Suboptimal Direct Digital Control of a Trickle-Bed Absorption Column

SUBOPTIMAL DIRECT DIGITAL CONTROL OF A TRICKLE-BED ABSORPTION COLUMN A. Cinar* and L. D. Durbin** ·Department of Chemical Engineering, Bogazici University, Bebek, Istanbul, Turkey UDepartment of Chemical Engineering, Texas A&M University, College Station, Texas 77843, U.S.A .

Abstract. For the selection of the best process variables to be used in a suboptimal feedback controller the design scheme proposed by Crosby and Durbin is further investigated and the selection procedure is improved. Furthermore, the suboptimal controllers formed by the selected variables are tested on-line on a trickle bed absorption column. The controllers used include the integral mode for the controlled variable plus the proportional mode for some selected state variables including the controlled variable. Substantial reductions in the performance index and peak overshoot are obtained by using the suboptimal controllers instead of conventional feedback controllers. Keywords.

Direct digital control; suboptimal control, chemical process control.

INTRODUCTION their ability to reduce a regression sum of squares error. The sum of squares of error criterion is defined as

Various methods have been proposed for selecting some of the state variables of a system for generating suboptimal feedback controllers (Longmuir and Bohn, 1967; Love and Lavi, 1968; Graham and Strauss, 1970).

(1 )

Applications of suboptimal control schemes to chemical processes are available in the literature. Gould, Evans and Kurihara (1970) designed a suboptimal control scheme to control a fluid catalytic crakcking unit and demonstrated the performance of the resulting control scheme by simulation. Lee, Koppel and Lim (1973) considered the suboptimal control of heat exchanger and countercurrent packedbed absorber models. Crosby and Durbin (1974, 1976) developed a design scheme and have applied this procedure to a packed trickle-bed column with carbon dioxide absorption. This paper concerned with further investigation and improvement of the selection scheme proposed by Crosby and Durbin (1974, 1976) and with testing the suboptimal controllers designed, on a trickle-bed absorption column. VARIABLE SELECTION PROCEDURE The optimal state and control trajectories which minimize a performance index are determined by using the Hamiltonian approach. Then, by using these trajectories along with statistical selection schemes, the relative importance of available states variables are determi ned. Statistical Selection Schemes The state variables are evaluated in terms of IS

Here, ufb(x) denotes the feedback control trajectory generated using the selected state variables x, and n denotes the total number of time increments. If the step-up procedure is used, at first the single variable regressions are carried out between the dependent variable Ui=u(t) and each of the state variables x. The particular state variable which gives the smallest value for the error sum of squares defined above is selected as the most important variable. Next, two-variable regressions are carried on the dependent variable . The subsets considered consist of the state variable selected and all other variables. The particular stilte variable which, together with the variable selected in the first stage, gives the smallest sum of squares error for the twovariable regressions is considered to be the second most important variable. Additional variables can be selected by extending this scheme. In the computational procedure first the sums of squares then the total correlation coefficients rij are calculated. The variable i with the largest absolute value of riu is selected as the most important variable. Next the partial correlation coefficients corrected for the state variable selected are calculated. In this manner the procedure can easily be extended to select more state variables.

16

A. Cinar and L. D. Durbin

Modification of Crosby's Variable Selection Procedure The first modification of Crosby's procedure is on the elimination of the effects of the load variable on all state variables . In the method proposed by Crosby the reasoning is as follows: The optimal state variable trajectories are a function of the optimal control trajectory. The explicit optimal control trajectory, u*, and the resulting optimal state variable trajectories, x, determine a unique relation b:tween u* and-~. Rather than taking u* as the lndependent variable and x as the dependent variable, U* can be taken-as the dep:ndent variable and ~ as the independent varlab1es. Then, the correlation coefficients are computed and variables for feedback control are selected. However, u* is not the only variable that effects the optimal state variable trajectories. The load change given to the system affects x as well and its effect has to be e1 iminated. -Thus, to see the iOlp1icit relationship between U* and x the correlation coefficients must be adjusted to eliminate the effect of the load change. The second modification is the elimination of forcing any variable into the pool of selected variables. The reasons are twofold. First, the v~riab1e force~ in may not be among the most lmportant varlab1es to be included in t~e controller. Secondly, the order of se1ectlon of the other variables may be altered. The constraint on the signs of the partial correlation coefficients of the controlled variable is modified. This modification will be discussed in detail in the controller design for the absorber section. EXPERIMENTAL EQUIPMENT In this work the laboratory absorber and the M~ta-4/1800 d~gita1 computer are used along wlth the requlred interface equipment to test and eva1 uate different control schemes. In the absorber, an air-C02 mixture and water are contacted countercurrent1y to cause the transfer of C02 fr~m the gas to the liquid. For all work carrled out the operation is isothermal and the pressure drop is negligible. she1~ of the trickle-bed absorber (Crosby ~urbln~ 1976; Clnar, 1976) is made of an a~umlnum plpe 1.83 m long, ~/ith an internal

The and

dlameter of 0.152 m. The shell has five sampling positions for the gas phase and four for the liquid phase (Fig. 1.). Tap water passes through a 12.7 mm orifice, a control valve and a rotameter before entering the co1u~n. To monitor. the gas phase responses contlnuous1y at varlOUS axial positions along the column, gas samples are drawn the absorber by probes inserted into the column at the sampling positions. The gas samples pass through thermal conductivity cells, a surge tank, a flow regulator, a rotameter and a vacuum surge tank. The output voltage of the thermal conductivity cell is a linear function of C02 ~oncentration up to a dry gas C02 concentratlon of 42 percent by volume. The time constant of the cell and the tUbing connecting

the prObe to the cell is about 1 s and is reg1igib1e compared to the time constant of the column which is about 75 s. The ~1eta-4/1800, manufactured by Digital Scientific Corporation emulates the IBM 1800 process control computer. The core memoTY of 16 K binary 16 bit words has a read-write cycle of 900 ns. The peripheria1 equipment consist of a card reader, a typewriter, a single drive disk unit, a CRT display unit and a digital input/output unit with direct memory access capacity. The D/A conversion unit is built around a 4 digit BeD D/A converter module. The A/D conversion system has 16 channels with differential inputs. The equipment used to close the control loop around the absorber is shown in Fig. 2. Signals from the sensors are amplified from 0-40 mV to 0-10 V before being sent to the multiplexer. The multiplexer and the A/D converter are synchronized for converting all incoming signals to digitized forms. The digitized values are then used by the computer to generate a control valve setting in the o to 9999 range. This setting is sent to the D/A converter. The 0 to +9.999 Vdc control voltage from the D/A converter generates a 4-20 mA dc signal by means of V/I converter. The current signal drives a I/P transducer. The . r~su1ting 3-15 psig pneumatic signal actlvltes the control valve that adjusts the water flow rate. The water flow rate is regulated by means of an equal-percentage 25 mm single port Fisher Controls control valve. COMPUTER PROGRAMS The data acquisition and control programs were developed for use with the IBM Time-Share Executive (TSX) programming system. The programs are written in FORTRAN IV and in IBM 1800 Assembler Language used in data input/output and in code conversion routines. The flowchart of the on-line control programs is given in Fig. 3. The on-line and off-line cold start core loads are named PCST and NPCST, respectively. RESTA is the main1ine error recovery core load. TSEND is the interrupt core load responding to a "console interrupt" and queues either COORD or PCST . COORD is the main1ine core load that coordinates data acquisition, computation of the controller output, data storage and outputting the con~ roller output to the D/A converter. The flowchart of the portion of COORD that handles the data acquisition is shown in Fig. 4. This ~etup eliminates the delay in servicing the lnterrupt generated by the interval timer, as long as COORD stays in core. A better way of e1iminat1ng the delay would have been to have an INSKEL interrupt subroutine. However, the size of the "Skeleton Executive" for the current system did not have enough space. A subroutine called by COORD is used to implement the control action. By modifying this subroutine, different control schemes can be generated. CLBRT is used for calibration of sens~rs and .LOGDT converts data to approriate physlca1 Unlts. An interrupt subroutine

Suboptimal Dire c t Digital Co ntrol

loaded with COORD was used to service the in terrupt caused by the interval timer. A sampling time of 1 s was used throughout the control action. Sampling data, computing the contro 11 er setting and output i ng it took very much less than 1 s. DESIGN OF SUBOPTIMAL CONTROLLERS FOR THE ABSORBER Model for the Absorber and the Control LOop. The blackflow cell model is used to describe the dynamic characteristics of the absorber. Twenty-six gas and liquid cell pairs were used to describe the packed absorber from the level of the bottom probe VSl to the top of the packing. The inlet gas stream to the model was then the gas stream at the level of VSl having the same concentration as samples drawn from this probe. The response obtained at VSl for a step increase of 19 to 38% by volume in C02 in the gas input to the column is used as the inlet gas stream to the absorber model. The cell numbers corresponding to the gas and liquid probe positions along the column are given in Table 1. In the control loop only the dynamics of the valve actuator are considered. Two-point implicit method is used in solving the model differential equations. The model equations and the general procedure for solving the model equations for the absorber and the control loop are outlined in detail elsewhere (Crosby and Durbin, 1976; Clnar, 1976). Open-Loop Optimization The absorber is controlled to keep the C02 concentration in the gas stream coming out of the absorption column constant by manipulating the inlet water flow rate. Both in modeling and in experimental work the C02 concentration in the gas sample at gas probe VS5, Cy VS5, is considered to be the C02 concentration in the exit gas stream. The performance index used is P.I.

=

105~tf

t(CyVS5 - C;VS5)2dt .

(2)

* Here, t denotes time and Cy.VS5 denotes the desired value (set point) for CyVS5 '

The open-loop optimization problem is to find the time trajectory of the control voltage that manipulates the control valve, which yields a minimum value for the performance index. The load change is an increase in the C02 concentration in the inlet gas stream from 19% to 38% by volume, at constant gas flow rate. In this study the optimal state variable and control voltage trajectories obtained by Crosby and Durbin (1976) are used. Variables Used in the Statistical Selection Schemes. Out of the 78 state variables defining the state of the backflow cell model with 26 cells for the absorber, only 52 of them are accessible. The concentrations of CO 2 dissolved in the water phase cannot be directly measured conveniently with a probe.

17

The optimal state trajectories for a step-up in the C02 concentration in the inlet gas stream show that the net change in the steady state of each response and the speeds of response increase for the cells in the lower portions of the column. The steady state effects of state variables must be filtered out. The filter used both by Crosby and Durbin (1976) and in this work is described by Buckley (1970). It is of the form ( S( s ) ) f ' l

=

_4>_s- S( s )

1

(3 )

4> S + 1

Here, 4> is the filter time constant, S is a state variable and s is a complex variable. Filter time constants of 72 sand 144 s, which are about twice and four times the time constant of the state variable corresponding to VS4 are used. The state variables used in the selection schemes are as follows: S 1·

=

(Cyl' }f'1 1. • t

S53

(4 )

( (CyVS5 o

S54

(CyVSl ) fi 1.

S55

=

*

CyVS5 - CyVS5 '

Here, Cyi is the C02 concentration in the gas phase and pHi is the hydrogen ion concentration in the 1iquid phase of the ith cell. CyVSl and Cy VS5 are the C02 concentrations at gas probes VSl and VS5. They correspond to the input to the absorber model and the controlled variable, respectively. The control function used in the statistical selection schemes is defined as the deviation of the control voltage from steady state. Two sets of available states are considered for the statistical selection schemes. In Set Set A, the complete set of available state variables,are included, In set B, the state variables corresponding to gas and liquid probe positions (S,1 ' SlO, S16, S21, S31, S37, S46, S52, S54, S55) are included. For all variables expect S4, riu had its highest value when there was no filter and its value decreased as the filter time constant decreased. Filtering makes S55 more oscillatory and since there is more "variation" S4 has a higher correlation coefficient on the control variable compared to S55. Consequently, the variable S4 was excluded from the set of available variables in the rest of this work. Evaluation of Variables Selected by the Step-Up Procedure. After the correlation coefficient matrix is obtained from the open-loop optimal state variable and control trajectories the effect of load change is el iminated. In other words rij.54 are computed before starting the selection of variables.

18

A. Cinar and L. D. Durbin

The results of the Step-Up, procedure are given in Tables 2, 3 and 4. In Table 2 the variables are listed in the order they are selected and the sign of their partial correlation coefficient at the stage they are selected is given. The integral action on the controlled variable, S53, is selected as the most important variable . With ~ = 72 s (Table 3), SlO, S3l and S55 are among the most important variables and with ~ = 144 s, SlO and S3l have high values of the partial correlation coefficients. All these variables belong to the top portion of the column. Here, S55 is selected as the second and SlO as the third most important variable for ~ = 72 s. For ~ = 144 s, SlO and S52 are the second and the third most important variable. S52 is the state variable corresponding to the bottom pH probe. Since it is almost at the same physical location as the bottom gas probe, its selection may indicate the stage at which S54 would have been selected. The variables selected for different subset sizes are summarized in Table 5. S52 is replaced by S54 in these subsets. In the procedure proposed by Crosby and Durbin (1976), S5~ was forced into the pool of selected varlables since it was the primary controlled variable . Both S55 and S53 were admitted to the pool of selected variables if they had a positive partial correlation coefficient on the control variable. In all cases considered in his work, S55 was forced in and had a positive correlation coefficient and S53 was not admitted since it always had a negative correlation coefficient. However, S53 was used in controlling the laboratory absorber. In this work, both of these points are modified. First, no variable is forced into the pool of selected variables here. Secondly, S4, S53 and S55 are admitted only if they have a negative partial correlation coefficient. In all cases considered, S53 was selected as the most important variable. The constraint on the signs of the partial correlation coefficients of S53 and S55 on the control variable is also alte r ed here. Crosby advocates that to ensure the stability of the control loop S53 and S55 must be positively correlated with the water flow rate. This is true for the closed-loop control system. When the C02 concentration at VS5 is increased, the water flow rate must be increased to bring the C02 concentration to the set point level. However, the data used in the statistical selection schemes are the result of open-loop optimization. When the water flow rate is increased , the C02 concentration at VS5 is reduced and vice versa . Thus, S53 and S55 are negatively correlated with the water flow rate. CONTROL OF THE ABSORBER Once the variables to be used in the suboptimal closed-loop control schemes are selected the individual controller gains have to be decided upon. An optimization program is used to tune the closed-loop controllers that are selected . The gains of the variables used

in the controller are to mlnlmlze the performance index given in Eq. (2). Powell 's method is used to locate the minumum of the performance index. Unidimensional searches are carried out by Coggin's method. Controller Tuning by Simulation For the digital simulation of the closed-loop system with the absorber model, a time increment of 1 s was used for the first 10 s. For the remaining transient time, the time increment was 5 s. The final time was 700 s for essentially attaining steady state after the step increase in the inlet gas stream. The operating conditions are the same as those used in the open-loop control programs. The most general form of the closed-loop controller used is given u

=

u* + KcS55 + KI S53 + t: Ki(Si)fil 1

(5)

Here, Si denotes any state variable included in the controller besides S53 and S55, and Ki the corresponding gain. Kc and KI are the gains for the proportional and integral actions on the controlled variable. The filtering can be implemented digitally as follows: (S 1,n . )f 1. ·l = [S 1,n o -So10 -(l/ ~ )(F.1,n + Si,n_1 6t/2)]

(1+6t/2~ )

(6)

where Fi,n = Fi,n-l+ ESi,n)fil.+ (Si,n_l)fil}t/2 (7)

Here, F. is the approximation of the integral of the Jelected variable for filtering over time. The subscript "n" denotes the nth time increment from zero time which is taken to be the time of the step changes to the absorber or model thereof. Load changes. The subsets used, the minimal values of the performance indices and the corresponding gains are given in Table 6. The load change in all cases corresponds to the response at VSl for a 19 to 38% by volume step change in· the inlet C02 concentration to the absorber. The controlled variable, S55, is plotted for different control schemes in Figs 5 to 7. The peak overshoots for different subsets used are summarized in Table 7. The transient for subset (SlO' S53) has almost the same rise time as that for conventional PI control but its peak overshoot is 2.4 times ~maller. The transient for subset (SlO, S53) dampens fasterl but as steady-state is approached, it is more oscillatory. The peak overshoots of the trajectories for the three-variable subsets vary from 0.160 to 0.072 in units of S55' For ~ = 72 s, the subset (SlO' S53' S55) causes S55 to drop considerably below the final steady state value during the early portion

SUbo ptima l Direc t Di g it a l Co ntro l

19

of the transient. The subset (553' 554, 555 ) reduces the dip in 555. The peak occurs much faster and consequently the performance index is reduced. However, the peak overshoot is increased to 0. 160 and 555 settles to the final steady state rather slowly. The subset (510' 553, 554) offers a good compromise If these results are compared with those indicated by the step-up procedure, the conclusion is that, it does not always indicate the best subsets. However, most of the subsets which gave good simulation results are selected by the 5tep-up procedure for one of the values of the filter time constant .

C02 gas concentration of the inlet gas stream from 19 to 38% C02 ~y vo l ume. For all of the stable cases,the step-down responses appeared to be controlled as weJl as the step-up changes . The controllers used,their gains,the performance inde x and the peak overshoot resulting from their use are listed in Table 9. For subsets (553,555)' (510,553,555) the gains computed by means of the tuning simulations were used . Furthermore,for subset (510,553, 555), the optimal gains proposed by Crosby and Durbin (1976) were used for ~ = 72 s. For ~ = 144 s , the gains were extrapolated using the optimal gains for ~ = 72 s and ~ = 216 s.

In order to indicate the effects of filter time constant ~ , the trajectories for 555 are plotted in Fig.8 for ~ = 72 s and ~ = 144 s. The subset (510, 553, 555) was used in the contro 1 ac t ion and 510 1s fil tered. As fi lter ~ime constant is increased the peak overshoot 1S reduced. However, convergence to the final steady state is slower for ~ = 144 s.

During the experimental runs both the water temperature and the water pressure in the water system of the building varied. Although the water temperature varied from day to day and from daytime to nighttime, the water pressure was varying within minutes. Thus, on the one hand, the variation in the water temperature made it hard to start a run with the same initial steady state as the tuning simulations. On the other hand, the fluctuations in the water pressure added an extra load change to the system. Even in this "realworld" situtation, the suboptimal on-line controllers performed quite successfully. The fluctions in the water flow rate were as large as 0.3 gallons per minute (0.019 ~ /s) .

5et-point changes. 50me further studies were made to note the effects of set-point changes upon the closed-loop simulation based upon the absorber model. In almost all of the cases that were considered, the optimal gains for load changes caused either highly oscillatory or unstable responses for set-point changes . Consequently, the controllers were tu~ed indepe~dently for a 2% change in the setp01nt. The 1nlet C02 concentration to the absorber was kept at 19% C02 by volume. All of the other operating conditions were the same as in the case for load changes. The results of these tests are given in Table 8 and in F~g. 9. Among two variable subsets PI control glves the better controlled response. The best subsets and the optimal gains for set-point variations are different from the best subsets and optimal gains for load changes. The computer can be used to store the subset a.nd gains to be used in each case and to use the correct setup for either type of change. On-Line Control of the Absorber. The control schemes are implemented by using a form similar to the velocity form(5mith, 1972) of PI controllers. 's un-un_l=Kc(en-en_l )+KI(en+en-l);r +K 1.l:i(5l· )f l·1 .

(8)

Here, en is the error at time t , ' S is the sampling time, and 5i denotes tRe state va~iable i used in the suboptimal controller. Kc' Kr and Ri are the controller gains used in the control of the absorber. To convert Kc, KI and Ki to Rc RI and Ri the calibrations of thermal conductivity cells and the column temperature are used. The filtering is implemented as given by Eqs . (6) and (7). Analysis of Experimental Runs. Conventional PI control and var10US suboptimal feedback controllers were used to control the laboratory absorber for load changes . The load change introduced was a step change in the

The controlled variable (555) is plotted in Fig. 10 for subsets (553,555) and (510, 553). It can be noted that the use of 510 with 553 improved both the peak overshoot and the overall shape of the transient curve. The performance index is reduced almost by a factor of three. Due to some technical limitations only subset (510, 553, 555) was tried among the three variable subsets. The control system was unstable when the gains obtained from the tuning simulations were used. This is probably due to both a high value of the gain for the integral action and to the positive feedback of the controlled va r iable. Next the optimal gains computed by Crosby and Durbin (1976) are used. For ~ = 72 s the response (Fig. 11) is nearly as good as the model response obtained by Crosby. Although the first half of the response for ~ = 144 s is the same as that for ~ = 72 s, the second half of the response is poor in that an offset occurs. Also, it was observed during the tuning simulations that for ~ = 144 s the steady state effects of the V54 response were not totally el iminated CONCL U5 ION5 The purpose of this work is to improve the design scheme proposed by Crosby and Durbin (1974, 1976) and demonstrate its usefulness by on-line digital control of an absorber. 50me radical changes were made in the procedure of variable selection for the control action. Thus, no variable is forced in the pool of selected variables. Also, the effect of the load change on the state variable trajectories is eliminated by adjusting the par-

A. Cinar and L. D. Durbin

20

tial correlation coefficient. As a result of these modifications, a combination of integral action at probe VS5 and proportional action at VS4 was selected as the best two-variable subset. Both the tunina simulations and the online control of the absorber proved that this controller as far superior to the conventional PI control. The selection of the integral action in the optimal controller and control of tubular processes using proportional action at an intermediate point and integral action at the exit are in agreement with the results of Shih (1970) and Koppe1, Kamman and Woodward (1970). The three-variable subset used in control of the absorber provided better control than the best two-variable subset. Noting that this subset (S10' S53, S55) was not the best three-variable subset, chances are that the best three-variable subset would improve the response even further. The simple suboptimal controllers used improved performance substantially. Investment in hardware not too prohibitive considering that an extra variable may improve the performance considerably. This fact motivates further study of sUboptimal feedback controllers for better control of other distributed parameter processes . ACKNOWLEDGMENT The authors wish to acknowledge the support of this work by the National Science Foundation as Grant GK-38555 administered by the Texas A&M Research Foundation. Also, A.Clnar wishes to express appreciation for the financial assistance from the Department of Chemical Engineering at Texas A&M University.

Crosby J.E. and L. D. Durbin (1976). Design and Application of State Variable Feedback Controllers for a Packed Trickle Bed Column with Carbon Dioxide Absorption. Proc. JACC, 171-184. Clnar, A. (1976). On-1 ine Suboptimal Feedback Control of a Trickle-Bed Absorption Column. Ph.D. Thesis, Texas A&M University. Gou1d,A.L., L.B. Evans and H. Kurihara (1970). Dynamic Control of a Fluid Catalytic Cracking Unit. Automatica, 6, 695-702. Graham,E .N. and J.C. Strauss (1970) . A Simplification Method for Suboptimal Controller Synthesis. Proc. JACC, 829-832. Koppe1, L.B., D.T. Kammon and J.L. Woodward (1970). Two-point Practical Control of a Class of Distributed Processes. Ind. En~. Chem. Fundam., 9, 198-205. -Lee, H. . , L. B. Koppe1 ana H.C. Lim (1973). Optimal Sensor Locations and Controller Settings for a Class of Counter current Processes. Ind.Eng.Chem . Process Des. Develop., 12, 36-41. Longmuir, A.G. and E.V. Bohn (1967). The Synthesis of Suboptimal Feedback Control Laws. IEEE Trans . Automat. Contr., AC-12 755-758. -Love, C.G. and A. Lavi (1968). Evaluation of Feedback Structures . Proc. JACC, 10811091 . Shih, V.P . (1970). Integral Action in the Opt imal Control of Linear Systems with Quadratic Performance Index. Ind. Eng. Chem. Fundam., 9, 35-37. Smith, C.L. (1972) . -Digital Computer Process Control, In, text, Scranton, Pa.

REFERENCES Buck1ey, P.S. (1970). Override Controls on a Chemical Reactor. Proc. of Twenty-Fifth Annual Symposium on Instrumentatlon for the Process Industries, 6-12. Crosby J.E. and L.D. Durbin (1974). Selection of Process Variable Feedback Control - A New Systematic Scheme. Sixth Annual Symp . on System Theory, Baton Rouge, Louislana, SeSSlOn Tp-3 . Table g. Set no .

4

Table 1.

Model cell numbers corresponding to gas and llquid probe positions

Gas Probe

Ce 11 Numer

Liquid Probe

Cell Number

VS5 VS4 VS3 VS2 VSl

4 10 16 20 27*

LSl LS2 LS3 LS4

5 11 20 26

*Inlet gas stre.., to the col""", .

Subsets used In the on-llne control of the absorber

Subset used

Gain (simulation)

Gain (absorber)

53 55

11.305 1108 . 2

0.00626 0 .613656

10 53

693.45 39 . 509

0 .88726 0. 0219

72

405.71

0.211

10 53 55

410 . 95 19.799 653.21

0 . 5258 0 .011 0 . 36171

72

341.01

0.183

10 53 55

383.24 18 .034 640.5

0 . 4094 0.010 0 . 355

144

1694.28

0.174

10 910 . 43 53 54 . 329 55 -370 .67 water teooperature 26 . 11°C.

1. 1649 0.030084 -0.20526

Filter time constant ;(s)

Per fonnance index

Peak overshoot

1182 .4

72

0 . 461

unstable

Suboptimal Direct Digital Control Table 3 .

Plrtlll co,""lItlon coefficients for Cue B C+ • 12

ParUll co,""latlon coefficient

r lu

r lu • 53

r lu •53 ,55

S,

21

corrected for 554 l

r lu •53 ,55,10

r lu •53 ,55,10,52

State variable noi 10

.53132

.41707

.64795

16

.44313

.12399

.42093

- . 44161

. 42356

21

.31273

-.35361

-.03351

-.68860

. 56057

Table 2 . Order of selection of state variables by the Step-Up procedure C5 is excluded and correlation coefficients. are c rrected for effect of 5 ) 54

3

+ • 72 s

+ • 144 s

Case A

Case B

Case A

Case 8

31

.54143

37

- . 52056

-.17062

- . 39257

.59854

. 27584

53 C-l

53 C-l

53 C-l

53 C-)

46

- .41751

.02983

0.29805

.56929

-.45706

25 C-l

55 (-)

3 C-l

10 C+l

52

-.31631

.38253

.11466

. 77619

53

-.97556

.42056

.63890

-.08747

- . 31754

- . 49905 -_ .............55-_ ..........-_ .......10795 - ................ - ...... -_ .... -- ...... _.... --- .. - .......... _---_ .. - -- ---- ........................................ Vartable selected

Table 4.

555

553

5

510

52

27 C+l

10 C+l

2 C+l

52 C+l

26 C+l

52 C+l

36 C+l

16 C+l

28 (-)

21 C+l

19 C+l

37 C-)

521

Partial co,""lltlon coefficients for Case 8 C+ • 144 s, co,""cted for 554 l

Partial correlation coefficient

r

lu

r lu •S3

10

.69648

. 77802

r lu • 53 ,10

r lu •5 3,10,52

r lu ' 5 3,10,52 ,16

State variable no i Table 7 •

Peak overshoOts for dl fferent subsets used

16

.62845

.60858

- . 39398

. 55912

21

. 50635

. 20867

- .67501

.28096

. 10589

Subset used

• • 72 s

31

. 70521

. 77670

.02075

- . 25620

- . 35503

53,55 10,53 53,54,55 10,53,55 10,53,54 10,53,54,65 Open-loop

0 . 280 0 . 119 0.160 0 . 101 0.072 0.051 0.001

Peak overshoot

37

-.69169

- . 60686

. 58096

. 28073

- .43707

46

- . 60771

- . 49859

. 53964

- . 54712

0.01355

52

- . 50905

-.17949

.11976

53

- . 96748

55

.......... _--- ......

• • 144

S

0.280 0 . 092 0.257 0.065 0.089 0.001

-.14056 - . 70893 - . 43599 _--_ .........28560 - .... - ...... ---------- .. -_ ..... _-------------- .. ------_ .... --------- .. - .... ----_ .....

Variable selected

Subsets of variables to be used in closed-loop Table 5 . control based on the selection by the SterUp procedure (corrected for effect of S54 .

f • 72 s

Subset si ze 2

3

4

Table 8 .

Table 6 .

•• 144 s

53,55 10,53

10,53 53,55

10,53,55 53,54,55 10.53.54

10,53,54 10,53 . 55

10.53,54.55

16.53.54,55

Controller evaluation for set-point changes

••

•• 72 s

Controller evaluation for load changes

••

12

s

• • 144 s P. I.

Gain

693.45 39.509

100 . 07

554.96 26.735

11.305 1108.2

390.36

11.305 1108.2

Subset used

Gai.n

10 53 53 55

8421.7

5.1284 -8.3885

53 54

4.8318 -67 . 058

10 53 54

481. 52 33.697 -53.179

29 . 067

456.74 20 . 951 -39 . 846

53 54 55

37 . 942 -158.97 1189.2

48 . 428

14.476 -81.444 1107.3

910 . 43 54 . 325 -370 . 67

72 . 098

876 . 25 49 . 362 -713.86

P. I. 98 . 831 390 . 36 10092.0

59 . 407

144 s

Subset used

Gain

P. I.

·Gain

P. I.

53 55

10.591 1382 . 3

0 . 17137

10 . 591 1382. 3

0 . 17137

10 53

2128.9 56 . 099

1.0724

1810 . 2 45 . 251

0. 82664

10 53 55 31 52 53

10 53 55

75.878 10 . 893 1421.0

0 . 16839

42 . 351 11.027 1441 . 6

0.16979

10 53 54

56 . 192 2136.2 -53 . 179

1.0725

45 . 026 1794 . 0 -53 . 179

0 . 82641

10 53 54 55

-19.899 -7.7444 SO. 936 633 . 32 47 . 531 -54 . 729 -981.90

636 . 57

14.227

313 . 09

59 . 056

A. Cinar and L. D. Durbin

22 Tap water

- - C.ll OOEUE

k

-

- - - C.ll TutU - - - - Call lEVEL

8

,,--Liquid spray ring

c:==='--=~---TOP

~

of packing

liquid sampling position>: LS1, LS2, LS3, LS4

Call VIAQ

C.llli4T(l

SCAN

Gas sampling positions:

VS1, VS2, VS3, VS4, VS>

Flowchart of on-line cont1"01

co~

load!>.

gas mi xture

~ A B C

o

E F G H I J K

F1 g. 1.

Si ze (in . ) 16.00 10.50 19 . 25 16 . 00 16.25 16 .00 2. 00 18.50 26.50 17.25 4.50

Size (11111) 406 . 4 266 . 7 488.9 406.4 412 . 7 406.4

31 • Subset (53,55) • Subset (10,53)

SO.8

9 673 . 1 438.1 114 . 3 ~69 .

Sch.... tic diagram of the packed-bed absorber. 23 l~et.·4/l800

dig; tal computer

100

200

400 300 Time. S

500

600

700

nurrber

nunber

I

I

Fig.S .

Optim.al gas trajectories at probe VSS (SSS) for two-variable subsets--closed.loop sirrulation.

OIA converter

AID converter

I

I

vo 1t.ge

vo lta g"

I Current pump I current

Subset (53.54,55)

•

Subset (10.53,54)

•

Subset (10,53.55)

w

'101

tage

I

I

Current to pneumati C transducer

Sign.l ampl Hiers

I P1

•

~

e ·0 ~ ~

I

,.,

>

u

voltage

Tap water

400

500

600

Time. s

Absorber 1----""

Fig. 1.

Sch .... tic diagram of the control loop .

F1 g. , .

Opt i rna 1 gas trajec tor1 es at probe VSS (S55) for three-vlrtab 1e subsets- -c l osed-l o.) tl

,t .. latien.

23

Suboptimal Direct Digital Control

..

Subset (10,53)

o

Subset (lO,53,54) Subset (10,53,54,55)

100

200

300

400

27r-------------------------------------~

700

600

500

T11ne. S

Fig . T.

Optt., 9as trajectories at probe VS5 {SS~ for dtffer-ent subset sizes closed-loop s;,..,lIt10n.

500

Time. Fig. 8.

5

Optimal gas trajectories at probe VS5 (S55) for subset (lO,53,55) It different. closed-loop stlnJlat1on.

35 .. Subset (lO,53)

..

33

o Subset (53,55), PI control

.....

31

1

29

...!:?

27

.... ~

> u,.,

25 23 21 100

400

Time. Fig . 10.

Fig... .

700

5

Transtent responses at gas probe VSS (controlled var1able) for two-variable subsets.

Flowchart of dat .. acqu i sition portion of I"ITt .

29

25.0,.------------------...,

..

27

.....

"

i"

.! <>

500

9. Opthntl gu traJector i es at probe VS5 (5 cl osed-l oop

25

600

Tlrre, S

fig.

26

) fo,. s.t - po1nt ch,l\ges 55 s ;....,l.ti on.

21 100

200

400

300

Time, s Fig. 11. Transient responses It gas probe VS5 (controlled variable) for subset (S10,S53,S55) '