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Transcript of Controller Design, PID Tuning and Troubleshootinglpre.cperi.certh.gr/auth/files/Process...
2/12/2014
1
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
Prof. Costas Kiparisides
2/12/2014
2
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
Prof. Costas Kiparisides
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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
Prof. Costas Kiparisides
2/12/2014
3
Ch
apte
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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
Prof. Costas Kiparisides
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)
Ch
apte
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Alternatives for Controller Design
Prof. Costas Kiparisides
2/12/2014
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Direct Synthesis
Internal Model Control
Prof. Costas Kiparisides
Standard Feedback Control System
Prof. Costas Kiparisides
2/12/2014
5
( 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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apte
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(11-3b)
G
Direct Synthesis Method
Prof. Costas Kiparisides
(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-
Prof. Costas Kiparisides
2/12/2014
6
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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apte
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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
Prof. Costas Kiparisides
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
Prof. Costas Kiparisides
2/12/2014
7
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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apte
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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-
Prof. Costas Kiparisides
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
Ch
apte
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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
Prof. Costas Kiparisides
2/12/2014
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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
Prof. Costas Kiparisides
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
Prof. Costas Kiparisides
2/12/2014
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Simulation Results
b. Incorrect model gain
Prof. Costas Kiparisides
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Internal Model Control (IMC)
Prof. Costas Kiparisides
2/12/2014
10
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apte
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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
Prof. Costas Kiparisides
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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
Prof. Costas Kiparisides
2/12/2014
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Ch
apte
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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
*
Prof. Costas Kiparisides
Ch
apte
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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( )
Prof. Costas Kiparisides
2/12/2014
12
IMC-Based PID Controller Settings
Prof. Costas Kiparisides
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
Prof. Costas Kiparisides
2/12/2014
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apte
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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
Prof. Costas Kiparisides
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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
Prof. Costas Kiparisides
2/12/2014
14
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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apte
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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
Prof. Costas Kiparisides
• 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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apte
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Tuning for Lag Dominant Models
Prof. Costas Kiparisides
2/12/2014
15
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.
Ch
apte
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Tuning for Lag Dominant Models
Prof. Costas Kiparisides
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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Tuning for Lag Dominant Models
11
Prof. Costas Kiparisides
2/12/2014
16
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
τ 1c Ch
apte
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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).
Prof. Costas Kiparisides
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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apte
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Example 3
Prof. Costas Kiparisides
2/12/2014
17
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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apte
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Example 3: Results
Prof. Costas Kiparisides
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.
Ch
apte
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Controller Tuning Methods
Prof. Costas Kiparisides
2/12/2014
18
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
Prof. Costas Kiparisides
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
Prof. Costas Kiparisides
2/12/2014
19
1 2
CK K
and1
2
1
2 CK K
Ch
apte
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Select Kc to give 5.04.0 (overshoot)
Step response of underdamped second-order process
Second Order Time Response
Prof. Costas Kiparisides
Ch
apte
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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
Prof. Costas Kiparisides
2/12/2014
20
Process Reaction Curve
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apte
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Step Test Method
Prof. Costas Kiparisides
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apte
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Typical Process Reaction Curves
Typical process reaction curves: (a) non-self-regulating process, (b) self-regulating process.
Prof. Costas Kiparisides
2/12/2014
21
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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apte
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Process Response
4. Obtain controller parametrs from tuning correlations
Ziegler-NicholsCohen-Coon ISE, IAE or ITAE optimal tuning relations
KM
Prof. Costas Kiparisides
Ch
apte
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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
Prof. Costas Kiparisides
2/12/2014
22
Ch
apte
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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)
Prof. Costas Kiparisides
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apte
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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
Prof. Costas Kiparisides
2/12/2014
23
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
Prof. Costas Kiparisides
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
Prof. Costas Kiparisides
2/12/2014
24
Ch
apte
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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
Prof. Costas Kiparisides
Ch
apte
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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
Prof. Costas Kiparisides
2/12/2014
25
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).
Ch
apte
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Disadvantages of Tuning Correlations
Prof. Costas Kiparisides
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apte
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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
Prof. Costas Kiparisides
2/12/2014
26
Graphical Interpretation of IAE
Prof. Costas Kiparisides
Ch
apte
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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, ,
Prof. Costas Kiparisides
2/12/2014
27
Controller Design Based on ITAE
Assuming a FOPTD Model
Prof. Costas Kiparisides
Ch
apte
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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.,
Prof. Costas Kiparisides
2/12/2014
28
Ch
apte
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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
. .
. . .
. .
. .
.
.
Prof. Costas Kiparisides
Ch
apte
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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
Prof. Costas Kiparisides
2/12/2014
29
Ch
apte
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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
Prof. Costas Kiparisides
Prof. Costas Kiparisides
2/12/2014
30
Ch
apte
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Experimental determination of the ultimate gain
The Continuous Cycling Method
Prof. Costas Kiparisides
Ch
apte
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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?
Prof. Costas Kiparisides
2/12/2014
31
Ch
apte
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Comparison of PID Controllers
Prof. Costas Kiparisides
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apte
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Relay Auto-Tuning (Astrom)
Kcu = 4d/(πα)
Prof. Costas Kiparisides
2/12/2014
32
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apte
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Continuous Cycling of Control Loop
Prof. Costas Kiparisides
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apte
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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
2/12/2014
33
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.
Prof. Costas Kiparisides
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.
Ch
apte
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On-line Controller Tuning
Prof. Costas Kiparisides