R
= setpoint value = root mean square error = time = sampling time = dead time = output variable = input variable = intermediate model variable = z-transform variable
R.M.S.E. t
T Td W
2
Y Z
G R E E KLETTERS ff,
= coefficients of nonlinear polynomials = measurement and system noise
P
E
SUPERSCRIPTS = constrained value = estimated value = transpose of a vector
rl;
A I
SUBSCRIPTS 0, 1, 2 , . . . , 1 2
TI
= model coefficients = load variable
= control variable
i m
= transfer function coefficients = nonlinear polynomial coefficients
literature Cited
Astrom, K. -4., Bohlin, T., Proceedings, I.F.A.C. Conference on
Self-Adaptive Control Systems, Teddington, England, 1965. Bollinger, R . E., Lamb, 11. E., ISD. Esc:. CHEX.FVNDAXEXTALS 1, 245 (1962). Briggs, P. A. N., Clarke, P. W.) Hammond, P. H., Control 12, S o . 117, 233 (1068). Harris, G., Lapidus, L., I n d . Eng. Chem. 69, 30.6 , 66 (1967). Kalman, 11. E., Trans. A.S.M.E. 80, 468 (1958). Narendra, K . S.,Gallman, P. G., I . E . E . E . Trans. Automatic Control AC-11, 546 (1966). RagazziEi, J. R., Franklin, G. F., “Sample Data Control Systems, 1lcGraw-Hill, New York, 1958. Rogers, A . E., Steiglitz, K., I.E.E.E. Trans. ilutoniatic Control AC-12, 544 (1967). Schulz, E. R., I . E . E . E . Trans. Automatic Control AC-13, 424 f,1-R~ A 8~ ) -,.
Shinskey, F. G., I . S . A . J . 61 (Sovember 1963). Steialitz, K., AIcBride, L. E., I.E.E.E. Trans. Aictoniatic Control At-10, 461 (1965). Young, P. C., Control 12, S o . 105, 931 (1968). RECEIVED for review January 9, 1969 ACCEPTED February 27, 1969 Work made possible through the computer facilities of the Institute of Computer Science, University of Toronto.
O P T I M A L FEEDFORWARD-FEEDBACK CONTROL OF DEAD T I M E S Y S T E M S H . H . W E S T ’ AND M . L. McGUlRE Uniuersity of Oklahoma, )Torman, Okla. 73069 Optimal composite feedforward-feedback controllers for multivariable chemical systems can be obtained by using the parametric expansion technique to solve the dynamic programming equations which were used to formulate the optimization problem. Separable random load disturbance signals are considered because of simplifications important in practical controller design. The Smith linear predictor configuration i s a natural result of this optimal design technique. This controller design method appears to be a powerful tool for a wide variety of process control problems.
IK REC‘EST
j~eai-sthe advantages of feeclforivvard contlol have attracted attention i n the chemical l)~’ncessiridustiies. Most investigators have conc~ludrtl that a cmibination nf feedforward and feed1)ack cont 1.01 \vould be the best configuration. E’eedfoi~nartlcontrol alone is inadequate for iiidustrial ~ ) ~ ‘ o c e s s clxmuse s of its seusitivity to ex iliatheiliatical niodcl (Haskins and Sliepcevic.h, 1965). Sevei,al investigators (Ihllinger and Laink), 1963; H a i ~ i s and Schector, 1963) have specified linear “mirror image” o r “ideal” feerlfor~vardcontrollers, while others (Haskins and Slieprevich, 1965; Luyben, 1968) have used nonlinear coritrollers. In all cases feedback controllers were added by r u t and try methods i i i order to reduce the output to a given level of attenuat,iori. There has been little concern with achieving sonic type of optimum rornposite controller. Recently Luerke and M r Guire (1968) defined an optimum
Preierit addl eis, L)epartment of Chemical Engineering, University of Pittsburgh, Pittsburgh, Pa. 15213
coniposite coiitrollt.i, througli tlie iise of I\’eiiier’s iiietliotls. Using the frequency doinain methods popularized by Yewton, ( + o d d , and Kaiser (1957), an optinial control configuration was specified. Reiner’s inethods are not able, to handle t’inie-varying 0 1 ’ niultivariable systems. Furthemore, the procedure requires the use of an intermediate function which does not allow identification of the separate feedhack and feedforward roniponent.s of the optirnuni composite controller. Denn (1968) used the results of Kalmari (1962) in his study of distributed systems and obtained a feedforward controller when distiir1)ance vectors were included in the state descriptioii. The results of llerriam (1964), who used a dynamic programming approach, ran be applied t’o the opt’inium composite control problem. The dynamic programming and variational calculus methods will furnish the same result, the difference lying in the derivation of the working equations. I n the present work, the parametric expansion method developed by Merriam was chosen. This technique is more VOL.
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powerful than frequency domain methods because it can be more easily applied to time-varying, multivariable systems.
Consider the general matrix formulation of the dynamical equations of a process
Using Equations 5 , 7 , and 8 the redefined parameter can be determined from the following matrix differentia! equation: S (p)=
where x ( t ) is the state variable, m(t) is the control signal, u ( t ) is the disturbance, and q ( t ) is the output. The matrices A, B, C, and D are, in general, continuous variables of time. To search for a n optimum controller, a scalar performance index is defined. A quadratic functional is chosen to provide the required convexity.
Jt
where T = terminal time boundary Q -= diagonal nonnegative definite weighting factor matrix W -:. diagonal positive definite weighting factor matrix The superscript used in this equation denotes the conditional mean. K h e n statistical dist,urbances are allowed, the use of the condit,ionalniean relieves the functions of their dependence on a particular ensemble. Statistical averaging by way of the conditional mean causes dependence of the performanee index on the measured values of the state a t t,he present' t'ime, t , hence the use of t in the superscript. The minimum scalar performance index must be singled out in order to use the dynamic prograinming approach. Define m(H)
111 niost optiiiiizatioii ~)robleiiis,this ~iiininiu~ii perfornialice itidex is assumed t o bc a quadratic function of the state variahle, with t,he result that a feedliack caontroller is specified. The paramet ric expansioii terhiiique (Ahriain, 1964) assumes that tlie niiniiiiuiii perforniance index is a con~bination of linear and quadratic functions of the state variable. Then the resulting optimal control equation is an elementary application of dynamic programming.
(t)mt
W - V j ( t ) -Y I C T K
(4 1
The parameters j ( t ) and K ( t ) are defined by the following equations. j (p)
-.
KC!PICTj ( p ) - BTj( p )
K ( p ) == K(p)CW-lCTK(p)
+K D m t
K(T)
- BTK(p)- K ( p ) B - A@AT
: :
(9 )
m t
=
U(Y,
om0'
3 q tis the measurable load disturbance signal. I&EC
With this new parameter, the optimal controller equation is more easily interpreted in terms of familiar control configurations. m(t)
FUNDAMENTALS
=
{ Y I C T s ( t ) } u ( t ) t- { !PLICTK(t))x(t)t
(10)
Since K ( t ) is a part of the coefficient matrix of the state variable vector in Equation 10, it can be identified with the feedback portion of the control signal. Similarly, the parameter s ( t ) is identified as part of the feedforward controller. Therefore, to obtain the optimal controller gains, Equations 6 and 9 must be solved. These boundary-value matrix differential equations are readily solved with a digital computer. The specification of t w o controller equations can be eliminated by the well-known (Koppel, 1968) niethod of incorporating the separable disturbance vectors within the state variable vector. Howevei , this technique increases the dimensionality of the system, causing computational difficulties. Chemical Process Regulator Problem
The preceding optimal controller syntheqis method is applicable t o time-varying linear process models. For purlposes of illurtration, this niethod i b applied to the firstolder time-delay tiansfer function which is so coii1moi1 in the chemical process models. The general transfer function of interest i-
I n order to use the optimal controller design procedure, it is necessary t o assume some statistical characterization for the load disturbance. A frequently encountered (Laning and Battin, 1956) spectral density was chosen
Csing Equation 10 the optimal controller equation for this process is
0; j (T)=- 0
m ( t ) = Qou(t)- Qcz(t+
Equation 5 is not in a convenient form for controller specification. If the conditional mean of the load disturbance can be put in a separated form, a more practical controller specification can be made. This limitation is not critical because most disturbances can be described by gaussian statist'ics which are a subset of the class of all separable signals. The separated load disturbance can be written as
254
(p, t )
(5 1 (61
where
+ KDU
s(T) 0
q ( t ) = Ax(t)
1-1
KC?F?-'CTs ( p ) - BTs( p )
B x ( t ) + C m ( t ) + Du(t) (1)
m (t)
(81
j ( t ) = s(t)mot
Basic Design Equations
X(t) =
this separable form a new feedforward parameter can be defined.
The controller parameters Equations 8 and 11.
KM Qc =
(7 ) With
-P
QD=--
QD
QD
and
QC
can be found from
+ dP + w (@/*I
[ + dPT+ Kn? (@/'*I
-0
u
+ d P 2+ K.??(@/*I KM
(13)
7)
I (15)
and Qc are, respectively, the feedforward and feedback
LOAD DISTURBANCE e U F E E DFOR WAR D CONTROLLER
DI STURBAN C E TRANSFER FUNCTION
CONTROL SIGNAL TRANSFER FUNCTl 0 N
WHEN MINOR FEEDBACK LOOP IS DELETED
I----
I
'F; PROCESS TIME DELAY
FEEDBACK
CONTROLLER MEAN SOUARE O U T P U T T )
--
p, ( I e
Figure 1 .
Block
Figure 2. Performance diagram of a first-order system showing points of instability T (S+P) )
diagram of
a
first-order
time-delay
system
controller gains. Both gains are functions of t'he ratio of the weighting factors, @/9. K h e n a very large stat'e variable weighting factor (a = 03 ) is chosen, the feedforward approaches the "ideal" or "invariant" controller, while the feedback function approaches infinity. Therefore, perfect control is achieved by ideal feedforward, infinite feedback or the limit of a combination of both. Perfect control is not achieved in practice because the feedforward function is sensitive to errors in the process model and infinite gain is unrealistic because of equipment dead zones and measurement noise. Equat,ion 13 is a function of z ( t T ) , which means a future value of the state variable is required. However, the impulse response equation can be expressed as a funct'ion of measurable quantities
+
z(t+
7 )= :
e(T)x(t)
+
[:re i t - v ) ( C m ( v )+ D u ( v ) ] 4
(16)
where e(;.)= fundamental mat,rix of Equation 1. Substituting Equation 16 into Equation 13 provides us with a physically realizable controller. Transforming this equation into the frequency domain and rearranging gives
Although this equation looks very complicated, it is easily simulated, as can be seen from the block diagram shown in Figure 1. The minor loop around the feedback cont,roller is similar to the Smith linear predictor. Buckley (1964) and Smith (1958) have reported experimental work with this type of configuration in the feedback loop which has the effect of "tuning out" the process delay time. While others have had to use cut and try met.hods t'o obtain the controller gains, Equation 17 is an analytical expression for the optimal controller gains. Figure 2 shows a plot of the mean square control effort, against the mean square output. Using the optimal controller presented in Equation 17 results in exactly the same
.I
.01
MEAN SQUARE OUTPUT,
ID
(E1
Figure 3. Performance diagram of third-order system and approximate first-order time-delay model
performance curve as obtained from the system with no dead time. If the controllers specified in Equation 17 are accurately simulated, there is no chance for the system to become unstable. If the minor feedback loop is neglected, a point of instability occurs. The performance diagram shown in Figure 2 indicates the range of controllability of various dead time systems when no provision for the minor feedback loop is made. The instability is the result of the exponential term in the denominator of the over-all transfer function. A s the feedback gain increases, it causes the exponential term to exert a larger influence until a point is reached where it becomes dominant. K h e n this occurs, the system poles are located in t8he right half plane of the frequency domain, indicating instability. The minor loop prevents t,he esponential term from reaching this point, of instability. Because of the coinput'ational difficulties in obtaining accurate higher order models, it is often desirable t o approximate higher order dynamics with a first-order time delay model. Therefore, a third-order mathematical model was simulated on the analog computer and an approximate time delay model was derived using step response characteristics and the approximation method of Luecke, Crosser, and VOL
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255
DISTURBANCE
-“\/l
VARIABLE-W. COOLANT F k W 0 RATE LB/MIN
-
11.
I
’
+20 OUTPUT V A R I A B L E T, WALL TEMPERATURE
1
-3
,
OF
O P T I M A L CONTROLLER USING 3 R D ORDER M E T H O D
UNCONTROLLED RESPONSE TO R A N D O M DISTURBANCE
Figure 4.
Response of third-order analog model to random disturbances
I
D l STURBANCE
OUTPUT
-
V A R I A B L E T, WALL TEMPERATURE O F
.CONTROLLER DERIVED FROM FIRST ORDER T I M E LAG MODEL MINOR LOOP DELETED
CONTROLLER DERIVED FROM L A G MODEL
FIRST ORDER T I M E
Figure 5. Response of third-order analog model with controllers calculated from the approximate first-order system
hlcGuire (1967) :
X(s)=
+ 0.329)Jl (s) - 0.207 (s + 0.569) L‘(s) (s -I- 0.117) (s 4-0.45) (s + 0.583)
0.156 (s
(18)
X ( s ) =-
0.195e-8d1 ( s ) - 0.499e-1.5SC( s ) (s
+ 0.117)
(19)
Optinial controllers were calculated for both matheniatical models using t,he previously defined design technique. Figure 3 shows the performance diagiain for both systeiiis. As one would expect, the third-order inode1 is iiiore efficient because it requires less control effort to attairi any given output attenuation. Figures 4 and 5 show the response of the system t o a stochastic disturbance. A decrease in controller perforinaiice was found when the minor feedback loop was deleted from the systein.
desi6,n proceduretems>40 it is a powerful tool for attacking iiiultivariable pro1)leiii~without the coniplesities of lioiiinteractiiig coiitroller 256
IbEC
FUNDAMENTALS
A B C D e (P) E [ z ( p ) , p] j (P) K(P)
= output matrix = system matrix = control signal matrix = disturbance matrix = scalar perforniance index = minimum scalar performance index = n-element feedforlvard vector = symmetric n X n feedback matrix
= disturbance gain = control signal gain = control vector = control signal in frequency domain = disturbance transfer function = control signal transfer function = output vector = feedback matrix = feedforward matrix = feedforward parameter = frequency in radians per unit time = time = terminal time boundary = disturbance vector = disturbance in frequency domain = state variable vector = state variable in frequency domain
GREEKLETTERS
P
= first-order pole
U
= variable of integration
P
= dummy time variable = time delay = output weighting factor = control weighting factor = autocorrelation function of u
7
Q,
!€-
e,,
literature Cited
Bollinger, R., Lamb, D. E., “1963 Joint Automatic Control Conference,” Preprint XVIII-4, University of Minnesota, hlinneapolis, 1963. Buckley, P., “Techniques of Process Control,” Wiley, New York, 1464
-I--.
Denn, hI. 11.,ISD.ESG. CHEX F U X D A M E S T A L S 7, 414 (1968). Harris, J. T., Schector, R. S.,ISD.ESG.CHEar. FLXDAMESTALS 2, 245 (1963). Haski&, D.‘ E., Sliepcevich, C. M., ISD.ESG. CHEX.F U S D ~ IIENTALS 4, 241 (1965). Kalman, R . E., Research Institute for Advanced Study, Baltimore, Nd., Tech. Rept. 62-18, (1962). Koppel, L., “Introduction to Control Theory,” Prentice-Hall, Englewood Cliffs, N. J., 1968. Lanning, J. H., Battin, R. H., ‘(RandomProcesses in Automatic Control,” McGraw-Hill, New York, 1956. Lnecke, R. H., Crosser, 0. K., lIcGuire, 31.L., Chem. Eng. Progr. 63, 60 (1967). Luecke, R . H., AIcGuire, 11. L., A.Z.Ch.E. J . 14, 181 (1968). Luyben, W.L., -4.Z.Ch.E. J . 14, 37 (1968). Merriam, C. W,, “Optimization Theory and the Design of Feedback Control Svstems.” NcGraw-Hill. New York. 1964. Newton, G.C., Child, L. A , , Kaiser, J. F., “Analytical Design of Linear Feedback Controls,” Wiley, New York, 1957. Smith, 0. J. RI., “Feedback Control Systems,” McGraw-Hill, Ken, York, 1958. RECEIVED for review December 23, 1968 ACCEPTEDFebruary 27, 1969 Work supported in part by the National Science Foundation under Grant GK-98.
CONTROL OF NONLINEAR STOCHASTIC S Y S T E M S J O H N H . SEINFELD, GEORGE R. GAVALAS, AND M Y U N G HWANG C’hemical Engineering Lnboralory, Colijornio Insfilzrfc o j Technology, Posrrrltwo, Cnlij. 91109
The control of nonlinear lumped-parameter dynamical systems subject to random inputs and measurement errors i s considered. A scheme i s developed whereby a nonlinear filter i s included in the control loop to improve system performance. The case of pure time delays occurring in the control loop i s also treated. Computations are presented for the proportional control on temperature of a CSTR subject to random disturbances.
temh nliicli one (1e;irc:s to control are .ubject to some degree of uncertainty. Even when the fundamental physical phenomena are kno~vii! the mathematical model may contain parameter.; d i o s e values are unknown, or the ten1 may be subject, to unknown random disturbances. In designing a control system the easiest approach is to neglect the randomness associated with inputs, assign certain nominal values to parameters, and base the design on classical deterministic theory. However, it is obvious that a design based on deterministic control theory becomes inadequate when the proces uncertainties become significant. The alternative is to consider the problem as one of control of a stochastic system. The control of stochastic systems is of significant theoretical and practical importance. -1large and elegant theory exists for the analysis of linear control systems subject to corrupting noise (.iris and Xmundson, 195Sb; Newton et al., 1957; Solodovnikov, 1960). Recently, solutions have been obtained for the optimal control of linear systems with white noise forcing and quadratic performance criteria (Aoki, 1967;
Iiiislnier. 1965; .\Iditch, 1968; Swortler, 1967). The structure of the optimal feedhack control in thii caw is a minimum variance (Iialman) filter followed by the optimal controller for the deterministic system. The optimal control of nonlinear systems n-it,h white noise inputs can be reduced t o the solution of a .set of nonlinear, integro-partial differential equations, Lvhich, as one might suspect, are almost iinpoi4ble to solve. The key problems in chemical 1)roce.s control involve nonlinear system. with noisy input>, the statktical properties of which are usually unknoivn. In addition, there are almost always delays in the control loop because of noninstantaneous control action and/or the time necejsary for the analysis of measurements. -4feasible way of handling such systems represents a challenging problem in chemical process control. The objectives of this paper are: to formulate a scheme by which a nonlinear system with unknown random inputs can be controlled; to extend this scheme to the case of time delays in the cont,rol loop; and to apply the scheme to the proportional control of a continuous stirred-tank reactor VOL.
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