CHE302 LECTURE XI CONTROLLER DESIGN AND PID …€¦ · CHE302 Process Dynamics and Control Korea University 11-2 CONTROLLER DESIGN • Performance criteria for closed -loop systems

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


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∞

+ ∆∫


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)


• 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


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=


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


• 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

=

=


• 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)


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.


DESIGN RELATIONS FOR PIDCONTROLLERS

• Cohen-Coon controller design relations– Empirical relation for ¼ decay ratio for FOPDT model


• 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

∞= ∫


• 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.


• 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


• 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= ∆


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 +


• 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

+= + + + +


• 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


• 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


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


• 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= = + −%


• 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


• 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

+= = + =

+ +


IMC based PID controller settings


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


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.


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.

Documents

CHE302 LECTURE XI CONTROLLER DESIGN AND PID …€¦ · CHE302 Process Dynamics and Control Korea University 11-2 CONTROLLER DESIGN • Performance criteria for closed -loop systems