Upload
others
View
15
Download
0
Embed Size (px)
Citation preview
CHE302 Process Dynamics and Control Korea University 11-1
CHE302 LECTURE XICONTROLLER DESIGN AND PID
CONTOLLER TUNING
Professor Dae Ryook Yang
Fall 2001Dept. of Chemical and Biological Engineering
Korea University
CHE302 Process Dynamics and Control Korea University 11-2
CONTROLLER DESIGN
• Performance criteria for closed-loop systems– Stable– Minimal effect of disturbance– Rapid, smooth response to set point change– No offset– No excessive control action– Robust to plant-model mismatch
• Trade-offs in control problems– Set point tracking vs. disturbance rejection– Robustness vs. performance
2 21 20, ,
min ( ( ) ( ))c I DK
w e w u dτ τ
τ τ τ∞
+ ∆∫
CHE302 Process Dynamics and Control Korea University 11-3
GUIDELINES FOR COMMON CONTROLLOOPS
• Flow and liquid pressure control– Fast response with no time delay– Usually with small high-frequency noise– PI controller with intermediate controller gain
• Liquid level control– Noisy due to splashing and turbulence– High gain PI controller for integrating process– Conservative setting for averaging control when it is used for
damping the fluctuation of the inlet stream
• Gas pressure control– Usually fast and self regulating– PI controller with small integral action (large reset time)
CHE302 Process Dynamics and Control Korea University 11-4
• Temperature control– Wide variety of the process nature– Usually slow response with time delay– Use PID controller to speed up the response
• Composition control– Similar to temperature control usually with larger noise and
more time delay– Effectiveness of derivative action is limited– Temperature and composition controls are the prime
candidates for advance control strategies due to its importanceand difficulty of control
CHE302 Process Dynamics and Control Korea University 11-5
TRIAL AND ERROR TUNING
• Step1: With P-only controller– Start with low Kc value and increase it until the response has a
sustained oscillation (continuous cycling) for a small set pointor load change. (Kcu)
– Set Kc = Kcu.
• Step2: Add I mode– Decrease the reset time until sustained oscillation occurs. ( )– Set .– If a further improvement is required, proceed to Step 3.
• Step3: Add D mode– Decrease the reset time until sustained oscillation occurs. ( )– Set .
(The sustained oscillation should not be cause by the controller saturation)
Iuτ3I Iuτ τ=
Duτ3D Duτ τ=
CHE302 Process Dynamics and Control Korea University 11-6
CONTINUOUS CYCLING METHOD
• Also called as loop tuning or ultimate gain method– Increase controller gain until sustained oscillation– Find ultimate gain (KCU) and ultimate period (PCU)
• Ziegler-Nichols controller setting– ¼ decay ratio (too much oscillatory)
– Modified Ziegler-Nichols setting
0.5PCU/8PCU /20.6KCUPID
-PCU /1.20.45KCUPI
--0.5KCUP
KCController
PCU/3PCU /20.2KCUNo overshoot
PCU/3PCU /20.33KCUSome overshoot
PCU/8PCU /20.6KCUOriginal
KCController
Iτ Dτ
DτIτ
CHE302 Process Dynamics and Control Korea University 11-7
• Examples
4.06.00.19No overshoot
4.06.00.31Some overshoot
1.56.00.57Original
KCController DτIτ
3.875.81.58No overshoot
3.875.82.60Some overshoot
1.455.84.73Original
KCController DτIτ
3.54( )
7 1
s
pe
G ss
−
=+
2( )
(10 1)(5 1)
s
pe
G ss s
−
=+ +
0.95
12CU
CU
K
P
=
=
7.88
11.6CU
CU
K
P
=
=
CHE302 Process Dynamics and Control Korea University 11-8
• Advantages of continuous cycling method– No a priori information on process required– Applicable to all stable processes
• Disadvantages of continuous cycling method– Time consuming– Loss of product quality and productivity during the tests– Continuous cycling may cause the violation of process
limitation and safety hazards– Not applicable to open-loop unstable process– First-order and second-order process without time delay will
not oscillate even with very large controller gain
=> Motivates Relay feedback method. (Astrom and Wittenmark)
CHE302 Process Dynamics and Control Korea University 11-9
RELAY FEEDBACK METHOD
• Relay feedback controller– Forces the system to oscillate by a relay controller– Require a single closed-loop experiment to find the ultimate
frequency information– No a priori information on process is required– Switch relay feedback controller for tuning– Find PCU and calculate KCU
– User specified parameter: d
– Use Ziegler-Nichols Tuning rules for PID tuning parameters
4CU
dK
aπ=
Decide d in order not to perturb thesystem too much.
CHE302 Process Dynamics and Control Korea University 11-10
DESIGN RELATIONS FOR PIDCONTROLLERS
• Cohen-Coon controller design relations– Empirical relation for ¼ decay ratio for FOPDT model
CHE302 Process Dynamics and Control Korea University 11-11
• Design relations based on integral error criteria– ¼ decay ratio is too oscillatory– Decay ratio concerns only two peak points of the response– IAE: Integral of the Absolute Error
– ISE: Integral of the Square Error
• Large error contributes more• Small error contributes less• Large penalty for large overshoot• Small penalty for small persisting oscillation
– ITAE: Integral of the Time-weighted Absolute Error
• Large penalty for persisting oscillation• Small penalty for initial transient response
0IAE ( )e t dt
∞= ∫
[ ]2
0ISE ( )e t dt
∞= ∫
IAE
0ITAE ( )t e t dt
∞= ∫
CHE302 Process Dynamics and Control Korea University 11-12
• Controller design relation based on ITAE forFOPDT model
• Similar design relations based on IAE and ISE forother types of models can be found in literatures.
CHE302 Process Dynamics and Control Korea University 11-13
• Example1 Example210( )2 1
seG ss
−
=+
0.977(0.859)(1/2) 1.690.169
c
c
KKK
−= =⇒ =
PI
0.680/ (0.674)(1/2) 1.081.85
I
I
τ ττ
−= =⇒ =
1.850.169ITAE
2.440.245ISE
2.020.195IAE
KcMethod Iτ
3.54( )7 1
seG ss
−
=+
Mostoscillatory
Trade-offs
CHE302 Process Dynamics and Control Korea University 11-14
• Design relations based on process reaction curve– For the processes who have sigmoidal shape step responses
(Not for underdamped processes)– Fit the curve with FOPDT model
– Very simple– Inherits all the problems of FOPDT model fitting
* / /S S u K τ= ∆ =( )( 1)
sKeG s
s
θ
τ
−
=+
/S K u τ= ∆
CHE302 Process Dynamics and Control Korea University 11-15
DIRECT SYNTHESIS METHOD
• Analysis: Given Gc(s), what is y(t)?• Design: Given yd(t), what should Gc(s) be?• Derivation
– If (Y/R)d = 1, then it implies perfect control. (infinite gain)– The resulting controller may not be physically realizable– Or, not in PID form and too complicated.– Design with finite settling time:
( ) 1 /( ) 1 1 1 /
OL cc
OL c
G G GY s Y RG
R s G G G G Y R = = ⇒ = + + −
Let OL m c v p cG K G G G G G= @
( / )1Specify ( / )
1 ( / )d
d cd
Y RY R G
G Y R
⇒ = −
c
1( / ) =
1dY Rsτ +
CHE302 Process Dynamics and Control Korea University 11-16
• Examples1. Perfect control (Kc becomes infinite)
2. Finite settling time for 1st-order process
3. Finite settling time for 2nd-order process
1 2
( ) and ( / ) 1( 1)( 1) d
KG s Y Rs sτ τ
= =+ +
1 1( ) (infinite gain, unrealizable)
( ) 1-1 ( )cG sG s G s
∞ = =
1( ) and ( / )
( 1) 1dc
KG s Y R
s sτ τ= =
+ +
1/( 1)1 1 1( ) 1 (PI)
( ) 1-1/( 1)c
cc c c
s sG s
G s s K s K sτ τ ττ τ τ τ
+ + = = = + +
1 2
1( ) and ( / )
( 1)( 1) 1dc
KG s Y R
s s sτ τ τ= =
+ + +
1 2 1 2
1 2 1 2
( ) 1( ) 1 (PID)
( ) ( )cc
G s sK s
τ τ τ ττ τ τ τ τ
+= + + + +
CHE302 Process Dynamics and Control Korea University 11-17
• Process with time delay– If there is a time delay, any physically realizable controller
cannot overcome the time delay. (Need time lead)– Given circumstance, a reasonable choice will be
– Examples1.
2. With 1st-order Taylor series approx. ( )
3.
( )/1
c s
dc
eY R
s
θ
τ
−
=+
( ) and ( / ) ( )( 1) 1
s s
d cc
Ke eG s Y R
s s
θ θ
θ θτ τ
− −
= = =+ +
/( 1)1 1 1( ) (not a PID)
( ) 1- /( 1) 1-
sc
c s sc c
e s sG s
G s e s K s e
θ
θ θ
τ ττ τ
−
− −
+ += = + +
1se sθ θ− ≈ −1 1 1
( ) 1 (PI)( ) ( )c
c c
sG s
K s K sτ τ
τ θ τ θ τ+ = = + + +
1 2
( ) and ( / ) ( )( 1)( 1) 1
s s
d cc
Ke eG s Y R
s s s
θ θ
θ θτ τ τ
− −
= = =+ + +
1 2 1 2 1 2
1 2 1 2
( 1)( 1) 1 ( ) 1( ) = 1 (PID)
( ) ( ) ( ) ( )cc c
s sG s s
K s K sτ τ τ τ τ τ
τ θ τ θ τ τ τ τ + + +
= + + + + + +
Physicallyrealizable
CHE302 Process Dynamics and Control Korea University 11-18
• Observations on Direct Synthesis Method– Resulting controllers could be quite complex and may not even
be physically realizable.– PID parameters will be decided by a user-specified parameter:
The desired closed-loop time constant ( )– The shorter makes the action more aggressive. (larger Kc)– The longer makes the action more conservative. (smaller Kc)– For a limited cases, it results PID form.
• 1st-order model without time delay: PI• FOPDT with 1st-order Taylor series approx.: PI• 2nd-order model without time delay: PID• SOPDT with 1st-order Taylor series approx.: PID• Delay modifies the Kc.
• With time delay, the Kc will not become infinite even for theperfect control (Y/R=1).
(1st order)( )c cK K
τ ττ τ θ
→+
1 2 1 2( ) ( )(2nd order)
( )c cK Kτ τ τ τ
τ τ θ+ +
→+
cτ
cτ
cτ
CHE302 Process Dynamics and Control Korea University 11-19
INTERNAL MODEL CONTROL (IMC)
• Motivation– The resulting controller from direct synthesis method may not
be physically unrealizable.– If there is RHP zero in the process, the resulting controller
from direct synthesis method will be unstable.– Unmeasured disturbance and modeling error are not
considered in direct synthesis method.
• Source of trouble– From direct synthesis method
( / )11 ( / )
dc
d
Y RG
G Y R
= −
Direct inversion of processcauses many problems
Resulting controller may havehigher-order numerator thandenominator
Process is unknown
CHE302 Process Dynamics and Control Korea University 11-20
• IMC– Feedback the error between the process output and model
output.– Equivalent conventional controller:
– Using block diagram algebra* ( )cC GP L P G E E R C C R C GP= + = = − − = − +% %
*
* *
( )
( ) /(1 )
c
c c
P G R C GP
P G R C G G
= − +
⇒ = − −
%%
* *( ) /(1 )c cC GG R C G G L= − − +%* * * *(1 ) (1 )c c c cGG G G C GG R G G L+ − = + −% %
* *
* *
(1 )1 ( ) 1 ( )
c c
c c
G G G GC R L
G G G G G G−
= ++ − + −
%% %
*
*1c
cc
GG
G G=
− %
* *If , (1 )c cG G C G GR G G L= = + −%
CHE302 Process Dynamics and Control Korea University 11-21
• IMC design strategy– Factor the process model as
• contains any time delays and RHP zeros and is specified sothat the steady-state gain is one
• is the rest of G.– The controller is specified as
• IMC filter f is a low-pass filter with steady-state gain of one• Typical IMC filter:
• The is the desired closed-loop time constant and parameter r isa positive integer that is selected so that the order of numerator ofGc
* is same as the order of denominator or exceeds the order ofdenominator by one.
G G G+ −=% %%G+%
G−%
* 1cG f
G −= %
1( 1)r
cf
sτ=
+cτ
Uninvertibles
CHE302 Process Dynamics and Control Korea University 11-22
• Example– FOPDT model with 1/1 Pade approximation
(1 / 2)(1 / 2)( 1)
K sG
s sθ
θ τ−
=+ +
%
1 / 2 (1 / 2)( 1)
KG s G
s sθ
θ τ+ −= − =
+ +% %
* 1 (1 / 2)( 1) 1( 1)
cc
s sG f
K sGθ τ
τ−
+ += =
+%*
*
(1 / 2)( 1) (PID)( / 2)1
cc
cc
G s sGK sG G
θ ττ θ
+ += =+− %
1 ( / 2) / 2/ 2
( / 2) / 2c I D
cK
Kτ θ τθ
τ τ θ ττ θ τ θ
+= = + =
+ +
CHE302 Process Dynamics and Control Korea University 11-23
IMC based PID controller settings
CHE302 Process Dynamics and Control Korea University 11-24
COMPARISON OF CONTROLLER DESIGNRELATIONS
• PI controller settings for different methods
2( )
1
seG s
s
−
=+
No modeling error 50% error in process gain
best
Somewhat robust
CHE302 Process Dynamics and Control Korea University 11-25
EFFECT OF MODELING ERROR
• Actual plant
• Approx. model
– Satisfactory for this case– Use with care
• All kinds of tuning method should be used forinitial setting and fine tuning should be done!!
2( )
(10 1)(5 1)
seG s
s s
−
=+ +
4.72( )
12 1
seG s
s
−
=+
%
As the estimated time delaygets smaller, the performancedegradation will be pronounced.
CHE302 Process Dynamics and Control Korea University 11-26
GENERAL CONCLUSION FOR PID TUNING
• The controller gain should be inversely proportional to theproducts of the other gains in the feedback loop.
• The controller gain should decrease as the ratio of timedelay to dominant time constant increases.
• The larger the ratio of time delay to dominant time constantis, the harder the system is to control.
• The reset time and the derivative time should increase asthe ratio of time delay to dominant time constant increases.
• The ratio between derivative time and reset time is typicallybetween 0.1 to 0.3.
• The ¼ decay ratio is too oscillatory for process control. Ifless oscillatory response is desired, the controller gainshould decrease and reset time should increase.
• Among IAE, ISE and ITAE, ITAE is the most conservativeand ISE is the least conservative setting.