Controller Design, PID Tuning and Troubleshootinglpre.cperi.certh.gr/auth/files/Process Control_Design of... · Chen and Seborg (discussed in textbook) methods are essentially identical

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November 2, 2014

Prof. Costas Kiparisides

Professor Costas Kiparissides

Department of Chemical Engineering,

Aristotle University of Thessaloniki,

Process Dynamics and Control

Subject: Controller Design and Tuning

Controller Design, PID Tuning and

Troubleshooting


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1. Stable

2. Quick responding

3. Adequate disturbance rejection

4. Insensitive to model, measurement errors

5. Avoids excessive controller action

6. Suitable over a wide range of operating conditions

Impossible to satisfy all 6 unless self-tuning. Use “optimum sloppiness"

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Desirable Controller Features


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Unit-step disturbance responses for the candidate controllers (FOPTD Model: k=1, θ= 4, τ= 20) and Gp=Gd

Unit-step Disturbance Responses


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Controller Design and Tuning

Direct synthesis : Specify servo transfer function required and calculate required controller. Assume real plant = plant model.

Internal Model Control: Morari et al. (1986). Similar to direct synthesis except that real plant and plant model can differ.

Tuning relations:

Cohen-Coon - 1/4 decay ratio

Designs based on ISE, IAE and ITAE

Frequency response techniques

Bode criterion (Ziegler-Nichols)

Nyquist criterion

Field tuning and re-tuning


1. Tuning correlations – most limited to 1st order plus

dead time

2. Closed-loop transfer function - analysis of stability

or response characteristics.

3. Repetitive simulation (requires computer software

like MATLAB and Simulink)

4. Frequency response - stability and performance

(requires computer simulation and graphics)

5. On-line controller cycling (field tuning)

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Alternatives for Controller Design


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Direct Synthesis

Internal Model Control


Standard Feedback Control System


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( G includes Gm, Gv)

1. Specify closed-loop response (transfer function)

2. Need process model, (= GPGMGV)

3. Solve for Gc,

GG

GG

Y

Y

C

C

sp

1

dspY

Y

1

1

s p dC

s p d

Y

YG

G Y

Y

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(11-3b)

G

Direct Synthesis Method


(12 6)1

s

sp cd

Y e

Y s

, c speed of response θ = process time delay in G

But other variations of (12-6) can be used (e.g., replace time delay with polynomial approximation)

•cc

1 1If θ = 0, then (12 - 3b) yields G = ) (12 - 5)

G τ s

cc c c

K τs + 1 τ 1For G = , G = = + (PI)

τs + 1 Kτ s Kτ Kτ s

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First-Order Plus Time-Delay Model

A reasonable choice for the desired closed-loop transfer function is a FOPTD model:

11

1111

11-


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θ

(12-10)τ 1

sKeG s

s

Substituting and rearranging gives a PI controller,

with the following controller settings: 1 1/ τ ,c c IG K s

1 τ, τ τ (12-11)

θ τc Ic

KK

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Specify closed-loop response as FOPTD (12-6), but

approximate - se - - s. 1

PI Controller for FOPTD Process

11

11-

11


Let

(11-3b)

1 21sp cd

sY

Y s

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(1 )2( )1 ( 1)(1 )2

s K sKeG s

s s s

2 11

2 2( ) 12 1c I D

c

KK

1 21 1 1 112 2

11 22 211

cc

c

c

θ- sθ θτs+ + s τs+ + sτ s+

G = • =θθ θ- sK - s K + τ s

-τ s+

(11-3a)

(11-30)

1

1

sp dc

sp d

Y

YG

G Y-

Y

PID Controller for FOPTD Process


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Use of FOPTD closed-loop response (12-6) and time delay approximation gives a PID controller in parallel form,

11 τ (12-13)

τc c DI

G K ss

where

1 2 1 21 2

1 2

τ τ τ τ1, τ τ τ , τ (12-14)

τ τ τc I Dc

KK

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Consider a second-order-plus-time-delay model,

θ

1 2

(12-12)τ 1 τ 1

sKeG s

s s

Second-Order Plus Time-Delay Model

11

11

11

11-


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

= 1, 3, and 10. Evaluate the controllers for unit step changes inboth the set point and the disturbance, assuming that Gd = G.Perform the evaluation for two cases:

a. The process model is perfect ( = G).

b. The model gain is = 0.9, instead of the actual value, K = 2. This model error could cause a robustness problem in the controller.

G

K

0.9

10 1 5 1

seG

s s

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

2

10 1 5 1

seG

s s

Example 1


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The controller settings for this example are:

3.75 1.88 0.682

8.33 4.17 1.51

15 15 15

3.33 3.33 3.33

τ 1c τ 3c 10c

2cK K

0.9cK K

τIτD

Note only Kc is affected by the change in process gain.

Direct Synthesis Controller Settings


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

and do not change, as indicated by Eq. 11-14.

τcτI τD

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Simulation Results

a. Correct model gain


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Simulation Results

b. Incorrect model gain


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Internal Model Control (IMC)


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Direct Synthesis From closed-loop transfer function

Isolate Gc

For a desired trajectory (C/R)d and plant model Gpm, controller is

given by

Not necessarily PID form

Inverse of process model to yield pole-zero cancellation

(often inexact because of process approximation)

Used with care with unstable process or processes with RHP zeroes

C

R

G G

G Gc p

c p

1

GG

CRC

Rc

p

1

1

G

G

CRC

Rc

pm

d

d

1

1


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Direct Synthesis 1. Perfect Control

– Cannot be achieved, requires infinite controller gain

2. Closed-loop process with finite settling time

– For 1st order Gp, it leads to PI control– For 2nd order, get PID control

3. Processes with delay θ

– Requires– Again, 1st order leads to PI control– 2nd order leads to PID control

C

R d

1

C

R sd c

1

1

C

R

e

sd

s

c

c

1

c


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IMC Controller Tuning

Closed-loop transfer function

CG G

G G GR

G G

G G GD

c p

c p pm

c p

c p pm

*

*

*

*( ) ( )1

1

1

In terms of implemented controller, Gc

GG

G Gc

c

c pm

*

*1

Gpm

Gp

R

-+

+-

++

D

CGc

*


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IMC Controller Tuning

1. Process model factored into two parts

where contains dead-time and RHP zeros, steady-state gainscaled to 1.

2. Controller

where f is the IMC filter

Based on pole-zero cancellation Not recommended for open-loop unstable processes Very similar to direct synthesis

G G Gpm pm pm

Gpm

GG

fcpm

* 1

fsc

r

1

1( )


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IMC-Based PID Controller Settings


Example 2: PID Design

PID Design using IMC and Direct Synthesis for the process

Process parameters: K=0.3, τ=30, θ=9

1. IMC Design: Kc=6.97, τI=34.5, τd=3.93

Filter

2. Direct Synthesis: Kc=4.76, τI=30

Servo Transfer function

G se

sp

s( )

.

9 0 3

30 1

fs

1

12 1

C

R

e

sd

s

9

12 1


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Example 2: Results Servo Response

IMC and direct synthesis give roughly same results

IMC not as good due to Pade approximation

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00

5

1 0

1 5

2 0

2 5

IMC

DirectSynthesis

y(t)

t


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Example 2: Results

Regulatory response

Direct synthesis rejects marginally disturbance more rapidly

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 01 5

2 0

2 5

3 0

3 5

4 0

y(t)

Direct Synthesis

IMC

t


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Model-based design methods such as DS and IMC produce PIor PID controllers for certain classes of process models, withone tuning parameter τc (see Table 11.1)

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1. > 0.8 and (Rivera et al., 1986)

2. (Chien and Fruehauf, 1990)

3. (Skogestad, 2003)

τ /θc τ 0.1τc

τ τ θc

τ θc

• Several IMC guidelines for have been published for the model (FOPTD) in Eq. 11-10:

c

How to Select τc?

DS and IMC Tuning Relations


• First- or second-order models with relatively small time delays

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 is very large.

• Fortunately, this problem can be solved in three different ways.

τI

θ / τ<<1

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Tuning for Lag Dominant Models


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Method 1: Integrator Approximation

*Approximate ( ) by ( )

1where * / .

s sKe K eG s G s

s sK K

• Then can use the IMC tuning rules (Rule M or N) to specify the controller settings.

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Method 2. Limit the Value of τI

• Skogestad (2003) has proposed limiting the value of :

1τ min τ ,4 τ θ (12-34)I c

τI

where τ1 is the largest time constant (if there are two).

Method 3. Design the Controller for Disturbances, Rather Set-point Changes

• The desired CLTF is expressed in terms of (Y/D)d,

rather than (Y/Ysp)d

• Reference: Chen & Seborg (2002)

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11


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Consider a lag-dominant model with θ / τ 0.01:

100

100 1sG s e

s

Design three PI controllers:

a) IMC

b) IMC based on the integrator approximation in Eq. 11-33

c) IMC with Skogestad’s modification (Eq. 11-34)

τ 1c

τ 2c

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Example 3

Evaluate the three controllers by comparing their performancefor unit step changes in both set point and disturbance. Assumethat the model is perfect and that Gd(s) = G(s).


Solution

The PI controller settings are:

Controller Kc

(a) IMC 0.5 100

(b) Integrator approximation 0.556 5

(c) Skogestad 0.5 8

I

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Example 3


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Comparison of set-pointresponses (top) anddisturbance responses(bottom) for Example 11.4.The responses for theintegrator approximation andChen and Seborg(discussed in textbook)methods are essentiallyidentical.

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Example 3: Results


Controller tuning methods in the time-domain can beclassified into two groups:

(a) Techniques based on a few characteristic points of thecontrolled system’s time response.

(b) Techniques based on the entire system’s time responseor integral error criteria.

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Controller Tuning Methods


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The Figure below can be viewed as the closed-loop response of acontrolled process. Important response characteristics include

settling time, % overshoot, rise time, decay ratio, etc.

Several methods based on 1/4 decay ratio have been proposed: Cohen-Coon, Ziegler-Nichols

Approaches Based on Time Response


Second order process with PI controller can yield second order closed-loop response

PI: or

Let τI = τ1, where τ1 > τ2

Canceling terms,

11)(

21

ss

KsG

s

1sK)s(G

I

ICC

s

11K)s(G

ICC

1 2 1C

sp C

KKY

Y s s KK

Check gain (s = 0)

Closed Loop Time Response


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1 2

CK K

and1

2

1

2 CK K

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Select Kc to give 5.04.0 (overshoot)

Step response of underdamped second-order process

Second Order Time Response


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Process Reaction Curve It is based on approximation of process using FOPTD model

1. Step in U is introduced2. Observe behavior ym(t)

3. Fit a first order plus dead time model

GpGc

Gs

1/s D(s)

Y(s)

Ym(s)

Y*(s)

U(s)

Manual Control

Y sKe

sm

s( )

1


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Process Reaction Curve

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Step Test Method


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Typical Process Reaction Curves

Typical process reaction curves: (a) non-self-regulating process, (b) self-regulating process.


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0 1 2 3 4 5 6 7 8-0.2

0

0.2

0.4

0.6

0.8

1

1.2

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Process Response

4. Obtain controller parametrs from tuning correlations

Ziegler-NicholsCohen-Coon ISE, IAE or ITAE optimal tuning relations

KM


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Ziegler-Nichols Tunings Controller Kc Ti Td

P-only )/)(/1( pKPI )/)(/9.0( pK 3.3

PID )/)(/2.1( pK 0.2 5.0

- Note presence of inverse of process gain in controller gain- Introduction of integral action requires reduction in controller gain- Increase gain when derivative action is introduced

Example:

PI: Kc= 10 I=29.97

PID: Kc= 13.33 I=18 d=4.5

G se

sp

s( )

.

9 0 3

30 1


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Z-N Tunings: Servo Response

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 02 0

2 5

3 0

3 5

4 0

4 5

5 0Z -N P I

Z -N P ID

D ire c t S y n t h e s is

t

y(t)


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Z-N Tunings: Regulatory Response

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 01 0

1 5

2 0

2 5

3 0

3 5

4 0

D i r e c t S y n t h e s i s

Z - N P ID

Z - N P I

Oscillatory with considerable overshoot

Tends to be conservative


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Cohen-Coon Tuning Relations

Designed to achieve 1/4 decay ratio

Fast decrease in amplitude of oscillation

Example:

PI: Kc=10.27 τI=18.54

PID: Kc=15.64 τI=19.75 τd=3.10

Controller Kc Ti Td

P-only ]3/1)[/)(/1( pKPI ]12/9.0)[/)(/1( pK

)/(209

)]/(330[

PID]

12

163)[/)(/1(

pK)/(813

)]/(632[

)/(211

4


Cohen-Coon Tuning

Cohen-Coon: Servo

– More aggressive/ Higher controller gains

– Undesirable response for most cases

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 02 0

2 5

3 0

3 5

4 0

4 5

5 0

5 5

C - C P ID

C - C P Iy(t)

t


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Cohen-Coon Tuning

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 05

1 0

1 5

2 0

2 5

3 0

3 5

4 0

C - C P I

C - C P ID

y(t)

t

Cohen-Coon: Regulatory

– Highly oscillatory

– Very aggressive


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Controller Ziegler-Nichols Cohen-Coon

Proportional CKK 3

1

CKK

Proportional +

Integral

33.3

9.0

I

CKK

2.20.1

33.033.3

083.09.0

I

CKK

Proportional +

Integral +

Derivative

5.0

0.2

2.1

D

I

CKK

2.00.1

37.0

813

632

270.035.1

D

I

CKK

ZN and Cohen-Coon Tuning


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1. Stability margin is not quantified.

2. Control laws can be vendor - specific.

3. First order + time delay model can be inaccurate.

4. Kp, and θ can vary.

5. Resolution, measurement errors decrease stability margins.

6. ¼ decay ratio not conservative standard (too oscillatory).

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Disadvantages of Tuning Correlations


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Integral Error Relations

1. Integral of absolute error (IAE)

2. Integral of squared error (ISE)

Penalizes large errors

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

Penalizes errors that persist

ITAE is most conservative

ITAE is preferred

ISE e t dt

( )2

0

ITAE t e t dt

( )0

IAE e t dt

( )0


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Graphical Interpretation of IAE


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ITAE Relations

Choose Kc, τI and τd that minimize the ITAE:

For a first order plus dead time model, solve for:

Design for Load and Set-point changes yield different

ITAE optimum.

ITAE

K

ITAE ITAE

c I d 0 0 0, ,


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Controller Design Based on ITAE

Assuming a FOPTD Model


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ITAE Relations

From table, we get

Load Settings:

Set-point Settings:

Example:

Y A KKB

c Id

Y A KK A BB

c Id

,

Ge

sGs

s

L

0 3

30 11

9.,


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ITAE Relations

Example (contd)

Set-point Settings:

Load Settings:

KK

K K

c

c

0 965 2 6852

2 6852 2 68520 3 8 95

930

0 85. .

. .. .

.

I

I

d

d

0 796 01465 0 7520

0 752030

0 7520 39 89

0 308 930 01006

01006 30194

930

0 929

. . .

. . .

. .

. .

.

KK

K K

c

c

1357 4 2437

4 2437 4 24370 3 1415

930

0 947. .

. .. .

.

I

I

d

d

0842 2 0474

2 047430

2 0474 14 65

0 381 930 01150

01150 34497

930

0 738

0 995

. .

. . .

. .

. .

.

.


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ITAE Relations

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 02 0

2 5

3 0

3 5

4 0

4 5

5 0

5 5

6 0

IT A E ( L o a d )

I T A E ( S e t p o i n t )

Servo Response

Design for load changes yields large overshoots


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ITAE Relations

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 00

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

I T A E ( S e t p o i n t )

IT A E ( L o a d )

Regulatory response

Tuning relations are based GL=Gp

Set-point design has a good performance for this case



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Experimental determination of the ultimate gain

The Continuous Cycling Method


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Ziegler-Nichols Controller Settings

Z-N controller settings are widely considered to be an "industry standard".

Z-N settings were developed to provide 1/4 decay ratio -- too oscillatory?


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Comparison of PID Controllers


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Relay Auto-Tuning (Astrom)

Kcu = 4d/(πα)


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Continuous Cycling of Control Loop


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Disturbance Rejection

Using ZN, CC, Lopez and IAE Minimization Criteriafor Selection of Controller’s Tuning Parameters

Minimization of IAE error provides the optimum controller’s parameters Prof. Costas Kiparisides

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Summary of Tuning Relations In all correlations, controller gain should be inversely proportional

to process gain, KPKVKM .

Reduce Kc, when adding more integral action

Increase Kc, when adding derivative action

To reduce oscillation, decrease KC and increase I .

Controller gain is reduced as increases

Integral time constant and derivative constant should increase

as increases (in general, )

Ziegler-Nichols and Cohen-Coon tuning relations yield aggressive

control with oscillatory response (requires detuning)

ITAE provides conservative performance (not aggressive)

d

I 0 25.


1. Controller tuning inevitably involves a tradeoff betweenperformance and robustness.

2. Controller settings do not have to be precisely determined. Ingeneral, a small change in a controller setting from its best value(for example, ±10%) has little effect on closed-loop responses.

3. For most plants, it is not feasible to manually tune eachcontroller. Tuning is usually done by a control specialist(engineer or technician) or by a plant operator. Because eachperson is typically responsible for 300 to 1000 control loops, it isnot feasible to tune every controller.

4. Diagnostic techniques for monitoring control systemperformance are available.

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On-line Controller Tuning


Documents

Controller Design, PID Tuning and Troubleshootinglpre.cperi.certh.gr/auth/files/Process Control_Design of... · Chen and Seborg (discussed in textbook) methods are essentially identical