Multivariable Control of a Catalytic Tubular Reactor Using Both Wiener-hopf Controller Design and Internal Model Controller Design Approaches

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MULTIVARIABLE CONTROL OF A CATALYTIC TUBULAR REACTOR USING BOTH WIENER-HOPF CONTROLLER DESIGN AND INTERNAL MODEL CONTROLLER DESIGN APPROACHES D. J. Kozub*,

J.

F. MacGregor* and Joseph D. Wright**

*Depl. uf ChemiCll/ Ellgilleering, McMaster University , Hamilton , Ontario, Canada L8S 4L7 *"Xerux Resmrch Cel/lre, Alississallga, Ontario, Canada ABSTRACT

controllers using input-output transfer function type models . The resulting controllers are then expressed directly in terms of past inputs and outputs, and the Internal Model structure of the controllers makes any subsequent analysis simpler, and provides a better understanding of the resulting controller.

A practical application of advanced model-based multivariate control to a pilot plant packed bed reactor carrying out highly exothermic butane hydrogenolysis reactions is presented . The system was first stabilized by using butane flow rate to control the reactor hotspot temperature. Propane production and butane conversion were then controlled using multivariate controllers which manipulated the hotspot temperature setpoint and hydrogen flow rate . An approximate linea r dynamic-stochastic model was developed using multivariate time series and process identification methods .

In this paper we present the design and results of a practical application of two multivariate controllers to a pilot plant tubular reactor carrying out the highly exothermic hydrogenolysis of butane over a nickel on silica gel catalyst. A favourable choice of input control variables is demonstrated which leads to an open-loop stable configuration through use of an internal hotspot controller. In this work we employ simple linear empirical input/output models developed from time series analysis and process identification techniques on data collected directly from the reactor . The two multivariate designs presented are constrained minimum variance control (LQ output) and internal model control. Both designs are well suited for output control using a pulse transfer function process model. The designs are straightforward for practical chemical process industrial application, and take full advantage of the true interactive nature of a multidimensional system to achieve good controller performance. Furthermore, they allow one to incorporate into the designs a considerable degree of robustness to mismatch between the model and the process.

Two multivariate controllers were applied . The first was a simple version of Internal Model Control ([MC) utilizing a stabilized approximation of the model inverse and a tunable exponential filter . The second was a constrained minimum variance controller design using pulse transfer function mod e ls to charact e rize the proces s dynamics and autoregressive-moving average models to characterize the stochastic disturba nces. The optimal control so lution was arrived at via a spect ral factorization so lution to the Wiener -Hopf equations. The re sults indicated that both designs, if well -tuned, provide adequate performance and robustness over a wide region of operation. Key words : Multivariate Control: Internal Model Control;

Linear Quadratic Optimal Control; Wiener -Hopf; Catalytic Reactor .

2. MULTIVARIATECONTROL 2.1 Multivariate Model Representation

1. INTRODUCTION

An important problem in many chemical processes is that of controlling the product stream compositions from catalytic reactors in which one is carrying out multiple reactions involving multiple species. A high degree of coupling among the product compositions due to the reaction stoichiometry makes the control problem a truly multivariable one. Since the reactions are often highly exothermic and have high activation energies, stability of the reactor is also a major concern. Furthermore, reaction kinetics are usually highly nonlinear in both reactant concentrations and temperature . A controller design strategy for such reactors must therefore be capable of handling complex interactions between input/output pairs and must be robust to nonlinearities across the operating region.

Frequency response representation of a multivariate process has become a popular structure for controller design which naturally results from input/output identifications . Multivariate controller design with pulse transfer functions, unlike single input/single output design, is hampered by messy matrix computations and the noncommuting property of matrices . A discrete, linear multivariate process can always be represented in a right matrix fraction form Y(t) = w(z-I)8(z-l)-l z -kU(t)

+ 9(z-I)(z-I)-IA(t)

(I)

which leads to some simplification in the controller design equations. Y(t) represents the vector of n observable outputs at sampling interval t. U(t) is the vector of m controllable inputs at sampling interval t . Then n X m matrix w(z -I) and the m X m matrix 8(z -1) are polynomial operators in the backward shift operator z -I with nonrational elements. 8(z -I) is diagonal with the coefficient of zO set to I. The minimum whole periods of delay is given by k. The zeros of the process are defined to be the roots of

Advanced optimal controller design using a state space model formulation of the multidime nsional problem has received much attention in control literature. Successful application of this theory has been demonstrated by Jutan, Tremblay, Wright, and MacGregor (1977), Sorensen, Jorgensen , and Clement (1980, a,b), and Wall man , Silva, and Foss (1979) to packed bed control problems using rigorous mechanistic modelling that led to convenient state space model forms . This approach to a packed bed reactor control problem was also found to be successful by MacGregor and Wong (1980) using empirical tr ans fer function mod e l identification followed by transformation to state space form . Optimal linea r -quadratic (LQ) controller des ign is usua lly associated with these state space techniques However, when one has measurements, or inferred values, of the outputs to be controlled, there are many adva nt ages to designing

Iw(z-I)I = 0

(2)

A multivariate process is invertible if all roots of equation 21ie inside the unit circle . The poles of the process are defined to be the roots of

18(l -1)1 = 0

(3)

A process is open-loop stable if all roots lie within the unit circle.

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.J.

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The second term on the right side of equation 1 represents the observable but uncontrollable contributions to the process outputs which are assumed to be stochastic. ACt) represents a vector of n independently distributed random shocks with covariance matrix E. An autoregressive integrated moving average model in a right matrix fraction form is used to characterize the nature of the process disturbances . The n x n matrices 8(z -I) and 4>(z - I) are polynomial operators in z-I having nonrational elements . The zO coefficient of both matrices is set to I and 4>(z -I) is diagonal. :-(z-l) matrix. We can also represent deterministic disturbances with equation 1 if Alt) is a vector of shocks having a probability distribution with non-zero values occurring only infrequently (~acGregor et aI. , 1984). Right matrix fraction representations are not unique . A simple and convenient method for transforming rational transfer function matrices into a right matrix fraction form is by choosing the i'th diagonal element of o(z -I) or 4>(z -I) to be the greatest common factor of the denominators of the i'th column of the rational transfer function matrix or ARI~A matrix respectively .

.J.

D. Wrighl

inverse. Fllz-I) and F2Iz- l ) are nXm disturbance and setpoint change filters . The optimal approximate inverse for the cost function given by equation 4 leads to Gclz-I)

= olz-I) flz-I)-I

(8)

where flz -I) is an m X m nonrational polynomial matrix in z-I having no zeros outside the unit circle . nz-I) is determined from the solution to the spectral factorization equation

+ oT(z)(l-Z)Q2{l-z-')0Iz-l) = rrCz)rCz-I)

(9)

The choice of Q2 equal to 0 with the special case of a square wIz-I) having all zeros inside the unit circle leads to minimum variance control with nz - t) = w(z -I). The optimal filters corresponding to minimization of equation 4 are dependent on the approximate inverse and on the disturbances and setpoint models respectively . Both filters can be expressed as F(Z-I) = T(z-I)8(z-I)-1

(10)

where T(z-I), a nonrational polynomial matrix, is determined by solving the set of linear equations obtained by equating coefficients of z in

2.2 Constrained Minimum Variance Control

This section presents the results of the linear quadratic optimal control problem using a spectral factorization solution of the Wiener-Hopf equations . Details of the solution are given by Wilson (1970) and Harris and ~acGregor I 19861.

with the z -0 matrix coefficient ofC(z) set to O.

The performance index chosen in the constrained minimum variance (CMV) or linear-quadratic (LQ) output controller design is the minimization of

Equations 5,6,7, and 8 can be combined to calculate control actions at each sampling interval as a function of past inputs and outputs as

wT(z)QI zk8(z-l)

= C(z)4>(z-l) + rr(z)T(z-l)

(11)

[nz-I) - £'I(z-I) w(z-I)] o(z-I) -I U(t) (4)

+ V UT (t)Q2 V U(t) + UT (t)Q3 U(t)} where Q" Q2, and Q3 are positive semi-definite weighting matrices, and e(t) = Yet) - Ysp(t). In equation 4 we may penalize the variance of the inputs about a mean level using weighting matrix Q3 or the change in the control action, given bv VCCt) = U(t)- U(t-l), with weighting matrix Q2 · Disturbances encountered in chemical processes are normally nonstationary with noticeable drifting output mean levels . Regulation of processes with changing mean levels theoretically requires infinite input variance. For this reason we choose to penalize the change in control action VU(t) = U(t) - U(t-l). It can be shown that if the disturbances are nonstationary and VU(t) is penalized that integral action will be present in the optimal controller design. For the case of the number of inputs exceeding the number of outputs we must penalize only the change in control action of a choice of inputs equal to the number of outputs in order to acquire reset action . The remaining inputs may be penalized about a mean input level using weighting matrix Q3· Harris and MacGregor (1986) show that the general LQ optimal feedback control solution can be expressed in the internal model controller form :

(12)

As an alternative, we may retain the separate block structure of figure 1. The advantage of a separate block structure is inherent reset windup protection since actual implementated U(t)'s are entered into the model predictor block . Another advantage of the separate block structure is that differences between process outputs and model predicted outputs can be recorded to keep track of the severity of mismatch. Constrained minimum variance controllers will always be stable for any choice of input weighting matrices provided no model mismatch is present. When model mismatch is present stability cannot be guaranteed for any choice of input weighting matrices . The robustness of constrained minimum variance controllers can be improved by increasing the magnitude of input penalty weights (detuning). Harris and MacGregor (1986) show that there exists a Q2 which is sufficiently large to guarantee robustness to arbitrary large modelling errors provided that Re {},,(j)[w(z = 1) nz = l)-I]}>O;j = 1,2, . . ,n, where },,(j)[A] is the j'th eigenvalue of A. It has been shown by Bergh and MacGregor (1986) , and Harris and MacGregor (1986) that constrained minimum variance controllers exhibit an excellent compromise between performance and robustness.

2.3 Internal Model Control

where (6) (7)

The control block diagram is shown in figure I . GmCz -I) is an n X m process model transfer function and Y sp(t) is the setpoint. GcCz-') is a square mXm matrix . When a multivariate system is square Gclz-I) can be physically related to an Invertible approximation of the process model

The concept of internal model control ([MC) is very simple and is easily applied to multivariate systems. The detailed development of this design is given by Garcia and Morari (l985,a,b) . In the IMC design we assume that disturbances and setpoints enter the control loop as noninteractive deterministic step changes and no stochastic disturbances are present. In this section the results of a simple IMC design approach for square multivariate systems will be presented which leads to a decoupled multivariate closed-loop response.

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Multi\'ariable Control of a Catalytic Tubular Reacto r

The IMC block diagram is represented by Figure 1. The closed-loop transfer functions relating the output and input response to disturbances and setpoint changes are Y(t) = ;>Ht) + G p(z-I)G c(z-I)(F2(Z-I)Y,p(t) - FI(z-l) :-.i(t))

(13)

U(t) = Gc(z-I)(F2(Z-I) Y,p(t) - FI(z-l) N(t»

(14)

when no model mismatch is present. These equations show that selection of a stable Gc(z -I) matrix will lead to closed-loop stability provided that the process is open-loop stable. In the IMC design independent selections of Gc(z -I), FI(z-I) , and F2(Z-I) are made, unlike in the constrained minimum variance controller des ign where FI(z-l) and F2(Z -I) are dependent on the specification of Gc(z-I) through equation 11 The advantage of the IMC design over conventional feedback controller designs operating on the error signal is that controller stability is addressed directly through a stable choice of Gc(z -I). Stability in conventional feedback blocks is examined indirectly through a complicated return difference equation. The only requirement for the specification of Gc(z -I) is that it leads to a stable controller. Hence the designer is free to make any stable selection. The optimal choice of Gc(z -I) depends on the objective function to be optimized and the properties of the multivariate process. The simplest design is that given by Garcia and Morari (1985, a,b) which leads to a decoupled closed-loop response . In this des ign Gc(z -I) is an approximate model inverse given by Gc(Z-I) = (I)(z-I) w(z-I)-I zk) V + I (z-I) v +2 (z -I)

(5)

The first expression in brackets on the right side of equation 15 is the inverse of the process model. V +I(z-I) is a diagonal matrix with each diagonal element of the form of .-,tj). This matrix eliminates prediction terms in the inverted process model. S(j) is equal to k plus the highest positive power of z in column j of the inverted w(z -I) matrix . This choice of V + I (z -I) leads to perfect decoupling with output y(j) having the minimum s(j) periods of delay in the closed-loop response. V + 2 (z -I) is a rational or nonrational polynomial in z -I which removes or replaces unwanted zeros in Iw(. -1)1 that will appear in the controller. Zeros of Iw(z -1)1 that are outside the unit circle yield an unstable controller. The designer can choose to simply cancel unstable zeroes or cancel and replace unstable ze roes by their image or any other desired values . Replacement of unstable zeros with their image can be shown to be optimal in a least squares sense, and arises naturally in the unconstrained CMV design (Q2 = Q3 = 0). Stable zeros of Iw(z-I)I located near (-1,0) lead to unfavourable input ringing which should be removed using v +2 (z -I). The selection of Gc(z-I), as given by equation 15, will lead to the closed loop output response Y(t) = N(t)

+ V +I(z-I) V+2(z-I)[F2(.-I) Y,p(t)

- FI(z-I)N(t)!

F(. -I)

= diag

I

1- f

ii_ I

)

1 - f .. •

(17)

11

where each filter parameter lies in the range of 0:5 fii < 1. Each filter parameter can be physically related to the closed-loop time constant of a setpoint change or disturbance rejection by f ..

"

= exp(-

~)

Tcl. .

(8)

11

where T, is the sampling interval and Tc!. is the closed· loop time constant for output i. A filter facto·~ corresponding to each output can be used to shape the closed-loop response and reduce the severity of input manipulations. It has been shown by Garcia and Morari (1985a) that there exists fii for each output which is sufficiently large to guarantee robustness to arbitrary large model mismatch provided Re{X(j)[Gp(' = 1) G m(. = 1) -I]} > 0 for j = 1,2, . ,n. The appealing feature of the IMC design is that a single, physically meaningful filter tuning factor for disturbance and setpoint changes for each output can be used for on-line tuning once a stable Gc(' -1) is specified. This is contrary to the constrained minimum variance design where the Wiener-Hopf design procedure would have to be resolved for any new choice of Q2 and Q3 . The decoupled IMC design is of the correct structure if the multivariate process is balanced. A multivariate process is balanced if the rows or columns of Gm(z -I) can be manipulated such that the minimum dead time in each row appears on the diagonal. High performance control of an unbalanced multivariate process is not possible with the decoupled Gc(.- I) design, especially if there is a large difference between the minimum dead time and the minimum settling time. The search for a least squares optimal inverse is far from trivial when the process is not balanced . An easy approach to solve this problem is to use the results of the constrained minimum variance controller design for a nonstationary noise model. The right matrix spectral factor, determined by equation 9, provides a stable least squares approximation to the inverse of w(.-I) and is a function of only the process model parameters. The control block in the IMC design is then given by equation 8. Severe harshness of the input movements from the model inversion and input ringing can be prevented by a suitable specification of the Q2 input weighting matrix . IMC filters can then be tuned on-line in the usual way.

3.. PILOT PLANT REACTOR SYSTEM The process under investigation is a pilot plant packed bed reactor carrying out highly exothermic butane hydrogenolysis reactions over a nickel on silica gel catalyst . The hydrogenolysis of butane can be represented by the following overall reactions:

(16)

when no model mismatch is present. Equation 16 shows that the IMC design requires that time delays and nonminimum phase characteristics become part of the closed-loop specification. If FI (z- l) and F2(Z- I) are removed from the co ntrol block diagram in figure I , the EVIC design using equation 15 will lead to deadbeat control actions for step changes. Deadbeat control actions can lead to harsh actuator manipulations and severe rippling in the true process output. It can also be shown that the stability of the closed· loop input and output transfer functions are very sensitive to mismatch between Gm(z -I) and G p(' - I) . The filter blocks serve to relax the specification of perfect control and deal with these two important problems. The simplest and most common choice for the filter blocks FI(.-I) and F 2(z-1) are diagonal (decoupledl first order exponential filters with a gain equal to I as given below

(19)

Only three of the reactions shown above are stoichiometrically independent, but in the usual operating region of the reactor only two are practically independent (MacGregor and Wong, 1980). Therefore, only two output or product species need to be controlled . The reactor consists of a 1 112 inch Schedule 80 stainless steel pipe mounted concentrically within a carbon steel pipe The inner pipe contains th(· ratd ly; t bed and the outer pipe form, d cooling jaeket for a heal lran,fer oil circulation loop The

288

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Kozub.

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F. MacGregor and

length of the catalyst bed is 89 cm . The feed gases , hydrogen and butane are mixed up stream and enter the reactor with back pressure set to 14 psig. Thermocouples positioned on the axis of the reactor provide temperature profile readings along the length of the bed. The region of operation is set by maintaining a constant wall temperature setpoint. On/off control of an air source to a shell and tube heat exchanger in the oil circulation loop is used to regulate wall temperature. An on-line gas chromatograph system sampling the effiuent provides concentration measurements at 3 minute intervals. The process control configuration of the pilot plant is shown in figure 2. Percent butane conversion and propane production (cc/sec) are the outputs controlled by a multivariate outer loop controller which computes control actions at three minute intervals from the gas chromatograph readings . The multivariate control variables are hydrogen flow rate setpoint (cc/sec) and hotspot setpoint (C). Hotspot setpoint is sent to an internal hotspot control loop which uses the butane flow rate setpoint to control the difference between the reactor wall temperature setpoint and the hottest temperature detected by the thermocouples in the bed . Control actions are taken by the hotspot loop at 30 second intervals. Temperature gradients exist in the reactor bed due to the highly exothermic reactions and the limited ability to remove heat from the center of the reactor through the reactor wall. A typical temperature profile along the axis of the bed is given in Figure 3. The purpose of the internal hotspot loop is to provide a safe open -loop stable configuration for multivariate control. Hydrogen and butane flow rate setpoints are determined by the multivariate control loop and the hotspot inner loop respect ively. The flow rate of the two feed streams are controlled once every 2 seconds by discrete PI controllers. The hotspot temperature controller is a "one-step optimal" LQ controller tuned by using the implicit self-tuned controller of Clarke and Gawthrop (1975). The self-tuning approach was used simply as a convenient way to tune the temperature controller. It was not used in a continuously adaptive mode. Since the response of hotspot temperature change to change in butane flow rate is very nonlinear, a logarithmic transformation of the hots pot temperature was used to linearize the temperature control loop. The controlled output after the linear transformation becomes Y(t)

= en(hotspot (t)) -

en(hotspot setpoint (t))

[n this work an empirical transfer function approach was selected to model the dynamics of the system based on input/output data. This technique eliminates the tremendous effort required in mechanistic modelling at the expense of losing some important insight into the process. Multivariate time series analvsis combined with process identification techniques were 'used to develop an empirical linear model. The details of this identification technique are covered in great depth by Wilson (1970) and Box and Jenkins (19701. This approach was used to identify a linear transfer function and noise model which approximates the dynamics of the process in the region of operation. Input/output data were generated by pseudo random binary perturbations of the input variables . Hydrogen flow rate was switched between 68 and 83 cc/sec. Hotspot setpoint was switched between 5 and 9 C above the wall temperature setpoint. Wall temperature setpoint was kept constant at 232 C during the data gathering. Two switching frequencies were used in the identification . A switching frequency of 6 minutes was taken in the first half of the data set to allow adequate identification of the fast output dynamics associated with changes to inlet concentration and flow rate . A 20 minute switching frequency was chosen in the second half to allow proper identification of the slow output dynamics with change in temperature. The input/output and time series data was used to identify an interactive model in the Wilson canonical form Y(t)

The pilot plant is operated through a distributed control network. The computer hardware consists of a DIGITAL VAXll1750 host computer connected to a DPM-23 Front End minicomputer . A VISTA 6000 gas chromatograph and a VISTA 401 GC control station is interfaced to the computer hardware. Communication between the user and the field is handled by real time software running in the VAX 111750 and DPM -23. 110 and low level control operations are performed in the DPM-23 Front End while high level control, operator communication, and data logging are performed by the VAX host.

= V(z-l) U(t) + N(t)

(22)

= 8(z-l) A(t)

(23)

(Z-I) N(t) where

Viz -1): n X m rational polynomial matrix in z-1 (z-I): is an n X n diagonal polynomial matrix in z-1 having nonrational elements and zO coefficient equal to I 8(z-I): is a full nXn polynomial matrix in z-1 having non rational elements and zO coefficient equal to I.

where

This non linear transformation was found to greatly improve the performance of the self-tun~ng regulator. The choice of the input magnitude constraint factor in the one-step optimal algorithm was used to provide a trade off between acceptable hotspot control to stabilize the system and the requirement of smooth variations of the butane feed rate. The second requirement was necessary since the end objective was to control the reactor outputs rather than hotspot temperature . Details of this controller design are given by Onderwater, MacGregor, and Wright (1986) .

D. Wright

will lead to four simultaneous, non-linear partial differential equations. Detailed knowledge of the reaction kinetics and process parameters together with lengthy mathematical model reduction operations are required to arrive at a useable model for control.

(20)

hotspot(t) = hotspot temperature (t) - wall temperature (t) (21)

J.

Maximum likelihood parameter estimation and diagnostic checking of resulting fitted model led to the approximate linear model Ylt)

= 0.5384z -1

1.8-6.643z -1 + 5.158z -2 1-1.042 -1 + 0.34122-

2

1- 1.2622 - 1 + 0.5203z - 2 0.15922 - 2 + O.1143z - 3

-0.7498 _ 3.39& -1 _ 1.012z - 2 1- 0.61432 - 1 U ( t-3)

+ 1 + 0.12&-1 0.2112z - 1

_0.12492-

1

1- 0.3904z - 1

I

(l - z [

-1

o

- 1

) 11-z

-1

)

1

where y( l ,t) :

4. MODEL IDE:-;TIFICATION A theoretical investigation of the process using fundamental material and energy balances can be performed to de\'elop a mechanistic dynamic model .J ulan (1976) has shown that this

propane production - 24 (mmollsec) y(2,t) : butane conversion - 29 .6% uO,t): (hydrogen flow rate· 75 cc/sec) X 10- 1 u(2 .t) : hotspot setpoint· 7 (C)

and the covariance malrix for the innovations Alt ) was

Al l )

(24)

Multiyariable Control of a Catalytic Tubular Reactor

,,_ = [ 0.134 "-

0.043

0 .043 0.200

I

(25)

This process model can be shown to display nonmlnlmUm phase response with respect to propane production. The model is not invertible and is not balanced with respect to time delays . Furthermore, there was evidence of a fairly high degree of process non linearity over the wide range of outputs and inputs to be considered in the control studies. Therefore, this model could only be considered to be a rough linear approximation to the true process dynamics, and in the design of the controllers robustness to process model mismatch was an important consideration.

289

closed-loop interaction can be observed when the final two setpoint changes were made where one of the two setpoints was held fixed . Figure 4 shows that input manipulations were smooth and well behaved during the entire control run . Figure 5 shows a control run on a new catalyst charge using the same Q2 = diag( 100,10) . The outputs, especially percent conversion, of the new charge are noticably noisier than those in figure 4. However, regulation and setpoint change performance was still good.

Constrained minimum variance and internal model controllers were designed using the model transformed into a right matrix fraction form . The control objective for both designs was to achieve good regulatory and good setpoint performance, while at the same time providing adequate robustness to the large degree of model mismatch that will be present due to nonlinearities and changing catalyst activity .

A more tightly tuned constrained minimum variance control run is shown in figure 6 with less constraint on the input variances using Q2 = diag(50,5L This control run was operated with a wall temperature of 233 C and all setpoint changes were raised 5 units relative to the previous run. The adjustment was made to compensate for significant observed changes in the catalyst activity . A high degree of oscillations were observed with the inputs and propane production. The problem became more severe when a setpoint change was made to a region of low propane production and high conversion. The unexpected poorer performance relative to the previous run can be attributed to loss of robustness by tighter tuning which made the controller more sensitive to model mismatch and nonlinearities.

5.1 Constrained Minimum Variance Control Runs

5.2 Internal Model Control Runs

The cost function used for designing constrained minimum variance controllers for the reactor was

A simple decoupled IMC controller was designed using the identified transfer function model in equation 24. A stable model inverse Gc(z-l) was designed according to equation 15. An unstable pole resulting from a non-invertible zero was reflected inside the unit circle , and a ringing pole arising from a negati ve real axis zero was shifted to the origin. Both these adjustments were accomplished using v + 2(Z - 1) in equation 15. Identical first order diagonal filters (17) were used for F 1(z-I) and F 2(z-I) . Details of the design procedure are outlined by Kozub (1986) .

5. CONTROL STUDIES

J = E{eT(t) Ql e(t) + VCT(t) Q2 V U(t)} (26) The model, given by equation 24, contains a commonly occurring nonstationary integrated moving average disturbance modeL A short open-loop run on the reactor showed that this disturbance led to substantial drifts in the conversion and propane production rate away from steady-state values. The controller was to be designed to compensate for this nonstationary stochastic disturbance, and to accommodate step changes in the setpoints of either output. The model for randomly occurring step changes in setpoints is given by

VY sp(t) = art)

(27)

where V = (1- z -1) L Since the structure of this model for setpoint changes is very close to that for the stochastic disturbances in the process it was considered sufficient to design the CMV controller for the IMA disturbance model in F2(Z-I) in the controller given by 24 and take Fdz- 1 ) equations 5, 6, and 7. This, of course, would be true if the disturbance models were identical (Harris and MacGregor, 1986). The CMV controller used was therefore the single degree of freedom controller

=

U(t) = H 1(z-I) [G m(z-I) U(t) - [Y(t) - Ysp(t)Jl

(28)

The form of the cost function and the disturbance model will lead to a controller with integral action. The output weighting matrix Ql was set to I for all controller designs and Q2 was adjusted to provide the required amount of input constraint and controller robustness. Figure 4 shows a control run on the same catalyst charge used for the model identification. An input constraint matrix of Q2 = diag(lOO ,10) was selected for this run. The dashed lines in figure 4 indicate operating setpoints. The abscissa on these figures and others to follow refers to the sampling interval, where each sampling interval represents 3 minutes. The solid line of the input setting figure represents hydrogen flow rate setpoint (cc/sec x 10 -1) and the dashed line represents hotspot setpoint (C) . Excellent regulation can be observed in the first 5 hour section of data in Figure 4. In the remaining part of the run a variety of conversion and propane production rate setpoint changes were made, some in the same direction, some in opposite directions and some where only one setpoint was changed. In all cases the setpoint change performance of the controller was good, and it appeared to be robust over the wide range of operating condition; ;tudied :"u ; 'g llificant

A control run with the filter constants fii tuned to diag(O.7,0.7) is shown in figure 7. This run was performed immediately following the CMV run in figure 5 with the new catalyst charge. The identical pattern of setpoint changes were made in all of these runs in order to facilitate comparisons. Setpoint change performance and regulation were good for the first two setpoints. The second setpoint change demanding high conversion with low propane production led to very poor controller performance. Large oscillations observed with the inputs and propane production suggest controller instability in this region of operation . Complete recovery by the controller can be seen when a setpoint change was made in propane production rate back to a more stable region of operation. In order to increase the robustness of this IMC controller to accommodate such setpoint changes the controller was detuned by using larger filter constants in F(z-I), namely F = diag(O.9,0.9) . The control run is shown in figure 8. Detuning led to substantial reduction in the variation of the input manipulations . Regulation still appeared to be good over the full region of operation. Improved robustness to the process nonlinearities can be observed by the stable controller performance when the setpoint changes were made to the region of high conversion and low propane production . No significant interaction was observed when the final setpoint change was made to propane production with conversion setpoint held fixed. The filter block was retuned to improve the setpoint change response of butane conversion . A control run with F = diag(0.9,0 .8) is shown in figure 9. This control run was performed at a higher wall temperature of 233 C and all setpoints were shifted 5 units higher relative to the previous runs . The process disturbance noise was noticed to be slightly higher with this change in operating conditions. Again, regulation was adequate over the entire region of operation although slightly increased deviation of propane production was observed in the region of low propane production and high butane convers iun Substantial improvement in the setpoint

290

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~Iac(;reg()r

and

J.

D. Wright REFERE:\CES

change response of butane con version can be observed in figure 9 relative to the previous controller , as a result of the smaller filter constant (08 vs . 0.91.

Bergh , L.G ., and J F . MacGregor (1986) . Constrained Minimum Variance Controllers: Internal Model Structure and Robustness Propertie s. submitted to Ind . Eng . Chem ., Proc. Des. Dev .

5.3 Discussion of Experimental Results

Box, G.E .P ., and G.M. Jenkins (1970). Time Series Analysis: Forecasting and Control. Holden Day, San Francisco.

The results indicate that the tightness of control for both controller design s was restricted by severe model mismatch . mainly due to severe process nonlinearities and changing catalyst properties. Trial and error adjustment of the tuning matrices led to robust designs with acceptable regulation and setpoint change performance . Improvement in the performance of these controllers would require use of a complicated adaptive mechanistic nonlinear model in the Gm(z - 1) block to eliminate some of the significant and time varying sources of model mismatch . The quality of performance of both controller designs was approximately the same in this investigation , although slightly improved propane production control was evident in the constrained minimum variance control run s Improvement of propane production control in the IMC design might have been achieved by further on-line adjustment of the filter block . The results also indicate that the inner hotspot control loop design provided a safe stable configuration for applica tion of good multivariate control. Even when marginally stable multivariate control actions were present, as in figures 6 and 7, the reactor did not runaway .

6. CONCLCSIONS An application ofmultivariate constrained minimum variance and internal model controller designs to composition control of a packed bed butane hydrogenolysis reactor was considered to be very successful. A simplified linear empirical input/output pulse transfer function model was used in these designs. Both linear designs, when well tuned, were robust to process nonlinearities and provided good regulatory and servo control over a wide region of operation. Performance of these controllers is limited by model mismatch between the approximate linear empirical model and the true nonlinear process, indicating that incorporation of a rigourous mechanistic model in the se controller designs might be desirable if tighter control were necessary . The multivariate control loop proved to be successful in stabilizing the reactor system for applicaton of safe multivariate control.

Clarke, D.W . and B.A .Gawthrop (1975). Self-Tuning Controllers. Proceedings lEE, 122, pp. 929-934. Garcia, C.E . and M. Morari (1985), Internal Model Control. 2. Design Procedure for Multivariate Systems. Ind . Eng. Chem. Process Des. Dev ., f.1, 472-484 . Garcia, C.E . and M. Morari (1985) . Internal Model Control. 3. Multivariable Control Law Computation and Tuning Guidelines . Ind. Eng. Chem. Process Des. Dev., 24, 484-494 . Harris , T .J . and J .F . MacGregor (1986). Design of Discrete Multivariable Linear Quadratic Controllers Using Transfer Functions. Submitted to AIChE Journal. Jutan, A. (1976) . State Space Modelling with Multivariate Stochastic Control of a Pilot Packed -Bed Reactor. Ph .D. Thesis , McMaster L:niversity , Hamilton, Canada. Jutan, A., J.P. Tremblay , J .F . MacGregor and J .D. Wright (1977). Multivariate Computer Control of a Butane Hydrogenolysis Reactor: Part Ill . On-line Linear Quadratic Stochastic Control Studies. AIChE Journal, ~, 5. Kozub , D.J. (1986). Advanced Model-Based Multivariate Control Design and Application to Packed Bed Reactor Control. M.Eng . Thes is , Dept. of Chem . Eng., McMaster University, Canada. MacGregor , J .F . and A.K.L . Wong (1980) Multivariate Model Identification and Stochastic Control of a Chemical Reactor . Technometrics, ~, 4, 435-464. MacGregor, J .F ., P .A. Tayor and J.D . Wright (1984) . Advanced Process Control , An Intensive Short Course on Digital Computer Techniques for Process Identification and Control. McMaster University, Hamilton, Ontario , Canada . Onderwater D., J .F . MacGregor and J . D. Wright (1986) :-.!onlinear Temperature Control of a Catalytic Tubular Reactor Using Self-Tuning Regulators . Submitted to Can . J . Chem. Eng. Sorensen, J .P ., S.B. Jorgensen and K. Clement (1980) Fixed Bed Reactor Kalman Filtering and Optimal Control - I Computational Comparisons of Discrete vs. Continuous Time Formulation. Chemical Engineering Society, lQ., 1223-1230, Pergamon Press Ltd. Sorensen, J.P., S.B. Jorgensen and K. Clement (1980) Fixed Bed Reactor Kalman Filtering and Optimal Control - II Experimental Investigation of Discrete Time Case with Stochastic Disturbances. Chemical Engineering Society, lQ., 1231-1236, Pergamon Press Ltd . Wall man , P . H . , J . M . Silva, and A .S . Foss (1979) Multivariable Integral Controls for Fixed Bed Reactors. Ind. Eng . Chem Fundam., ~ 4, 392-399. Wilson, G .T . (1970 ) Modelling Linear Systems for Multivariate Control. Ph .D. Thesis, University of Lancaster, England .

Multi\'ariable Control of a Catalnic Tubular Reactor

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