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References Gerrens, H. On Selection of Polymerization Reactors. Ger. Chem. Eng. 1981,4, 1-13. U.S. Patent 4,221,883, Sept 9,1980; assigned to Dow Chemical Company. Duerksen, J. H.; Hamielec, A.; Hodgins, J. W. Polymer Reactors and Molecular Weight Distribution. AIChE J. 1967, 13, 108. Hui, A. W.; Hamielec, A. Thermal Polymerization of Styrene at High Conversions and Temperatures. An Experimental Study. J. Appl. Polym. Sci. 1972, 16, 749. Husain, A,; Hamielec, A. Thermal Polymerization of Styrene. J . Appl. Polym. Sci. 1978, 22, 1207. Khac-Tien, N.; Flaschel, E.; Renken, A. The Thermal Bulk Polymerization of Styrene in a Tubular Reactor. Polymer Reaction Engineering; Hanser Publishers: Germany, 1980. Mendelson, R. A. A Method for Viscosity Measurement of Concentrated Polymer Solutions in Volatile Solvents at Elevated Temperatures. J . Rheology 1979,23(5), 545. Mendelson, R. A. Concentrated Solution Viscosity Behavior at Elevated Temperatures-Polystyrene in Ethylbenzene. J . Rheology 1980,24(6), 765. Metzner, A. B.; Taylor, J. S. Flow Patterns in Agitated Vessels. AIChE J . 1960, 6(1), 109. Dickey, D. S. Program Chooses Agitator. Chem. Eng. 1984, 91(1), 73. Boundy, R. H.; Boyer, R. F. Styrene: Its Polymers, Copolymers, and Derivatives, Part I; Hafner Publishing co.: Darien, CT, 1970. Goldstein, R. P.; Amundson, N. R. An Analysis of Chemical Reactor Stability and Control-Xa. Chem. Eng. Sci. 1965,20, 195. Jaisinghani, R.; Ray, W. H. On the Dynamic Behavior of a Class of Homogeneous Continuous Stirred Tank Polymerization Reactors. Chem. Eng. Sci. 1977, 32, 811. Schmidt, A. D.; Clinch, A. B.; Ray, W. H. The Dynamic Behaviour of Continuous Polymerization Reactors-111. Chem. Eng. Sci. 1984, 39(3), 419. Schmidt, A. D.; Ray, W. H. The Dynamic Behavior of Continuous Polymerization Reactors-I. Chem. Eng. Sci. 1981,36, 1401. Choi, K. Y. Analysis of Steady State of Free Radical Solution
(17) (18) (19) (20) (21) (22) (23) (24)
(25) (26) (27) (28) (29) (30)
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Polymerization in a Continuous Stirred Tank Reactor. Polym. Eng. Sci. 1986, 26(14), 975. Knorr, R. S.; O’Driscoll, K. F. Multiple Steady States, Viscosity, and High Conversion in Continuous Free-Radical Polymerization. J. Appl. Polym. Sci. 1970, 14, 2683. Henderson, L. S. Stability Analysis of Polymerization in Continuous, Stirred-Tank Reactors. Chem. Eng. Prog. 1987,83(3), 42-50. Nagata, S. Miring Principles and Applications; Halsted Press: New York, 1975. Oldshue, J. Fluid Mixing Technology; McGraw-Hill: New York, 1983. Rase, H. F. Chemical Reactor Design for Process Plants; John Wiley: New York, 1977; Vol. 1. Van Heerden, C. Autothermic Processes, Properties and Reactor Design. Znd. Eng. Chem. 1953, 45(6), 1242. Aris, R. Introduction to the Analysis of Chemical Reactors; Prentice-Hall: Englewood Cliffs, NJ, 1965. Beckman, J. Design of Large Polymerization Reactors. Polymerization Kinetics and Technology;Advances in Chemistry Series 128; American Chemical Society: Washington, DC, 1973. Luyben, W. L. Stability of Autorefrigerated Chemical Reactors. AIChE J. 1966, 12(4), 662-668. Blanks, R. F.; Stokes, R. L. Mixing of Viscous Fluids with Turbine Agitators. Chem. Eng. World 1972, 7(3), 65-69. Mueller, A. C. Criteria for Maldistribution in Viscous Flow Coolers. Paper HE1.4, Presented at the 5th International Heat Transfer Conference, Tokyo, 1974; pp 170-174. Streiff, F. A. Statische Wiirmeubertragungsaggregate. Wiirmeubertragung bei der Kunstoffaufbereitung;VDI Verlag: Dusseldorf, BRD, 1986; pp 241-275. Joosten, G. E.; Hoogstraten, H. W.; Ouwerkerk, C. Flow Stability of Multitubular Continuous Polymerization Reactors. Ind. Eng. Chem. Process Des. Dev. 1981,20(2), 177-182. Khac-Tien, N.; Streiff, F.; Flaschel, E.; Renken, A. Motionless Mixers for the Design of Multitubular Polymerization Reactors. Presented at the Annual Meeting of the American Institute of Chemical Engineers, San Francisco, 1984. Received for review October 3, 1988 Accepted June 30, 1989
Combining Infrequent and Indirect Measurements by Estimation and Control Bengt E. V. Lennartson Control Engineering Laboratory, Chalmers University of Technology, S-412 96 Gothenburg, Sweden
In the process industry, it is often difficult to measure product qualities as often as desired for control purposes. Indirect information like temperature and pressure is therefore mostly used to control the process. Sampled measurements, for example, from laboratory analysis, are, however, often available. A combination of these infrequent samples with the indirect on-line measurements in an estimator is then a natural approach. This idea is investigated for the control of a continuous stirred tank reactor (CSTR). The temperature is then given every sampling interval, but the concentration measurements are obtained a t infrequent sampling instants, after a time-consuming analysis. The results give some guidelines about the benefits of the proposed estimation procedure.
1. Introduction In a process industry, it is often difficult to measure product qualities continuously for control purposes. Quantities that are closely related to these primary product characteristics are mostly used to inform the control system about the process behavior. This type of indirect or inferential control often works satisfactorily. A typical example from chemical engineering is distillation control, where a column temperature is used as an inferential measure to the product composition. Sometimes, the correlation between the measured quantities and the product qualities is not sufficient. For
example, a process disturbance can give rise to a significant deviation from the desired product composition but only cause a small change in the measured process variable. Sampled measurements of the primary quantities are often available from laboratory analysis or instrumentation close to the process. Typical examples are gas chromatographs and mass spectrometers. Shortcomings in the inferential control are today overcome by performing manual supervisory control based on these sampled data and trends in the current inferential measurements. A natural approach is to use a Kalman filter (Anderson and Moore, 1979) to estimate the process conditions based
0888-588518912628-1653$01.50/0 0 1989 American Chemical Society
1654 Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989
on the continuously available measurements that are sampled as often as desired. To improve the estimate, the sampled measurements of the primary product quantities are included when they are available. Sometimes this occurs at a slower rate than desired, which results in multirate measurement sampling. The purpose of this paper is to investigate the performance of this proposed estimation principle for the control of a continuous stirred tank reactor (CSTR).The results will give some guidelines for the significance of multirate sampling. To be able to investigate the control performance for different sampling periods, we will assume that the process plant can be described by a continuous-time model driven by stochastic disturbances. Applying sampled data control, the controlled system variance can vary considerably during the sampling periods (Lennartson and Soderstrom, 1989). Thus, the continuous-timeaverage variance will also be introduced as an appropriate measure of the closed-loop behavior. 2. Statement of the Problem Consider the stochastic, continuous-time system
dx(t) = Ax@) dt
+ B u ( t ) dt + dw(t)
(2.1)
where x ( t )is an n-dimensional state vector, u ( t )a control signal of dimension m,and w ( t ) a Wiener process with incremental covariance R, dt. The output y is assumed to be measured at equidistant sampling instants as y(kh) = C(kh)x(kh) e(kh) (2.2)
u ( t ) = u(kh)
< kh + h
(2.6) More specifically, we will assume that the system (2.1) is controlled by constant feedback from optimal estimated states (a Kalman filter); Le.,
u(kh) = -Lk(Izh/kh)
(2.7)
where all available measurements up to time kh are used in the filter estimate k(kh/kh). Due to the periodic outputs (2.2) and (2.5), the closed loop system will in steady state be a cyclostationary process. This means that the state covariance
P,(t) = E X ( t ) X T ( t )
(2.8)
will be a periodic function over the period T = rh. Hence, the time average covariance (2.9) gives an appropriate measure of the system performance. The aim of this paper is to discuss further control law (2.7) and to investigate the average covariance (2.9). The results will be applied to the control of a CSTR system combining infrequent and indirect measurements. 3. Multirate Control In this section, we will show how controller (2.7) can be designed. Introduce the notation W t ) = eAt
+
where h is the sampling period, C(kh) a time-varying output matrix and e(kh)the measurement error composed of discrete-time white noise with zero mean and a timevarying covariance matrix Ee(kh)eT(kh)= R2(kh). The disturbances w(.) and e ( . ) are further assumed to be mutually independent. The time variation of the matrices C and & in (2.2) will now be used to introduce the concept of multirate measurement sampling (Glasson, 1983; Broussard and Halyo, 1984; Lennartson, 1986). This means that sensors are sampled at different rates. The reason is that some measurements are not available at the desired sampling rate. For simplicity, consider a system with two sampling rates (extension to the multirate case is obvious). Some of the measurements are then only available at a slower rate corresponding to a sampling interval rh. The output matrix C(kh) in (2.2) then becomes
Izh 6 t
v ( t , ~=) i t + T @ ( t + ~ - s )dw(s)
U t ) = I t0@ ( s ) Bds
R,(t) = f @ 0 ( ~ ) R , @ ~ (ds s)
(3.1) System (2.1) can then, at the sampling instants, be described as x(kh+h) = Fx(kh) + Gu(kh) v(kh,h) (3.2)
+
where F = iP(h),G = I'(h),and v(kh,h)is a discrete-time white noise with covariance Rl(h) (Astrom, 1970). 3.1. State Feedback Control. The feedback matrix L in (2.7) can be designed to optimize the well-known linear quadratic criterion
(3.3) The solution is then given by the Riccati equation (Astrom, 1970)
S = FTSF + Q1 - FTSG(GTSG+ Q2)-'GTSF
(
[e,] for k = lr + i
i = 1, ..., r - 1
and
where 1 is an arbitrary integer. The related measurement covariance R&h) is assumed to be R,(kh) =
[ iJ tRzl
for k = lr
for k = Ir
(2.4)
+i
(3.4)
i = 1, ..., r - 1
We note that the dimension of the output y will vary periodically and that the matrices C and Rz are periodic with a period T = rh; i.e., C(kh+T) = C(kh) R,(kh+T) = R,(kh) (2.5) Finally, it is assumed that u (t) is piecewise constant over the the shorter sampling period, which means that
L = (GTSG + Q2)-'GTSF
(3.5)
According to the separation theorem (Astrom, 1970),this feedback matrix, together with the Kalman filter in section 3.2, gives the optimal solution to the minimization of criterion (3.3). As an alternative, the feedback matrix L can be designed by a pole-placement technique. By use of the sampling period h as a discrete-time unit, the closed-loop poles for system (3.2) and for (2.7) are specified by the eigenvalues of the matrix (F - GL); see Lennartson (1986). 3.2. Kalman Filter with Multirate Measurement Sampling. In a Kalman filter, different sampling rates are easily handled. At each sampling instant, all accessible measurements are used to update the state estimate. The
Ind. Eng. Chem. Res., Vol. 28, No. 11, 1989 1655 periodic matrices (2.3) and (2.4) can just be considered as time-varying matrices. A Kalman filter is then given as follows (Anderson and Moore, 1979): %(kh+h/kh) = Ff(kh/kh) Gu(kh) (3.6)
+
%(kh+h/kh+h) = %(kh+h/kh) + K(kh+h)[y(kh+h)- C(kh+h)%(kh+h/kh)](3.7)
K(kh+h) = P(kh+h)CT(kh+h)X [C(kh+h)P(kh+h)CT(kh+h)+ R,(kh+h)]-l (3.8) P(kh+h) = FP(kh)FT + R,(h) - FP(kh)CT(kh)X (3.9) [C(kh)P(kh)CT(kh)+ RZ(kh)]-lC(kh)P(kh)FT Due to the periodicity of matrices C and R2 in (2.5),the Riccati equation (RE, eq 3.9) will under weak conditions (Bittanti and Gaurdabassi, 1986) converge to a periodic solution. It can be obtained by iterating (3.9) step by step until P(kh+rh) = P(kh). There are, however, more efficient methods to reach this periodic solution. An algebraic Riccati equation (ARE) that involves the system behavior during one period can easily be determined (Lennartson, 1988). This makes it possible to use standard algorithms for ARE, such as eigenvector methods (Vaughan, 1970; Laub, 1979) or doubling formulas; see, e.g., Anderson and Moore (1979).
4. Time-Average Performance Now it will be shown how the time-average state covariance (2.9) can be computed. 4.1. Time-Average State Covariance. The intersample state covariance (2.8) integrated over the period T = rh (eq 2.9) can be divided into a sum of r integrals as 1 t-1 h P,(kh) = L]khfrhP,(t) dt = --C P,(kh+ih+s) ds rh kh rh;=n o
1
To continue, we need an expression for the intersample covariance P,(kh+ih+s) for 0 < s < h. Integrating the system equation (2.1) from t = kh ih to t = kh + ih + s, assuming sampled-data control (2.6),gives directly, for O
