1

Design of Feedback

Controllers

2

• Design of Feedback Controllers

We notice that different controllers have different effects on the

response of the controlled process. Thus the first design question

arises:

Question 1: What type of feedback controller should be used to

control a given process?

Question 2: How do we select the best values for the adjustable

parameters of a feedback controller?

Question 3: What performance criterion should we use for the

selection and the tuning of the controller?

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There are a variety of performance criteria we could use, such

as:

- Keep the maximum deviation (error) as small as possible.

- Achieve short settling times.

- Minimize the integral of the errors until the process has

settled to its desired set point, and so on.

Simple Performance Criteria

Consider two different feedback control systems producing

the two closed-loop responses shown in Figure 16.2

Figure 16.2 Alternative closed-

loop responses.

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If our criterion for the design of the controller had been

Return to the desired level of operation as soon as possible

Then, clearly, we would select the controller which gives the

closed-loop response of type A. But if our criterion had been

Keep the maximum deviation as small as possible

or

Return to the desired level of operation and stay close to it in the

shortest time

we would have selected the other controller, yielding the closed-

loop response of type B.

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The simple performance criteria are based on some

characteristic features of the closed-loop response of a system.

The most often quoted are:

- Overshoot

- Rise time (i,e., time needed for the response to reach the

desired value for the first time)

- Settling time (i.e., time needed for the response to settle

within ±5% of the desired value)

- Decay ratio

- Frequency of oscillation of the transient

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From all the performance criteria above, the decay ratio

has been the most popular by the practicing engineers.

Specifically, experience has shown that a decay ratio

1

4

C

A

is a reasonable trade-off between a fast rise time and a

reasonable settling time. This criterion is usually known as

the one-quarter decay ratio criterion.

7

Example 16.1 : Controller Tuning with the One-Quarter Decay

Ratio Criterion

Consider the servo control problem of a first-order process with PI

controller. It can be easily shown that the close-loop respnse is given by

the following equation, when Gm = Gf = 1:

2 2

1( ) ( )

2 1

I

sp

sy s y s

s s

where

I p

p cK K

and

1(1 )

2

I

p c

p p c

K KK K

We notice that the closed-loop response is second-order.

8

2

2exp( )

1decay ratio

Therefore, for our problem we have

2

12 (1 )

2 1exp

411 (1 )

4

Ip c

p p c

Ip c

p p c

K KK K

K KK K

After algebraic simplifications we take

2

12 (1 ) ln( ) (16.1)

4 (1 ) 4

I

p c

p p c I p c

K KK K K K

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Equation (16.1) has two unknowns: Kc and I. Let Kp=0.1 and p=10.

Then, we find the following solutions:

1 10 30 50 100

0.153 0.464 0.348 0.258 0.153

c c c c c

I I I I I

K K K K K

and so on.

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is the deviation (error) of the response

from the desired set point.

1. Integral of the square error (ISE), where

2. Integral of the absolute value of the error (IAE), where

3. Integral of the time-weighted absolute error (ITAE), where

• Time-Integral Performance Criteria

2

0

( )ISE t dt

(16.2a)

0

( )IAE t dt

(16.2b)

0

( )ITAE t t dt

(16.2c)

Note that )()()( tytyt SP

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Select the type of the controller and the values of its adjusted

parameters in such a way as to minimize the ISE, IAE, or

ITAE of the system’s response. The following are some

general guidelines:

If we want to strongly suppress errors, ISE is better than IAE

because the errors are squared and thus contribute more to the

value of the integral.

For the suppression of small errors, IAE is better than ISE

because when we square small numbers (smaller than one) they

become even smaller.

To suppress errors that persist for long times, the ITAE

criterion will tune the controllers better because the presence of

large t amplifies the effect of even small errors in the value of

the integral.

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• Select the Type of Feedback Controller

Which one of the three popular feedback controllers should be

used to control a given process?

The question can be answered in a very systematic manner as

follows:

1. Define an appropriate performance criterion (e,g,. ISE, IAE, or

ITAE)

2. Compute the value of the performance criterion using a P, or PI,

or PID controller with the best settings for the adjusted

parameters , , .c I DK and

3. Select that controller which gives the “best” value for the

performance criterion.

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1. If possible, use simple proportional controller.

Simple proportional controller can be used if (a) we

can achieve acceptable offset with moderate values

of Kc or (b) the process has an integrating action for

which the P control does not exhibit offset.

2. If a simple P controller is unacceptable, use a

PI. A PI controller should be used when

proportional control alone cannot provide

sufficiently small steady-state errors (offsets)

3. Use a PID controller to increase the speed of the

closed-loop response and retain robustness

We can adopt the following rules in selecting the most

appropriate controller.

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Example 16.3 Selecting the Type of Controller for Various Processes

Let us discuss various processes that are to be controlled by

feedback control systems. We will address primarily the question of

selecting the appropriate type of feedback controller.

1. Liquid-level control: proportional control alone is satisfactory.

2. Gas pressure control: Usually, we want to maintain p within a

certain range around a desired value, thus making a

proportional controller satisfactory for out purpose.

3. Vapor pressure control: For such systems with fast response, a

PI controller is satisfactory. For the system that the situation is

different, a PID controller should be selected because it will

provide enough speed and robustness.

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Figure 16.5 Pressure control loops: (a) direct effect, fast

response; (b) indirect effect, slow response.

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4. Flow control: a PI controller is satisfactory because it

eliminates offsets and retains acceptable speed of

response.

5. Temperature control: a PID controller would be the

most appropriate, because it can allow high gains for

faster response without undermining the stability of the

system.

6. Composition control: a PID controller should be the

most appropriate.

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There are two general approaches we can use for tuning a

controller.

1. Use simple criteria such as the one-quarter decay ratio,

minimum settling time, minimum largest error, and so

on.

2. Use time integral performance criteria such as ISE, IAE,

or ITAE

3. Use semiempirical rules

• Controller Tuning

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Process reaction curve method, developed by Cohen and Coon.

Cohen and Coon observed that the response of most processing

units to an input change, had a sigmoidal shape (see Figure 16.7),

which can be adequately approximated by the response of a first-

order system with dead time.

Figure 16.7 “Opened” control loop.

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Figure 16.8 (a) Process reaction curve; (b) its approximation with a first-

order plus dead-time system

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τ = B/S, where S is the slope of the sigmoidal response at

the point of inflection

( )( ) ( ) ( ) ( )

( )

( )( )

( ) 1

d

m

PRC f p m

t s

m

PRC

y sG s G s G s G s

c s

y s KeG s

c s s

which has three parameters: static gain K, dead time td , and

time constant τ

( )

( )

output at steady state BK

input at steady state A

td = time elapsed until the system

responded

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The results of their analysis are summarized below.

1. For proportional controllers, use

11

3

d

c

d

tK

K t

2. For proportional-integral controllers, use

10.9

12

d

c

d

tK

K t

30 3 /

9 20 /

d

I d

d

t

t

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3. For proportional-integral-derivative controllers, use

1 4

3 4

d

c

d

tK

K t

32 6 /

13 8 /

d

I d

d

tt

t

4

11 2 /D d

dt

In such a case the Cohen-Coon settings should be viewed

only as first guesses needing certain on-line correction.

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• Ziegler-Nichols Tuning Technique

The Ziegler-Nichols tuning technique is a closed-loop

procedure. It goes through the following steps:

1. Bring the system to the desired operational level

(design condition)

2. Using proportional control only and with the feedback

loop closed, introduce a set point change and vary the

proportional gain until the system oscillates

continuously.

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Figure 14. The Effect of Kc on close-loop response.

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Controller Kc τI(min) τD(min)

Proportional (P) Ku / 2 - -

Proportional-integral (PI) Ku / 2.2 Pu / 1.2 -

Proportional-integral-

derivative (PID)Ku / 1.7 Pu / 2 Pu / 8

3. Using the values of Ku and Pu , Ziegler and Nichols

recommended the following settings for feedback controllers:

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Typical Response of Feedback Control Systems

ลกษณะของผลการตอบสนองจะขนอยกบคา KC, และ ทใชและลกษณะของกระบวนการดวย กลาวโดยทวไปคอ

I D

- การเพมคา KC ท าให ผลการตอบสนองเรวขน แตถาสงเกนไปอาจท าใหผลการตอบสนองเกดการแกวงหรอไมเสถยร จงควรเลอกใชคาทเหมาะสม

- การเพมคา จะท าให ก าจดคา Offset ได แตถาสงมากเกนไปจะท าใหผลการตอบสนองชาลงมากภายหลงมโหลดรบกวนหรอเซตพอยทเปลยน

I

- การเพมคา จะชวย ลดทงเวลาการตอบสนองและการแกวง แตถาคาสงมากเกนไปจะขยายสญญาณรบกวน (Noise) และสงผลใหผลการตอบสนองเกดการแกวงมากขน

D

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Figure 15. Effect of gain on the closed-loop response of second-

order systems with proportional control.

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Figure 16. Effect of reset time on the closed-loop response of first-

order systems with integral action only.

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Figure 17. Effect of gain on the closed-loop response of first-order

systems with PID control.