CHAPTER 12: Controller Design, Tuning, & Troubleshooting Anis Atikah Ahmad [email protected]

CHAPTER 12: Controller Design, Tuning, & Troubleshooting

Anis Atikah [email protected]

Steps in Process Control

Outline Performance Criteria for Closed-Loop Systems PID Controller Design:

Model-Based Design Method Direct Synthesis Method Internal Model Control (IMC)

Controller Tuning Relations IMC Tuning Relations Tuning Relations Based on Integral Error Criteria Miscellaneous Tuning Method

Controllers with Two Degrees o Freedom On-Line Controller Tuning

Continuous Cycling Method Guidelines for Common Control Loops Troubleshooting Control Loops

Performance Criteria for Closed-Loop Systems

Unit-step disturbance responses of FOPTD model & PI controller

Which of the following provides the best response????

Performance Criteria for Closed-Loop Systems The function of a feedback control system is

to ensure that the closed loop system has desirable dynamic and steady-state response characteristics.

Ideally, we would like the closed-loop system to satisfy the following performance criteria:

Performance Criteria for Closed-Loop Systems

PID Controller Settings

PID controller settings can be determined by a number of alternative techniques:

1. Direct Synthesis (DS) method

In the Direct Synthesis (DS) method, the controller design is based on a process model and a desired closed-loop transfer function.

Consider the block diagram of a feedback control system in Figure 12.2.

The closed-loop transfer function for set-point changes is:

mpvc

pvcm

sp GGGG

GGGK

Y

Y

1

Eq. 12-1

1. Direct Synthesis (DS) method [cont.]

For simplicity, let and assume that Gm = Km.

Then Eq. 12-1 reduces to:

Rearranging and solving for Gc gives an expression for the feedback controller:

GG

GG

Y

Y

c

c

sp 1

mpv GGGG

Eq. 12-2

Eq. 12-3a sp

spc YYG

YYG

1


Equation 12-3a cannot be used for controller design because the closed-loop transfer function Y/Ysp is not known.

Also, it is useful to distinguish between the actual process G and the model, that provides an approximation of the process behavior.

A practical design equation can be derived by replacing the unknown G by , and Y/Ysp by a desired closed-loop transfer function, (Y/Ysp)d:

dsp

dsp

cYYG

YYG

1~ Eq. 12-3b

G~


Note that the controller transfer function in (12-3b) contains the inverse of the process model.

For processes without time delays, the first-order model in Eq. 12-4 is a reasonable choice,

Where τc is the desired closed loop time constant.

Because the steady-state gain is one, no offset occurs for set-point changes.

dsp

dsp

cYYG

YYG

1~ Eq. 12-3b

1

1

sY

Y

cdsp Eq. 12-4


By substituting (12-4) into (12-3b) :

Solving for Gc , the controller design equation becomes:

The term provides integral control action and thus eliminates offset.

Design parameter provides a convenient controller tuning parameter that can be used to make the controller more aggressive (small ) or less aggressive (large ).

dsp

dsp

cYYG

YYG

1~1

1

sY

Y

cdsp

Eq. 12-5

sGG

cc

1~1

sc1

cc

c

Direct Synthesis (DS) method [cont.]

If the process transfer function contains a known time delay θ, a reasonable choice for the desired closed-loop transfer function is:

The time-delay term in (12-6) is essential because it is physically impossible for the controlled variable to respond to a set-point change at t = 0.

Combining Eqs. 12-6 and 12-3b :

Eq. 12-6

1

s

e

Y

Y

c

s

dsp

1

s

e

Y

Y

c

s

dsp

dsp

dspc

YYG

YYG

1~


Combining Eqs. 12-6 and 12-3b

Gives:

The following derivation is based on approximating the time-delay term in the denominator of (12-7) with a truncated Taylor series expansion:

1

s

e

Y

Y

c

s

dsp

dsp

dsp

cYYG

YYG

1~

sc

s

c es

e

GG

1~1

Eq. 12-7

se s 1 Eq. 12-8


Substituting (12-8) into the denominator of Eq. 12-7

and rearranging gives

Note that this controller also contains integral control action.

sc

s

c es

e

GG

1~1

se s 1

se

GG

c

s

c

~1

Eq. 12-9


1. First-Order-plus-Time-Delay (FOPTD) ModelConsider the standard FOPTD model,

Substituting Eq. 12-10 into Eq. 12-9 and rearranging gives a PI controller;

with the following controller settings:

1

~

s

KeG

s

Eq. 12-10

se

GG

c

s

c

~1

1

~

s

KeG

s

sKG

Icc

11

cc KK

1

I Eq. 12-11


2. Second-Order-plus-Time-Delay (FOPTD) ModelConsider a second-order-plus-time-delay model,

Substitution into Eq. 12-9 and rearranging gives a PID controller;

with the following controller settings:

11

~

21

ss

KeG

s

Eq. 12-12

se

GG

c

s

c

~1

ss

KG DI

cc 1

1

c

c KK 211

21 I

11

~

21

ss

KeG

s

21

21

D

Eq. 12-13

Eq. 12-14

Example 12.1

Use the DS design method to calculate PID controller settings for the process:

Consider three values of the desired closed-loop time constant: .

Evaluate the controllers for unit step changes in both the set point and the disturbance, assuming that Gd = G.

Repeat the evaluation for two cases: a. The process model is perfect ( = G). b.The model gain is incorrect, = 0.9, instead of the actual value, K = 2.

10,3,1c

15110

2

ss

eG

s

G~

K~

Example 12.1- Solution

Use the DS design method to calculate PID controller settings for the for two cases:

se

GG

c

s

c

~1

15110

2~

ss

eG

s

se

e

ssG

c

s

sc

2

15110

s

ssG

cc

2

15110

ss

KG DI

cc 1

1

s

ssK

I

IDIc

12

sss

c

2

11550 2

Comparing with standard PID controller;

Thus;

15I

comparing

33.31550 D 1215 ccK


The controller settings are as follows:

15I 33.31550 D 1215 ccK

19.015 ccK

(a)For K =2 (b)For K =0.9

The values of Kc decrease as τc increases, but the values of τI

and τD and do not change


Simulation results for (a)

( = G), G~

2~ K

As τc increases, the responses become more sluggish

Increasing τc


Simulation results for (b)

( = 0.9). K~

As τc increases, the responses become more sluggish

Increasing τc


Which one is better???WHY?

2~ K 9.0

~ K

c

c KK 211

2. Internal Model Control (IMC)

The Internal Model Control (IMC), similar to DS method, is based on an assumed process model and leads to analytical expressions for the controller settings.

The IMC method is based on the simplified block diagram shown in Fig. 12.6b.

The model response is subtracted from the actual response Y, and the difference, is used as the input signal to the IMC controller,

Y~

YY~

Fig. 12.6

Internal Model Control (IMC)

The two block diagrams are identical if controllers Gc and Gc* satisfy the relation:

GG

GG

c

cc ~1 *

*

Fig. 12.6

Eq. 12.16

Any IMC controller is equivalent to a standard feedback controller Gc, and vice versa.


The following closed-loop relation for IMC can be derived as follows:

For the special case of a perfect model, Eq. 12.17 reduces to

DGGG

GGY

GGG

GGY

c

csp

c

c~

1

~1

~1 *

*

*

*

Eq. 12.17

GG~

DGGGYGY cspc

~1 ** Eq. 12.18


The IMC controller is designed in two steps:

1.The process model is factored as

where contains any time delay and right half

plane zeros.

2. The controller is specified as:

Where f is a low-pass filter with steady state gain of one and:

fG

Gc

~1*

Eq. 12.19 GGG~~~

Eq. 12.20

G~

rcsf

1

1

Eq. 12.21


For an ideal case where the process is perfect ( ), substituting the IMC

controller transfer function in closed loop relation,

gives

Thus, the closed-loop transfer function for set-point changes (D=0) is:

fG

Gc

~1*

Eq. 12.22

Eq. 12.23

GG~

DGGGYGY cspc

~1 **

DGffYGY sp ~

1~

fGY

Y

sp~

Example 12.2

Use the IMC design method to design a FOPTD model. Assume that f is specified by eq 12.21 with r=1, and consider 1/1 Padé approximation for the time delay term

FOPTD Model:

According to 1/1 Padé approximation,

Substituting into FOPTD Model;

Eq. 12.24a

Eq. 12.25

1

~

s

KeG

s

s

se s

21

21

12

1

21

~

ss

sKG

Example 12.2 cont.

Using 2 steps in IMC Controller design;

STEP 1: Factor this model as

Model:

Thus,

GGG~~~

Eq. 12.26 12

1

21

~

ss

sKG

sG2

1~

contains any time delay and right half plane zeros

12

1~

~~

ss

K

G

GG

Eq. 12.27

Example 12.2 cont.


STEP 2: Specify the controller into

Thus, substituting and ,

Thus,

Eq. 12.28

12

1

~

ss

KG

fG

Gc

~1*

rcsf

1

1

1

12

1*

sK

ssG

cc

Example 12.2 cont.


STEP 2: Specify the controller into

Thus, substituting and ,

Thus,

Eq. 12.28

12

1

~

ss

KG

fG

Gc

~1*

rcsf

1

1

1

12

1*

sK

ssG

cc

Example 12.2 cont.

The equivalent controller Gc can be obtained from eq 12.16;

And rearranged into PID controller;

Eq. 12.30

sK

ssG

c

c

2

12

1

GG

GG

c

cc ~1 *

*

12

121

cc KK

2I

12

D

Eq. 12.29

*Type of controller (PI or PID) depends on time-delay approximation. Repeating this derivation for Taylor series approximation gives a standard PI controller.

3. Controller Tuning Relations3.1. IMC Tuning Relations

Table 12.1


Lag-dominant models (θ/τ<<1)

• First- or second-order models with relatively small time delays (θ/τ<<1 ) are referred to as lag-dominant models.

• The IMC and DS methods provide satisfactory set-point responses, but very slow disturbance responses, because the value of τI is very large. Fortunately, this problem can be solved in three different ways.

1. Approximate the lag-dominant model by integrator-plus-time delay model.

Then apply IMC tuning relation in Table 12.1. (Case M or N)

s

eKsG

s

*

KK

*



2. Limit the value of τI

For lag-dominant models, the standard IMC controllers for first-order and second-order models provide sluggish disturbance responses because τI is very large.

As a remedy, Skogestad (2003) has proposed limiting the value of :

cI 4,min 1



3. Design the controller for disturbance rejection, rather than set-point tracking.

For example, develop an extension of the DS approach based on closed-loop transfer function for disturbance.

(Y/D)d, rather than (Y/Ysp)d

Example 12.4

Consider a lag-dominant model withθ/τ =0.01:

Design four PI controller;

(a)IMC (τc=1)(b)IMC (τc=2) based on the integrator approximation(c)IMC (τc=1) with Skogestad’s modification(d)Direct synthesis method for disturbance rejection. The controller settings are Kc=0.551 and τI=4.91

Evaluate the four controllers by comparing their performance for unit step changes in both set point and disturbance.

ses

sG

1100

100~

Example 12.4-solution

The PI controller settings are:

Kc τI

IMC 0.5 100

Integrator approximation

0.556 5

Skogestad 0.5 8

DS (disturbance) 0.551 4.91

Example 12.4-solution

IMC controller provides an excellent set-point response, while the other three controllers have significant overshoots and longer settling times.

Set-point response

Disturbance response

However, the IMC controller produces an unacceptably slow disturbance response owing to its large τI value.

3. Controller Tuning Relations

3.2 Tuning Relations Based on Integral Error Criteria

Controller tuning relations have been develop that optimize the closed-loop response for a simple process model & a specified disturbance or set-point change. The optimum settings minimize an integral error criterion. Three popular integral error criteria are:

a. Integral of the absolute value of the error (IAE)

a. Integral of squared error (ISE)

a. Integral of the time-weighted absolute error (ITAE)

dtteIAE

0

dtteIAE

0

2

dttetIAE

0



IAE value

Graphical interpretation of IAE



3. Controller Tuning Relations3. 3 Miscellaneous Tuning Relations

• Hägglund and Åström tuning relations

• Skogestad tuning relations

G(s) Kc τI

s

Ke s

1

s

Ke s

K35.0

KK 28.014.0

7

100

8.633.0

Condition Kc τI τD

81

K215.0

21

81

8

85.0 21

K 28

21

21

2

2

8

8

Important Notes on Controller Design & Tuning Relations

1. Kc 1/K where K = KvKpKm .

2. / ↑ Kc ↓ (more time delay → poorer performance)

3. / ↑ I and D ↑ ( D / I = 0.1 -0.3 )

4. When integral control action is added to a proportional-only controller, Kc ↓

The addition of derivative action Kc ↑ (stability margin increased)


Controllers with two degree of freedom

The specification of controller settings for standard PID controller typically requires a tradeoff between set-point tracking and disturbance rejection.

Fortunately, two simple strategies can be used to adjust the set-point and disturbance responses independently. These strategies referred to as controllers with two degrees of freedom.

First strategy: set point changes are introduced gradually rather than as abrupt step changes . Eg: the set-point can be ramped or “filtered” by passing it through first order transfer function:.

1

1*

sY

Y

fsp

sp

Controllers with two degree of freedom

Second strategy: adjusting the set-point response, based on a simple modification of the PID control law:

tm

DI

cmspc dt

dydtteKtytyKptp

0

**1

Set-point weighting factor, 0<β<1

As β increases, the set point response become faster but exhibits more overshoot. When β =1, the modified control law reduces to standard PID control law.

Example 12.6 For the first-order-

plus delay model of Example 12.4, the PI controller with DS-d settings provided the best disturbance response. Can set-point weighting significantly reduce the overshoot without adversely affecting the settling time?

Example 12.6

Set-point weighting with ß =0.5 provides a significant improvement, because the overshoot is greatly reduced and the settling time is significantly decreased.

4. On-line Controller Tuning4.1 Continuous Cycling Method

Step 1. After the process has reached steady state (at least approximately), eliminate the integral and derivative control action by setting τD to zero and τI to the largest possible value.

Step 2. Set Kc equal to a small value (e.g., 0.5) and place the controller in the automatic mode.

Step 3. Introduce a small, momentary set-point change so that the controlled variable moves away from the set point.

- Gradually increase Kc in small increments until continuous cycling occurs.- The term continuous cycling refers to a sustained oscillation with a constant amplitude. - The numerical value of Kc that produces continuous cycling (for proportional-only control) is called the ultimate gain, Kcu. - The period of the corresponding sustained oscillation is referred to as the ultimate period, Pu.

4. On-line Controller Tuning

4.1 Continuous Cycling Method

Step 4. Calculate the PID controller settings using the Ziegler-Nichols (Z-N) tuning relations in Table 12.6.

Step 5. Evaluate the Z-N controller settings by introducing a small set-point change and observing the closed-loop response. Fine-tune the settings, if necessary.



Figure 12.12 Experimental determination of the ultimate gain Kcu.



Table 12.6. Controller settings based on the continuous cycling method

Guidelines for Common Control Loops

1. Flow Rate

- Flow control loops are characterized by fast response with essentially no time delay- For flow control loops, PI control is generally used.- The presence of recurring high-frequency noise discourages the use of derivative action because it amplifies the noise.

2. Gas Pressure

- PI controllers are normally used with only a small amount of integral control action (тI is large)- Derivative action is normally not needed because the process response times are usually quite small compared to those of other process operations.

Guidelines for Common Control Loops

3. Temperature

- General guidelines are difficult to state because of the wide variety of processes and equipment involving heat transfer and different time scales.-PID controller are commonly employed to provide more rapid responses than can be obtained with PI controllers.

4. Liquid Level

- Standard P or PI controllers are commonly used for level control.- Because offset is not important in averaging level control, it is reasonable to use a proportional-only controller.

If a control loop is not performing satisfactorily, then troubleshooting is necessary to identify the source of the problem.

Based on experience in the chemical industry, Buckley (1973) has observed that a control loop that once operated satisfactorily can become either unstable or excessively sluggish for a variety of reasons that include:

a. Changing process conditions, usually changes in throughput rate. b. Sticking control valve stem c. Plugged line in a pressure or differential pressure transmitter. d. Fouled heat exchangers, especially reboilers for distillation columns. e. Cavitating pumps (usually caused by a suction pressure that is too low).

Troubleshooting Control Loops

The starting point for troubleshooting is to obtain enough background information to clearly define the problem. Many questions need to be answered:

1.What is the process being controlled? 2.What is the controlled variable? 3.What are the control objectives? 4.Are closed-loop response data available? 5.Is the controller in the manual or automatic mode? Is it reverse or direct acting? 6.If the process is cycling, what is the cycling frequency? 7. What control algorithm is used? What are the controller settings? 8. Is the process open-loop stable? 9. What additional documentation is available, such as control loop summary sheets, piping and instrumentation diagrams, etc.?

After acquiring this background information, the next step is to check out each component in the control loop.

Troubleshooting Control Loops

Documents

CHAPTER 12: Controller Design, Tuning, & Troubleshooting Anis Atikah Ahmad [email protected]