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Comparison and application of DDC algorithms for a heat exchanger
Automatica, Vol. 12, pp. 393--402. Peqpunon Press, 1976. Printed in Great Britain
Comparison and Application of DDC Algorithms for a Heat Exchanger* H. U N B E H A U E N , t
C H R . SCHMIDi" and F. B ( S T T I G E R t
Seven DDC-algorithms, applied to the temperature control o f a heat exchanger in a comparative study, show which controller is the most suitable one. summmTmln recent years, several algorithms for Direct Digital Control (DDC) have been proposed in literature. Although some of these, such as PID or cascade controllers, are very commonly used in industrial applications, the more recent ones like optimal state feedback controllers using an observer or parameter adaptive controllers have rarely been applied to a real plant. The primary difficulty behind this application has been perhaps the lack of testing such algorithms on a pilot plant. Moreover, there has been no serious attempt to make a comparative study of the merits of such algorithms for an existing plant under actual operating conditions• In this paper, seven DDC algorithms are applied to the temperature control of a heat exchanger. These algorithms are: PID, cascade, compensation (pole assignment), deadbeat, half-proportional, adaptive and optimal state feedback controller using an observer. The system performance and sensitivity with respect to changes of the plant parameters, disturbances and set point variations are investigated for the heat exchanger using these algorithms. The results indicate that the more sophisticated algorithms, e.g. optimal state feedback, compensation and adaptive controllers, requiring more computer time and memory, yield relatively less improvement when applied to a low-order plant than do the simple algorithms such as PlD or cascade. It was deduced that the PID controller with anti-windup is the most suitable one.
This leads directly to the p r o b l e m of sensitivity of a control s y s t e m to its operating conditions. The increasing application of digital c o m p u t e r s for process control permits the implementation of various control algorithms using c o m p u t e r programs. Many DDC algorithms h a v e been p r o p o s e d for process c o m p u t e r s during the past few y e a r s [ I - 5 ] , however, there are hardly any c o m p a r a t i v e investigations on the efficiency of these algorithms for practical applications. T h e r e f o r e , this p a p e r c o m p a r e s s o m e of the most important DDC algorithms with respect to p e r f o r m a n c e , efficiency and sensitivity as applied to the t e m p e r a t u r e control of a heat exchanger. 2. DYNAMIC BEHAVIOUR OF THE HEAT EXCHANGER The process used for testing the different control algorithms is a pilot plant representation of a high-pressure s t e a m superheater shown in Fig. 1. It consists of an electrically heated brass tube (¢~ 4.5 ram) with c o m p r e s s e d air as the flow medium. H e r e , the object is to regulate the output t e m p e r a t u r e O~ of the process stream. The two t e m p e r a t u r e s 0 , and OA are m e a s u r e d by t h e r m o c o u p l e s while three input variables
1. INTRODUCTION MANY chemical and p o w e r plants have to be controlled such that their r e s p o n s e s are tolerable under various operating conditions with variations in plant p a r a m e t e r s and disturbances. H o w e v e r , a controller which is optimal for one load level m a y in fact provide p o o r or even unstable responses under other conditions. With this in mind, it is desirable to know, whether a controller which was designed f o r one special operating point is also a p p r o p r i a t e for others.
dist ur bonce himting system A
*Received 9 July 1974; revised 18 February 1975; revised 25 August 1975; revised 12 February 1976. The original version of this paper was presented at the 4th IFAC/IFIP International Conference on Digital Computer Applications to Process Control which was held in Zurich, Switzerland during March, 1974. The published Proceedings of this IFAC Meeting may be ordered from: IFAC/IFIP Conference, Gloriastrasse 35, CH-8006, Zurich, Switzerland. This paper was recommgnded for publication in revised form by associate editor H. A. Spang HI. tLehrstuhi f~r Elektrische Steuerung und Rngelung, Ruhr-Univer~tlit Bochum, !)-4630 Bochum, Universitlitsstr. 150, IC 3, W. Germany.
Fro. 1. Schematic diagram of the direct digital computer control of the heat exchanger. 393
AUTO. VeL 12, No. $--A
394
H. UNnEH^UES et al.
(variations in the flow rate AM, in the heating rate AQ and in the inlet temperature AOE) all influence the variation AO^ in the outlet temperature. Step changes of Aft;/ and AQ can be obtained directly by means of a valve and the electric heating element, respectively. Rapid variations of &OB, however, are only possible by step changes AQs of a separate efficient electric preheater, the disturbance heating system shown in Fig. 1. Figure 2 shows a detailed block diagram of the heat exchanger system. The most important normalized transfer functions are
sufficiently accurate and much easier to deal with. The identification method used to obtain the models was numerical Fourier transformation of the transient response and approximation of the Bode plot by a rational transfer function. This model does not take the nonlinearities of the actuating element into account. The nonlinearities are of a static type and have only a small effect on the system responses. 3. DDC ALGORITHMS
In the following subsections, seven DDC algorithms which are used in this paper are briefly described.
2.22
Gos(s)G.~(s) = (I + 370s) (I + 118s)
(l)
0.73 G , . ( s ) = 1 + 232s
(2)
0.5 GQ~(s) = (1 + 59s) 2
(3)
3.1 PID control algorithm (I) For implementation on a process computer, the transfer function of the PID controller with a low-pass filtered differential channel given by _.2'{u(t)}.~{e(t)}= Ks • .(1 + ~ ,1s + l ~ VT~s \ ~s )
Os (s ) -
0.285 GQ.(s) = (1 + 20s) (1 + 100s)"
They were found experimentally by measuring and analysing input-output relationships for a flow rate of 35% of the maximum value. In order to find the optimal controller settings of the cascade control system, a process model described by (1)--(4) is used. For synthesizing or tuning the other control algorithms, the following reduced model transfer function was proved to be satisfactory AOA(S) 0.65 Gs(s) = AQ(s) (1 + 2 5 s ) ( ! +213s)"
(5)
Although the process is actually a distributed parameter system with time delay, this type of model has not been used because the delay time is insignificant and lumped parameter models are
Y
I~'-J
Ik-~
l'
-
--:u !1 ~']'---lo',=j=
cascaded control system
Fio. 2. Block diagram of the direct dillital control of the heat (~xchal)S(~.
must be transformed to a discrete form
~{u(k)) d o + d , z - ' +d~z -2 G s ( z ) - - ~'{e(k)} = 1 + c,z-' + c2z -2"
(7)
where u(t) denotes the control signal, e(t) is the output error defined as e(t) = w(t) - y(t), w(t) is the reference input, and Kt, T~, To and Tv are controller gain, integral time constant, derivative time constant, and filter time constant, respectively. The coefficients c, and d~ in (7) are functions of parameters Ks, 7"i, To, Tv and the sampling period T. In order to prevent reset windup in this controller, the integral action is reduced to zero (TI ~0o) if the control signal exceeds the upper or lower bounds of the actuator. In sequel, the controller without antiwindup will be referred to as Ia while the controller with anti-windup is denoted by lb. Applying this control algorithm to the reduced model given by (5), there are only marginal optimum values for the controller parameters Ks and Tt, [6, 7]. If the behaviour of the actuating element is taken into consideration, it can be seen that these parameters cannot be chosen to be greater than Ks = 15.4 and To = 10.0 sec. Then, Tt ffi 38.0 sec and Tv ffi 13.5 sec are optimal values in the sense of minimizing the ISEcriterion for step disturbances with a sampling period of T = 5.0 sec.
_&
CmlsV
....
(6)
(4)
3.2 Cascade control algorithm (II) This control system consists of a master controller (PI action) with the transfer function
Comparison and application of DDC algorithms for a heat exchanger
(8)
395
compensation network Co
GK = P(s)
and an inner-loop controller with the transfer function
Gs, = Ks,.
(9)
These two controllers are cascaded as shown in Fig. 2. Like the PID-controller, this algorithm has been used with and without anti-windup and will be referred to as controller IIb and Ila, respectively. The algorithm was tuned for step disturbances • Qs by a simulation study using the model given by (1)-(4) and the system described in Fig. 2. The final parameters used in the experiments are gain factors Km~ = 3.1, Ks2 = 7.7, integral time constant Tn = 60 sec and sampling period T ffi 5 sec. 3.3 Compensator control algorithm (III) Considering an nth-order transfer function zo+ z~s + . •. + z,s'
Z(s)
G,(s) = no+ n,s +'-: ~ -+-M'-" n,s = N(s)'
i ' . L--
~ .
.
.
.
.
.
Zls) l
GsIs}--N~- i .
.
.
I. . . . . . .
|
J .-]
Fro. 3. Block diagram of the compensator controller.
(II)
where P ( s ) and Q(s) are similarly polynomials of degree m. The coefficients po, p,, q,, i = 1, 2. . . . . m are determined from the characteristic equation of the closed-loop system, given by m+R
II (s - s,) = 0,
is connected in series with the closed-loop as shown in Fig. 3. For the practical realization of the controller components Gs(s) and GK(s), they are described in state-space notation with input-signals y(t) and w(t) and output signal u(t). The design of this control algorithm is based on the simplified model given by (5). The closed-loop system will then be of fourth order. The poles s, = s2 = s3 = - 0 . 0 5 and s , = - 0 . 1 are chosen with regard to the upper and lower bounds of the actuating element.
(10)
of the plant where ! < n and Z(s) and N ( s ) denote polynomials of degrees 1 and n, respectively, a controller is to be designed such that the closed-loop system has a prescribed distribution of poles s, in the s-plane (pole assignment). The transfer function of the controller is assumed to be of the form
po+ p,s + . . . + p = s = " = P(s) Gs(s)ffiqo+q,s+..+q=s ~-~
(13)
(12a)
I-I
or, alternately using (I0) and (I I),
3.4 Adaptive control algorithm (IV) The over-all behaviour of a closed-loop system with time varying plant parameters p,(t) can be specified, to some extent arbitrarily, by the model transfer function, o,,[s,
g(t,)] O.[s. igt)]
Ou(s) = 1 + Gs [s, ~t~)] Os [s, p(t)]
where Gs[S,p(t)] is the plant transfer function with parameter vector IKt), and Gs Is, ~(t,)] is the transfer function of the adaptive controller with the corresponding estimated parameter vector ~(t~) at discrete time tk. The variations of the parameter vectors p(t) and ~(t~) are assumed to be slow in comparison with the plant dynamics. The result of an on-line identification of the unknown plant can be used directly for synthesizing the controllers transfer function
1
G,,(s)
G , [ s . # ( t , ) ] ffi G s [ s . # ( t , ) ] " l - G , , ( s ) "
N ( s ) Q ( s ) + Z ( s ) P ( s ) = 0.
(14)
(]5)
(12b)
A comparison of the coefficients of these two characteristic equations leads to a linear system of simultaneous algebraic equations for the controller coefficients [8]. For m = n, this system has a solution with qo as a free parameter. If qo ffi 0 (integral action), the control behaviour under disturbances is improved. In order to compensate the zeros of the closed-loop transfer function for reference input, an additional
In this way, an adaptive control system is obtained, whose dynamic behaviour follows approximately the given model. Therefore, the task of the adaptive controllers is to compensate Gs[s,~t)] with llGs[S,~(tk)] as closely as possible. In this case, both feedback signals of the closed-loop system, according to Fig. 4, are equal and thus the overall system follows the model (14). Also in the case of incomplete compensation, both feedback-loops accomplish
396
H. UNBEHAUEN
et al.
~(k ) = ~(k - l) + K(k )[y*(k )-mT(k )fJ(k - I)]
---
I P(k) = ~[P(k - l)-K(k)mT(k)P(k
uitk),~
glt k)
- I)]
(18)
J : K(k) = P(k - l)m(k)[l + mT(k)P(k - l)m(k)]-'.
I
processcomputer FIG.4. Block diagram of the adaptive control system.
significant improvements in the over-all system behaviour. Thus, the adaptive control system [9] is very efficient. For on-line identification the method of linear filtering[9] is used. In this case, the input signal u ( t ) and the output signal y(t) are each filtered by a linear filter Gp(s). If the dynamic behaviour of the process is approximated by a third-order model, Gsu[s, Ks(t), a.(t ), a,(t), a,(t)]. Ks(t) 1 + a,(t)s + a2(t)s2+ a,(t)s 3
(16)
then a set of equations for obtaining the process parameters will result from the filter output signals u*(t) and y*(t):
Rules for the choice of the inital conditions of P(0) and ~(0) and the weighting factor ~ can be found in[10]. In order to calculate the filtered signals u*(t) and y*(t), the transfer function of the filters $
G,,(s)=(l + Ss)(l + 25s),
was realized by the process computer. In the same way, the control signal u ( t ) can be obtained using the model transfer function 1 G u ( s ) = (1 + 20s)'"
(20)
3.5 Half-proportional control algorithm (V) This nonlinear half-proportional (HP) control algorithm can be described analytically by the following relationships [ l I] u(k)= u(k - I)+ Au(k)
K s ( t ) u * ( t ) = y*(t) + a , ( t ) : * ( t ) + a2(t):*(t) + a,(t)g*(t)
(19)
(21a)
with
KmAe (k) for [e (k)J - le (k - 1) > 0 (increasing error) Au(k) =
0 for le(k)]- le(k - 1)1 < 0 (decreasing error)
(21b)
Kc[e(k ) - P, sgn(e(k ))] for e(k ) = e(k - 1) . . . . .
or in vector notation: y*(t) = mT(t)l~t)
le(k)l> P,
and (17)
with roT(t) = [- :*(t),- 9*(t),- 7*(0, u*(t)] and pT(t) = [a,(t), a~(t), a3(t), Ks(t)]. From (17) a system of simultaneous algebraic equations for the parameter vector p for k discrete time intervals is obtained. This system can be solved recursively for each value of k ---tJT
e(k - N ) and
Ae(k) = e(k)-e(k - I).
(21c)
Here, the gain factor KR represents the intrinsic parameter of the controller, whereas the factor Kc, the error tolerance P, and N are of minor importance. The optimal controller setting Km was obtained for the reduced model (5) under step disturbances of AQ, minimizing the IAEcriterion. The optimal values for the parameters were found to be K , = 8.47, Kc -- 3.08, P, -- 0.05 and N ~ 5 with T = 25 soc[12, 13]. 3.6 Deadbeat response control algorithm (VI) This method is based on the computation of a piecewise constant control signal for a system of
397
Comparison and application of DDC algorithms for a heat exchanger n th order, so that after n sampling instants, the response to a unit step w will reach its final value without any overshoot[14]. The control signal is obtained for the reduced second-order model (5) by
and the performance index must be modified as follows
u ( k ) = hoe(k) + h , e ( k - 1) + h2e(k - 2) + g , u ( k - 1)+(1 - g , ) u ( k - 2 )
where in fact v(t)--,O as t --,~. Using the output error as a state variable
(22)
with gt = y ( T ) l w . For computing the coefficients h,, the following system of n + 1 equations
[ = f : [xT(t)Qx(t) + v2(t)] dt
(26)
(27)
x,(t) = e(t),
the state vector converges to zero, x ( t ) - , 0 as t ---)00.
ho + h, + h2 = l / K s ho + h t e TIT' + h2 e 2TIT' --- 0
(23)
For the design of the optimal state feedback controller according to (25)-(26), the reduced model (5) is unsatisfactory. A fourth-order model
ho + h, cTIT=+ h2 e 2TIT2 = 0
must be solved. The sampling period T is the only design parameter of the algorithm and must be chosen for some prescribed reference step input with respect to the actuator bounds. Accordingly, the investigations on the heat exchanger had to be conducted with T -- 75 sec. 3.7 Optimal state feedback controller with observer (VlI) An optimal linear state feedback controller for a single .input system is designed to minimize a quadratic performance index
I = f: [xT(t)Qx(t)+ u'(t)]dt
bo + b,s + b2s 2 G s ( s ) = a o + a , s + a 2 s ' + a , s 3 + s 4'
including the description of the long-range behaviour, proved to be adequate. Considering the additional integral term of (25) as part of the process, the following state-space representation can be found
x(t) = Ax(t) + by(t)
(29a)
e (t) = cTx(t)
(29b)
with
(24)
I
00 0- b o A--0
where x ( t ) is the state vector of order n and Q is
a positive semi-definite symmetric weighting matrix. This optimization problem has a solution only if the criterion (24) is finite i.e. if u ( t ) - , O and x(t)-,0 as t - , ® . For a system without integral action and with constant disturbances or setpoint changes an integrator must be introduced as shown in Fig. 5115]. Thus, the control signal is given by
(28)
0 -
ao
1- b l 0
0- b , 1
0 0
0
0
I
-
at
-
02
-
1 , a3
(30a)
bT=[0
0 0 0
I],
(30b)
cT=[I
0 0 0 0].
(30c)
The optimal state feedback controller v(t)
=
rTx(t)
1
u(t) ul0)|
=
u(0)+
f0 v0")d~',
(25)
was determined from the solution of the algebraic matrix Riccati equation
~'c~ess
WbbTW - WA - ATW - Q = 0
(32)
r T = bTW.
(33)
with ,_.
II ~
2 r _°ee_ss..c _°~ P_ut_. ~ .
~tk|varmble
.
.
~
.
.
.
reduced
FIG. 5. Reduced obeefver and optimal state feedback controllet for the heat exchanger.
For the identification of the unknown state variables x2(t) to x,(t), a reduced Luenberger observer[16] of fourth order is constructed, whose eigenvalues are larger than those of the plant. The weighting matrix Q was found by
398
H. UNBEHAUEN et al.
simulation studies in order to get good results for setpoint changes as well as step disturbances. Figure 5 shows the complete schematic diagram of the optimal state feedback control system. Most of the algorithms described are based on design procedures leading to continuous controllers. In order to obtain a discrete form of these controllers for process computer implementation, an approximate z-transformation, for the PID controller and the cascade controller, and a discretization procedure by power series expansion of the system matrix, (for the compensator controller III and adaptive controller IV), were used. These controllers can only operate with small sampling periods T, because the effects of the zero-order hold are not considered in the model or design procedure. Both the deadbeat response and the HP controller, on the other hand, are designed directly as discrete algorithms including the zero-order hold in the model. Therefore, the sampling period T, which is used as an additional design parameter, can be considerably larger than in other cases.
coincide with the assumptions under which the algorithms were designed. The tests were carried out for step changes of flow rate M preheating Qs heating rate Q, (control signal) setpoint and for disturbances due to pseudo random binary signals (PRBS) in the flow rate M both at the 35 and the 80% operating points. In Table 1 the essential results of the experiments are summarized according to the normalized maximum overshoot ym~/y (oo) and to the settling time T3~. For step disturbances the final value of the transient function of the plant was chosen as y(o0) and for reference input, y(~) is the final value of the unit-step response of the closed-loop system. The settling time T3~ is defined as the time at which the response y(t) attains and, from that point on, remains within -+3% of the final value y(oo). In addition to the step responses of the output temperature AOA after disturbances of AM, AQ, and AQs, the behaviour of the heat exchanger under stochastic disturbances is of considerable interest. Consequently, the rate of air flow was disturbed by a PRBS signal (m-sequence: N . = 3 1 , At, = 33.3sec) with a +15/-10% range around the operating points. The term ym~/y*, in Table I characterizes the ratio of the maximum values of
4. EXPERIMENTAL RESULTS
The described DDC algorithms were implemented on an HP2100 process computer with 32K core storage and 16-bit words. Each controller was designed for one specific operating condition only. In order to investigate the sensitivity of the algorithms, they were also tested under conditions which did not completely
T~SLE 1. Normalized maximum overshoot and settling time of the step response of the heat exchanger under disturbances and set-point changes for all investigated control algorithms for the operating level 35~. The numbers in parenthesis represent the results for the operating level 80~. The underlined items indicate the condition which the controllers were designed for under disturbances step input AM = + 15% (+ 13%) No.
DDC algorithm
~[%]
T~dsec]
AQs = +27% (+ 13%)
PRBS
AQ, = + 17% (+ 17%)
AM = + 15%/- 10%
t%1 T.secl 1%1 ..lseol
Y'~
la PID
3.4 (2.4)
300 (0)
2.3 (6.6)
0 (250)
7.._66 (18.7)
12.5 (160)
47.8 (15)
cascade
3.4 (2.4)
80 (0)
I ._~6 (I.0)
0 (0)
6.4 (15.6)
270 (220)
50.4 (20)
0 (350) 600 1350) 2000 ( > 1000) 1400 (1300) 800 (450)
10.3 (15.6) 16.7 (43.7) 15...4 (37.5) 26.9 443.5) 23.1 (43.7)
150 (130) 500 1350) 820 (600) 600 (800) 280 (450)
46.1 (15) 47.8 (40) 52.2 (50) 43.5 (100) 43.5 (30)
Ib lla lib III
compensator
IV
adaptive
V
half proportional
VI
deadbeat response
VII
state feedback
3.4 (4.8) 8 (4.8) 20 (19) 29.0 (22.8) 14.3 (14.2)
300 2.3 (220) (8.4) 150 3.3 {300) (8.4) > 1000 18.3 ( > 1000) (33.4) 1050 19 (!150) (37.7) 680 10 (470) (18.8)
under set point changes Aw = 6.5°C
Y"y(®) - Y(®)[%]
T,dsec]
114.3 (75.0) 14.2 (3.6) 60.0 (44.6) 14.2
550 (3 I0) 300 (I 20) 500 (230) 410
(8.9)
(21o)
0 (0) 7./ (8.9) 89 (75) O (1) 28.6 (17.8)
210 (2"-~) 300 (350) 725 (475) 200 (750) 4~0 (330)
399
Comparison and application of DDC algorithms for a heat exchanger
disturbances &J~r, &Qs and .AQ, on the uncontrolled heat exchanger is shown at operating points 35 and 80%. Trill' indicates a strong loaddependent behaviour of the process, which is very important for the performance and stability of the closed-loop system. Thus, the control algorithms designed for the operating point 35% were also tested for the operating point 80%. At these two operating levels, step responses for the setpoints were also measured. Figures 7-9 show the transient responses of the controlled output temperature at the 35% operating point for all DDC algorithms after step disturbances of AM, AOs and &Q,. Tests carried out at the 80% operating point led essentially to the same conclusions. The step responses of the output temperature after setpoint changes are shown in Figs. 10-11 for the operating levels of 35 and 80%, respectively.
the controlled to the uncontrolled responses. The underlined items in Table 1 indicate theoPerating condition for which each control algorithm has been designed. From the comparison of the results in each column, it is obvious that the algorithms generally give best results when operating under the particular design conditions. For this comparison, both the settling time T,,, and the relative maximum overshoot yr,~/y(°°) must be considered. Some of the controllers still give good results if these conditions are changed. In Fig. 6a, b and c the influence of the l*C]4 ~ I ~ . . _ , . . . . ~
1,00
600
80O
80°/*
I°cl~ ~ "
~
(b)
5. EVALUATION OF THE RESULTS o
z0o
~o0
60o
8oo
ldoo
t ~sI
~OIj-
~so% O
~
.
•
5.1 Evaluation of the control experiments Following the summary of experimental resuits in Table 1, the PID controller (I) is particularly good for all step disturbances Ajar, AQs and AQ, and also yields satisfactory results for setpoint changes Aw using an anti-windup controller. In the case of random disturbances, however, the results are relatively poor. This is
•
soo 8oo ,~] FiG. 6. Step responses of the uncontrolled heat exchanger for the two operating levels 35 and 80% flow rate: (a) step change of &M, (Io)step change of &Qs(~-&OE),(c) step change of AQ,. o
2oo
~oo
0,
--
_/
-- ~
soo
300
.-~
..........
oo I
-'1
6oo
'
i-iv
....
700
'
"~
........
.
•
"
\
v, v,,
-6
~
s
"
Mj-:z3 e
Fro. 7. Transient response of the output temperature AO^ of the heat exchanger to a step disturbance of for all investigated D D C algorithms I - V I I (operating level 35%).
A~/
[°C)8 AaA 6 ~i
/ /
200
l - Iv
~
/ ~ - - = =........... - - - - -~_-_-. - - - -_~-..-"~.=........................ = - ~ - = . . . ~ . _ Z -. ~0
200
300
1.00
500
600
t, OS...r.= z 2
v VlVl!
t
700
I
~ 800
l
--....... 900
tls]
FIO. $. Transient response of the output temperature AS^ of the heat exchanger to a step disturbance of AOs ( "- &OE) for all investigated D D C algorithms I - V I I (operating level 35%).
I°C]t ~OA 61
~ ~ ' ~ u n c o n t r o l l e d I
- III
response
AQ1 r =Z 1
IV
.... , ~ S t ) (. ].-.~. .. tls]; SO0 660 700 Fro. 9. Transientresponseof the outputtemperature&@,~of the heat exchang=rto a step disturbanceof AQ, for all invest~atedDDC alsorithmsI- VII (eperati~level3~F~). 0 ~ ~ ' - ~1~:) ~ = , ~ . ~ . = . z300 , ~ = = _~bO ---
4OO
et al.
H . UNBEHAUEN
~t.I AOAI"CI
Ia
~-~. 12
//
/
vll
z
\
"..
.# y. Ib
35%
"-..
~
--.. ~
l,
Aw=6,5°C
.
°. .
.
.
..
..
..
.
.
~
.
..~:
O.
~
0
lO0
200
3b0
460
500
600
760 t[sI
FIo. 10. Transient response of the output teml~rature AOA of the heat exchanger to a set-point change Aw for all investigated DDC algorithms I - VII in the operating level 35%.
AaAI°C) 1~
80%
12
I a...,"'~.
l01
Z.(II a"('-, /V
viii/ ,) / / 0
Aw=6,5°C
Ib"., ,,,,v,,b
1()0
,-. .....
"-,
v,
200
3()0
400
5/)0
600
700 t [s]
FIo. 1I. Transient response of the output temperature AO~ of the heat exchanger to a set-point change Aw for all investigated DDC algorithms I - VII in the operating level 80~.
partly due to the non-minimum phase characteristic of the output temperature for load disturbances as shown in Fig. 6a. It is also well known[6], that PID controllers tuned for step disturbances are not really suitable for random disturbances. A simple proportional controller would give better responses for this case. The behaviour of the cascade controller (II) is similar to that of the PID controller. The maximum overshoot for this controller is slightly better and the responses are faster due to the auxiliary signal y , and the inner loop dynamics given by GQ,(S) in (3), see Figs. 7-11. Control results with the compensator algorithm (III) are comparable to those of the PID and cascade controller. The response to a setpoint change, shown in Fig. 10, matches the desired closed-loop response with insignificant error. This error is caused by simpification of the model as given by (5). Because of the additional compensation element GK(s) corresponding to (13) which acts as a lag on the setpoint changes, the actual control loop is relatively fast and thus also well suited to step disturbances. The adaptive algorithm (IV) which is also a compensator, is not supplied with such an
additional compensation element and, therefore, has a long settling time for disturbance control, as shown in Fig. 9. However, the behaviour of this algorithm is heavily dependent upon the accuracy of the identified process model. This is due to the fact that is must continuously follow a model of the plant, thus the system response is independent of load changes. Moreover, since the identification speed is limited, considerable transient model errors caused by load changes, temporarily, lead to a worse response. In the examples shown in Figs. 10 and I l, for controller IV, those errors are appreciably decreased. For the simple half-proportional controller (V), the overall performance is very poor, especially for slow disturbances like AQs shown in Fig. 8. Therefore, this controller proved to be unsuitable for this particular application and care must be taken when applying it to similar control problems. The deadbeat response controller (VI) shows very good control behaviour for setpoint changes, but only at the operating point for which it has been designed. In the case of disturbances, however, its performance is not satisfactory. This was primarily due to the
401
Comparison and application of DDC algorithms for a heat exchanger
(III) algorithms consists of solving a linear system of three and five simultaneous equations given by (23) ahd (12), respectively. For the design of the optimal state feedback controller (VII) an observer must be designed and the matrix Riccati equation (32) must be solved. Considering the design parameters which can be chosen freely, the deadbeat response algorithm has the least requirements because only the sampling period T has to be specified. More detailed specifications are needed for the compensator design, where four closed-loop poles are required. In the adaptive control algorithm (IV) three model time constants must be chosen to compromize between fast response and the limitations of the control signal. In this algorithm, four additional time constants of the linear filters must be assumed without further knowledge of the process dynamics and the necessity of an exact process model. The optimal state feedback controller, however, needs an exact model in a special state-space form and has a large number of design parameters which must be chosen properly. The eigenvalues of the state observer are influenced by a heuristic procedure in this particular case. In addition the effect of the weighting matrix Q on the control behaviour is not completely obvious, and it should always be chosen by a simulation study. The amount of computer usage for the implementation of these algorithms can be compared directly in Table 2, showing the core storage and computing time requirement for each algorithm. The numerical values are typical for the implementation on the HP2100 process computer using FORTRANIV programs and only apply to the mathematical part of the control programs, i.e. excluding organization tasks like data acquisition, program scheduling and data output. The computing time and core storage for algorithms la, V and VI are the least among the seven controllers. These algorithms are represented by simple mathematical structure requiring few computational steps. The mathematical models of the anti-windup in PID (Ib) and cascade controller (llb) require a larger number
necessity of a long sampling period and the limitation of the reaction velocity. Therefore, the output error caused by disturbances settles slowly which is similar to the dynamic response of the plant. The state feedback controller (VII) proved to be rather insensitive to different operating conditions. The reason for this is due to the fact that the observer provides measured values of all states and, moreover, the weighting matrix Q penalizes any further variations in the states. The performance of the output error e(t) is also dependent upon matrix Q, giving rise to a faster response and less overshoot. The DDC algorithms tested in this investigation are designed for deterministic signals. The experiments using PRBS signals show that they are not suitable in the case of random disturbances. Therefore, if in practical applications the dominant characteristic of the disturbances is random, these algorithms should not be used or must be retuned, e.g. for the PID and cascade controller.
5.2 Design and implementation requirements The performance of a DDC algorithm always has to be balanced against the design and implementation requirements. One group of algorithms, the PID (I), cascade (II) and the HP algorithm (V), can be designed according to normalized controller setting specifications [6, 7, 11-13]. For the cascade algorithm in particular, rough controller adjustments can be found on the basis of the optimal PID controller. An improved design can be obtained either by a simulation study or by tuning the controller on site. The variation of any parameter (KR, Th TD, Tv and T) then requires the recomputation of the discrete controller coefficients. The other group of algorithms is characterized by design procedures based directly on a detailed mathematical model of the plant. These procedures are systematic and are usually performed by a computer program. The design of the deadbeat response (VI) and the compensator
TABLE 2. Computing time and core storage for the implementation of the DDC algorithms on a HP2100 process computer algorithm core storage in 16-bit words computing time T~ in millisec sampling time T in sec
relative computing time
la
lb
lla
lib
III
IV
V
VI
VII
! 14
168
216
371
360
926
153
114
284
1.2
1.5
3.1
3.3
13.6
56.9
!.3
1.2
8.7
5
5
5
5
5
4
25
75
5
0.062
0.066
0.272
i.42
0.005
0.002
0.174
0.024 0.03
H. UNBEHAUEN et al.
402
of logical decisions and operational steps, hence the core storage is increased by 50%. The optimal state feedback controller (VII), although very expensive in design, does not need much storage and requires even less computing time than the compensator algorithm (III), in which the additional compensation element Gg (s) also has to be computed. The computations of the adaptive algorithm (IV) include the identification, the decision process and the modification of the controller parameters, leading to the maximum amount of core storage and computing time requirements.
(lb) is the best to control the heat exchanger under different operating conditions. If an auxiliary signal is available in special applications, the PID can be changed to cascade controller which can slightly improve the dynamic behaviour. These results can also be extended to similar processes such as preheaters, evaporators, superheaters and others. In the case of predominantly random disturbances, all control algorithms fail. These difficulties can be overcome by using a stochastic controller design. This will be a deserving extention to the studies in comparative investigations of D D C algorithms.
6. CONCLUSIONS
The load dependence of the heat exchanger is characterized by decreasing gain and time constants for increasing load M. Therefore, the load change, from 35 to 80%, does not seriously effect the stability of the closed-loop system and in fact in some test cases gives even better results. This is a good example of the well known rule[6], that controllers for processes with parameters that change significantly due to load conditions, or other factors, should be designed for the worst case, i.e. for the situation with the highest gain and the largest time constants. This investigation shows that for the control of single input, single output systems subject to different operating conditions, the PlD, cascade and compensator control algorithms yield best results as far as the systems response, sensitivity to parameter variations, design requirement, the amount of computer storage and computation time are concerned. Other relatively simple control algorithms, e.g. HP and deadbeat response algorithms proved to be extremely sensitive to changes in the operating conditions and less efficient under step disturbances. This is generally valid for all DDC algorithms operating with a low sampling rate. Therefore, algorithms of that kind can only be used in certain applications, even though they are very promising with regard to the design requirements and the amount of computation. The application of adaptive algorithms is relatively inexpensive in the design procedure but seems to be justified only in the case of large parameter changes in the process dynamics or of the operating conditions because of the excessive computer requirement. In comparison, the computation of a control signal for an optimal state feedback controller is less expensive. Its large design requirement, however, can only be justified for a complicated multivariable control system. From the results of this investigation, it follows that the PlD controller with anti-windup
Acknowledgements--This research work was supported by the "Bundesministerium ffir Forschung und Technologie" of the Federal Republic of Germany under the PDV-project "ProzeBlenkung mit DV-Anlngen" of the 2nd DV-program. The authors wish to thank the reviewers, Prof. M. Jamshidi of Pahlavi University, Shiraz, Dan, and Dr. M. Searle of the Polytechnic of Central London for making a number of useful suggestions to improve the paper. REFERENCES [1] W. D. T. DAvies: Control algorithms for DDC. Instrum. Pract. 21, 70--77 (1967). [2] C. L. Sgrrtl: D/g/ta/Computer Process Control. Intext Educational, Scranton (1972). [3] J. Cox, L. J. HELLUMS,T. J. W:LL!~lS, R. S. BANKS G. J. KINK: A practical spectrum of DDC chemical process algorithms. J'-ISA 13, 65--72 (1966). [4] R. R. DE BOLT and B. E. POWELL: A natural 3-mode-controller algorithm for DDC. J.ISA 13, 43--47
(19~). [5] R. SCHILBACH,M. PRANDITand W. WERF.,it: Procedures
for realizing linear DDC algorithms (in German). Regdungstechnik 16, 337-344 (1968). [6l H. UNBEHAUEN:Stability and optimal tuning of linear and nonlinear controllers in single-loop systems, with different types of processes with proportional and integral action (in German). VDI-Verlng Dfisseldorf, Fortschr. Ber. VDI-Zeitschrift, 8, No. 13 (1970). [7] A. M. LOpEZ,P. W. MU~LLand C. L. SMITH:Tuning PI and PID di6ital controllers, lnstrums Control Syst. 42, 89-95 (1969). [8] W. WEBER: A systematic way for design of linear and adaptive control systems(in German). Elektrotech. Z. A, rag, 138-144 (1967). [9] W. SCHAUFEt,BEROF.R:Model-adaptive control systems (in German). Dissertation, ETH Ziirich (1969). [10] P. C. YOUNG: Applying parameter estimation to dynamic systems, Parts I and II. Control Engineering 16, 119-125 (1969) No. 10; No. 11, pp. 118-124. [11] V. FERNER: The half-proportional controller--a new promising type (in German). D/e Technik !1, 811 (1956). [12] M. ElCltNER: Design of a DDC system for control temperature and power of a steam generator (in German). Dissertation, Universit&t Stuttlpu't (1973). [13] M. EIC~INBR:DDC Design based o n a digital simulation technique. IFAC/IFlP-Symposium, Part IV-5, pp. I-9. Helsinki (1971). [14] E. Juav: Sampled-data Control Systems. Wiley, New York (1958). [15] M. A'rtlANS: On the design of PID controllers using optimal linear regulator theory. Automatica 7, 643-647
0971). [16] D. G. LUENanOmt: Observing the state of a linear system. Trans. IEEE MIL S, 74-80 (1964).
Comparison and Application of DDC Algorithms for a Heat Exchanger* H. U N B E H A U E N , t
C H R . SCHMIDi" and F. B ( S T T I G E R t
Seven DDC-algorithms, applied to the temperature control o f a heat exchanger in a comparative study, show which controller is the most suitable one. summmTmln recent years, several algorithms for Direct Digital Control (DDC) have been proposed in literature. Although some of these, such as PID or cascade controllers, are very commonly used in industrial applications, the more recent ones like optimal state feedback controllers using an observer or parameter adaptive controllers have rarely been applied to a real plant. The primary difficulty behind this application has been perhaps the lack of testing such algorithms on a pilot plant. Moreover, there has been no serious attempt to make a comparative study of the merits of such algorithms for an existing plant under actual operating conditions• In this paper, seven DDC algorithms are applied to the temperature control of a heat exchanger. These algorithms are: PID, cascade, compensation (pole assignment), deadbeat, half-proportional, adaptive and optimal state feedback controller using an observer. The system performance and sensitivity with respect to changes of the plant parameters, disturbances and set point variations are investigated for the heat exchanger using these algorithms. The results indicate that the more sophisticated algorithms, e.g. optimal state feedback, compensation and adaptive controllers, requiring more computer time and memory, yield relatively less improvement when applied to a low-order plant than do the simple algorithms such as PlD or cascade. It was deduced that the PID controller with anti-windup is the most suitable one.
This leads directly to the p r o b l e m of sensitivity of a control s y s t e m to its operating conditions. The increasing application of digital c o m p u t e r s for process control permits the implementation of various control algorithms using c o m p u t e r programs. Many DDC algorithms h a v e been p r o p o s e d for process c o m p u t e r s during the past few y e a r s [ I - 5 ] , however, there are hardly any c o m p a r a t i v e investigations on the efficiency of these algorithms for practical applications. T h e r e f o r e , this p a p e r c o m p a r e s s o m e of the most important DDC algorithms with respect to p e r f o r m a n c e , efficiency and sensitivity as applied to the t e m p e r a t u r e control of a heat exchanger. 2. DYNAMIC BEHAVIOUR OF THE HEAT EXCHANGER The process used for testing the different control algorithms is a pilot plant representation of a high-pressure s t e a m superheater shown in Fig. 1. It consists of an electrically heated brass tube (¢~ 4.5 ram) with c o m p r e s s e d air as the flow medium. H e r e , the object is to regulate the output t e m p e r a t u r e O~ of the process stream. The two t e m p e r a t u r e s 0 , and OA are m e a s u r e d by t h e r m o c o u p l e s while three input variables
1. INTRODUCTION MANY chemical and p o w e r plants have to be controlled such that their r e s p o n s e s are tolerable under various operating conditions with variations in plant p a r a m e t e r s and disturbances. H o w e v e r , a controller which is optimal for one load level m a y in fact provide p o o r or even unstable responses under other conditions. With this in mind, it is desirable to know, whether a controller which was designed f o r one special operating point is also a p p r o p r i a t e for others.
dist ur bonce himting system A
*Received 9 July 1974; revised 18 February 1975; revised 25 August 1975; revised 12 February 1976. The original version of this paper was presented at the 4th IFAC/IFIP International Conference on Digital Computer Applications to Process Control which was held in Zurich, Switzerland during March, 1974. The published Proceedings of this IFAC Meeting may be ordered from: IFAC/IFIP Conference, Gloriastrasse 35, CH-8006, Zurich, Switzerland. This paper was recommgnded for publication in revised form by associate editor H. A. Spang HI. tLehrstuhi f~r Elektrische Steuerung und Rngelung, Ruhr-Univer~tlit Bochum, !)-4630 Bochum, Universitlitsstr. 150, IC 3, W. Germany.
Fro. 1. Schematic diagram of the direct digital computer control of the heat exchanger. 393
AUTO. VeL 12, No. $--A
394
H. UNnEH^UES et al.
(variations in the flow rate AM, in the heating rate AQ and in the inlet temperature AOE) all influence the variation AO^ in the outlet temperature. Step changes of Aft;/ and AQ can be obtained directly by means of a valve and the electric heating element, respectively. Rapid variations of &OB, however, are only possible by step changes AQs of a separate efficient electric preheater, the disturbance heating system shown in Fig. 1. Figure 2 shows a detailed block diagram of the heat exchanger system. The most important normalized transfer functions are
sufficiently accurate and much easier to deal with. The identification method used to obtain the models was numerical Fourier transformation of the transient response and approximation of the Bode plot by a rational transfer function. This model does not take the nonlinearities of the actuating element into account. The nonlinearities are of a static type and have only a small effect on the system responses. 3. DDC ALGORITHMS
In the following subsections, seven DDC algorithms which are used in this paper are briefly described.
2.22
Gos(s)G.~(s) = (I + 370s) (I + 118s)
(l)
0.73 G , . ( s ) = 1 + 232s
(2)
0.5 GQ~(s) = (1 + 59s) 2
(3)
3.1 PID control algorithm (I) For implementation on a process computer, the transfer function of the PID controller with a low-pass filtered differential channel given by _.2'{u(t)}.~{e(t)}= Ks • .(1 + ~ ,1s + l ~ VT~s \ ~s )
Os (s ) -
0.285 GQ.(s) = (1 + 20s) (1 + 100s)"
They were found experimentally by measuring and analysing input-output relationships for a flow rate of 35% of the maximum value. In order to find the optimal controller settings of the cascade control system, a process model described by (1)--(4) is used. For synthesizing or tuning the other control algorithms, the following reduced model transfer function was proved to be satisfactory AOA(S) 0.65 Gs(s) = AQ(s) (1 + 2 5 s ) ( ! +213s)"
(5)
Although the process is actually a distributed parameter system with time delay, this type of model has not been used because the delay time is insignificant and lumped parameter models are
Y
I~'-J
Ik-~
l'
-
--:u !1 ~']'---lo',=j=
cascaded control system
Fio. 2. Block diagram of the direct dillital control of the heat (~xchal)S(~.
must be transformed to a discrete form
~{u(k)) d o + d , z - ' +d~z -2 G s ( z ) - - ~'{e(k)} = 1 + c,z-' + c2z -2"
(7)
where u(t) denotes the control signal, e(t) is the output error defined as e(t) = w(t) - y(t), w(t) is the reference input, and Kt, T~, To and Tv are controller gain, integral time constant, derivative time constant, and filter time constant, respectively. The coefficients c, and d~ in (7) are functions of parameters Ks, 7"i, To, Tv and the sampling period T. In order to prevent reset windup in this controller, the integral action is reduced to zero (TI ~0o) if the control signal exceeds the upper or lower bounds of the actuator. In sequel, the controller without antiwindup will be referred to as Ia while the controller with anti-windup is denoted by lb. Applying this control algorithm to the reduced model given by (5), there are only marginal optimum values for the controller parameters Ks and Tt, [6, 7]. If the behaviour of the actuating element is taken into consideration, it can be seen that these parameters cannot be chosen to be greater than Ks = 15.4 and To = 10.0 sec. Then, Tt ffi 38.0 sec and Tv ffi 13.5 sec are optimal values in the sense of minimizing the ISEcriterion for step disturbances with a sampling period of T = 5.0 sec.
_&
CmlsV
....
(6)
(4)
3.2 Cascade control algorithm (II) This control system consists of a master controller (PI action) with the transfer function
Comparison and application of DDC algorithms for a heat exchanger
(8)
395
compensation network Co
GK = P(s)
and an inner-loop controller with the transfer function
Gs, = Ks,.
(9)
These two controllers are cascaded as shown in Fig. 2. Like the PID-controller, this algorithm has been used with and without anti-windup and will be referred to as controller IIb and Ila, respectively. The algorithm was tuned for step disturbances • Qs by a simulation study using the model given by (1)-(4) and the system described in Fig. 2. The final parameters used in the experiments are gain factors Km~ = 3.1, Ks2 = 7.7, integral time constant Tn = 60 sec and sampling period T ffi 5 sec. 3.3 Compensator control algorithm (III) Considering an nth-order transfer function zo+ z~s + . •. + z,s'
Z(s)
G,(s) = no+ n,s +'-: ~ -+-M'-" n,s = N(s)'
i ' . L--
~ .
.
.
.
.
.
Zls) l
GsIs}--N~- i .
.
.
I. . . . . . .
|
J .-]
Fro. 3. Block diagram of the compensator controller.
(II)
where P ( s ) and Q(s) are similarly polynomials of degree m. The coefficients po, p,, q,, i = 1, 2. . . . . m are determined from the characteristic equation of the closed-loop system, given by m+R
II (s - s,) = 0,
is connected in series with the closed-loop as shown in Fig. 3. For the practical realization of the controller components Gs(s) and GK(s), they are described in state-space notation with input-signals y(t) and w(t) and output signal u(t). The design of this control algorithm is based on the simplified model given by (5). The closed-loop system will then be of fourth order. The poles s, = s2 = s3 = - 0 . 0 5 and s , = - 0 . 1 are chosen with regard to the upper and lower bounds of the actuating element.
(10)
of the plant where ! < n and Z(s) and N ( s ) denote polynomials of degrees 1 and n, respectively, a controller is to be designed such that the closed-loop system has a prescribed distribution of poles s, in the s-plane (pole assignment). The transfer function of the controller is assumed to be of the form
po+ p,s + . . . + p = s = " = P(s) Gs(s)ffiqo+q,s+..+q=s ~-~
(13)
(12a)
I-I
or, alternately using (I0) and (I I),
3.4 Adaptive control algorithm (IV) The over-all behaviour of a closed-loop system with time varying plant parameters p,(t) can be specified, to some extent arbitrarily, by the model transfer function, o,,[s,
g(t,)] O.[s. igt)]
Ou(s) = 1 + Gs [s, ~t~)] Os [s, p(t)]
where Gs[S,p(t)] is the plant transfer function with parameter vector IKt), and Gs Is, ~(t,)] is the transfer function of the adaptive controller with the corresponding estimated parameter vector ~(t~) at discrete time tk. The variations of the parameter vectors p(t) and ~(t~) are assumed to be slow in comparison with the plant dynamics. The result of an on-line identification of the unknown plant can be used directly for synthesizing the controllers transfer function
1
G,,(s)
G , [ s . # ( t , ) ] ffi G s [ s . # ( t , ) ] " l - G , , ( s ) "
N ( s ) Q ( s ) + Z ( s ) P ( s ) = 0.
(14)
(]5)
(12b)
A comparison of the coefficients of these two characteristic equations leads to a linear system of simultaneous algebraic equations for the controller coefficients [8]. For m = n, this system has a solution with qo as a free parameter. If qo ffi 0 (integral action), the control behaviour under disturbances is improved. In order to compensate the zeros of the closed-loop transfer function for reference input, an additional
In this way, an adaptive control system is obtained, whose dynamic behaviour follows approximately the given model. Therefore, the task of the adaptive controllers is to compensate Gs[s,~t)] with llGs[S,~(tk)] as closely as possible. In this case, both feedback signals of the closed-loop system, according to Fig. 4, are equal and thus the overall system follows the model (14). Also in the case of incomplete compensation, both feedback-loops accomplish
396
H. UNBEHAUEN
et al.
~(k ) = ~(k - l) + K(k )[y*(k )-mT(k )fJ(k - I)]
---
I P(k) = ~[P(k - l)-K(k)mT(k)P(k
uitk),~
glt k)
- I)]
(18)
J : K(k) = P(k - l)m(k)[l + mT(k)P(k - l)m(k)]-'.
I
processcomputer FIG.4. Block diagram of the adaptive control system.
significant improvements in the over-all system behaviour. Thus, the adaptive control system [9] is very efficient. For on-line identification the method of linear filtering[9] is used. In this case, the input signal u ( t ) and the output signal y(t) are each filtered by a linear filter Gp(s). If the dynamic behaviour of the process is approximated by a third-order model, Gsu[s, Ks(t), a.(t ), a,(t), a,(t)]. Ks(t) 1 + a,(t)s + a2(t)s2+ a,(t)s 3
(16)
then a set of equations for obtaining the process parameters will result from the filter output signals u*(t) and y*(t):
Rules for the choice of the inital conditions of P(0) and ~(0) and the weighting factor ~ can be found in[10]. In order to calculate the filtered signals u*(t) and y*(t), the transfer function of the filters $
G,,(s)=(l + Ss)(l + 25s),
was realized by the process computer. In the same way, the control signal u ( t ) can be obtained using the model transfer function 1 G u ( s ) = (1 + 20s)'"
(20)
3.5 Half-proportional control algorithm (V) This nonlinear half-proportional (HP) control algorithm can be described analytically by the following relationships [ l I] u(k)= u(k - I)+ Au(k)
K s ( t ) u * ( t ) = y*(t) + a , ( t ) : * ( t ) + a2(t):*(t) + a,(t)g*(t)
(19)
(21a)
with
KmAe (k) for [e (k)J - le (k - 1) > 0 (increasing error) Au(k) =
0 for le(k)]- le(k - 1)1 < 0 (decreasing error)
(21b)
Kc[e(k ) - P, sgn(e(k ))] for e(k ) = e(k - 1) . . . . .
or in vector notation: y*(t) = mT(t)l~t)
le(k)l> P,
and (17)
with roT(t) = [- :*(t),- 9*(t),- 7*(0, u*(t)] and pT(t) = [a,(t), a~(t), a3(t), Ks(t)]. From (17) a system of simultaneous algebraic equations for the parameter vector p for k discrete time intervals is obtained. This system can be solved recursively for each value of k ---tJT
e(k - N ) and
Ae(k) = e(k)-e(k - I).
(21c)
Here, the gain factor KR represents the intrinsic parameter of the controller, whereas the factor Kc, the error tolerance P, and N are of minor importance. The optimal controller setting Km was obtained for the reduced model (5) under step disturbances of AQ, minimizing the IAEcriterion. The optimal values for the parameters were found to be K , = 8.47, Kc -- 3.08, P, -- 0.05 and N ~ 5 with T = 25 soc[12, 13]. 3.6 Deadbeat response control algorithm (VI) This method is based on the computation of a piecewise constant control signal for a system of
397
Comparison and application of DDC algorithms for a heat exchanger n th order, so that after n sampling instants, the response to a unit step w will reach its final value without any overshoot[14]. The control signal is obtained for the reduced second-order model (5) by
and the performance index must be modified as follows
u ( k ) = hoe(k) + h , e ( k - 1) + h2e(k - 2) + g , u ( k - 1)+(1 - g , ) u ( k - 2 )
where in fact v(t)--,O as t --,~. Using the output error as a state variable
(22)
with gt = y ( T ) l w . For computing the coefficients h,, the following system of n + 1 equations
[ = f : [xT(t)Qx(t) + v2(t)] dt
(26)
(27)
x,(t) = e(t),
the state vector converges to zero, x ( t ) - , 0 as t ---)00.
ho + h, + h2 = l / K s ho + h t e TIT' + h2 e 2TIT' --- 0
(23)
For the design of the optimal state feedback controller according to (25)-(26), the reduced model (5) is unsatisfactory. A fourth-order model
ho + h, cTIT=+ h2 e 2TIT2 = 0
must be solved. The sampling period T is the only design parameter of the algorithm and must be chosen for some prescribed reference step input with respect to the actuator bounds. Accordingly, the investigations on the heat exchanger had to be conducted with T -- 75 sec. 3.7 Optimal state feedback controller with observer (VlI) An optimal linear state feedback controller for a single .input system is designed to minimize a quadratic performance index
I = f: [xT(t)Qx(t)+ u'(t)]dt
bo + b,s + b2s 2 G s ( s ) = a o + a , s + a 2 s ' + a , s 3 + s 4'
including the description of the long-range behaviour, proved to be adequate. Considering the additional integral term of (25) as part of the process, the following state-space representation can be found
x(t) = Ax(t) + by(t)
(29a)
e (t) = cTx(t)
(29b)
with
(24)
I
00 0- b o A--0
where x ( t ) is the state vector of order n and Q is
a positive semi-definite symmetric weighting matrix. This optimization problem has a solution only if the criterion (24) is finite i.e. if u ( t ) - , O and x(t)-,0 as t - , ® . For a system without integral action and with constant disturbances or setpoint changes an integrator must be introduced as shown in Fig. 5115]. Thus, the control signal is given by
(28)
0 -
ao
1- b l 0
0- b , 1
0 0
0
0
I
-
at
-
02
-
1 , a3
(30a)
bT=[0
0 0 0
I],
(30b)
cT=[I
0 0 0 0].
(30c)
The optimal state feedback controller v(t)
=
rTx(t)
1
u(t) ul0)|
=
u(0)+
f0 v0")d~',
(25)
was determined from the solution of the algebraic matrix Riccati equation
~'c~ess
WbbTW - WA - ATW - Q = 0
(32)
r T = bTW.
(33)
with ,_.
II ~
2 r _°ee_ss..c _°~ P_ut_. ~ .
~tk|varmble
.
.
~
.
.
.
reduced
FIG. 5. Reduced obeefver and optimal state feedback controllet for the heat exchanger.
For the identification of the unknown state variables x2(t) to x,(t), a reduced Luenberger observer[16] of fourth order is constructed, whose eigenvalues are larger than those of the plant. The weighting matrix Q was found by
398
H. UNBEHAUEN et al.
simulation studies in order to get good results for setpoint changes as well as step disturbances. Figure 5 shows the complete schematic diagram of the optimal state feedback control system. Most of the algorithms described are based on design procedures leading to continuous controllers. In order to obtain a discrete form of these controllers for process computer implementation, an approximate z-transformation, for the PID controller and the cascade controller, and a discretization procedure by power series expansion of the system matrix, (for the compensator controller III and adaptive controller IV), were used. These controllers can only operate with small sampling periods T, because the effects of the zero-order hold are not considered in the model or design procedure. Both the deadbeat response and the HP controller, on the other hand, are designed directly as discrete algorithms including the zero-order hold in the model. Therefore, the sampling period T, which is used as an additional design parameter, can be considerably larger than in other cases.
coincide with the assumptions under which the algorithms were designed. The tests were carried out for step changes of flow rate M preheating Qs heating rate Q, (control signal) setpoint and for disturbances due to pseudo random binary signals (PRBS) in the flow rate M both at the 35 and the 80% operating points. In Table 1 the essential results of the experiments are summarized according to the normalized maximum overshoot ym~/y (oo) and to the settling time T3~. For step disturbances the final value of the transient function of the plant was chosen as y(o0) and for reference input, y(~) is the final value of the unit-step response of the closed-loop system. The settling time T3~ is defined as the time at which the response y(t) attains and, from that point on, remains within -+3% of the final value y(oo). In addition to the step responses of the output temperature AOA after disturbances of AM, AQ, and AQs, the behaviour of the heat exchanger under stochastic disturbances is of considerable interest. Consequently, the rate of air flow was disturbed by a PRBS signal (m-sequence: N . = 3 1 , At, = 33.3sec) with a +15/-10% range around the operating points. The term ym~/y*, in Table I characterizes the ratio of the maximum values of
4. EXPERIMENTAL RESULTS
The described DDC algorithms were implemented on an HP2100 process computer with 32K core storage and 16-bit words. Each controller was designed for one specific operating condition only. In order to investigate the sensitivity of the algorithms, they were also tested under conditions which did not completely
T~SLE 1. Normalized maximum overshoot and settling time of the step response of the heat exchanger under disturbances and set-point changes for all investigated control algorithms for the operating level 35~. The numbers in parenthesis represent the results for the operating level 80~. The underlined items indicate the condition which the controllers were designed for under disturbances step input AM = + 15% (+ 13%) No.
DDC algorithm
~[%]
T~dsec]
AQs = +27% (+ 13%)
PRBS
AQ, = + 17% (+ 17%)
AM = + 15%/- 10%
t%1 T.secl 1%1 ..lseol
Y'~
la PID
3.4 (2.4)
300 (0)
2.3 (6.6)
0 (250)
7.._66 (18.7)
12.5 (160)
47.8 (15)
cascade
3.4 (2.4)
80 (0)
I ._~6 (I.0)
0 (0)
6.4 (15.6)
270 (220)
50.4 (20)
0 (350) 600 1350) 2000 ( > 1000) 1400 (1300) 800 (450)
10.3 (15.6) 16.7 (43.7) 15...4 (37.5) 26.9 443.5) 23.1 (43.7)
150 (130) 500 1350) 820 (600) 600 (800) 280 (450)
46.1 (15) 47.8 (40) 52.2 (50) 43.5 (100) 43.5 (30)
Ib lla lib III
compensator
IV
adaptive
V
half proportional
VI
deadbeat response
VII
state feedback
3.4 (4.8) 8 (4.8) 20 (19) 29.0 (22.8) 14.3 (14.2)
300 2.3 (220) (8.4) 150 3.3 {300) (8.4) > 1000 18.3 ( > 1000) (33.4) 1050 19 (!150) (37.7) 680 10 (470) (18.8)
under set point changes Aw = 6.5°C
Y"y(®) - Y(®)[%]
T,dsec]
114.3 (75.0) 14.2 (3.6) 60.0 (44.6) 14.2
550 (3 I0) 300 (I 20) 500 (230) 410
(8.9)
(21o)
0 (0) 7./ (8.9) 89 (75) O (1) 28.6 (17.8)
210 (2"-~) 300 (350) 725 (475) 200 (750) 4~0 (330)
399
Comparison and application of DDC algorithms for a heat exchanger
disturbances &J~r, &Qs and .AQ, on the uncontrolled heat exchanger is shown at operating points 35 and 80%. Trill' indicates a strong loaddependent behaviour of the process, which is very important for the performance and stability of the closed-loop system. Thus, the control algorithms designed for the operating point 35% were also tested for the operating point 80%. At these two operating levels, step responses for the setpoints were also measured. Figures 7-9 show the transient responses of the controlled output temperature at the 35% operating point for all DDC algorithms after step disturbances of AM, AOs and &Q,. Tests carried out at the 80% operating point led essentially to the same conclusions. The step responses of the output temperature after setpoint changes are shown in Figs. 10-11 for the operating levels of 35 and 80%, respectively.
the controlled to the uncontrolled responses. The underlined items in Table 1 indicate theoPerating condition for which each control algorithm has been designed. From the comparison of the results in each column, it is obvious that the algorithms generally give best results when operating under the particular design conditions. For this comparison, both the settling time T,,, and the relative maximum overshoot yr,~/y(°°) must be considered. Some of the controllers still give good results if these conditions are changed. In Fig. 6a, b and c the influence of the l*C]4 ~ I ~ . . _ , . . . . ~
1,00
600
80O
80°/*
I°cl~ ~ "
~
(b)
5. EVALUATION OF THE RESULTS o
z0o
~o0
60o
8oo
ldoo
t ~sI
~OIj-
~so% O
~
.
•
5.1 Evaluation of the control experiments Following the summary of experimental resuits in Table 1, the PID controller (I) is particularly good for all step disturbances Ajar, AQs and AQ, and also yields satisfactory results for setpoint changes Aw using an anti-windup controller. In the case of random disturbances, however, the results are relatively poor. This is
•
soo 8oo ,~] FiG. 6. Step responses of the uncontrolled heat exchanger for the two operating levels 35 and 80% flow rate: (a) step change of &M, (Io)step change of &Qs(~-&OE),(c) step change of AQ,. o
2oo
~oo
0,
--
_/
-- ~
soo
300
.-~
..........
oo I
-'1
6oo
'
i-iv
....
700
'
"~
........
.
•
"
\
v, v,,
-6
~
s
"
Mj-:z3 e
Fro. 7. Transient response of the output temperature AO^ of the heat exchanger to a step disturbance of for all investigated D D C algorithms I - V I I (operating level 35%).
A~/
[°C)8 AaA 6 ~i
/ /
200
l - Iv
~
/ ~ - - = =........... - - - - -~_-_-. - - - -_~-..-"~.=........................ = - ~ - = . . . ~ . _ Z -. ~0
200
300
1.00
500
600
t, OS...r.= z 2
v VlVl!
t
700
I
~ 800
l
--....... 900
tls]
FIO. $. Transient response of the output temperature AS^ of the heat exchanger to a step disturbance of AOs ( "- &OE) for all investigated D D C algorithms I - V I I (operating level 35%).
I°C]t ~OA 61
~ ~ ' ~ u n c o n t r o l l e d I
- III
response
AQ1 r =Z 1
IV
.... , ~ S t ) (. ].-.~. .. tls]; SO0 660 700 Fro. 9. Transientresponseof the outputtemperature&@,~of the heat exchang=rto a step disturbanceof AQ, for all invest~atedDDC alsorithmsI- VII (eperati~level3~F~). 0 ~ ~ ' - ~1~:) ~ = , ~ . ~ . = . z300 , ~ = = _~bO ---
4OO
et al.
H . UNBEHAUEN
~t.I AOAI"CI
Ia
~-~. 12
//
/
vll
z
\
"..
.# y. Ib
35%
"-..
~
--.. ~
l,
Aw=6,5°C
.
°. .
.
.
..
..
..
.
.
~
.
..~:
O.
~
0
lO0
200
3b0
460
500
600
760 t[sI
FIo. 10. Transient response of the output teml~rature AOA of the heat exchanger to a set-point change Aw for all investigated DDC algorithms I - VII in the operating level 35%.
AaAI°C) 1~
80%
12
I a...,"'~.
l01
Z.(II a"('-, /V
viii/ ,) / / 0
Aw=6,5°C
Ib"., ,,,,v,,b
1()0
,-. .....
"-,
v,
200
3()0
400
5/)0
600
700 t [s]
FIo. 1I. Transient response of the output temperature AO~ of the heat exchanger to a set-point change Aw for all investigated DDC algorithms I - VII in the operating level 80~.
partly due to the non-minimum phase characteristic of the output temperature for load disturbances as shown in Fig. 6a. It is also well known[6], that PID controllers tuned for step disturbances are not really suitable for random disturbances. A simple proportional controller would give better responses for this case. The behaviour of the cascade controller (II) is similar to that of the PID controller. The maximum overshoot for this controller is slightly better and the responses are faster due to the auxiliary signal y , and the inner loop dynamics given by GQ,(S) in (3), see Figs. 7-11. Control results with the compensator algorithm (III) are comparable to those of the PID and cascade controller. The response to a setpoint change, shown in Fig. 10, matches the desired closed-loop response with insignificant error. This error is caused by simpification of the model as given by (5). Because of the additional compensation element GK(s) corresponding to (13) which acts as a lag on the setpoint changes, the actual control loop is relatively fast and thus also well suited to step disturbances. The adaptive algorithm (IV) which is also a compensator, is not supplied with such an
additional compensation element and, therefore, has a long settling time for disturbance control, as shown in Fig. 9. However, the behaviour of this algorithm is heavily dependent upon the accuracy of the identified process model. This is due to the fact that is must continuously follow a model of the plant, thus the system response is independent of load changes. Moreover, since the identification speed is limited, considerable transient model errors caused by load changes, temporarily, lead to a worse response. In the examples shown in Figs. 10 and I l, for controller IV, those errors are appreciably decreased. For the simple half-proportional controller (V), the overall performance is very poor, especially for slow disturbances like AQs shown in Fig. 8. Therefore, this controller proved to be unsuitable for this particular application and care must be taken when applying it to similar control problems. The deadbeat response controller (VI) shows very good control behaviour for setpoint changes, but only at the operating point for which it has been designed. In the case of disturbances, however, its performance is not satisfactory. This was primarily due to the
401
Comparison and application of DDC algorithms for a heat exchanger
(III) algorithms consists of solving a linear system of three and five simultaneous equations given by (23) ahd (12), respectively. For the design of the optimal state feedback controller (VII) an observer must be designed and the matrix Riccati equation (32) must be solved. Considering the design parameters which can be chosen freely, the deadbeat response algorithm has the least requirements because only the sampling period T has to be specified. More detailed specifications are needed for the compensator design, where four closed-loop poles are required. In the adaptive control algorithm (IV) three model time constants must be chosen to compromize between fast response and the limitations of the control signal. In this algorithm, four additional time constants of the linear filters must be assumed without further knowledge of the process dynamics and the necessity of an exact process model. The optimal state feedback controller, however, needs an exact model in a special state-space form and has a large number of design parameters which must be chosen properly. The eigenvalues of the state observer are influenced by a heuristic procedure in this particular case. In addition the effect of the weighting matrix Q on the control behaviour is not completely obvious, and it should always be chosen by a simulation study. The amount of computer usage for the implementation of these algorithms can be compared directly in Table 2, showing the core storage and computing time requirement for each algorithm. The numerical values are typical for the implementation on the HP2100 process computer using FORTRANIV programs and only apply to the mathematical part of the control programs, i.e. excluding organization tasks like data acquisition, program scheduling and data output. The computing time and core storage for algorithms la, V and VI are the least among the seven controllers. These algorithms are represented by simple mathematical structure requiring few computational steps. The mathematical models of the anti-windup in PID (Ib) and cascade controller (llb) require a larger number
necessity of a long sampling period and the limitation of the reaction velocity. Therefore, the output error caused by disturbances settles slowly which is similar to the dynamic response of the plant. The state feedback controller (VII) proved to be rather insensitive to different operating conditions. The reason for this is due to the fact that the observer provides measured values of all states and, moreover, the weighting matrix Q penalizes any further variations in the states. The performance of the output error e(t) is also dependent upon matrix Q, giving rise to a faster response and less overshoot. The DDC algorithms tested in this investigation are designed for deterministic signals. The experiments using PRBS signals show that they are not suitable in the case of random disturbances. Therefore, if in practical applications the dominant characteristic of the disturbances is random, these algorithms should not be used or must be retuned, e.g. for the PID and cascade controller.
5.2 Design and implementation requirements The performance of a DDC algorithm always has to be balanced against the design and implementation requirements. One group of algorithms, the PID (I), cascade (II) and the HP algorithm (V), can be designed according to normalized controller setting specifications [6, 7, 11-13]. For the cascade algorithm in particular, rough controller adjustments can be found on the basis of the optimal PID controller. An improved design can be obtained either by a simulation study or by tuning the controller on site. The variation of any parameter (KR, Th TD, Tv and T) then requires the recomputation of the discrete controller coefficients. The other group of algorithms is characterized by design procedures based directly on a detailed mathematical model of the plant. These procedures are systematic and are usually performed by a computer program. The design of the deadbeat response (VI) and the compensator
TABLE 2. Computing time and core storage for the implementation of the DDC algorithms on a HP2100 process computer algorithm core storage in 16-bit words computing time T~ in millisec sampling time T in sec
relative computing time
la
lb
lla
lib
III
IV
V
VI
VII
! 14
168
216
371
360
926
153
114
284
1.2
1.5
3.1
3.3
13.6
56.9
!.3
1.2
8.7
5
5
5
5
5
4
25
75
5
0.062
0.066
0.272
i.42
0.005
0.002
0.174
0.024 0.03
H. UNBEHAUEN et al.
402
of logical decisions and operational steps, hence the core storage is increased by 50%. The optimal state feedback controller (VII), although very expensive in design, does not need much storage and requires even less computing time than the compensator algorithm (III), in which the additional compensation element Gg (s) also has to be computed. The computations of the adaptive algorithm (IV) include the identification, the decision process and the modification of the controller parameters, leading to the maximum amount of core storage and computing time requirements.
(lb) is the best to control the heat exchanger under different operating conditions. If an auxiliary signal is available in special applications, the PID can be changed to cascade controller which can slightly improve the dynamic behaviour. These results can also be extended to similar processes such as preheaters, evaporators, superheaters and others. In the case of predominantly random disturbances, all control algorithms fail. These difficulties can be overcome by using a stochastic controller design. This will be a deserving extention to the studies in comparative investigations of D D C algorithms.
6. CONCLUSIONS
The load dependence of the heat exchanger is characterized by decreasing gain and time constants for increasing load M. Therefore, the load change, from 35 to 80%, does not seriously effect the stability of the closed-loop system and in fact in some test cases gives even better results. This is a good example of the well known rule[6], that controllers for processes with parameters that change significantly due to load conditions, or other factors, should be designed for the worst case, i.e. for the situation with the highest gain and the largest time constants. This investigation shows that for the control of single input, single output systems subject to different operating conditions, the PlD, cascade and compensator control algorithms yield best results as far as the systems response, sensitivity to parameter variations, design requirement, the amount of computer storage and computation time are concerned. Other relatively simple control algorithms, e.g. HP and deadbeat response algorithms proved to be extremely sensitive to changes in the operating conditions and less efficient under step disturbances. This is generally valid for all DDC algorithms operating with a low sampling rate. Therefore, algorithms of that kind can only be used in certain applications, even though they are very promising with regard to the design requirements and the amount of computation. The application of adaptive algorithms is relatively inexpensive in the design procedure but seems to be justified only in the case of large parameter changes in the process dynamics or of the operating conditions because of the excessive computer requirement. In comparison, the computation of a control signal for an optimal state feedback controller is less expensive. Its large design requirement, however, can only be justified for a complicated multivariable control system. From the results of this investigation, it follows that the PlD controller with anti-windup
Acknowledgements--This research work was supported by the "Bundesministerium ffir Forschung und Technologie" of the Federal Republic of Germany under the PDV-project "ProzeBlenkung mit DV-Anlngen" of the 2nd DV-program. The authors wish to thank the reviewers, Prof. M. Jamshidi of Pahlavi University, Shiraz, Dan, and Dr. M. Searle of the Polytechnic of Central London for making a number of useful suggestions to improve the paper. REFERENCES [1] W. D. T. DAvies: Control algorithms for DDC. Instrum. Pract. 21, 70--77 (1967). [2] C. L. Sgrrtl: D/g/ta/Computer Process Control. Intext Educational, Scranton (1972). [3] J. Cox, L. J. HELLUMS,T. J. W:LL!~lS, R. S. BANKS G. J. KINK: A practical spectrum of DDC chemical process algorithms. J'-ISA 13, 65--72 (1966). [4] R. R. DE BOLT and B. E. POWELL: A natural 3-mode-controller algorithm for DDC. J.ISA 13, 43--47
(19~). [5] R. SCHILBACH,M. PRANDITand W. WERF.,it: Procedures
for realizing linear DDC algorithms (in German). Regdungstechnik 16, 337-344 (1968). [6l H. UNBEHAUEN:Stability and optimal tuning of linear and nonlinear controllers in single-loop systems, with different types of processes with proportional and integral action (in German). VDI-Verlng Dfisseldorf, Fortschr. Ber. VDI-Zeitschrift, 8, No. 13 (1970). [7] A. M. LOpEZ,P. W. MU~LLand C. L. SMITH:Tuning PI and PID di6ital controllers, lnstrums Control Syst. 42, 89-95 (1969). [8] W. WEBER: A systematic way for design of linear and adaptive control systems(in German). Elektrotech. Z. A, rag, 138-144 (1967). [9] W. SCHAUFEt,BEROF.R:Model-adaptive control systems (in German). Dissertation, ETH Ziirich (1969). [10] P. C. YOUNG: Applying parameter estimation to dynamic systems, Parts I and II. Control Engineering 16, 119-125 (1969) No. 10; No. 11, pp. 118-124. [11] V. FERNER: The half-proportional controller--a new promising type (in German). D/e Technik !1, 811 (1956). [12] M. ElCltNER: Design of a DDC system for control temperature and power of a steam generator (in German). Dissertation, Universit&t Stuttlpu't (1973). [13] M. EIC~INBR:DDC Design based o n a digital simulation technique. IFAC/IFlP-Symposium, Part IV-5, pp. I-9. Helsinki (1971). [14] E. Juav: Sampled-data Control Systems. Wiley, New York (1958). [15] M. A'rtlANS: On the design of PID controllers using optimal linear regulator theory. Automatica 7, 643-647
0971). [16] D. G. LUENanOmt: Observing the state of a linear system. Trans. IEEE MIL S, 74-80 (1964).