Control of Nonlinear Stochastic Systems

zyxwvutsr zyxwv zyxwvu zyxwvu zyxw zyxwv zyxwvutsrq zyxwvuts zyxwvutsrq = 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 7 = time delay Q, = output weighting factor = control weighting factor = autocorrelation function of u !€- 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). “Optimization Theory and the Design of FeedMerriam, C. W,, back 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. 0 NO. 2 M A Y 1969 257 (CSTR) and compare the performance of deterministic and stochastic control when the reactor is subjected to random disturbances. The first alternative is to neglect the stochastic nature of the process entirely and rely upon the inherent property of feedback control to decrease the sensitivity of the entire loop to disturbances. I t is expected that in the presence of substantial disturbances the controller n-ould experience difficulty in regulating the system. The next, alternative is to filter the system output in sonie manner before the output signal is sent' to the controller. h simple R-C filter could be used to smooth the output' signal; ho\Tever, such a filter incorporates no information on the nature of the system. K h a t is desired is not merely to sniooth the output signal but to use this signal to estimate the state of the system. -4s noted above, the optimal feedback control of a linear plant with white noise disturbances and a quadratic performance index can be segmented into a Kalman filter followed by the optimal deterministic controller. Consider a nonlinear system with random inputs and measurement errors for which we desire to estimate the actual state of the system. Optimal least square state estimates for such a system can be obtained from the solution of a set of nonlinear differential equations, termed the nonlinear filtering or sequential estimation equations (Xthans et al., 1968; Detchniendy and Sridhar, 1966; Gavalas et a/., 1969). The estimation equations use the process observations as input, their solution providing continuous estimates of the actual state of the process. K h e n the process and output are linear, t,hese equations are called the Kalman filter (Kalnian and Bucy, 1961). I n this papei' tlie schenie i i i which a filter--i.e., a procesh control coniputer which integrates the estimation equation-- . is incorporated prior to the controller in tlie control loop is examiiied. The filter 1)rovides ail estimate of the actual state of the pi'ocess at) each instant, which, in ternis of rontrd, i h the quantity of most interest. First tlie proposed schenies, iiicliiding tlic case of 1lure tinic delays in the control loop, arc' presented with the aid of lilock diagrams. The appropriate equation:: for the filter arc derived in each case. Then tlic proportional rontrol of a C'STR subject to random inpiit distidiaiices and ~iieas~ire~iient errors i? considered. The responrc of the C:STH with cm>tant gain and no filter is compared to that with a filter i n the cwiitrol loop. zyxwvuts zyxwvuts zyxwvutsr zyxwvutsrqpon zy ~ General Problem I_ ... XI11 - I - , I .... \ I U(t) Figure 1. Feedback control of a system subject to disturbances 258 I&EC FUNDAMENTALS Figure 3. Feedback control of a system with filtering No time delays y ( t ) , the random inputs to the process by the n-vector c ( t ) , and the measurement errors by the ni-vector n(t). The controller output is represented hy the r-vector u ( t ) . If a pure tinie delay-e.g., transportation lag--of magnitude crl exist-: i n the observation and a pure titlie delay of niagnitutle cr2 z Figure 1 sliow the cuqtomary feedback control of a dynamical process. The state of the system is represented by the n-vector x ( t ) , the observations or output by the m-vector 4 SYSTEM 1 Figure 2. Feedback control of a system subject to disturbances and pure time delays exists in the control actioii, the situation in depictetl in Figure 2. The control objective is to niaintain the Ptate in spite of changes in the input. There two types of input disturbances which affect iiifrequent tlisturhances due to changes in the fe or flow rate, and high frequency disturbances, the characteristic time of which is milch smaller than the characteristic tinie of the system. Both types of disturbance. normally occur in practice; however, if the aniplitude of the high frequency noise is small, the convent,ional control schemes in Figures 1 and 2 should be successful in regulating the system. When the aniplitude of the high frequency noise approaches the same order of magnitude as the aniplitude of the low frequency upsets, the performanee of the conventional may be poor. In addition to noisy inputs, the output nieasurenients invariably contain random errors, which wise typically as a result of inaccuracies in the measuring inrtruments. K e will study the effect of both types of random disturbances on the performance of the controlled Figures 3 and 4 correspond to Figures 1 and 2 but include a filter after the measuring element. It is the comparative performance of Figures 1 and 3 and Figures 2 and 4 that we wish to consider. Let us no\y formulate mathematically the situations depicted in Figures l to 4. For the scheme in Figure l the system is governed by zyxwvu k0) = fCt, x(t), u(t)l+ and the output is (1 1 zyx zyxwvutsrqpon Since the output at time t is related to the state at time t - al, the filter produces an estimate of the state of the system at t - a1, ? ( t - a1). The control action at time t , u(t), depends on the filter output at time t , or ? ( t - al). Thus, u ( t ) = g[? (t - al)]. Then Equation 1 becomes in this case (11) k ( t ) = f { t , x ( t ) , gC?(t- ad11 E O ) + “OLLERt Q(t-al-ad zyxwvutsrqpo [7. zyxwvutsrqponm + The filter equations are, correspondingly, J a2 Feedback control of a system with filtering Figure 4. P(T) = f ( 7 , Ei(7), gCS(7 - adl} PhxTQ(y(t) - hI7,?(7)1) Time delays in observation and control + PfxT+ P(hxTQ{y- h[7,? (12) zyxwvutsrqp zyxwvutsrqponm zyxwvutsr zyxwvutsrq P ( 7 ) = f,P It is assumed that E[t,(t)] = E [ n ( t ) ] = 0. Equations 1 and 2 are valid if the noisy inputs are additive or of low amplitude so that a linearization about expected values can be carried out. The state x ( t ) , the output y ( t ) , and the control u ( t ) are all random variables because ( t ) and n ( t ) are random variables. The control is a prescribed function of the output, u ( t ) = g[y(t)]. For the case depicted in Figure 2 the systeni is governed by Equation 1 with the output Y ( 1 ) = hCt, x (t - a1)I-k n ( t ) (3) while the control action is u ( t ) = g[y ( t - az)]. The schemes of Figures 3 and 4 have a filter after the observation element. For a system described by 6 = w,P+ \\liere )xP4-R-’ (13) where 7 = t - al. Although Equations 12 and 13 are integrated in real time with input as the current observation y ( t ) , the result is the state estimate a t t - al. For a delay a2 in the controller, u ( t ) = g[?(t - a ~ ) ] . The system is governed by f i t , x ( t ) , gL-?(t - a2)Il = +E@) (14) with output given by Equation 2. Since y ( t ) is related to x ( t ) , the filter output, is ? ( t ) . The filter is described by Equations 9 and 10 with f[t, ?, g ( ? ) ] replaced by f ( t , 2, g[? (t - a2)]}. Two papers have appeared on the subject of filtering systems with time delays. Kwakernaak (1967) extended t,he Kalman filter to linear systems with multiple time delays. Koivo and Stoller (1968) derived filter equations for a filter placed outside a control loop involving a pure time delay. Since we have assumed that the control output is precisely known-i.e., g (x)-this function can be directly used in place of u ( t ) in the filter equations. I n practice, if the controller action cannot be represented precisely, the resulting u ( t ) can be measured and sent directly t>othe filter simply as a known function of time. zyxwvutsrq k ( t ) = WCt, x ( t ) l + E ( t ) (4 1 Y (0== hCt, x ( t ) l (5 1 + n(t) the differential equations which constitute the least square filter are (Detchmendy and Sridhar, 1966) P (T)]} 1 w ( t ,2 ) + Ph,*Q[y - h(t, ?)I PwXT+P(hXTQ[y- h ( t , ? ) ] } . P + (6 1 R-l (7) ? is the estimate of the actual system state x, P is an n X n symmetric niatris mhich in the linear case is the covariance niatriv of tlir e3tiniate error, Q is an m X 777 symnietiic matrix which in the lineal case is the covariance matrix of n, and R is ai1 n X 71 syninietric matrix nhich in the linear case ia the covariance niatris of €. h, and w, are the appropriate ,Jacobian niatrice>-e.g., (13h,/az,)?. The initial ronditions for Equations 6 and 7 are x ( 0 ) and P ( 0 ) . These quantities are taken as the expected initial state of the systeni and the covariance of this estimate, respectively. If no a priori inforniation i- known, these values are chosen arbitrarily, For the svhenie of Figure 3 the systeni and observations are given by Equations 1 and 2. Since the filter has been inierted, u ( t ) = g[? (t)]--Le., the control now depends on the filter output, the current state estimate? ( t ) . Thus, Equation 1 becomes X ( t ) = f ( t , x(t), g C W l 1 €(t) (81 + We wish to consider the performance of a CSTR with proportional control on temperature in each of the schemes of Figures 1 to 4. Consider the dynamical equations of a CSTR with an evothermic first-order reaction and heat removal by a coil or jacket. V (dclds) = q (c, - c ) - Vhoe--EiRTc p V C p ( d T / d s ) = qC,(T, - T ) + (-AH)Vk,e-”iRTc- D ( T - T,) (15) (16) Defining the dimensionless variables, t = p qs/v 4 = c/co = 111 = (Vk,/q) EpC,/(-AH)c,R (17) zyxwvutsrq The filter equations are + PhxTQ[y - h ( t , ?)I P = f,P + PfXT + P(hxTQ[y - h(t,?)]}xP + R-l P = f[t, Control of a CSTR Subject to Random Inputs ?, g (?)I $ = pC,T/(-AH)c, (9 ) (10) where f, indicates (aj,/az,)a. R e consider the delays a1 and a2 separately, as later we wish to consider the individual effect of each. When the sole delay is that in the observation with magnitude a1, the state is governed by Equation 1 with the output given by Equation 3. 0 = D/pqCp Equations 15 and 16 become &/dt d+/dt = +G = -$ 1 - 9 - exp + CP - (7/$)14 exp CP - (7/+)14 - 0 M - + e ) (18) (19) If feedback proportional control on temperature is used to manipulate the flow rate of coolant, the dimensionless heat transfer coefficient, 0, can be expressed as ( h i s and VOL. 8 NO, 2 M A Y 1 9 6 9 259 e= zyxwv zyxwvu zyxwvut zyxwvu zyxwvutsrq zyxw zyxwv zyxwvutsrqponmlk + :1 ++ zyxwvutsrqponmlk Amundson, 1958a) $ 2 $s WCT 08 k (IF/ - $cr 111. - I < $5 - $8) $8 $8 $cT (20) $CT where k is the proportional gain, 0, the steady-state heat transfer coefficient, $s the desired reactor temperature, and k+cp a constant corresponding to half range of the coolant flow valve. The object of control is to maintain the outlet teniperature, $, a t $,q. As a low frequency inlet disturbance we will consider a step change in the inlet temperature $ o a t t = 0. High frequency fluctuations in inlet concentration and temperature enter the right-hand sides of Equations 18 and 19 additively. Thus, we add the random variables $1 ( 1 ) and t~7( t ) , d d d t = 1- 4 d$/.ldt = -$ + $1 - e($ + - ex11 CP - + exp CP - (Y/$)+ (7/$)14 +q(t) (22) (23 1 where 6 in Equation 20 now depends on y ( t ) rather than $ ( t ) . If there exists a pure h i e delay of magnitude a1 in the observation, Equation 23 is replaced by Y(t) = $ ( t - a11 1: ++ m(t k*cT - az) - (241 7 (t) a2 in the controller, + - a?)2 $s $cr - a 2 )- +s I < $cr Y ( t - az) I $s Y(t ' (25) $CT The original steady state of the react,or corresponds to $ = $s and 0 = 0, (no control). The following paran1eter.s are used: O,? = 1, $C = 1.7Elj k$cT = 1, /3 = 25, y = 50, $ o ( t < 0 ) = 1.75. For t,hese parameters there are three steady states for the CSTR with no control, The initial steady state was chosen as +8 = 0.5 and+,?= 2.0, t,he unstable steady state. The inlet, temperature $o for i > 0 is taken as 1.85. For a particular value of the gain b the new steady state (s) can be computed. The difference of $ (t-+ cc ) from $, is the offset. In this study k = 20 was used, for which there are three steady states, the unstable one resulting in the smallest offset. K-e wish to compare the performance of the CSTR in the schemes of Figures 1 to 4. Thus, it is necessary to siniulate each of the situations depicted by means of computer experinients. I n particular, the dynamical and measurenientj noise must be siniulated by appropriate expressions. The response of the CSTK with 110 noise and 110 time delay can be obtained from the solution of Equations 18 to 20 with 4(0) = &, $ (0) = The response of the CSTR with no noise and a delay a2 in the loop can be obtained from the solution of Equations 18, 19, and 25 with ~ ( -t c y z ) simply replaced by $ (t - a ~ ) .With no filter the location of the pure time delay in the loop is immaterial. These responses are referred to as the deterministic responses. T o simulate the noisy dynamics of Figure 1, Equations 20 to 22 are integrated with 4 (0) = &, $ (0) = $,, and tl(t) = dl cosw1t (26) E2 260 ( t ) = -42cos wpt I&EC FUNDAMENTALS 1 I 2 I 3 I I 5 4 I Figure 5. Transient response of CSTR and filter with controller off where the values dl = 1.5, 1 2 = 1.0,w1 = 20x, and w2 = 40x are used. To produce the noisy observation, ( t ) from Equation 22 is used in Equation 23 with 7 ( t ) as a norinally distributed random variable with zero mean and variance of 0.1. To siniulate the response when a pure time delay a1 exists, Equations 21 and 22 are integrated with Equations 24 and 26. M'he11 the delay a2 exist.3, Equations 21 and 22 are integrated with Equations 23, 25, and 26. The3e responses are referred to as the unfiltered responses. Sext, we wish to simulate the response of the entire loop when a filter is placed after the observation. If we let zT = (51, 5 % )= (4, +) and tT = ($1, $ 2 ) ) Equations 21 and 22 can be ivrit,ten in the general form of Equation 1, where n = 2, ?n = 1, and r = 1. The filter output is 51, $2, PII,PIZ, and P22 (P21 = P y l ) . Q is a scalar and R is taken as a diagonal matrix with elements Rll and R??. Since the state equations are nonlinear, no direct statistical interpretation can be ascrilwl to Q and R. As mentioned previously, h o v ever, in the linear case Q and R are the covariances of q and 6. So if we have some a priori knowledge as to these covariances, these values represent reasonable choices for Q and R even in the nonlinear case. From the rewlts of an earlier study (Rellman et al., 1966) it is apparent that the performance of the filtering equations depends significantly on the choices of x ( O ) , P(O),Q, and R . I n order to examine the convergence of the filter equations, Equations 21 to 23 were considered with 0 = 8, (no control)-Le., the pure transient response of t,he CSTR to a step change in $0 in the presence of dynamical and ineasiireinent noise. Since a t f = 0 we know that the ten1 is a t (qS, x2,,) = (& $ s ) ! the most reasonable choice for [q(0), x2 ( O ) ] is (4s,q 8 ) , Several cases mere examined in which P (0), Q, and R were varied. One example is shown i n Figure 5 , where Q = 1, R11 = 5, R22 = 10, P11, = 1, P I Z=~ 1, P z 2 0 = 4. The true and eitimated values are almod identical over the entire time of integration. Other cases not shown converged more or less the same as in Figure 5; however, if Q is too small convergence is not obtained. These values are used in the remainder of the study. + zyx + For a pure time delay of magnitude Equation 20 is replaced by e = e, 1 t (21) The measured output from the reactor is the temperature, $> which may, in general, have a noisy coniponent q ( t ) , Y(t) = $ ( t ) 1 Computational Simulation First n e consider the control schemes of Figures 1 and 3 (no time delays). The cimulation of the scheme in Figure 1 has been de3cribed. For the filtered system (Figure 3) the state, observation, and filter (Equatioiir 2, 8, 9, and 10) are solved simultaneously from f = 0. The outlet temperature 2.3 2.2 XP 2.I zyxwvutsrqpon zyxwvutsrqpon 2.0 zyxwvutsr xL LFILTERED I I 1 2 I 3 I t 4 I 1 5 6 LDETERM I N ISTIC Figure 6. Comparison of filtered and unfiltered responses with no time delays I 2' LFILTERED X2 3 t I I 4 5 xz I 6 Figure 8. Comparison of filtered and unfiltered responses with c y 2 = 0.1 I I I I I 2 I I I I 1 zyxwvutsrqpon I 3 I t 4 1 5 I 6 I zyxwvutsr zyxwvutsrqponm Figure 7. Comparison of filtered and unfiltered responses with a1 = 0.1 responses are shown in Figure 6. The deterministic x ~ ( t is) the response of the reactor temperature with no noise. The unfiltered x2 ( t ) is the response of the reactor temperature corresponding to Figure 1. The filtered 2 2 ( t ) is the response corresponding to Figure 3. The comparison of interest is between the filtered and unfiltered x z ( t ) . R e see that performance has been substantially improved in the filtered case. S e s t we consider the case of time delays (Figures 2 and 4 ) . As noted, we treat a1and a2 separately to compare the effect of the time delay location. The deterministic ~ ( t is ) the response of the reactor with no noise. The unfiltered response for a1= 0 and a2 = 0 is obtained by the simultaneous integration of Equations 21 and 22 with Equations 20 and 24 from t = 0. The filtered response is obtained by integration of Equations 3 and 11 to 13. The responses for a1 = 0.1 and a2 = 0 are shown in Figure 7 . It appears that the unfiltered response is exhibiting unstable or limit cycle behavior while the filtered response is not. We discuss this matter suhsequently. The estimation of cy1 = 0, cy? = 0.1 is depicted in Figure 8. The deterministic respoiise for a1 = 0.1 and a:,= 0 is obviously identical to that for a1= 0, a? = 0.1. The locatio11 of the delay i n the loop is seen t'o have some effect on the filtered response. When a pure time delay occurs in the observation, the combined effect of the delay and the observation noise causes larger oscillations in the riy)onse thau when a delay of the same magnitude occurs in the controller. In the former case the filter produces estimates delaycd by al, ? (t - a ) . The responses for a1 = 0, I I I I 2 3 1 I I I 4 , 5 6 J Figure 9. Comparison of filtered and unfiltered responses with a:, = 0.1 5 0.15 are presented in Figure 9. Whereas in Figure 8 the deterministic 2 2 experiences decaying oscillatioiis, now with a2 increased to 0.15, the deterministic 2 2 undergoes sustained oscillations. It has been shown that limit cycle behavior is obtained for certain combinations of lz and a in the deterministic system with proportional control (Seinfeld, 1969). The unfiltered 5 2 exhibits the same oscillatory behavior as in Figure 8. A comparison of the deterministic and unfiltered 2 2 in Figure 9 shows that that noise makes the oscillations more severe. The filtered x:, is kept more closely to the deterministic 22. cy2 = Effect of Noise on Control of CSTR I n each of the above cases, both dynamical and measurement noise has been considered. Dynamical noise enters the process as inputs and the differential equations as random forcing terms. The CSTR acts a3 a natural filter as long as the principal frequeiicy band of the power spectrum of the noise is much greater than the characteristic frequency of the CSTR (the reciprocal of the time constant, q / V ) . Thus, in the absence of observational errors, the unfiltered reactor response with high frequency dynamical noise is not too different from the response with no noise at all. If the frequency of the dynamical noise approaches the characteristic frequency of the system, the dynamical noise affects the system like an additional disturbance in the input. The VOL. 0 NO. 2 M A Y 1969 261 1. zyxwvut srqponmlkjihgfedcba zyxwvutsrqponml I I I I 1 DETERMINIST IC xz I I Figure 10. I I 2 3 r-dimensional vector function ni-dimensional vector function performance index proportional gain n X n covariance matrix flow rate, cu. ft./niin. m X m weighting matrix gas constant, B.t.u./lb. mole-OR. n X n weighting matrix time variable, min. time variable temperature O R . r-dimensional control vector volume of reactor, cu. it. n-dimensional vector function n-dimensional st,ate vector m-dimensional output vector I x z W I T H OBSERVATION - zyxwvutsrq X2 W I T H DYNAMIC ERFOR ONLY1 I 1 4 5 6 Effect of dynamical and observation noise on zyxwvuts zyxwvutsrq zyxwvu zyxwvutsrqponm zyxwv CSTR, a2 = 0.1 effects of dynamical and measurement noise are depicted in Figure 10 for the case of no filter and a1 = 0, CYZ = 0.1. The deterministic z2( t ) is the same as in Figure 8. The response x2 for dynamical error only is seen to be close to the deterministic x2, confirming the natural filtering characteristic of the CSTR. The response xz with observational error oiilv is seen to be much more violent. The x2 curve with both dynaniical and measurement noise is the same as the unfiltered 22 in Figure 8. It is obvious that the measurement noise is the key factor in stochastic control. I n this example, the sinusoidal measurement noise has caused the entire loop to enter a limit cycle, whereas in the absence of tliih noise the system is driven to the desired state. For other coinbinations of k and a?,the noise could cause the system to go t>oone of the stable steady states. It is in a case of this type t)liat> filtering is of most usefulness. GREEKLETTERS a,a1,a2 = time lags = coiistant defined in Equation 13 P = constant defined in Equation 13 Y E = n-dimensional noise vector n = m-dimensional noise vector e = dimensionless heat transfer coefficient P == fluid density 7 =: time 4 =: dimensionless concentration = dimensionless temperature w1, wa = noise frequencies + SUBSCRIPTS C --- coolant fluid CI' -- coolaiit rate 0 :. initial or inlet -= steady state SUPERSCRIPT Summary A The object of t,liis work has I)een to present and examine schemes for the control of noisy iionlinear dynaniical systems. The addition of a filter significantly improves performaiicc when the amplitude of noise is large. The actual choice of whether or not to include a filter depends 011 the tradr-off 1)etwee:i the improved performance of regulation and the cost of coniputer use. If pure time delays beconic large, one might try to cornpensate by placing a predictor after t,he filter. For tlxaniple, if the filter output is ( t - a l ) , the predictor \vould integrate ,ten1 equations from t = f - a1 to t = t to produce x ( t ) a t each iiihtant. This scheine was actually tried in this study. The increased perforniance was not commensurate with the additional computing iequii,etiieiits, and for moderate time lags a predictor is probably unnecessary. Nomenclature Ax, A2 = C = concentration, lb. moles/cu. CP D = E f 262 error amplitudes ft. specific heat of reaction mixture, B.t.u./lb.-OF. = over-all heat transfer coefficient, B.t.u./min.-°F. = activation energy, B.t.u./lb. mole = n-dimensional vector function l&EC FUNDAMENTALS = estimated value literature Cited Aoki, AI., "Optimization of Stochastic Systems," Academic Press, New York, 1967. Aris, K., Amundson, S . It., Chern. Eng. S e i . 7, 121 (1958a). Aris, K., dmundson, S . R., Chem. Eng. Sci. 9, 250 (1958b). Athans, M,,\Vishner, R. P., Bertolini, A., Joint Automatic Control Coiiference, Session 11, Ann Arbor, Mich., 1968. Bellmarl, H. I(agimada, H. H., Kalubn, R. E., Sridhar, K., J . Astronaut. SCZ., 13(3), 110 (1966). Detchmendy, D. M., Sridhar, R., J . Basic Eng. 88D,362 (1966). (;avalas, G , R., Seinfeld, J . I{., Sridhar, K., J . Hnsic Eng., submitted for publication, 1969. Kalman, R. E., Hucy, R. S., J . Basic Eng. 83D, 95 (1961). Koivo. A . J.. 'Stoller. It. I,., Joint AAutomatic Control Conference, .kin Irbor, hlich., p. 116, 1968. Kushner, H. J., J . Math. Anal. A p p l . 11, 78 (1965). Kwakernaak, 11.) I.E.E.B. Tmns. Aufomatic Control AC-12,(2), 169 (1967). Meditch. J. S., Boeirlg- Research Laboratories, Doc. Dl-82-0693, April 1968. Sewton, G. C., Gould, L. A., K;iser, J. F., "dnalytical Design of Linear Feedback Controls, Xiley, Sew York, 1987. Seinfeld, J. H., Intern. J . Control, in press, 1969. Solodovnikov, V. V., "Introductjpn to the Statistical Analysis of Automatic Control Systems, Dover, Sew York, 1960. Sworder, D. D., Intern. J . Control 6(2), 179 (1967). RECEIVED for review November 20, 1968 ACCEPTED January 16,1969 %;.j Work supported in part by Xational Science Foundation Grant GK-3342.