EXAMPLE 17.6

SOLUTION

17.5 Direct Synthesis for Design of Digital Controllers 463

A process is modeled in continuous time by a second-order-plus-time-delay transfer func-tion with K = 1, TL = 5, and T2 = 3. For At = 1, the discrete-time equivalent (with zero-order hold) is

c _ (bi + b2z-l)z-N-1 1 H- a\z~~l H- d2Z~2 (17-67)

where ai = -1.5353, a2 = 0.5866, b\ = 0.0280, bi = 0.0234, and iV = 0 (cf. Eq. 7-36 to 7-40). For Dahlin's controller with X = At = 1, plot the response for a unit change in set point at t = 5 for 0 < r < 10 using Simulink.

The Simulink diagram for the example was shown earlier in Fig-17.9. Using Eq. 17-63, the de-sired closed-loop transfer function for 0 = 0 (N = 0) and X = At is (Y/Ysp)d = 0.632Z"1/ (1 0.368z_1). Applying (17-64), we see that the formula for the controller is

GDC 1 + aiz-1 + ax'2 0.632?

_ -k _

- i

b \ z - l + b2Z-2 1 - z

Substituting the numerical values for au #2, b\, and bi, the controller is

1 - 1.5353Z-1 + 0,5866?-* 0.632 ^ D C 0.0280 + 0.0234?"1 1 - z~l

(17-68)

(17-69)

When this controller is implemented, an undesirable characteristic appears, namely, intersam-pie ripple. Figure 17.11*7 shows the response y and ZOH output p to a unit step change in set point at / = 5. Although the response does satisfy y(k) = 1 at each sampling instant (At = 1)

T 1 1 r T r

(a) Dahlin's controller

01 - i 0 20

50

P 0

0

i 1 1 1 1 r 1 r~r

5Ql i i i * J I L 10 20

IP1 m

f f r ~ [ ~ r - r T t

(6) Modified Dahlin's controller

20

/> 0

-20 0

1 1 1 1 1 1 1 1 1 1 1

1 1 f 1 1 1 I

10 Time

20

Figure 17-11 Comparison of (a) ringing and (b) nonringing Dahlin's controllers for a second-order process (X = 1), Example 17.6 (y controlled var iab le ,= controller output after zero-order hold).

17.5 Direct Synthesis for Design of Digital Controllers 465

-10

30 10 20 Time

Figure 17.12 Comparison of closed-loop response for a second-order process in Example 177 using (a) nonringing Dahlin's controller (X = 1), (b) backwards difference PID controller and (c) Vogel-Edgar controller (y = controlled variable, p ~ controller output after zero-order hold).

is set equal to one to provide the best response for a set-point change. When the BD approxi mation s - (1 z~ l)/Af is substituted into the PID controller transfer function,

GBD = 4.1111 3.1486 - 5.054U-* + 2.027Q;"2

1.7272 - 2.4444Z"* + 0.7222s"2 (17-73)

The closed-loop response is shown in Fig. 17.126. When the BD-PID digital controller is de-signed for this system, there is no need to correct for ringing as is required for Dahlin's con-troller. This is true for wide ranges of X and Af that have been investigated.

Figure 17.12 shows the closed-loop response for the Vogei-Edgar controller (GVE) for the same second-order model. This controller is

GVE - 0-6321 1 - 1.5353z-1 + 0.5866g-2

0.0514 - 0JB66Z"1 - 0D148?"2 (17-74)

The tuning parameter A is selected to be 0.368 (Xj= 1). For this second-order system, the con-trolled variable response for GVE is superior to Goc* If a time delay is added to the model, the comparative performance of GVE and GDC is still the same because both controllers uti-lize the same form of the Smith predictor.

Note that the controller parameters in (17-72) to (17-74) have been reported with four deci-mal points in order to avoid roundoff errors in the control calculations. H

Vogel and Edgar (1988) have shown that their controller satisfactorily handles first-order or second-order process models with positive zeros (inverse response) or negative zeros as well as simulated process and measurement noise. Many higher-order process models can be successfully controlled with GVE. Neither GVE nor GDC **re suitable for unstable process models, however. The robustness of the Vogel-Edgar controller is generally better than Dahlin's controller when model errors are present GVE

17,6 Minimum Variance Control 467

Sheet-making processes for producing paper and plastic film or sheets are common examples (Feather-stone et al.,2000).

The Minimum Variance Control (MVC) design method generates the form of the feedback control law, as well as the values of the controller parameters. As for the Direct Synthesis and Internal Model Control design methods, the MVC method results in PI or PID controllers for simple transfer function models (MacGregor, 1988; Ogunnaike and Ray, 1994; Box and Luceno, 1997). Although MVC tends to be quite aggressive, the design method can be modified to be less aggressive (Bergh and MacGregor, 1987). Because Minimum Variance Control is a limiting case on actual controller performance, it provides a useful bench-mark for monitoring control loop performance (Harris and Seppala, 2002); see Chapter 21.

The starting point for the M V C design method is the following discrete transfer function model:

The disturbance D(z) can be written as a zero, mean white (e.g., Gaussian) noise signal, a(z)t and a dis-turbance transfer function Gd(z):

In previous discussions on treating disturbances, we focused on deterministic changes in the distur-bance such as step changes. Here we will consider the four alternative disturbance models shown graphically in Fig. 17.13. These disturbances are persistent (as a result of the random component) but may also exhibit features such as dynamics, drift, or trending. A typical process disturbance seldom will be random but will depend on past values of the disturbance. These models can be constructed by starting with an input a{z) that is a white noise sequence. This input passes through a dynamic element such as a first-order transfer function or an integrating transfer function. The output D(z) is an auto-correlated disturbance to the process.

Table 17.2 gives the most important time-series models that are commonly encountered in industrial process control, including statistical process control applications (see Chapter 21). Stationary distur-bance models (a) and (b) have a fixed mean; that is, the sum of deviations above and below the line is equal to zero, but case (a) rarely occurs in industrial processes. Nonstationary disturbance models (c)

Y(z) = G(z)U(z) + D(z) (17-79)

D(z) = Gd{z)a{z) (17-80)

Figure 17,13 Four models for d(k): (a) stationary white noise disturbance; (b) stationary autoregressive disturbance; (c) nonstationary disturbance (random walk); (d) integrated (nonstationary) moving-average disturbance (adapted from Box and Luceno, 1997).

Exercises 469

Featherstone, A, P., J. G. VanAntwerp, and R. D. Braatz, Identifi-cation and Control of Sheet and Film Processes, Springer-Verlag, London, 2000.

Franklin, G. R, D. J. Powell, M. L, Workman, and X D. Powell, Digital Control of Dynamic Systems, 3d ed., Addison-Wesley, Reading, MA, 1997.

Garcia, C. E., and M Morari, Internal Model Control, 1. A Unify-ing Review and Some New Results. lECProc Des. Dev., 21, 308 (19S2).

Harris, T. J., and C. T. Seppala, Recent Developments in Con-troller Performance Monitoring and Assessment Techniques, Chemical Process Control VI , AIChE Symp. Ser., 98, No. 326, 208 (2002).

Isermann, R., Digital Control Systems, 2d ed., Springer-Verlag, New York, 1989,

Ljung, L., System Identification: Theory for the User, 2d ed,, Pren-tice Hall, Upper Saddle River, NJ, 1999.

MacGregor, J. F., On-Line Statistical Process Control, Chem. Engr, Prog., 84,21 (October 1988).

McConneil, E., and D. Jernigan, Data Acquisition, Sect. 117, p. 1795, in The Electronics Handbook, J. C Whitaker (Ed.), CRC Press, Boca Raton, FL, 1996.

Middleton, R, H., and G. C. Goodwin, Digital Control and Identi-ficat'tonA Unified Approach, Prentice Hall, Englewood Cliffs, NJ, 1990.

Ogata, K., Discrete-Time Control Systems, 2d ed., Prentice Hali, Englewood Cliffs, NJ, 1994.

Ogunnaike, B. A., and W. H. Ray, Process Dynamics, Modeling, and Control, Oxford University Press, New York, 1994.

Oppenheim, A. V., and R. W. Shafer, Discrete Time Signal Process-ing, 2d ed., Prentice Hall, Englewood Cliffs, NJ, 1999.

Phillips, S. F., and D. E. Seborg, Adaptive Control Strategies for Achieving Desired Temperature Control Profiles during Process Startup, Ind. Eng. Chem. Res., 27,1434 (1987).

Seborg, D. E., T, F. Edgar, and D. A. Mellichamp, Process Dynam-ics and Control, 1st ed., Wiley, New York, 1989.

Vogel, E, G., and T, F, Edgar, An Adaptive Pole Placement Con-troller for Chemical Processes and Variable Dead Time, Comp. Chem. Engr.^U, 15(1988).

Wellons, M. C, and T. F. Edgar, The Generalized Analytical Pre-dictor, Ind. Eng. Chem. Res.s 26,1523 (1987).

Zafiriou, E., and M. Morari, Digital Controllers for SISO Systems: A Review and a New Algorithm, int. 7. Control, 42t 885 (1985).

EXERCISES

17.1 The mean arterial pressure P in a patient is sub-jected to a unit step change in feed flow rate F of a drug. Normalized response data are shown below. Previous experience has indicated that the transfer function,

P(s) _ 5 F(s) 10s + 1

provides an accurate dynamic model. Filter these data using an exponential filter with two different values of a, 0.5 and 0.8. Graphically compare the noisy data, the filtered data, and the analytical solu-tion for the transfer function model for a unit step input.

Time Time (min) P (min) P

0 0 11 3.336 1 0.495 12 3.564 2 0.815 13 3.419 3 1.374 14 3.917 4 1.681 15 3.884 5 1.889 16 3.871 6 2.078 17 3.924 7 2.668 18 4.300 8 2.533 19 4.252 9 2.908 20 4.409

10 3.351

17.2 Show that the digital exponential filter output can be written as a function of previous measurements ym(k) and the initial filter output yr(0).

17.3 A signal given by

ym{t) = t + 0.5 sin(f 2) 1 is to be filtered with an exponential digital filter over the interval 0 ^ t ^ 20. Using three different values of a (0.8, 0.5, 0.2), find the output of the f i l -ter at each sampling time. Do this for sampling peri-ods of 1,0 and 0.1. Compare the three filters for each value of A/.

17.4 The following product quality data ym were obtained from a bioreactor, based on a photometric measure-ment evaluation of the product:

Time ym (min) (absorbance)

0 0 1 1.5 2 0.3 3 1.6 4 0.4 5 1.7 6 1.5 7 2.0 8 1.5

Exercises 471

Determine whether the controlled system is stable by calculating the response to a set-point change using Simulink,

17.14 Determine how the maximum allowable digital con-troller gain for stability varies as a function of At for the following system:

G p ( - S ^ (55 + l)(s + 1)

GC = IQ Gm = 1

Use At = 0.01, 0.1, 0.5, and trial and error (with Simulink) to find the maximum Kc for each At; Kc can range between 10 and 1200. What do you conclude about how sampling period affects the allowable con-troller gain?

17.15 A temperature control loop includes a second-order overdamped process described by the discrete trans-fer function.

( , _ (0,0826 + 0.0368Z-*);;-1 U { z > (1 - 0.894?- ,)(l - 0.295Z"1)

and a digital PI controller

Find the maximum controller gain Kan for stability by trial-and-error.

17-16 A digital controller is used to control the liquid level of the storage tank shown in Fig. E17.16. The control valve has negligible dynamics and a steady-state gain, Kv = 0.1 ft 3/(min)(mA). The level transmitter has a time constant of 30 s and a steady-state gain of

4 mA/ft, The tank is 4 ft in diameter. The exit flow rate is not directly influenced by the liquid level; that is, if the control valve stem position is kept constant, #3 # f(h). Suppose that a proportional digital con-troller and a digital-to-analog converter with 4 to 20 mA output are used. If the sampling period for the analog-to-digital converter is At = 1 min, for what values of controller gain Kc is the closed-loop system stable? Use Simulink and trial values for Kc of 10, 50, and 90, Will offset occur for the proportional controller after a change in set point?

17.17 The block diagram of a digital control system is shown in Fig. E17.17, The sampling period is At = 1 min. (a) Design the digital controller Gc(z) so that the

closed-loop system exhibits a first-order response to a unit step change in the set point (after an ap-propriate time delay).

(b) Will this controller eliminate offset after a step change in the set point? Justify your answer.

(c) Is the controller physically realizable? Justify your answer,

(d) Design a digital PID controller based on the ITAE (set-point) method in Chapter 12 and ex-amine its performance for a step change in set point. Approximate the sampler and zero-order-hold by a time delay of 6 = Atf2.

17.18 The exit composition a of the blending system in Fig. E17.18 is controlled using a digital feedback con-troller. The exit stream is automatically sampled every minute, and the composition measurement is sent from the composition transmitter (AT) to the digital controller. The controller output is sent to the ZOH device before being transmitted to the control valve.

Figure E17.17

Exercises 473

Liquid T * OUt W

Liquid in iv

I I I

C ^ - w. Steam

H trap Condensate

Figure E17.20

where the time constant has units of seconds and the primes denote deviation variables. The control valve and temperature transmitter have negligible dynam-ics and steady-state gains of Kv = 0.2 Ib/s/mA and Km = 0.25 mA/R Design a minimal prototype con-troller (i.e., Dahlin's controller with X = 0) that is physically realizable and based on a unit step change in the set point. Assume that a zero-order hold is used and that the sampling period is At = 2 s.

17.21 A second-order system G with K = 1, TI 6, and T2 4 is to be controlled using the Vogel-Edgar con-troller with \ 5 and Af = 1. Assuming a step change m.yspi calculate the controlled variabley(k) forfe = 0,1, . . . , 25 and plot y(k) and the controller output p{k).

17.22 Compare PID ( ITAE for set-point changes) and Dahlin controllers for At - 1, \ 1, and G(s) = 2e"V(10s + 1), For the I T A E controller, approximate the sampler and Z O H by a time delay equal to Af/2, Adjust for ringing, if necessary. Plot the closed-loop responses for a set-point change as well as the con-troller output for each case.

17.23 For a process, G(s) = 1.25^-^/(5^ + 1), derive the equation for Dahlin's controller with At and X 1

and plot controller output p{k) for a set-point change. Does ringing occur?

17.24 Compare the Dahlin and Vogel-Edgar controllers for G(s) = l/[(2s + l)(s + 1)] and X = At = 1. Does either controller ring? Derive the resulting difference equations for the closed-loop system y(k) related to ysp(k). Does overshoot occur in ei-ther case?

G

0

17.25 Design a digital controller for the liquid level in the storage system shown in Fig. E17.25* Each tank is 2.5 ft in diameter. The piping between the tanks acts as a linear resistance to flow with R = 2 min/f t 2 . The liquid level is sampled every 30 J . The digital controller also acts as a zero-order hold device for the signal sent to the control valve. The con-trol valve and level transmitter have negligible dy-namics. Their gains are Kv - 0.25 ft 3 /min/mA and Km = 8 mA/f t , respectively. The nominal value of qi is 0.5 f t 3 /min. (a) Derive Dahlin's control algorithm based on a

step change in set point. (b) Does the controller output exhibit any oscillation? (c) For what values of \ is the controller physically

realizable?

91

>

h

\

{

2

i

93 X

h I

LT L C 4 i I

92

Figure E17.25

Chapter

Multiloop and Multivariable Control

T A B L E OF CONTENTS

18.1 Process Interactions and Control Loop Interactions

18.1.1 Block Diagram Analysis

18.1.2 Closed-Loop Stability

18.2 Pairing of Controlled and Manipulated Variables

18.2.1 Bristol's Relative Gain Array Method

18.2.2 Calculation of the R G A

18.2.3 Methods for Obtaining the Steady-State Gain Matrix

18.2.4 Measure of Process Interactions and Pairing Recommendations

18.2.5 Dynamic Considerations

18.2.6 Extensions of the RGA Analysis

18.3 Singular Value Analysis

18.3.1 Selection of Manipulated Variables and Controlled Variables

18.4 Tuning of Multiloop PID Control Systems

18.5 Decoupling and Multivariable Control Strategies

18.5.1 Decoupling Control

18.5.2 General Multivariable Control Techniques

18.6 Strategies for Reducing Control Loop Interactions

18.6.1 Selection of Different Manipulated or Controlled Variables

Summary

In previous chapters, we have emphasized control problems that have only one controlled variable and one manipulated variable. These problems are referred to as single-input, single-output (SISO) or single-loop control problems. But in many practical control problems typically a number of variables

475

18.1 Process Interactions and Control Loop Interactions 477

feedback control strategy, consisting of two PI controllers, is to be used. This control system, re-ferred to as a multiloop control system because it employs two single-loop feedback controllers, raises several questions. Should the composition controller adjust WA and the flow controller ad-just WB* or vice versa? How can we determine which of these two multiloop control configurations will be more effective? Will control loop interactions generated by the process interactions cause problems?

In the next section, we consider techniques for selecting an appropriate multiloop control configura-tion. If the process interactions are significant, even the best multiloop control system may not provide satisfactory control. In these situations there are incentives for considering multtvariable control strate-gies such as decoupling control (Section 18.4) and model predictive control (Chapter 20), But first we examine the phenomenon of control loop interactions.

PROCESS INTERACTIONS AND CONTROL LOOP INTERACTIONS

A schematic representation of several SISO and MIMO control applications is shown in Fig, 18,2, For convenience, it is assumed that the number of manipulated variables is equal to the number of con-trolled variables. This allows pairing of a single controlled variable and a single manipulated variable via a feedback controller. On the other hand, more general multivariable control strategies do not make such restrictions (see Chapter 20). MIMO control problems are inherently more complex than SISO control problems because process interactions occur between controlled and manipulated vari-ables. In general, a change in a manipulated variable, say m, will affect all of the controlled variables yi, y2, . .. yn- Because of the process interactions, the selection of the best pairing of controlled and ma-nipulated variables for a multiloop control scheme can be a difficult task. In particular, for a control problem with n controlled variables and n manipulated variables, there are n\ possible multiloop con-trol configurations.

Disturbances

I I I I u * A Process

(a) Single-input, single-output process with multiple disturbances

Disturbances

(6) Multiple-input, multiple-output process (2 x 2)

Disturbances

^ ^ ^ ^

(c) Multiple-input, multiple-output process (n x n) F i g a r e i g , 2 S i S 0 and M I M O control problems.

18.1 Process Interactions and Control Loop Interactions 479

U,

1

Gpl2 Gpl2

[a) 1-1/2-2 controller pairing

U-

U-

^G

Gp2l Gp2l

Gp22

(6) 1-2/2-1 controller pairing Figure 18.3 Block diagrams for 2 X 2 multiloop control schemes.

The control loop interactions in a 2 X 2 control problem result from the presence of a third feedback loop that contains the two controllers and two of the four process transfer functions (Shinskey, 1996). Thus, for the 1-1/2-2 configuration, this hidden feedback loop contains Gcit G&, Gp\t, and GP2i, as shown in Fig, 18.4. A similar hidd en feedback loop is also present in the 1-2/24 control scheme of Fig. 18.36. The third feedback loop causes two potential problems:

1. It usually destabilizes the closed-loop system, 2. It makes controller tuning more difficult. Next we show that the transfer function between a controlled variable and a manipulated variable

depends on whether the other feedback control loops are open or closed. Consider the control system in Fig. 183a. If the controller for the second loop Ga is out of service or is placed in the manual mode with the controller output constant at its nominal value, then Uz - 0. For this situation, the transfer function between Y\ and Ui is merely Gpn:

- Gpn (Y2 - f/2loop open) (18-7)

If both loops are closed, then the contributions to Y\ from the two loops are added together:

Yi = Gpu Ui + Gpn Ui (18-8)

18.1 Process Interactions and Control Loop Interactions 481

Table 18.1 Controller Settings for Example 18.1

Controller Pairing 17 (min)

xD - R 0.604 16.37 XB-S -0.127 14.46

Assume that the controller settings are based on the I T A E tuning method for set-point changes in Chapter 12.

SOLUTION Table 18,1 shows the single-loop I T A E settings, and Fig. 18.5 shows simulation results for set-point changes for each controlled variable. The I T A E settings provide satisfactory set-point re-sponses for either control loop when the other controller is in manual (solid line). However, when both controllers are in automatic, the control loop interactions produce very oscillatory responses especially in XB (dashed line). McAvoy (1981) has discussed various approaches for improving the performance of the niultiloop controllers. See Exercise 18.1 for a similar M I M O control problem where the loops also exhibit oscillations.

1.4

1.4

1.2

i r 1 i i xB loop in manual Both loops in automatic

10 15 20 25 Time (min)

30 35 40

1 r 1 I I xD loop in manual

Both loops in automatic

15 20 25 Time (min)

30 35 40 Figure 18.5 Set-point responses for Exampli 18.1 using I T A E tuning.