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Measurement optimization in optimal process control
Automatica, Vol. 6, pp. 705-714. Pergamon Press, 1970. Printed in Great Britain.
.Measurement Optimization in Optimal Process Control* Optimalisation de la mesure dans la eommande optimale des proc6d6s MeBoptimierung bei der optimalen ProzeBsteuerung OI-ITI4MI43aI.[I4~I r I 3 M e p e H I 4 ~ B OIITHMaYlbHOM
ynpaBsmHHHn p o t ~ e c c a M r t
A. SANOI- and M. TERAOI"
Process parameters, cost functions, disturbances and precision of measurements affect the optimum timing of the discrete measurements. The number and precision of the measurements can be selected suitably for specified control performances. Summary--In connection with the direct digital control of industrial processes, the optimum timing of measurements has been studied in an effort to reduce the required number and the precision of the measurements. This paper gives the optimum timing for stable and unstable processes and discusses the influence of the cost function, measurement noises, and process disturbances on the measurement timing. The cost functions which are quadratic in the control and the terminal error, or steady state error, are evaluated by the use of the linear estimation theory and the stochastic optimization technique. The optimum timing of measurements for process control systems subject to external disturbances has almost uniform intervals when the optimization in the relatively long control interval are considered. This paper presents a compromise between the number and the precision of measurements for some specified control performance to minimize the total required information quantity.
Concerning the optimum timing of observations, KUS~NER [1] discussed the discrete unstable linear process with unknown initial states and without any disturbances. MEIER, PESCHON and DRESSLER [2] have formulated the measuring adaptive problem in which control is available over not only the process but also the measurement subsystem. Their formulation is general, but it is limited to discrete systems. BARON and KLEINMAN [3] recently extended the work of Meier et al. to the investigation of the optimal sampling period (it should perhaps be designated as an observation duration) of the human operator with continuous measurements in a two dimensional case. The attention in the present paper is mainly directed to obtaining the analytical results which represent the effect of various parameters of continuous system with discrete measurements on the optimum timing not shown in the previous papers. The optimum timing of measurements is strongly influenced not only by the process constants, the form of the cost function and its parameters, but also by the precision of measurements and the disturbance signal to the processes. When the emphasis is put on the transient state of the start up in batch processes, in which the process output reaches the specified final value in relatively short time compared with the response time of the processes, the measurement optimization can be regarded as the transient optimization problem. At the same time, the optimization in a relatively long interval should be considered with an emphasis on the steady state after the process output has reached the specified final value. Such a steady state optimization is especially required in the direct digital control [4]. In this case, especially if some degradation of the cost performance can be allowed, the sampling rate and the precision of measurements should be selected to reduce the total required information quantity.
INTRODUCTION IN ORDER to achieve efficient computing control systems for industrial processes, it is important to investigate the measurement optimization so as to minimize the amount of information processing. The problem is to find the minimum required precision and the optimum timing e f the measurements. This becomes more important when the measurements are expensive to make. This paper discusses the optimum timing of measurements in the case when the control process is a linear time-varying continuous system disturbed by white Gaussian noises and the output measurements are discrete impulse modulated data with the restrictions on the precision and the total number of measurements, and it presents the analytical solutions for several particular cases. * Received 6 October 1969; revised 18 March 1970. The original version of this paper was presented at the 4th IFAC Congress which was held in Warsaw, Poland, during June 1969. It was recommended for publication in revised form by associate editor L Meier. t Department of Mathematical Engineering and Instrumentation Physics, University of Tokyo, Tokyo, Japan. 705
706
A. SANO and M. TERAO
This paper shows some practical results for both transient and steady state measurement optimization. The optimum control and the optimum timing of measurements are determined independently to minimize the cost function with quadratic form on some assumptions so that the timing of measurements can be specified a priori. The measurement optimization can be represented by deterministic factors such as process constants and the parameters of cost functions as well as by statistical factors which are the variance of measurement error and the initial uncertainty normalized by the variance of external disturbances. The two types of cost functions are considered for the transient optimization. One is quadratic in the control and terminal error, and the other is quadratic in the control and state error. Some illustrative examples are shown in the following sections. PROBLEM STATEMENT
The dynamics of controlled processes is generally described by the following linear time-varying noise perturbed differential equation
2(t) = A (t)x(t) + B(t)u(t) + v(t)
measurement pulses which minimizes the cost function with the following quadratic form l
a =½E[x'(r)fx(r)] +
+ u'(t)R(t)u(t)}dt
i5)
where F and Q(t) are n x n nonnegative definite matrices and R(t) is an r x r positive definite matrix. E V A L U A T I O N O F T H E O P T I M U M COST PERFORMANCE
The minimum value of the cost function can be evaluated by the use of the linear estimation theory and stochastic optimization techniques. In this section the derivation of the optimum cost performance as a function of optimum timing of measurements is outlined briefly. The problem of minimizing the cost function (5) subject to (1), (2), (3) and (4) may be decoupled to the problem of determining the minimum variance linear estimate of x(t) and the problem of stochastic optimum control u°(t), using the conditional expected value 2(t) with knowledge of a set of measurements [5][6] where
(1)
where x(t) is an n-state vector, u(t) is an r-control vector and v(t) is an n-white Gaussian noise vector with zero mean and covariance
f {x'(t)9_(t)x(t)
do
2(t)=E[x(t)ly(tl), y(t2) . . . . . . )'(t,): t;<_t].
(6)
Therefore, the minimum cost of (5) can be derived by the choice of the optimum timing of measurements {ti} and the optimum control tt°(t) as
(2)
Ev(t)v'(t') = V(t)b(t- t').
d° =rain " min {q}
The state of the process is estimated from a set of noise corrupted discrete impulsive measurements given by
y(t3 = M(ti)x(ti) + w(t3,
i=1,2 .....
k
(3) where y(tl) is an m-measurement vector and w(ti) is the white Gaussian measurement error with zero mean and covariance
Ew(t3w'(tj) = W(ti)6 u.
(4)
The uncertainty of the initial state is normally distributed with mean value Xo and covariance matrix Po- For the sake of simplicity, the random variables x(0), v(t) and w(t) are independent of each other. The number and the precision of possible measurements are usually limited because of physical or economical constraints so that it is important to find the optimum timing of the k
J = minFd, +min,121
u(t)
',t;l L
(7)
u(t)
where
J, = ½E[(x( T ) - 2(T))'F(x(T) - 2(T))] + ½Elf 2 (x(t)-2(t))'O(t)(x(t)--~(t')dtl,(8)
[
J2=½E *'(r)f.~(r)+
f'
{.~'(t)O_(O.rc(t)
0
+.'(,)R(t)u(t)Id,/
(9)
The Ja of equation (8), that is the penalty for the estimation error, is rewritten by denoting covariance matrix of estimation error with P(t)= E(x(I) - 2(t))(x(t)- 2(0)' as
J1 =½tr[FP(T)] +
tr[Q(t)P(O]dt.
(I O)
o
The conditional expected value 2(0 and its covariance P(t) are derived by the following equations using Kalman's linear estimation theory [7] [8]. If there is no measurement in an interval
Measurement opumlzation in optimal process control It+_1, tT] where t+_1 and t;- denote the instant after the observation at ti-1 and before the observation at tl respectively, then
t~_~ +
.f(t) = A( t)£(t) + B( t)u( t) ,
the dynamic programming approach in the stochastic optimal control leads the Bellman's equation [6][111
t,
P(t)=A(t)P(t)+P(t)A'(t)+ V(t), t+_t <_t<_tf" (12) and if there is one measurement and ~(t) and P(t) are corrected at t~, then [9]
~(t~) = 5c(t7)+ K(t,)[y(t,)-.P(h-)],
t = t, (13)
where
~(t;)=M(ti)~.(t~), and
P(t +) = P(ty) - K(ti)M(ti)P(tT ),
t = fi
(14)
707
u(t) L
i= t j, I
+ {A(t).~(t) + B(t)u(t)}'O/~ + ½{g'(t)Q(t)x(t) + u'(t)R(t)u(t)}l
(19)
where zll(t ) is the j, /-element of the covariance matrix Z(t) given by equation (18). The equation (19) slightly differs from the equation in the case of continuous measurement as treated in Ref. [6] [10][I 1] etc. in that the first term in the right side is discrete additive. The approach assuming the solution of the form
J°(£(t), t) = ½£'(t)S(t)P~(t)+ ½U(t)
where
K(ti) = P(tT)M'(ti)[M(fi)P(tT)M'(ti) + W(ti)]- 1, (15) and K(fi) is the optimum gain of the correction at t~ in the discrete Kalman filter [7]. Combining equation (11) with (13) for .f(t), and equation (12) with (14) for P(t),
(20)
yields the following differential equations for S(t) and U(t). Thus the optimum control is determined from equations (19) and (20)
u°(t) = -- R- l(t)B'(t) ~jo ~2(t) = -- R- l(t)B'(t)S(t)~(t).
(21)
tl<_t
.~¢(t)=A(t).~(t)+B(t)u(t)+ ~ z(ti)5(t--ti),
(16)
i=1
Substituting equations (20) and (21) into (19), we have
QS;t
P(t) = A(t)P(t) + P(t)A'(t)- ~ K(ti)M(t)P(ti)
S(t) = - S(t)A(t)- A'(t)S(t) + S(t)B(t)R - 1
i=1
5(t ti) + V(t)
where z(tl)=K(ti)[y(ti)-P(tT) ] and this term is easily shown to be a white random variable with zero mean and covariance
Ez(t~)z (tj)=Z(ti)fij=P(t i )M (h)K (ti)6U. (18) t
t~
--
!
!
The estimation error term equation (8) is given by the use of the solution P(t) for the initial condition P(0)=Po. On the other hand, the optimum control u°(t) and the optimum performance J°(~(0), 0) can be obtained, regarding equation (16) as the noise perturbed control process equation with the quadratic performance index (9). Defining the optimum performance of equation (9) as
J~(2(t), 0 = rain ½EV£'(T)F2(T) .(t)
(t)B'(t)S(t)- Q (t),
(17)
- -
L
+ f~ (£'(t)Q(t)¢(t)+ u'(t)R(t)u(t))dt 1,
(22)
t~
(](t)=- ~ tr[S(t)Z(t,)]6(t-q)
(23)
i=1
with boundary conditions S(T)= F and U(T)=0. Therefore, the optimum performance jo is given, substituting the solution of equations (22) and (23) into (20), as
s°(~(0), o) =½~,s(O)~o k
+½ ~. tr[K(t~)M'(fi)P(ti)S(t,) ]. (24) i=l
As a result, the original optimum performance equation (7) is reduced to the following, using equations (10) and (24) and omitting the term which does not affect the timing {t~},
jo = min~ ½tr[FP( T)-I + ½It tr[ Q(t)P( t)Jdt {tdk
go
+½,=1 ~ tr[K(t,)M'(t,)P(t,)S(t,)]].
(25)
708
A. SkNO and M. TERAO
The first and the second terms denote the cost due to the state uncertainty caused by external disturbances and measurement error, and the third term denotes the cost of the k times of control corrections based on the k discrete measurements. The following sections illustrate several examples in a scalar case.
be easily derived as follows. I~ a < 0 , a stable process having time constant of 1/a, the both first and second terms in equation (28) for k = 1 increase as the tl increases, and so the optimum t~ is always zero. If a > 0 , an unstable process having a pole in the right half plane, the optimum t I is obtained by differentiating equation (28) for k = l with respect to t~. As a result.
SOME ILLUSTRATIONS F O R T R A N S I E N T O P T I M I Z A T I O N PROBLEM
tl=0,
In this section it is shown that the optimum timing of measurements is derived analytically in a scalar case. The analytical results clarify the effects of disturbances, measurement noises, process constant and other parameters on the optimum timing. Two types of cost performance, (1) Q = 0 , F = 1 and (2) Q = 1, F = 0 in equation (5) are considered.
1. The case: Q =0, F = t The cost function under consideration is quadratic in the control effort and terminal error. Let A(t)=a, B(t)=M(t)=l.O, R(t)=r, V(t)=V and W(t)= W in equations (1)-(5), then the minimum cost function of equation (25) in the scalar case is reduced to so = minFkP(T ) , . ~
S(fi)PZ(tf-)]
tl
a<0
= T - ~aln[x/2ar(1 + e2ar/2) + 1],
O
a>0.
(29b)
In Fig. 1, the optimum timing is shown against the process parameter a for various values of £ and r in the case of T = 1.0. It is interesting to note that although the optimum timing for a < 0 is always the initial time, the timing for a > 0 becomes later and approaches to T/2 for large a, as indicated also by KUSHNER [1]. Further, Fig. 1 shows that the optimum timing becomes later as the precision of measurement and the weighting coefficient r decrease and it seems to be consistent with our intuition. 50F
(26)
(29a)
r=o'o1 1
a
/X=O'01 0"I 1 10
where S(t) is the solution of the Riccati differential equation (22) given by
s(t) =
2at
1 + ( 2 a r - 1)e - 2 a r T - t ) "
\ ~\ \ \ \ / / j
(27) 5
1.1. The case with no disturbances. The case in which there is only an initial uncertainty is discussed. Normalizing j0 of equation (26) by the variance of the initial uncertainty Po and manipulating P(tZ) using equation (17), the optimum performance is given as
d°/Po =min
"~e2aT
:
/ -
/
/
{l+(2ar-1)e-2"{r-o}(~e2"ts+Ay~e2,0+AI \j=l
/\i=1
A
(28) where 2 = W/Po is the ratio of the variance of measurement error to the initial uncertainty. The cost performance equation (28) is to be minimized with respect to t . i = 1 , 2 . . . . . k. If k = 1, the case of a single measurement, the optimum timing may
/
/
/
/
/
/
I
/
0"10
0.2
I
/
/ /
/ f
/ 1
/
/
/ I
\
I
/ l
/
] 1
\
\
l I
77
!
are4at'
(
\
0'5
k
/ .,10
N
/
F
.x=o.ol .0.1
I
0 "/.
I
l
0'6
,/1
/ / I
I
0'8
J
1"0 t~
FIG. 1. Optimum timing of the single measurement for the process with no disturbances: ( F = 1.0, Q = 0.0).
1.2. The case with disturbances. It is conceivable that the control processes are always perturbed by external disturbances. The following discussion is focused on the case when the measurements
Measurement optimization in optimal process control are quite pre~se or the disturbance is quite large compared with the initial uncertainty and the measurement error. If k = l , then the optimum performance of equation (26) normalized by the variance of disturbances is given calculating P(ti') of equation (17) subject to disturbances and assuming that the measurement is quite precise, as
l-_e 2-, +e2°r J°/V--min/t, L 4ae r{(2at/+l)e2°'l-1} ] 4 2{1 + ( 2 a r - 1)e -2°(r-'')}
(30)
where t/=Po/V. Therefore the optimum timing of one measurement pulse may be uniquely determined differentiating equation (30) as
short time constant, it approaches the terminal time T asymptotically. For an ordinary stable process, if the initial uncertainty is negligible, i.e. t/--0, then the timing is near the terminal time of the control interval as represented by the solid lines in Fig. 2; however, if an initial uncertainty is present, i.e. if= 10, then the timing occurs earlier as represented by the dashed lines in Fig. 2. If the initial uncertainty ff is quite large, the measurement is to be made at the initial part of the interval. When the measurement has arbitrary precision, it is not so simple to derive the optimum timing analytically. In this case the optimum tl can he obtained numerically, and Fig. 3 shows the optimum timing against process constant for several measurement grades of precision normalized by the variance of disturbance (~ = W~ V).
?/
t, = T - l l n [ :[: x/2ar{2a r - 1 + (2at/+ l)e 2aT} -(2ar-1)],
aX0.
(31)
//
t
/
// /
The optimum timing obtained analytically by equation (31) is shown in Fig. 2 against the process 7
709
' ';// // /11
a
2
I "
0
~
/
t
I
I
i
__.J"
1"0 tl
-I
3
= 0"01
0"I
2
-2
1
-3
0
-4
-I
3 !
I
,~=I0 0
F]o. 3. Optimum timing of the error corrupted single measurement for the process with disturbances: (F= ].o, Q= o.o).
-2 -3 -4 FIG. 2. Optimum timing of the quite precise single
measurement for the process with disturbances: (F= |.0, Q=O.O). parameter a for various weighting coefficient • and initial uncertainty r/. It is observable that for large positive a, very unstable process, the optimum timing approaches the middle of the control interval and for large negative a, a stable process having a
2. The case: Q=I, F = 0 In this section the cost function is assumed to be quadratic in the control and in the state error. If the control interval [0, T] is large enough compared with the process parameter a, the control procedure can be viewed as in the steady state. Let A(t)=a, B(t)=M(t)=l.O, R(t)=r, V(t)=V and W(t)= W, then the optimum performance (25) is given by
Ud
U do
z~I P(tF)+ W ]
(32)
710
A. SANO and M. TERAO
where S(t) is the solution of the Riccati differential equation,
S(t)
l - - e - 2#(T-t)
, O.CI
fl - a + (fl + a ) e - 2~{T-,/'
0-I 1
---
;0
fl=~!+a 2.
(33)
If the control optimization interval T is considered to be large enough, then equation (33) becomes 0.5
S-r(fl+a).
(34) X=
2.1. The case with no disturbances. It requires more than one measurement to optimize in the steady state. However, for some analytical investigation, it may be convenient to consider the case o f k = 1. The optimum timing for arbitrary k can be obtained by the use of the simple hill climbing method since there is only one local optimum in this case. Normalizing j o in equation (32) by Po we obtain the following for k = 1,
L
r(fl+a)e4~"l
o.ol
0'~ r=OOt
0'10
I
1
2
(35)
+ 2(e 2 ' ' + 2) ]
a<0,
1 q=
[~/ In
FIG, 4. Optimum timing of the single measurement for the process with no disturbances: (F=0.0, Q = 1 '0).
j°/V:minFl~(42(t~-)+
(36a)
Figure 4 shows the solution given by equation (36) against the process parameter a > 0 for various 2 and r in the case of T = 5. When the penalty paid for the steady state error is more than that paid for the control corrections, that is, the weighting coefficient r decreases, the optimum t I approaches
a>O.
(e2afq~t-ti)--l)
{t,} L 4 a b = o \
2e 2~r ] 2z-+2ar(,B+a)+ 1 2 , a > O . (36b)
q=lln[x/22+2e2"r-2],
Z,
t~
v}±, ~" s~:(t:)-I
where 2 = W/Po. The optimum timing of t~ may be obtained by differentiating equation (35) with respect to tl and the result is ta=0,
3
2.2. The case with e vternal disturbances. The cost performance (32) subject to external disturbances is rewritten as
--e 2"q +2(e 2 a t - 1) J°/P° = min[eg"U 4a(e 2"t' + 2) t~
10 01
(37)
Figure 4 shows that the optimum timing does not depend on so much r, but on the measurement precision, 2. The more precise the measurement, a smaller 2, the earlier the measurement should be made.
i=lfP(ti
(38)
) q'- k ]
where ~(ti+) = P(t +)/1/, t1= P(O)/V, and ~ = W/V. For arbitrary q and ¢, the problem is much too elaborate to obtain the optimum {ti} analytically, so that numerical computation by a digital computer is more practical. The results for k = 1 are summarized in Fig. 5 and for k = 5 in Fig. 6. For k = l , the optimum timing of measurements is plotted for various initial uncertainty q and measurement precision 4. When the controlled process is stable (a<0), the optimum timing is rather sensitive to the initial uncertainty, and it approaches the initial part of the control interval as q increases. On the other hand, for tile unstable process (a > 0) the optimum timing approaches the middle of the interval for any q and 3. In the case where the process is not so much subject to the initial uncertainty but to the large external disturbances, the optimum timing t~ is obtained analytically (q =0. =0)
tl = T _ lln[2ar(fl + a) + 1] 2
4a
(39)
Measurement optimization in optimal process control
a
7
are to be observed since the optimum timing of measurements strongly depends on the selected state variables. In connection with this problem we are currently investigating the optimum location of measurements in the distributed parameter systems.
//Ill
i/tll
/
J lll
'oo
tilt lt]iz/"
1.,
711
O P T I M U M SELECTION OF THE SAMPLE-RATE A N D THE PRECISION OF MEASUREMENTS
3
In the last section it was established that the timing of measurements in steady state optimization subject to external disturbances has uniform intervals. Therefore, if the control interval T is large enough compared with the process response, the estimation error given by equation (17) becomes periodic with sampling interval A. Then the cost performance of equation (38) is reduced to
2
,-Y,"/II".
I
I
0
5
tl
P(t)
-1
-2
•
-3 "
_
.
i// i.i//ll
"/"
//I
~=0"01
/
~ i
1 ,o
do= r-.oolimJ°/Tv=l{A(dpx+l)(e2"a-1)-I
t -t
FIG. 5. Optimum timing of the single measurement for the process with disturbances: (F=0.0, Q = 1.0).
(40)
where
7 6
S~b~ 2A(q~2+ 4)
}
~b2 = ½(C + x/C 2 + D), t2
t~
C = (~a+ ~e 2°A-I), D = 2~(e2~A-aI)
5
IV/V
3 2
k=5 '7= 0"01
4=o.1 1
0
I
I
4
5
t
-2 -3
FiG. 6. Optimum timing of five measurements for the process with disturbances: (F=0.0, Q = 1.0).
and it is approximately the middle of the interval. For arbitrary k it is recognized that k measurements are to be taken at almost uniform intervals as shown in Fig. 6. These results are computed numerically using a k-dimensional hill climbing method. If the control process is a multivariable system, it is important to determine which state variables
and ~ = presents the precision of measurements. A denotes the sampling interval and so A -1 corresponds to the sample-rate. In connection with the D D C (direct digital control) for the process it is significant to reduce the total required information quantity of measurements as long as the specified control performance is satisfied [4]. The quantity of information which relates the number and the precision of measurements is reduced remarkably if some degradation of the control performance can be allowed. Figures 7 and 8 show the required sample-rate and the precision of measurements for specified performances for a = 1 and a = - 1 respectively. It may be observed that there is a tradeoff between the sample rate A -1 and the precision ~ for some specified performance. The near optimum performances are normalized by the ideal performance J~a which is obtained when measurements are not constrained, and it is calculated from equation (40) by A- t ~ ov and ~ - 1 ~ o o . Jia=
lim
lim
A - l(e2aA
jo= -
lim
1)(2aS + 1) - 2a _ S 8a 2 2
(41)
712
A. SANO and M. TERAO lOO ~-I
a:l r=l
lo
- - -
. . . . . . .
":"~"' .............
i t
0"~
,
......
-~,
l
0"2
I
I0
I00
L~ 1
I000
Fic. 7. Required sample rate A-I and the precision
of measurements for specified control performances: (unstable process with disturbances).
/
,:
0.2 . . . . . i 0'2
\
, ..\.,I\ , \ .
I
I0
100
1000
Fio. 8. Required sample-rate A-1 and the precision of measurements for specified control performances: (stable process with disturbances). The value J u of equation (41) is also the limit of equation (40) when only the sample-rate A-1 increases. So, to improve the performance as much as possible it is desirable to increase the sample-rate rather than the precision of measurements. It may be noted in Fig. 7 that more information quantity is required to optimize an unstable process than a stable process. There are several methods to determine the optimum A and ~ so that specified control performance is satisfied. For instance, a tradeoff between the sample-rate and the precision can be made through the following discussion minimizing the total required information capacity.
In case of computer control, the measurement data should be quantized for the analog to digital conversion and the precision of the measurements corresponds to the word-length of the quantizcd measurement data so that the word-length, samplerate tradeoffs become a problem [12]. The quantization error is considered to be equivalent to white noise having uniform distribution with variance q2/12 where q is the level of the quantization step and it corresponds to the measuremeant noise W approximately through the central limit theorem. The precision of measurement can be denoted by the binary word-length b(bits) instead of £ and ~ corresponds to ~b approximately, since
Measurement optimization in optimal process control the full range of quantization, which is plus and minus three times of the standard deviation of disturbances, may be specified as 6,/V where q = 6 x/V/2 b and V is the variance of disturbances. Using this correspondence, Fig. 8 can be reduced to Fig. 9 which shows the tradeoff between the precision (bits) and the sample-rate (A -1) of measurements.
I 2
~
6
8 10 20 ~1 ( samples/unit time )
/*0
FIG. 9. Optimum selection of the sample-rate and the quantization.
If a suboptimum cost performance is specified, the selection of the optimum sample-rate and quantization of measurements should be made along the broken line minimizing the total required information quantity (bits/unit time) [4] which is represented by the straight lines shown in Fig. 9. CONCLUSION
In connection with computing control, measurement optimization has been discussed for the case where the control process is a linear continuous system disturbed by noises subject to the cost function which consists of the control cost and the terminal error or state error. For a stable, scalar system, the optimum timing for the measurement is always in the initial part of the control interval if the external disturbances are absent. In the presence of disturbances, however, the optimum timing strongly depends on the form of cost function. For a very unstable process, the measurements should be made in nearly uniform intervals independent of the initial uncertainty, disturbances and the parameters of the cost function. The steady state optimization which is especially required for direct digital control has been discussed in terms of a tradeoff between the sample-rate and the precision of measurements for various near optimum performance. These discussions have been rather general in nature, and they have included not only stable processes but also unstable ones such as those typified by fermentation processes or nuclear
713
reactors and in addition, the methods may be applied to the measurement optimization of multivariable processes. REFERENCES [1] H. J. Kusm, r~g: On the optimal timing of observations for linear control system with unknown initial states. IEEE Trans. Aut. Control AC-9, 144-145 (1964). [2] L. MEn~, J. I~CHON and P. M. D ~ s s L ~ : Optimal control of measurement subsystems. IEEE Trans. Aut. Control AC-12, 528-536 0967). [3] S. B~a~oN and D. L. Kt~I~rMA~q: The human as an optimal controller and information processor. IEEE Trans. on A I M S MMS-10, 9-16 (1969). [4] M. TERAO: Quantization and sampling selection for efficient DDC. Instrum. Tech. 14, 49-55 0967). [5] J. J. FLO~wrrN: Partial observability and optimal control. J. Electron. Control 9, 65-80 0960). [6] W. M. Wo~,~M: Stochastic problem in optimal control. IEEE Inter. Cony. Rec. 11, pt. 2, 114-124 (1963). [7] R. E. KALMAN: A new approach to linear filtering and prediction problems. Trans. ,4SME, J. bns. Engng, ser. D 82, 35-45 (1960). [8] R. E. KALMANand R. S. Bucv: New results in linear filtering and prediction theory. Trans. A S M E , J. bas. Engng, ser D 83, 528-536 (1961). [9] S. S. L. CnANO. Optimum filtering and control of randomly sampled systems. IEEE Trans. ,4ut. Control AC-12, 537-546 (1967). [10] J. J. FLORENTIN" Optimal control of continuous time, Markov stochastic systems. J. Electron. Control 10, 473-488 (1961). [11] H. J. K u ~ : Optimal stochastic control. IRETrans. Aut. Control AC-7, 120-122 (1962). [12] R. N. LrrcEaAgGEg and G. K. L. CmEN: Word-length sample-rate tradeoffs in computer control systems. J A C C Preprints, 124-130 (1964).
R~hmmt---En liaison avec la commande numtrique directe des p r o c e s s industriels, la cadence optimale des mesures a 6t6 6tudite dans le but de rtduire le hombre et la prtcision ntcessaires des mesures. Le prtsent article donne la cadence optimale pour des proced~ stables et instables et discute l'influence de la fonction de coot, des bruits dans la mesure et des perturbations dans le proc~d6 sur la cadence des mesures. Les fonctions de cofit qui sont quadratiques dans la commande et l'erreur finale, oil erreur en regime 6tabli, sont 6valutes en utilisant la thtorie lintaire d'estimation et la technique d'optimalisation stochastique. La cadence optimale des mesures pour des systtmes de commande de proctd~ soumis /l des perturbations ext~rieures posstde des intervalles de temps presque uniformes lorsqu'on considtre l'optimalisation dans un intervalle de temps de commande relativement long. Le prc~sent article propose un compromis entre le nombre et la prh:ision des mesures pour une certaine performance sptcifi~ de commande afin de minimaliser la quantit6 totale d'information ntcessaire.
Zusammenfassung--In Verbindung mit der direkten digitalen Regelung industrieller Prozesse wurde die optimale Zeitsteuerung von Messungen studiert, um die geforderte Zahl und Priizision der Messungen zu reduzieren. Angegeben wird die optimale Zcitsteuerung fOr stabile und instabile Prozesse und der Einflul3 der Kostenfunktion, des MeBrauschens und der ProzeBsttrungen auf die Zeitsteuerung diskutiert. Die Kostenfunktionen, die im Regel- und Endfehler quadratisch sind, oder der statische Fehler, werden auf Grund der linearen Schtitztheorie und der stochastischen Optimierungstechnik ausgewertet.
714
A . SANO a n d M . TERAO
Die optimale Zeitsteuerung von Messungen fiir die Prozel~steuerung, die fiul3eren St6rungen unterworfen ist, hat beinahe gleichmfil3ige lntervalle, wenn die Optimierung in dem relativ langen Steuerungsintervall betrachtet wird. Vorgeschlagen wird ein KompromiB zwischen der Zahl und der Genauigkeit der Messungen und zwar fiir cinige Ffillc des Steuerungsverhaltens, um die insgesamt erforderliche Informationsmenge zu minimieren.
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.Measurement Optimization in Optimal Process Control* Optimalisation de la mesure dans la eommande optimale des proc6d6s MeBoptimierung bei der optimalen ProzeBsteuerung OI-ITI4MI43aI.[I4~I r I 3 M e p e H I 4 ~ B OIITHMaYlbHOM
ynpaBsmHHHn p o t ~ e c c a M r t
A. SANOI- and M. TERAOI"
Process parameters, cost functions, disturbances and precision of measurements affect the optimum timing of the discrete measurements. The number and precision of the measurements can be selected suitably for specified control performances. Summary--In connection with the direct digital control of industrial processes, the optimum timing of measurements has been studied in an effort to reduce the required number and the precision of the measurements. This paper gives the optimum timing for stable and unstable processes and discusses the influence of the cost function, measurement noises, and process disturbances on the measurement timing. The cost functions which are quadratic in the control and the terminal error, or steady state error, are evaluated by the use of the linear estimation theory and the stochastic optimization technique. The optimum timing of measurements for process control systems subject to external disturbances has almost uniform intervals when the optimization in the relatively long control interval are considered. This paper presents a compromise between the number and the precision of measurements for some specified control performance to minimize the total required information quantity.
Concerning the optimum timing of observations, KUS~NER [1] discussed the discrete unstable linear process with unknown initial states and without any disturbances. MEIER, PESCHON and DRESSLER [2] have formulated the measuring adaptive problem in which control is available over not only the process but also the measurement subsystem. Their formulation is general, but it is limited to discrete systems. BARON and KLEINMAN [3] recently extended the work of Meier et al. to the investigation of the optimal sampling period (it should perhaps be designated as an observation duration) of the human operator with continuous measurements in a two dimensional case. The attention in the present paper is mainly directed to obtaining the analytical results which represent the effect of various parameters of continuous system with discrete measurements on the optimum timing not shown in the previous papers. The optimum timing of measurements is strongly influenced not only by the process constants, the form of the cost function and its parameters, but also by the precision of measurements and the disturbance signal to the processes. When the emphasis is put on the transient state of the start up in batch processes, in which the process output reaches the specified final value in relatively short time compared with the response time of the processes, the measurement optimization can be regarded as the transient optimization problem. At the same time, the optimization in a relatively long interval should be considered with an emphasis on the steady state after the process output has reached the specified final value. Such a steady state optimization is especially required in the direct digital control [4]. In this case, especially if some degradation of the cost performance can be allowed, the sampling rate and the precision of measurements should be selected to reduce the total required information quantity.
INTRODUCTION IN ORDER to achieve efficient computing control systems for industrial processes, it is important to investigate the measurement optimization so as to minimize the amount of information processing. The problem is to find the minimum required precision and the optimum timing e f the measurements. This becomes more important when the measurements are expensive to make. This paper discusses the optimum timing of measurements in the case when the control process is a linear time-varying continuous system disturbed by white Gaussian noises and the output measurements are discrete impulse modulated data with the restrictions on the precision and the total number of measurements, and it presents the analytical solutions for several particular cases. * Received 6 October 1969; revised 18 March 1970. The original version of this paper was presented at the 4th IFAC Congress which was held in Warsaw, Poland, during June 1969. It was recommended for publication in revised form by associate editor L Meier. t Department of Mathematical Engineering and Instrumentation Physics, University of Tokyo, Tokyo, Japan. 705
706
A. SANO and M. TERAO
This paper shows some practical results for both transient and steady state measurement optimization. The optimum control and the optimum timing of measurements are determined independently to minimize the cost function with quadratic form on some assumptions so that the timing of measurements can be specified a priori. The measurement optimization can be represented by deterministic factors such as process constants and the parameters of cost functions as well as by statistical factors which are the variance of measurement error and the initial uncertainty normalized by the variance of external disturbances. The two types of cost functions are considered for the transient optimization. One is quadratic in the control and terminal error, and the other is quadratic in the control and state error. Some illustrative examples are shown in the following sections. PROBLEM STATEMENT
The dynamics of controlled processes is generally described by the following linear time-varying noise perturbed differential equation
2(t) = A (t)x(t) + B(t)u(t) + v(t)
measurement pulses which minimizes the cost function with the following quadratic form l
a =½E[x'(r)fx(r)] +
+ u'(t)R(t)u(t)}dt
i5)
where F and Q(t) are n x n nonnegative definite matrices and R(t) is an r x r positive definite matrix. E V A L U A T I O N O F T H E O P T I M U M COST PERFORMANCE
The minimum value of the cost function can be evaluated by the use of the linear estimation theory and stochastic optimization techniques. In this section the derivation of the optimum cost performance as a function of optimum timing of measurements is outlined briefly. The problem of minimizing the cost function (5) subject to (1), (2), (3) and (4) may be decoupled to the problem of determining the minimum variance linear estimate of x(t) and the problem of stochastic optimum control u°(t), using the conditional expected value 2(t) with knowledge of a set of measurements [5][6] where
(1)
where x(t) is an n-state vector, u(t) is an r-control vector and v(t) is an n-white Gaussian noise vector with zero mean and covariance
f {x'(t)9_(t)x(t)
do
2(t)=E[x(t)ly(tl), y(t2) . . . . . . )'(t,): t;<_t].
(6)
Therefore, the minimum cost of (5) can be derived by the choice of the optimum timing of measurements {ti} and the optimum control tt°(t) as
(2)
Ev(t)v'(t') = V(t)b(t- t').
d° =rain " min {q}
The state of the process is estimated from a set of noise corrupted discrete impulsive measurements given by
y(t3 = M(ti)x(ti) + w(t3,
i=1,2 .....
k
(3) where y(tl) is an m-measurement vector and w(ti) is the white Gaussian measurement error with zero mean and covariance
Ew(t3w'(tj) = W(ti)6 u.
(4)
The uncertainty of the initial state is normally distributed with mean value Xo and covariance matrix Po- For the sake of simplicity, the random variables x(0), v(t) and w(t) are independent of each other. The number and the precision of possible measurements are usually limited because of physical or economical constraints so that it is important to find the optimum timing of the k
J = minFd, +min,121
u(t)
',t;l L
(7)
u(t)
where
J, = ½E[(x( T ) - 2(T))'F(x(T) - 2(T))] + ½Elf 2 (x(t)-2(t))'O(t)(x(t)--~(t')dtl,(8)
[
J2=½E *'(r)f.~(r)+
f'
{.~'(t)O_(O.rc(t)
0
+.'(,)R(t)u(t)Id,/
(9)
The Ja of equation (8), that is the penalty for the estimation error, is rewritten by denoting covariance matrix of estimation error with P(t)= E(x(I) - 2(t))(x(t)- 2(0)' as
J1 =½tr[FP(T)] +
tr[Q(t)P(O]dt.
(I O)
o
The conditional expected value 2(0 and its covariance P(t) are derived by the following equations using Kalman's linear estimation theory [7] [8]. If there is no measurement in an interval
Measurement opumlzation in optimal process control It+_1, tT] where t+_1 and t;- denote the instant after the observation at ti-1 and before the observation at tl respectively, then
t~_~ +
.f(t) = A( t)£(t) + B( t)u( t) ,
the dynamic programming approach in the stochastic optimal control leads the Bellman's equation [6][111
t,
P(t)=A(t)P(t)+P(t)A'(t)+ V(t), t+_t <_t<_tf" (12) and if there is one measurement and ~(t) and P(t) are corrected at t~, then [9]
~(t~) = 5c(t7)+ K(t,)[y(t,)-.P(h-)],
t = t, (13)
where
~(t;)=M(ti)~.(t~), and
P(t +) = P(ty) - K(ti)M(ti)P(tT ),
t = fi
(14)
707
u(t) L
i= t j, I
+ {A(t).~(t) + B(t)u(t)}'O/~ + ½{g'(t)Q(t)x(t) + u'(t)R(t)u(t)}l
(19)
where zll(t ) is the j, /-element of the covariance matrix Z(t) given by equation (18). The equation (19) slightly differs from the equation in the case of continuous measurement as treated in Ref. [6] [10][I 1] etc. in that the first term in the right side is discrete additive. The approach assuming the solution of the form
J°(£(t), t) = ½£'(t)S(t)P~(t)+ ½U(t)
where
K(ti) = P(tT)M'(ti)[M(fi)P(tT)M'(ti) + W(ti)]- 1, (15) and K(fi) is the optimum gain of the correction at t~ in the discrete Kalman filter [7]. Combining equation (11) with (13) for .f(t), and equation (12) with (14) for P(t),
(20)
yields the following differential equations for S(t) and U(t). Thus the optimum control is determined from equations (19) and (20)
u°(t) = -- R- l(t)B'(t) ~jo ~2(t) = -- R- l(t)B'(t)S(t)~(t).
(21)
tl<_t
.~¢(t)=A(t).~(t)+B(t)u(t)+ ~ z(ti)5(t--ti),
(16)
i=1
Substituting equations (20) and (21) into (19), we have
QS;t
P(t) = A(t)P(t) + P(t)A'(t)- ~ K(ti)M(t)P(ti)
S(t) = - S(t)A(t)- A'(t)S(t) + S(t)B(t)R - 1
i=1
5(t ti) + V(t)
where z(tl)=K(ti)[y(ti)-P(tT) ] and this term is easily shown to be a white random variable with zero mean and covariance
Ez(t~)z (tj)=Z(ti)fij=P(t i )M (h)K (ti)6U. (18) t
t~
--
!
!
The estimation error term equation (8) is given by the use of the solution P(t) for the initial condition P(0)=Po. On the other hand, the optimum control u°(t) and the optimum performance J°(~(0), 0) can be obtained, regarding equation (16) as the noise perturbed control process equation with the quadratic performance index (9). Defining the optimum performance of equation (9) as
J~(2(t), 0 = rain ½EV£'(T)F2(T) .(t)
(t)B'(t)S(t)- Q (t),
(17)
- -
L
+ f~ (£'(t)Q(t)¢(t)+ u'(t)R(t)u(t))dt 1,
(22)
t~
(](t)=- ~ tr[S(t)Z(t,)]6(t-q)
(23)
i=1
with boundary conditions S(T)= F and U(T)=0. Therefore, the optimum performance jo is given, substituting the solution of equations (22) and (23) into (20), as
s°(~(0), o) =½~,s(O)~o k
+½ ~. tr[K(t~)M'(fi)P(ti)S(t,) ]. (24) i=l
As a result, the original optimum performance equation (7) is reduced to the following, using equations (10) and (24) and omitting the term which does not affect the timing {t~},
jo = min~ ½tr[FP( T)-I + ½It tr[ Q(t)P( t)Jdt {tdk
go
+½,=1 ~ tr[K(t,)M'(t,)P(t,)S(t,)]].
(25)
708
A. SkNO and M. TERAO
The first and the second terms denote the cost due to the state uncertainty caused by external disturbances and measurement error, and the third term denotes the cost of the k times of control corrections based on the k discrete measurements. The following sections illustrate several examples in a scalar case.
be easily derived as follows. I~ a < 0 , a stable process having time constant of 1/a, the both first and second terms in equation (28) for k = 1 increase as the tl increases, and so the optimum t~ is always zero. If a > 0 , an unstable process having a pole in the right half plane, the optimum t I is obtained by differentiating equation (28) for k = l with respect to t~. As a result.
SOME ILLUSTRATIONS F O R T R A N S I E N T O P T I M I Z A T I O N PROBLEM
tl=0,
In this section it is shown that the optimum timing of measurements is derived analytically in a scalar case. The analytical results clarify the effects of disturbances, measurement noises, process constant and other parameters on the optimum timing. Two types of cost performance, (1) Q = 0 , F = 1 and (2) Q = 1, F = 0 in equation (5) are considered.
1. The case: Q =0, F = t The cost function under consideration is quadratic in the control effort and terminal error. Let A(t)=a, B(t)=M(t)=l.O, R(t)=r, V(t)=V and W(t)= W in equations (1)-(5), then the minimum cost function of equation (25) in the scalar case is reduced to so = minFkP(T ) , . ~
S(fi)PZ(tf-)]
tl
a<0
= T - ~aln[x/2ar(1 + e2ar/2) + 1],
O
a>0.
(29b)
In Fig. 1, the optimum timing is shown against the process parameter a for various values of £ and r in the case of T = 1.0. It is interesting to note that although the optimum timing for a < 0 is always the initial time, the timing for a > 0 becomes later and approaches to T/2 for large a, as indicated also by KUSHNER [1]. Further, Fig. 1 shows that the optimum timing becomes later as the precision of measurement and the weighting coefficient r decrease and it seems to be consistent with our intuition. 50F
(26)
(29a)
r=o'o1 1
a
/X=O'01 0"I 1 10
where S(t) is the solution of the Riccati differential equation (22) given by
s(t) =
2at
1 + ( 2 a r - 1)e - 2 a r T - t ) "
\ ~\ \ \ \ / / j
(27) 5
1.1. The case with no disturbances. The case in which there is only an initial uncertainty is discussed. Normalizing j0 of equation (26) by the variance of the initial uncertainty Po and manipulating P(tZ) using equation (17), the optimum performance is given as
d°/Po =min
"~e2aT
:
/ -
/
/
{l+(2ar-1)e-2"{r-o}(~e2"ts+Ay~e2,0+AI \j=l
/\i=1
A
(28) where 2 = W/Po is the ratio of the variance of measurement error to the initial uncertainty. The cost performance equation (28) is to be minimized with respect to t . i = 1 , 2 . . . . . k. If k = 1, the case of a single measurement, the optimum timing may
/
/
/
/
/
/
I
/
0"10
0.2
I
/
/ /
/ f
/ 1
/
/
/ I
\
I
/ l
/
] 1
\
\
l I
77
!
are4at'
(
\
0'5
k
/ .,10
N
/
F
.x=o.ol .0.1
I
0 "/.
I
l
0'6
,/1
/ / I
I
0'8
J
1"0 t~
FIG. 1. Optimum timing of the single measurement for the process with no disturbances: ( F = 1.0, Q = 0.0).
1.2. The case with disturbances. It is conceivable that the control processes are always perturbed by external disturbances. The following discussion is focused on the case when the measurements
Measurement optimization in optimal process control are quite pre~se or the disturbance is quite large compared with the initial uncertainty and the measurement error. If k = l , then the optimum performance of equation (26) normalized by the variance of disturbances is given calculating P(ti') of equation (17) subject to disturbances and assuming that the measurement is quite precise, as
l-_e 2-, +e2°r J°/V--min/t, L 4ae r{(2at/+l)e2°'l-1} ] 4 2{1 + ( 2 a r - 1)e -2°(r-'')}
(30)
where t/=Po/V. Therefore the optimum timing of one measurement pulse may be uniquely determined differentiating equation (30) as
short time constant, it approaches the terminal time T asymptotically. For an ordinary stable process, if the initial uncertainty is negligible, i.e. t/--0, then the timing is near the terminal time of the control interval as represented by the solid lines in Fig. 2; however, if an initial uncertainty is present, i.e. if= 10, then the timing occurs earlier as represented by the dashed lines in Fig. 2. If the initial uncertainty ff is quite large, the measurement is to be made at the initial part of the interval. When the measurement has arbitrary precision, it is not so simple to derive the optimum timing analytically. In this case the optimum tl can he obtained numerically, and Fig. 3 shows the optimum timing against process constant for several measurement grades of precision normalized by the variance of disturbance (~ = W~ V).
?/
t, = T - l l n [ :[: x/2ar{2a r - 1 + (2at/+ l)e 2aT} -(2ar-1)],
aX0.
(31)
//
t
/
// /
The optimum timing obtained analytically by equation (31) is shown in Fig. 2 against the process 7
709
' ';// // /11
a
2
I "
0
~
/
t
I
I
i
__.J"
1"0 tl
-I
3
= 0"01
0"I
2
-2
1
-3
0
-4
-I
3 !
I
,~=I0 0
F]o. 3. Optimum timing of the error corrupted single measurement for the process with disturbances: (F= ].o, Q= o.o).
-2 -3 -4 FIG. 2. Optimum timing of the quite precise single
measurement for the process with disturbances: (F= |.0, Q=O.O). parameter a for various weighting coefficient • and initial uncertainty r/. It is observable that for large positive a, very unstable process, the optimum timing approaches the middle of the control interval and for large negative a, a stable process having a
2. The case: Q=I, F = 0 In this section the cost function is assumed to be quadratic in the control and in the state error. If the control interval [0, T] is large enough compared with the process parameter a, the control procedure can be viewed as in the steady state. Let A(t)=a, B(t)=M(t)=l.O, R(t)=r, V(t)=V and W(t)= W, then the optimum performance (25) is given by
Ud
U do
z~I P(tF)+ W ]
(32)
710
A. SANO and M. TERAO
where S(t) is the solution of the Riccati differential equation,
S(t)
l - - e - 2#(T-t)
, O.CI
fl - a + (fl + a ) e - 2~{T-,/'
0-I 1
---
;0
fl=~!+a 2.
(33)
If the control optimization interval T is considered to be large enough, then equation (33) becomes 0.5
S-r(fl+a).
(34) X=
2.1. The case with no disturbances. It requires more than one measurement to optimize in the steady state. However, for some analytical investigation, it may be convenient to consider the case o f k = 1. The optimum timing for arbitrary k can be obtained by the use of the simple hill climbing method since there is only one local optimum in this case. Normalizing j o in equation (32) by Po we obtain the following for k = 1,
L
r(fl+a)e4~"l
o.ol
0'~ r=OOt
0'10
I
1
2
(35)
+ 2(e 2 ' ' + 2) ]
a<0,
1 q=
[~/ In
FIG, 4. Optimum timing of the single measurement for the process with no disturbances: (F=0.0, Q = 1 '0).
j°/V:minFl~(42(t~-)+
(36a)
Figure 4 shows the solution given by equation (36) against the process parameter a > 0 for various 2 and r in the case of T = 5. When the penalty paid for the steady state error is more than that paid for the control corrections, that is, the weighting coefficient r decreases, the optimum t I approaches
a>O.
(e2afq~t-ti)--l)
{t,} L 4 a b = o \
2e 2~r ] 2z-+2ar(,B+a)+ 1 2 , a > O . (36b)
q=lln[x/22+2e2"r-2],
Z,
t~
v}±, ~" s~:(t:)-I
where 2 = W/Po. The optimum timing of t~ may be obtained by differentiating equation (35) with respect to tl and the result is ta=0,
3
2.2. The case with e vternal disturbances. The cost performance (32) subject to external disturbances is rewritten as
--e 2"q +2(e 2 a t - 1) J°/P° = min[eg"U 4a(e 2"t' + 2) t~
10 01
(37)
Figure 4 shows that the optimum timing does not depend on so much r, but on the measurement precision, 2. The more precise the measurement, a smaller 2, the earlier the measurement should be made.
i=lfP(ti
(38)
) q'- k ]
where ~(ti+) = P(t +)/1/, t1= P(O)/V, and ~ = W/V. For arbitrary q and ¢, the problem is much too elaborate to obtain the optimum {ti} analytically, so that numerical computation by a digital computer is more practical. The results for k = 1 are summarized in Fig. 5 and for k = 5 in Fig. 6. For k = l , the optimum timing of measurements is plotted for various initial uncertainty q and measurement precision 4. When the controlled process is stable (a<0), the optimum timing is rather sensitive to the initial uncertainty, and it approaches the initial part of the control interval as q increases. On the other hand, for tile unstable process (a > 0) the optimum timing approaches the middle of the interval for any q and 3. In the case where the process is not so much subject to the initial uncertainty but to the large external disturbances, the optimum timing t~ is obtained analytically (q =0. =0)
tl = T _ lln[2ar(fl + a) + 1] 2
4a
(39)
Measurement optimization in optimal process control
a
7
are to be observed since the optimum timing of measurements strongly depends on the selected state variables. In connection with this problem we are currently investigating the optimum location of measurements in the distributed parameter systems.
//Ill
i/tll
/
J lll
'oo
tilt lt]iz/"
1.,
711
O P T I M U M SELECTION OF THE SAMPLE-RATE A N D THE PRECISION OF MEASUREMENTS
3
In the last section it was established that the timing of measurements in steady state optimization subject to external disturbances has uniform intervals. Therefore, if the control interval T is large enough compared with the process response, the estimation error given by equation (17) becomes periodic with sampling interval A. Then the cost performance of equation (38) is reduced to
2
,-Y,"/II".
I
I
0
5
tl
P(t)
-1
-2
•
-3 "
_
.
i// i.i//ll
"/"
//I
~=0"01
/
~ i
1 ,o
do= r-.oolimJ°/Tv=l{A(dpx+l)(e2"a-1)-I
t -t
FIG. 5. Optimum timing of the single measurement for the process with disturbances: (F=0.0, Q = 1.0).
(40)
where
7 6
S~b~ 2A(q~2+ 4)
}
~b2 = ½(C + x/C 2 + D), t2
t~
C = (~a+ ~e 2°A-I), D = 2~(e2~A-aI)
5
IV/V
3 2
k=5 '7= 0"01
4=o.1 1
0
I
I
4
5
t
-2 -3
FiG. 6. Optimum timing of five measurements for the process with disturbances: (F=0.0, Q = 1.0).
and it is approximately the middle of the interval. For arbitrary k it is recognized that k measurements are to be taken at almost uniform intervals as shown in Fig. 6. These results are computed numerically using a k-dimensional hill climbing method. If the control process is a multivariable system, it is important to determine which state variables
and ~ = presents the precision of measurements. A denotes the sampling interval and so A -1 corresponds to the sample-rate. In connection with the D D C (direct digital control) for the process it is significant to reduce the total required information quantity of measurements as long as the specified control performance is satisfied [4]. The quantity of information which relates the number and the precision of measurements is reduced remarkably if some degradation of the control performance can be allowed. Figures 7 and 8 show the required sample-rate and the precision of measurements for specified performances for a = 1 and a = - 1 respectively. It may be observed that there is a tradeoff between the sample rate A -1 and the precision ~ for some specified performance. The near optimum performances are normalized by the ideal performance J~a which is obtained when measurements are not constrained, and it is calculated from equation (40) by A- t ~ ov and ~ - 1 ~ o o . Jia=
lim
lim
A - l(e2aA
jo= -
lim
1)(2aS + 1) - 2a _ S 8a 2 2
(41)
712
A. SANO and M. TERAO lOO ~-I
a:l r=l
lo
- - -
. . . . . . .
":"~"' .............
i t
0"~
,
......
-~,
l
0"2
I
I0
I00
L~ 1
I000
Fic. 7. Required sample rate A-I and the precision
of measurements for specified control performances: (unstable process with disturbances).
/
,:
0.2 . . . . . i 0'2
\
, ..\.,I\ , \ .
I
I0
100
1000
Fio. 8. Required sample-rate A-1 and the precision of measurements for specified control performances: (stable process with disturbances). The value J u of equation (41) is also the limit of equation (40) when only the sample-rate A-1 increases. So, to improve the performance as much as possible it is desirable to increase the sample-rate rather than the precision of measurements. It may be noted in Fig. 7 that more information quantity is required to optimize an unstable process than a stable process. There are several methods to determine the optimum A and ~ so that specified control performance is satisfied. For instance, a tradeoff between the sample-rate and the precision can be made through the following discussion minimizing the total required information capacity.
In case of computer control, the measurement data should be quantized for the analog to digital conversion and the precision of the measurements corresponds to the word-length of the quantizcd measurement data so that the word-length, samplerate tradeoffs become a problem [12]. The quantization error is considered to be equivalent to white noise having uniform distribution with variance q2/12 where q is the level of the quantization step and it corresponds to the measuremeant noise W approximately through the central limit theorem. The precision of measurement can be denoted by the binary word-length b(bits) instead of £ and ~ corresponds to ~b approximately, since
Measurement optimization in optimal process control the full range of quantization, which is plus and minus three times of the standard deviation of disturbances, may be specified as 6,/V where q = 6 x/V/2 b and V is the variance of disturbances. Using this correspondence, Fig. 8 can be reduced to Fig. 9 which shows the tradeoff between the precision (bits) and the sample-rate (A -1) of measurements.
I 2
~
6
8 10 20 ~1 ( samples/unit time )
/*0
FIG. 9. Optimum selection of the sample-rate and the quantization.
If a suboptimum cost performance is specified, the selection of the optimum sample-rate and quantization of measurements should be made along the broken line minimizing the total required information quantity (bits/unit time) [4] which is represented by the straight lines shown in Fig. 9. CONCLUSION
In connection with computing control, measurement optimization has been discussed for the case where the control process is a linear continuous system disturbed by noises subject to the cost function which consists of the control cost and the terminal error or state error. For a stable, scalar system, the optimum timing for the measurement is always in the initial part of the control interval if the external disturbances are absent. In the presence of disturbances, however, the optimum timing strongly depends on the form of cost function. For a very unstable process, the measurements should be made in nearly uniform intervals independent of the initial uncertainty, disturbances and the parameters of the cost function. The steady state optimization which is especially required for direct digital control has been discussed in terms of a tradeoff between the sample-rate and the precision of measurements for various near optimum performance. These discussions have been rather general in nature, and they have included not only stable processes but also unstable ones such as those typified by fermentation processes or nuclear
713
reactors and in addition, the methods may be applied to the measurement optimization of multivariable processes. REFERENCES [1] H. J. Kusm, r~g: On the optimal timing of observations for linear control system with unknown initial states. IEEE Trans. Aut. Control AC-9, 144-145 (1964). [2] L. MEn~, J. I~CHON and P. M. D ~ s s L ~ : Optimal control of measurement subsystems. IEEE Trans. Aut. Control AC-12, 528-536 0967). [3] S. B~a~oN and D. L. Kt~I~rMA~q: The human as an optimal controller and information processor. IEEE Trans. on A I M S MMS-10, 9-16 (1969). [4] M. TERAO: Quantization and sampling selection for efficient DDC. Instrum. Tech. 14, 49-55 0967). [5] J. J. FLO~wrrN: Partial observability and optimal control. J. Electron. Control 9, 65-80 0960). [6] W. M. Wo~,~M: Stochastic problem in optimal control. IEEE Inter. Cony. Rec. 11, pt. 2, 114-124 (1963). [7] R. E. KALMAN: A new approach to linear filtering and prediction problems. Trans. ,4SME, J. bns. Engng, ser. D 82, 35-45 (1960). [8] R. E. KALMANand R. S. Bucv: New results in linear filtering and prediction theory. Trans. A S M E , J. bas. Engng, ser D 83, 528-536 (1961). [9] S. S. L. CnANO. Optimum filtering and control of randomly sampled systems. IEEE Trans. ,4ut. Control AC-12, 537-546 (1967). [10] J. J. FLORENTIN" Optimal control of continuous time, Markov stochastic systems. J. Electron. Control 10, 473-488 (1961). [11] H. J. K u ~ : Optimal stochastic control. IRETrans. Aut. Control AC-7, 120-122 (1962). [12] R. N. LrrcEaAgGEg and G. K. L. CmEN: Word-length sample-rate tradeoffs in computer control systems. J A C C Preprints, 124-130 (1964).
R~hmmt---En liaison avec la commande numtrique directe des p r o c e s s industriels, la cadence optimale des mesures a 6t6 6tudite dans le but de rtduire le hombre et la prtcision ntcessaires des mesures. Le prtsent article donne la cadence optimale pour des proced~ stables et instables et discute l'influence de la fonction de coot, des bruits dans la mesure et des perturbations dans le proc~d6 sur la cadence des mesures. Les fonctions de cofit qui sont quadratiques dans la commande et l'erreur finale, oil erreur en regime 6tabli, sont 6valutes en utilisant la thtorie lintaire d'estimation et la technique d'optimalisation stochastique. La cadence optimale des mesures pour des systtmes de commande de proctd~ soumis /l des perturbations ext~rieures posstde des intervalles de temps presque uniformes lorsqu'on considtre l'optimalisation dans un intervalle de temps de commande relativement long. Le prc~sent article propose un compromis entre le nombre et la prh:ision des mesures pour une certaine performance sptcifi~ de commande afin de minimaliser la quantit6 totale d'information ntcessaire.
Zusammenfassung--In Verbindung mit der direkten digitalen Regelung industrieller Prozesse wurde die optimale Zeitsteuerung von Messungen studiert, um die geforderte Zahl und Priizision der Messungen zu reduzieren. Angegeben wird die optimale Zcitsteuerung fOr stabile und instabile Prozesse und der Einflul3 der Kostenfunktion, des MeBrauschens und der ProzeBsttrungen auf die Zeitsteuerung diskutiert. Die Kostenfunktionen, die im Regel- und Endfehler quadratisch sind, oder der statische Fehler, werden auf Grund der linearen Schtitztheorie und der stochastischen Optimierungstechnik ausgewertet.
714
A . SANO a n d M . TERAO
Die optimale Zeitsteuerung von Messungen fiir die Prozel~steuerung, die fiul3eren St6rungen unterworfen ist, hat beinahe gleichmfil3ige lntervalle, wenn die Optimierung in dem relativ langen Steuerungsintervall betrachtet wird. Vorgeschlagen wird ein KompromiB zwischen der Zahl und der Genauigkeit der Messungen und zwar fiir cinige Ffillc des Steuerungsverhaltens, um die insgesamt erforderliche Informationsmenge zu minimieren.
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