a set of matrices fa. Bur Ape Boils what
conditions guarantee the existence of the gain matrices
KK, and K, in order that the states of the plant and
P
of the reference model are the same, ie such that
lim e(t) = 97?
tre me
423
It is easily shown [11] that “perfect model
following" is only achieved if the gain matrices satisfy
the following equations
=o (aay
- AL + BS
BoE, 7 Ba TO (25)
And solution of these equations for K
possible if and only if
-B Bt =
@ - BL Bt) BL eo (26)
I - BL BY a =
( fp BY,) (AL > A) = 0 7)
te if the matrix (I - B, Bt) is either
(4) null, or
4 si
(44) orthogonal to B, and to (a, - AQ).
Bt, is the Penrose generalised inverse of B, given by
T -1 i,t
Bt = (Bo B Be
p 7 (8 B,) i. (28)
These conditions were first derived by Erzberger [12].
Equivalent conditions for "perfect model following" are [13]
xank B) = rank , = rALO .
pT rank (By y BL) = rank (BL, A, ~ AL) (29)
Several methods of determining, « , andk, for
perfect model following have been studied in the control
literature [14],
424
4.2 sD model as an LMFC system
The information vector of conventional system
dynamics, eqn 13, may be modified into
(30)
Thus the closed-loop SD model may be represented as shown
in Fig.8, It differs from the conventional LMFC system
only in the following respects:
(4) the exogenous inputs are reference inputs to
the reference model, u, = wy and
(44) some policy functions may also drive the
reference model.
The SD model-following system may thus be described
by
=A + +
Bp" Ap tpt BpEt Pye oy
Xe Ag Xn t Bg Bt OE (32)
with
Br-Sagqetew. (33)
where
6, eG, . (34)
het BL = (8) - BL), and Dd, = (D, - D,) (35)
2VWeeeee
425
Then it is easy to show that for "perfect model-following"
6, 1 G, and G, must satisfy the equations
A, - A, +B, (G6) - Gi) = 0 ET)
mp "ppm
Dd, - BL G, = 0 (37)
mn Bp Sw
Hence the Erzberger conditions for "perfect SD model-
following"
I- BB *) DO =0 38
¢ p Bp 2 De (38)
(BBL) (AL aL) = 0 (39)
SD model-following may be further recognised as
corresponding to the so-called real model-following in
control engineering, In such LMFC schemes, a reference
model is made physically part of the control system.
4.3 Policy algorithm
The various methods of determining K |, K_ and K,
P
satisfying the Erzberger conditions that have been
developed in control engineering also apply to the sD
policy design problem, In particular we present the
“linear equivalent policy" method [15] based on variable~
structure system theory.
This choice reflects our desire to exploit high-gain
feedback design for maximum "passive" adaptivity - the
additional complexity-of “active" adaptive control can
426
only be justified after the full capabilities of the
high-gain feedback design have been explored.
4.3.1 Variable-structure system [16, 17]
The basic idea of variable-structure control systems
is to steer a system
z(t) = aCe) 2(t) + BCE) w(t) + mCE) y(t) (40)
from some arbitrary initial state, z(t,) = 2,, to the
origin of state space, using controls which change their
structure depending on the values of some switching
functions, 5,(z):
uy if 8, (z) > 0
(41)
if 8, (2) <0
La ly coop m
Bach 5,(z) = 0 is a switching plane.
A striking property of such systems is that under
certain conditions, the state can be driven to the
intersection of the switching planes and then slide along
the intersection to the origin, This behaviour is depicted
in Pig. 9 for a second-order, single-input system. The
latter motion is known as sliding motion.
It has been shown that during sliding motion the
system satisfies
(=o, 42 m (42)
8,(z) = 0,
23/seeee
427
It is claimed that variable-structure control systems
with sliding motion are insensitive to plant parameter
variations and to noise disturbances,
4.3.2 Linear equivalent policy
It must be noted that the variable-structure control
of eqn 41 is a fast-switching control which may be
intolerable in certain situations. However, a smooth
control, called "linear equivalent control", ensuring
sliding motion for a time-invariant system, exists, and is
derived as follows:
Consider
xT ata, ut Dy (43)
= BL ut De yw (44)
and @ = x, ~ x. (45)
Then
g-aet+ ray) a tBu yy. (46)
Switching planes in error space are
e=0 (47)
where C is known as switching matrix,
In the sliding mode the system satisfies
se) = 4g)
1.84 ch eg (49)
24/eseee
428
6.3 Example and discussions
or
ee Consider the production/raw materials system with
numerical values as in Ref, 3. The policy matrices are
determined here using the reduced system, eqn. 7.
defined as the value of u(t) which satisfies eqn.50 a fh
B , = ;
ensuring sliding motion. P 1 Lo.
thus -0.25 ° Fo °]
~ 4 ~ ALT . Be ,
ws ‘* - - 51 z ° -0.25 °
ay (eae [>s e+, a - Oy ‘| (51) L |
dee. .
ur - Ge - Get Ge (52)
Bs p Xp” Sn £* Sy ¥
where
6, = PIAL ~ AL) (53)
The Erzbérger conditions, eqns. 38 and 39 are satisfied.
= 2x. (54)
n
bi 1
o,= PD, (55) In the general case there is as yet no systematic
er method of choosing the switching matrix, C. In this
Po = (CB) ¢ (56)
L particular example, though, the following observations
lead to an algorithm which is independent of Cc, if C is
This policy is analytically straight forward and
non-singular:
computationally simple. However, there are no systematic
methods available so far for choosing the switching matrix, rt 3, and ¢ are square matrices of the came order, and
C. such methods are presently being studied. ~
b) BL is involutory, i.e. it is its own inverse.
25/eecee 26/...66
429 430
and the policy matrices are:
0.25 0 -0,25 0
c= Ge . and
ae 0.75 -0.25 P| -0.75 0,25
0.5
rh.8
Fig. 10 shows the resulting influence diagram, and
Pig, 11 the simulation results for the plant without delay.
It should be noted that "perfect model following" is
achieved in this case - something that none of the previous
methods achieved, Note that in the noiseless case the
overall model may be considerably simplified by. removing
the SURPLUS ORDER terms in the policies. Fig. 12 shows the
results for the plant with delay. In this case "perfect
model following" is achieved only in the production sector ~
due to the decoupled nature of the reference model, the
plant approximation affects only the raw materials sector.
However, the production policy that gives zero
surplus order is sinusoidal but, in practice, less
oscillatory policies are to be preferred, The raw materials
ordering policy is also sinusoidal, as in options III and
Iv.
All this indicates that perhaps the SD policy design
problem would be more appropriately re-defined as’an LMFC
problem with constrained policies, ‘This is the subject of
another study,
AW sees
6. Conclusions
It has been shown how, starting with the influence
diagram, an SD model can be transformed into an analogue
computer flow diagram and then into a continuous-time
state-space mathematical model. ‘he SD policy design
problem was then formulated as a model-following problem
in modern control engineering, thereby providing policy
design a proper and firmer basis in an adaptive control
framework. A variable-structure policy algorithm was
derived. The numerical simulation results shows that
"perfect model following" is achieved when the plant is
accurately modelled. Deterioration of performance results
with plant model approximation,
There are several other directions in which the work
outlined in this paper could be developed: The feedback
(i.e. sensitivity) properties of the solution have yet to
be investigated. Model-following with nonlinear reference
models is an area completely unexplored - intuitively one
feels some difficult policy problems in managed systems
could be treated as such, And, of course, the adaptive
control framework adopted here is by no means the only set
of results that the vast and rich subject of automatic
control offers.
432
Acknowledgement
One of us (RKA) wishes to acknowledge his
sincere gratitude to Barclays Bank International
for providing the money that sustained him through
a period of study leave, and to the Royal society
for awarding and administering this sponsorship.
This work could not have been done without their
support.
433
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3
synthesis techniques", Proc. IEE, Vol. 117, No. 3, 28
5s
pp. 623-627, 1970. ca)
K.K.D. Young, "Design of variable structure model-
following control systems", IEEE Trans Autom. Control,
Vol. AC-23, pp. 1079-1085, 1978.
V.I. Utkin, "Variable structure system with sliding
modes", IEEE Trans Autom, Control, Vol. AC-22, 2 w
& $
pp. 212-222, 1977. 5 3
Q 3+——#e
225: we
A.S.I. Zinober, O.M.E, El-Ghezawi and s.A. Billings, go za
frame} =
"Multivariable variable-structure adaptive model-
following control systems", IEE Proc., Vol. 129,
Pt. D, No. 1, pp. 6-12, 1982.
SOCIAL .ECONOMIC
REAL WORLD
PROCESS
QUALITATIVE
JANALYSIS.
FLOW DIAGRAM OF SYSTEM DYNAMICS
Fig. 1
g
Fig.2 ORIGINAL SYSTEM INFLUENCE DIAGRAM
| \
Hl Vi Yao Ay
i Ven ee
arr) SS; ee
(hy “semeere
26 WEEK SINE VAVE FO? ORIGINAL SYSTEM
MODEL OF PRODUCTION/RAW MATERIAL STABILISATION
438
FUNCTION
ANALOGUE COMPUTER CIRCUIT
COEFFICIENT
MULTIPLIER
qd
INPUT 6 OUTPUT.
OUTPUT = q * INPUT, a20
439
INPUT,
SUMMING :
AMPLIFIER INPUT) oe
UTPUT bs INPUT,
ol = t A
i
INPUT, = gf
INTEGRATION : oureur
(ACCUMMULATION) INPUT) o——————
(RATES) (LEVEL)
INPUT — “ DP
T——* OUTPUT
SMOOTHED LEVEL IRATE) ° (LEVEL)
& = ‘/AVERAGING TIME
FIRST ORDER, INEUE: emit ley ot OUTPUT
EXPONENTIAL DELAY, (RATE) tRaTe)
t= Yooet
INPUT
4 OUTPUT
THIRD ORDER i
EXPONENTIAL DELAY] _
¥ = 3/DDEL
TABLE .
REFERENCE MODEL
* f
i
.
z
<
s
oo
é|
j i
ZL
x “¢
~
a z ~
fon
‘
4 l
3 > x
440
FIG.4: OPEN-LOOP SD MODEL
od ea eee
fh a a
{t TIME ony stmnatea re
if NO 07mm wed ORE RATE
Lee {AR PROUETION ATE
pur {AMO AY EATER ARIAS RATE
4 cee
st
Br
i 4
rr So a ro
{ii TIME ane srenatea roe
ij !Levos (are stenos
il 1 -08L06 eo oesreeD exe pe
ens ay paren
aes i besiaed may niteia,sToces
OPTION JI
Fig. 5+ MODEL OF PRODUCTION/RAW MATERIAL STABILISATION
441
jl
It TIME om simnaren rine
ie unre ev oxen asre
Ley me bronco te
Rae Wat Rav meta Ue, eae
poy a er .
a a se 7
TIME wae srmzaren rime 7 fo foo,
FiGb:
OPTION IV MATRIX 1
MODEL OF PRODUCTION/RAV MATERA! STABILISATION
442
vey
Xp 4X
Ku PLANT =O
z g
&
Kp — Km
Km
Fig. 7: LINEAR MODEL-FOLLOWING CONTROL (LMEC) SYSTEM
WAISAS 941 NV SV TSGOW SOINVNAG WISAS <8 514
Wo
Wg — do
INVId
ax
4 Ms,
hi O- O- we
7300W :
wx aay ad
3l
Lf few ON
cme a ee a J
Fig. 10 LINEAR MODEL-FOLLOWING SD MODEL
22
3OvdS 3IVIS NI NOLOW ONIGIS 6't
445
* o=(zIs
FIG. I
sof
e
x a 10. 20. Ey 40. 50, Li 70.
It ine
| fhng Ud) NEY 08228 RAE
Lape ‘ih passion nice
“RMS HA RU MBN rr RE
LINEAS EQUIVALENT POLICY: PLANT WITHOUT DELAY
MODEL CF PRODUCTION, RAW MATERIAL STABILISATION
447
1
mii
a OF
|! TIME
yg (ye ve een me
LAR tna Raven a. are
2000,
!
Ht
1750, T ‘
lf
1
1500, +
ppp 8 8
SiisiSiees
©
Tool
BS
esee
i
FAG [22 Linear EQUIVALENT POLICY
HODEL OF PRODUCTION/RAW MATERIAL STABILISATION
448
List of principal symbols
No
BLOG
PR
DBLOG
AOR
TAOR
COVERO
so
RMS
RMOR
RMAR
DDEL
APR
TAPR
COVERM
DRMS
SRM
TABL
TARMS
NEW ORDER RATE
ORDER BACKLOG
PRODUCTION RATE
DESIRED ORDER BACKLOG
AVERAGE ORDER RATE
ORDER AVERAGING TIME
WEEKS OF AVERAGE ORDER IN: DBLOG
SURPLUS ORDER
RAW MATERIAL STOCK
RAW MATERIALS ORDER RATE
RAW MATERIAL ARRIVAL RATE
RAW MATERIAL DELAY
AVERAGE PRODUCTION RATE
PRODUCTION AVERAGING TIME
WEEKS OF AVERAGE PRODUCTION IN DRMS
DESIRED RAW MATERIALS STOCK
SURPLUS RAW MATERIALS
TIME TO ADJUST BACKLOG
TIME TO ADJUST RAW MATERIAL STOCK
plant state vector
reference model or desired state vector
state generalised error vector
exogenous inputs vector
policy vector
vector of smoothed levels
449
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