The fundamentals of the structural diagram method for distributed parameter systems (DPSs) are presented and reviewed. An example is given to illustrate the application of this method for control design

Hartley, Tom T.

Lei, Xia

English

NASA Technical Reports Server

WP15
Temperature
17:00
Control in Continuous Furnace by
Lei Xia and Tom T. Harfley
Department of Electrical Engineering
The University of Akpon, Akron, OH 44325-3904
F¸ -_.
N92 '
Structural Diagram Method
Abstract
In this paper, the fundamentals of the structural
diagram method [1] for Distributed Parameter Systems
(DPS) are presented and reviewed. An example is given to
illustrate the application of this method for control design.
Acknowledgement
This research was partially supported by the Advanced
Control Technology Branch of NASA Lewis Center via
Grant NAG 3-904.
I. Introduction
For lumped systems, practical control methods are
developed based either on the transfer function or the state
space representation of the system [3]. Although there are
many inherent connections between these two methods, there
exists many differences between them in terms of
mathematical analysis tools, design methods, and
engineering interpretation, etc. For DPS, howevcr, control
methods are predominantly based on the state space
representation of the system [2]. It seems that uhis approach
parallels that developed for lumped systems, but it is
difficult to understand for practicing engineers, and to
irr, plcmcnt a physical controller without an intensive
functional analysis background [2]. The structural diagram
method for the DPS (the counterpart of transfer function
method for lumped systems), on the other hand, should be
easier for practicing control engineers due to its simplicity,
convenience, and clarity of representation of the systems.
Also, the popularity of this method should be enhanced by
the fact that control engineers are already familiar with
complex variable methods from their experience in dealing
with lumped system.
In this paper, the fundamental concepts of the structural
theories are reviewed in an attempt to explore the
methodological value of a generalization of the structural
theory of DPS for practical control design.
II. Structural Diagram Method
The structural diagram method developed in [1] is
briefly reviewed here.
1. Distributed signal
In DPS, a distributed signal depends not only on time
but also on certain spatial variables. It then has ',he form
w(x,t), xeD, where D is a spatial domain. The Laplace
transform of this signal with respect to time is w(x,s).
2. Distributed block
U(x,s)_Y(y,s)
In order to describe the input/output relationship of a DPS,
a distributed block, an analog to the transfer function of a
lumped system, has the form W(y.x,s), xeDl, yaD2 for a
stationary process (the coefficients in a PDE do not change
with time), where DI and De are the domain for the distributed
input and output signal respectively. Therefore, the
relationship between any two places in the space is reflected
by inserting in their position in W(y,x,s). This is then the
Laplace transfer of the Green's function of the given DPS. The
relationship of the input and output is not simply a product of
the input signal and transfer function _ in the lumped system,
but instead, a composition of the two, which is defined by:
f
Y(y,s) = W(y.x,s)_U(x,s) = l_ W(y,_:,s)U(x,s)dx
2. Connecting blocks
A complicated DPS may involve more than one block,
The operations among them then need to be clarified. The
following three kinds of operations often arise in control
design.
Connections in series.
L,. , .._[,,,. I,x,slL_t ...... .x,,_l_--, :,_)
_,x,S,--,.- i,, .(x V_,v -1 _ _ ", , .
If two blocks are connected in series as shown above, the
overall block is equivalent to:
/,
W(y,x,s) = Wl(x_,x,s)_W2(xt,x,s) = I Wl(xl,x,s)W2(y,xt,s)drl
),,
This is again, different from the lumped system where the
overall transfer function for two systems connect_ in series
is simply the product of two individual transfer functions. In
addition, the operation of composition is non-commutative in
__lseneral. If both block functions are known, the overall block
obtained after conducting the conventional integral
" operation.
Closed loop s'stem.
U(x's) 7"-q_Wlt---.-._V2(x,y,s)(Y'X'S) I_ _-Y(y,s)
Based on the block operation introduced before, it is easy to
show that the closed loop transfer function W(y,x,s) has the
following general form:
I
https://ntrs.nasa.gov/search.jsp?R=19920013254 2020-03-17T12:00:53+00:00Z
W(y.x,s)= W2t(y.x,s)@W(y:c,s)+ Wl(y.x,s)
W21(y.x,s)= Wl(y.x,s)@W2(Y:C,s)
For specialtypesof Wl(y.x,s)and W2(x,y,s),W(y,x,s)may
be simplifiedgreatly.
Lumped Regulator
Following figure shows that the interaction between a
DPS and a lumped parameter regulator is represented by two
S functions. The closed loop wansfer function from U(x,s) to
Y(a,s) is simplified to:
Wl(a,x,s)
W(a,x,s) = l.lt(s)i ° Wl(a,x_)dr
U(x,s)_
_J_"-Y(y,s)
4. Transfer function of DPS
The transfer function of DPS is closely related to the
attached boundary condition, as weU as given PDE. This is
what differentiates a DPS and lumped system. The transfer
functions of many PDEs with popular boundary conditions
are given in [1]. For a general PDE with arbitrary boundary
conditions, one needs to find the new transfer function by
solving the given PDE with its boundary conditions.
III. An Example
The following figure shows a furnace of length l=1
v, hich heats a continuous stxip of material passing ,&a_:gh
the furnace with speed v=l.
Furnace N4,/////] "IV[// . //_ .,,Matefi-A
_d ---_v r A_" being heated
_lllllllllltlllllllltllllllllllllllllll
Let the temperature field of the furnace be uniformly
distributed in space. Suppose the temperature distribution in
the heated strip isQ(x,t). Then for a normalized system, u(t)
and Q(x,t) are connected by a first order linear partial
differm_tLal equation
_W2(x,t).. _(+_.__+a(x,t) = u(t), O_-z_-I, t->0
at 3x
with boundary condition Q(O,t)=d(t).
Two specific control tasks can now be considered. One
is to regulate the temperature at the output of the furnace.
This is considered first below. The other task is to heat the
material to a specific spatial temperature distribution.
For the first control task, the control equation has the
form U(s)=W(s)[Q(I,t)-Q*(1 ,t)]. Then the control diagram is:
The system written in standard form is:
c)Q(x,t!._, °_Q(x't) +Q(x,t) = u(t)+S(x)d(t)
8t &
_,, - Q* (1 ,s)
F__ Q(1 ,s)
The open loop transfer function W(y.x,s) for the given plant
and boundary conditions is easily derived or found by looking
in the table in [1] to be
W(y,_,s) -- e 4*+t_r_
The closed loop transfer function from d(t) to Q(1 ,t) is
W (1,0,s)
Wl(s)=
1+ll(s) W(l.x,s)d_
0
(s+1)e_'*o
= (s + 1) +ll(s)(l -e _'* t:)
The closed loop transfer function from Q*(t) to Q(l,t) is
f'It(s) W(1.x,s)dx
JOW2(s) =-
1
I+11(S)io W(1 ,x,s)dx
ll(s)(l-e _'*J))
=(s + l )+l f(sX l-e ¢..I))
The response of Q(l,t) is then
Q(1,s) = Wi(s)C_.d(s)+W:(s)_*(l,s)
i2,_c ch._ac_cris'2c eq,_tion of uhc s)stcm is
_s) = (s + l )+l l(s)( l-e'_'z))
based on which, we can carry on control analysis and design.
All this can be done using conventional control techniques,
such as root locus, Bode plot, etc [3].
Consider a proportional controller where lf(s)=k. The
poles of the closed loop system are shown in Fig. 1. Notice
that there is an infinite number of open loop zeros and one
open loop pole at s=-I that is cancelled by the zero at the
same place. It is seen that for all positive k, the closed loop
system is stable. The closed-loop poles for a general PID
controller are shown in Fig.2 where
H(s) = Kill + I_.L-+Tas)
T_s
Although it is more tedious to implement control
analysis for DPS using conventional techniques, DPS obey
the same rules that a lumped system does. Those rules
include:
(i) The damping ratio and stability decrease with the increase
of Kp, system has a faster response;
(ii) Integral control reduces or eliminates steady-st,am errors at
the expense of reduced stability;
(iii) Derivativecontrolimprovessystem stability.
It should be pointed out that while all of these control
techniques are familiar to most of control engineers, it is the
structural diagram method that makes it possible to apply
this body of knowledge to DPS.
The step responses for proportional control where (1)
k=.5 and two cases of PID control whea'c (2) Kp=.01. Ti=l;6,
Td=2, (3) Kp=.05, Ti=]l]O, Td=l am shown in Fig. 3. The
simulations were performed on the transfer functions using
Euler's method to replace s (s=(z-l)lT) and e'S=r ',, where
n=l[T. The time step used is .001 sec.
The other classical approaches, such as lead and lag
compensators add little significant information.
Another more complex control problem is to regulat_
the temperature distribution along the whole x direction.
For this case, the chosen error signal has the following
form:
I
e(s) = I (Q(y,s) - Q*(y,.r))dy
The control signal is then U(s) = W(s)e(s), where W(s) is
the lumped compensator. Using the same *techniques as
above, the sysu_m response can be regulated to the desize.d
specifications for a reasonably given expected signal Q*(y,s)
(obviously, Q*(y,s)=constant is not a reasonable choice
because of the uniform distribution assumption of control
signal u(t)). Fig.4 shows the steady states of the
temperature distribution of the material being "cooled" for
two cases of control design.
This system will be considered further in a later paper
where robust control methods [4] and functional observers
will be considered.
IV. Summary
It has been shown that the structural theory for DPS
has many properties similar to those of lumped systems, as
well as some substantial differences. "T/w..seare mostly related
to having non-rational transfer functions. Unfortunately,
each particular type of PDE has a correstxmding non-rational
function of the complex variable s (eg. J(x,s), sinh(x,s), etc).
Thus each particular system requires unique special study to
determine its closed loop poles as root locus methods will
not generally be available.
V. References
[1] Butkovskiy, A. G., "Structural Theory of Distributed
Systean', Ellis Horwood Limited, 1983
[2] Cm'min, R.F., and Pritchard, AJ., "In/'mit_ Dimensional
Syswan Theory', Springer-Verlag, 1978
[3] Kuo, B. C., "Automatic Control Theory', Prentice Hall,
Inc, 1982
[4] Maciejowski, J2CL, "Mtthivariabl¢ Feedback Design',
Addison-Wesley Publislu'ng Company, 1989.
k --,_ k -.-_O
k.--_ =.._ 0
k....,,oo r
k -+** Y k -_
k Im(s)---'-"--
v=j
•2:rj Re(s)
_=j
Fig. I Closed-loop Poles of Proportional Control
k_=, .._ k--,O
k-=_o r k_9
k ----_ ""- k--%0
[_(s)-
_Jrj
._uj
•2_j Re(s)
4_rj
Fig. 2 Closed-loop Poles of PID Control
1.4 r .
1.2
_=
::
 0ry .........
0-4 I
z 0:II
F
0lt
............................... o.. .
<2)
..."i:::::::::::::::::::::::::::::::::::--::-:--:::-::-..::.:_.... ... .....!
:" (1) [
" 1
I
(1) k=o.5
................. (2) Kp=OI. T,=I'6. Tf.2 I
(3) Kp=.05. T,=///O. T,e=l t
0 $ 10 15 _0 L_
Time (see.)
Fig. 3 Step Response of Closed-loop System
3O
4 t ' , . .
J_ .............:'..-._==..... , ..... _2__¢.=.5,r,=.50,r.,_ .......j
. ............... ...................
0
o _1 0.: o.3 0., az 0.6 07 0., _
Normalized Position
Fig. 4 Steady States of Temperature Distribution


Temperature control in continuous furnace by structural diagram method

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