CHE302 LECTURE VIII DYNAMIC BEHAVIORS OF … LECTURE VIII DYNAMIC BEHAVIORS OF CLOSED- ... control 1. Symbol size may vary according to the user's needs and ... Control Engineering

CHE302 Process Dynamics and Control Korea University 8-1

CHE302 LECTURE VIIIDYNAMIC BEHAVIORS OF CLOSED-

LOOP CONTOL SYSTEMS

Professor Dae Ryook Yang

Fall 2001Dept. of Chemical and Biological Engineering

Korea University


BLOCK DIAGRAM REPRESENTATION

• Standard block diagram of a feedback control system

– Process TF: MV (M) effect on CV (X2, part of Y)– Load TF: DV (L) effect on CV (X1, part of Y)– Sensor TF: CV (Y) is transferred to measurement (B) – Actuator TF: Controller output (P) is transferred to MV (M)– Controller TF: Controller output (P) is calculated based on error (E)– Calibration TF: Gain of sensor TF, used to match the actual var.

Gp(s)

GL(s)

Gv(s)Gc(s)

Gm(s)

Km(s)MP Y

B

E

L

R X2

X1

%R

ProcessActuatorController

Sensor

Calibration

Load

+

+-+

Comparator


• Individual TF of the standard block diagram– TF of each block between input and output of that block– Each gain will have different unit.

• [Example] Sensor TF• Input range: 0-50 l/min• Output range: 4-20 mA

• Dynamics: usually 1st order with small time constant

– Block diagram shows the flow of signal and the connections– Schematic diagram shows the physical components connection

Gm(s)Y B

[l/min] [mA]

20 4Gain, 0.32 [mA/(l/min)]

50 0mK−

= =−

Electrical signal

Pneumatic signal

TT Temperature Transmitter

FC Flow Controller

LI Level Indicator

( )1

mm

m

KG s

sτ=

+


P&ID

• Piping and Instrumentation diagram– A P&ID is a blueprint, or map, of a process.– Technicians use P&IDs the same way an architect uses

blueprints.– A P&ID shows each of the instruments in a process , their

functions, their relationship to other components in the system.– Most diagrams use a standard format, such as the one

developed by ISA (Instrumental Society of America) or SAMA (Scientific Apparatus Makers Association).


Common connecting lines

Connection to process, or instrument supply:

Pneumatic signal:

Electric signal:

Capillary tubing (filled system):

Hydraulic signal:

Electromagnetic or sonic signal (guided):

Internal system link (software or data link):

Source: Control Engineering with data from ISA S5.1 standard


General instrument or function symbols

Primary location accessible to operator

Field mounted Auxiliary location accessible to operator

Discrete instruments

Shared display, shared control

Computer function

Programmable logic control

1. Symbol size may vary according to the user's needs and the type of document.2. Abbreviations of the user's choice may be used when necessary to specify location.3. Inaccessible (behind the panel) devices may be depicted using the same symbol but with a dashed horizontal bar.Source: Control Engineering with data from ISA S5.1 standard


Identification lettersFirst letter Succeeding letters

Measured or initiating var. Modifier Readout or passive func. Output function Modifier

A Analysis Alarm

B Burner, combustion User's choice User's choice User's choice

C User's choice Control

D User's choice Differential

E Voltage Sensor (primary element)

F Flow rate Ration (fraction)

G User's choice Glass, viewing device

H Hand High

I Current (electrical) Indication

J Power Scan

K Time, time schedule Time rate of change

Control station

L Level Light Low

M User's choice Momentary Middle, interm.

N User's choice User's choice User's choice User's choice

O User's choice Orifice, restriction

P Pressure, vacuum Point (test connection)

Q Quantity Integrate, totalizer

R Radiation Record

S Speed, frequency Safety Switch

T Temperature Transmit

U Multivariable Multifunction Multifunction Multifunction

V Vibration, mechanical analysis Valve, damper, louver

W Weight, force Well

X Unclassified X axis Unclassified Unclassified Unclassified

Y Event, state, or presence Y axis Relay,compute,convert

Z Position, dimension Z axis Driver, actuator

Source: Control Engineering with data from ISA S5.1 standard




CLOSED LOOP TRANSFER FUNCTION

• Block diagram algebra

• Transfer functions of closed-loop system

X1(s) X2(s) Xn(s)LU(s) Y(s)2 1

( ) ( ) ( ) ( )( ) n

Y s X s X s X sU s

= L

X1(s)

X2(s)

U1(s)

U2(s)Y(s)

1 1 2 2( ) ( ) ( ) ( ) ( )Y s X s U s X s U s= +

2 ( ) ( ) ( ) ( ) ( )p v cX s G s G s G s E s= ( ) ( ) ( ) ( ) ( )m mE s K s R s G s Y s= −

2( ) ( ) ( ) ( )LY s G s L s X s= + ( ) ( ) ( ) ( ) ( ) ( ) ( )L p v cY s G s L s G s G s G s E s= +

( ) ( ) ( ) ( ) ( ) ( )( ( ) ( ) ( ) ( ))L p v c m mY s G s L s G s G s G s K s R s G s Y s= + −

(1 ( ) ( ) ( ) ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( )m p v c L m p v cG s G s G s G s Y s G s L s K G s G s G s R s+ = +


• For set-point change (L=0)

• For load change (R=0)

• Open-loop transfer function (GOL)

– Feedforward path: Path with no connection backward– Feedback path: Path with circular connection loop– GOL: feedback loop is broken before the comparator

• Simultaneous change of set point and load

( ) ( ) ( )( )( ) 1 ( ) ( ) ( ) ( )

m p v c

m p v c

K G s G s G sY sR s G s G s G s G s

=+

( ) ( )( ) 1 ( ) ( ) ( ) ( )

L

m p v c

Y s G sL s G s G s G s G s

=+

( ) ( ) ( ) ( ) ( )OL m p v cG s G s G s G s G s@

( ) ( ) ( ) ( )( ) ( ) ( )

1 ( ) 1 ( )m p v c L

OL OL

K G s G s G s G sY s R s L s

G s G s= +

+ +


MASON’S RULE

• General expression for feedback control systems

– Assume feedback loop has negative feedback.– If it has positive feedback, should be . – In the previous example, for set-point change

– For load change,

1f

e

YX

π

π=

+

product of the transfer function in the path from X to Y

product of all transfer function in the entire feedback loopf

e

π

π

≡

≡

1 eπ+ 1 eπ−

( ) ( ) ( )f m c v pK G s G s G sπ = X R Y Y= = ( )e OLG sπ =

( ) ( ) ( )( )( ) 1 ( )

m p v c

OL

K G s G s G sY sR s G s

=+

X L Y Y= = ( )f LG sπ = ( )e OLG sπ =


• Example 1

– Inner loop:

– TF between R and Y:

– TF between L1 and Y:

Gc2 G1 G2 G3Km1

Gm2

Gc1

Gm1

R Y

L2L1

+ + + +++

--

Inner loop

X2

X1

1 22 1

2 1 21c

m c

G GX X

G G G=

+

1 21 3 2 1

2 1 21c

f m cm c

G GK G G G

G G Gπ =

+1 2

1 3 2 12 1 21

ce m c

m c

G GG G G G

G G Gπ =

+

1 3 2 1 2 1

2 1 2 1 3 2 1 2 11m c c

m c m c c

K G G G G GYR G G G G G G G G G

=+ +

3 2

1 2 1 2 1 3 2 1 2 11 m c m c c

G GYL G G G G G G G G G

=+ +


• Example 2

– Inner loop:

– TF between R and Y:

– TF between L and Y:

1 2 1 2( )E R G G M R G M G M= − − = − +

G1

G2

GcR

L

+ +

+

+

--

YE M

121c

fc

GG

G Gπ =

−

121c

c

GM X

G G=

−

R +

-

+ G1Gc

L

+

+

+ Y

G2

MEInner loop

X1

121c

ec

GG

G Gπ =

−

1 1

2 1 1 21 1 ( )c c

c c c

G G G GYR G G G G G G G

= =− + + −

2 2

2 1 1 2

1 11 1 ( )

c c

c c c

G G G GYL G G G G G G G

− −= =− + + −


PID CONTROLLER REVISITED

• P control

• PI control

• PID control

– Ideal PID controller: Physically unrealizable– Modified form has to be used.

( )( ) ( )

( )c cP s

p t p K e t KE s

= + → =L

0

1 ( ) 1 ( 1)( ) ( ) ( ) (1 )

( )

t Ic c c

I I I

P s sp t p K e t e d K K

E s s sτ

τ ττ τ τ

+= + + → = + =

∫ L

0

2

1( ) ( ) ( )

( ) 1 ( 1) (1 )

( )

t

c DI

I D Ic D c

I I

dep t p K e t e d

dt

P s s sK s K

E s s s

τ τ ττ

τ τ ττ

τ τ

= + + + →

+ +

= + + =

∫ L


• Nonideal PID controller– Interacting type

– Comparison with ideal PID except filter

– These types are physically realizable and the modification provides the prefiltering of the error signal.

– Generally, and typically .– In this form, is satisfied automatically since algebraic

mean is not less than logarithm mean.

2( 1)( ) I D I

c cI

s sG s K

sτ τ τ

τ+ +

=

* ** *

* *

( 1) ( 1)( ) (0 1)

( 1)I D

c cI D

s sG s K

s sτ τ

βτ βτ

+ += <

+=

* * ** * 2 * * * **

* * * * * *

( )( ( ) 1) 1 11

( ) ( )c I DD I I D D I

cI I I D I D

Ks sK s

s sτ ττ τ τ τ τ τ

τ τ τ τ τ τ ++ + +

= + + + + * * * * *

* ** * *

( ), ,

( )c I D D I

c I I D DI I D

KK

τ τ τ ττ τ τ τ

τ τ τ+

= = + =+

Filtering effect

I Dτ τ≥I Dτ τ≥ 4I Dτ τ≈


• Block diagram of PID controller– Nonideal interacting type PID

– Removal of derivative kick (PI-D controller)

– Removal of both P & D kicks (I-PD controller)

Kc

E(s)1

11I sτ +

11

D

D

ss

τβτ

++

P(s)++

PI control

1 1 11 1 11

1

I I

I I

I

s ss s

s

τ ττ τ

τ

+ += =

+ −−+

( 1) ( 1)( ) ( )

( 1)I D

cI D

s sP s K E s

s sτ τ

τ βτ+ +

=+

( 1) ( 1) ( 1)( ) ( ) ( )

( 1)I D I

cI D I

s s sP s K Y s R s

s s sτ τ τ

τ βτ τ + + +

= − +

( 1) ( 1) 1( ) ( ) ( )

( 1)I D

cI D I

s sP s K Y s R s

s s sτ τ

τ βτ τ + +

= − +


• Other variations of PID controller– Gain scheduling : modifying proportional gain

where

– Nonlinear PID controller• Replace e(t) with e(t) | e(t) |.• Sign of error will be preserved but small error gets smaller and

larger error gets larger.• It imposes less action for a small error.

GS GSc cK K K=

for (lower gap) ( ) (upper gap)1.

1 otherwise GapGS K e t

K≤ ≤

=

2. 1 ( )GSGSK C e t= +

3. is decided based on some strategyGSK


DIGITAL PID CONTROLLER

• Discrete time system– Measurements and actions are taken at every sampling

interval.– An action will be hold during the sampling interval.

• Digital PID controller– using

00

1

( ) ( ) (Rectangular rule)

( ) ( )( ) (Backward difference approx.)

nnt

ii

n n

e d t e t

e t e tde tdt t

τ τ=

−

= ∆

−=

∆

∑∫

1

0

( ) ( )( ) ( ) ( ) (Position form)

nn n

n c n i DiI

e t e ttp t p K e t e t

tτ

τ−

=

−∆= + + + ∆ ∑

1

1 21

( ) ( ) ( )

( ) 2 ( ) ( ) = ( ) ( ) ( ) (Velocity form)

n n n

n n nc n n n D

I

p t p t p t

e t e t e ttK e t e t e t

tτ

τ

−

− −−

∆ = −

− +∆− + + ∆


– Most modern PID controllers are manufactured in digital form with short sampling time.

– If the sampling time is small, there is not much difference between continuous and digital forms.

– Velocity form does not have reset windup problem because there is no summation (integration).

– Other approximation such as trapezoidal rule and etc. can be used to enhance the accuracy. But the improvement is not substantial.

• For discrete time system, z-transform is the counterpart of Laplace transform. (out of scope of this lecture)

1

01

1 2 3

( ) ( )( ) (Trapezoidal rule)

2( ) 3 ( ) 3 ( ) ( )( )

(Interpolation formula)

nnt

i i

i

n n n n

e t e te d t

e t e t e t e tde tdt t

τ τ −

=

− − −

−= ∆

+ − −=

∆

∑∫


CLOSED-LOOP RESPONSE OF 1ST ORDER SYSTEM

• Process

– Assume

1 2dh h

A q qdt R

ρ ρ ρ ρ= + −

2

( )( )

( ) 1 1p

pH s R K

G sQ s RAs sτ

= = =+ +

1

( )( )

( ) 1 1p

LH s R K

G sQ s RAs sτ

= = =+ +

GPGcR

Q1

+ ++

-

HE Q2

GL

( ) ( ) 1v mG s G s= =

( ) ( ) ( )1 1

c p L

c p c p

G G GH s R s L s

G G G G= +

+ +

MVDV

CV

SP

R

Sensor and actuator dynamics are fast enough to be ignored and gains are lumped in other TF.


• P control for set-point change (L=0)

– Closed-loop gain and time constant

– Steady-state behavior of closed-loop system

( ) ( >0)c c cG s K K=

/( 1) /(1 )( )( ) (closed-loop TF)

( ) 1 /( 1) ( /(1 )) 1c p c p c p

CLc p c p

K K s K K K KH sG s

R s K K s K K s

ττ τ

+ += = =

+ + + +

, (1 ) (1 )

c pCL CL

c p c p

K KK

K K K Kτ

τ= =+ +

1, lim 1 ( ( ) ( ), no offset)(1 ) c

c pCL CLK

c p

K KK G H s R s

K K →∞= < = =

+

Closed-loop response will not reach to set point (offset)

Infinite controller gain will eliminate the offset

Higher controller gain results faster closed-loop response: shorter time constant

1Steady-state offset = ( ) ( ) 1

1CLc p

r h KK K

∞ − ∞ = − =+


• P control for load change (R=0)



( ) ( >0)c c cG s K K=

/( 1) /(1 )( )( ) (closed-loop TF)

( ) 1 /( 1) ( /(1 )) 1p p c p

CLc p c p

K s K K KH sG s

L s K K s K K s

ττ τ+ +

= = =+ + + +

, (1 ) (1 )

pCL CL

c p c p

KK

K K K Kτ

τ= =+ +

0, lim 0 (disturbance is compensated)(1 ) c

pCL CLK

c p

KK G

K K →∞= > =

+

Disturbance effect will not be eliminated completely (offset)

Infinite controller gain will eliminate the offset

Higher controller gain results faster closed-loop response: shorter time constant

Steady-state offset = 0 ( ) 01

pCL

c p

Kh K

K K− ∞ = − = −

+


• PI control for load change (R=0)

– Closed-loop gain, time constant, damping coefficient


( ) ( 1) / ( >0)c c I I cG s K s s Kτ τ= +

2

/( 1)( )

1 ( 1) /( 1) / (1 )p p I

CLc p I I I I c p c p

K s K sG s

K K s s s s K K s K K

τ ττ τ τ τ τ τ

+= =

+ + + + + +

(1 )1, , /

2c pI I

CL CL CL Ic c p c p

K KK

K K K K Kτ ττ

τ ζ τ τ+

= = =

0lim ( ) 0 (disturbance is compensated for all cases)CLs

G s→

=

IτcK


– As Kc increases, faster compensation of disturbance and less oscillatory response can be achieved.

– As decreases, faster compensation of disturbance and less overshooting response can be achieved.

– However, usually the response gets more oscillation as Kcincreases or decreases. => very unusual!!

– If there is small lag in sensor/actuator TF or time delay in process TF, the system becomes higher order and these anomalous results will not occur. These results is only possiblefor very simple process such as 1st order system.

– Usual effect of PID tuning parameters• As Kc increases, the response will be faster, more oscillatory.• As decreases, the response will be faster, more oscillatory.• As increases, the response will be faster, less oscillatory when

there is no noise.

Iτ

Iτ

Iτ

Dτ


CLOSED-LOOP RESPONSE OF INTEGRATING SYSTEM

• Process

– Assume

1 2 3( )dh

A q q qdt

ρ ρ ρ= + −

3

( ) 1( )

( )p

H sG s

Q s As= = −

1

( ) 1( )

( )L

H sG s

Q s As= =

GPGcR

Q1 +Q2

+ ++

-

HE Q3

GL

( ) ( ) 1v mG s G s= =

( ) ( ) ( )1 1

c p L

c p c p

G G GH s R s L s

G G G G= +

+ +

Sensor and actuator dynamics are fast enough to be ignored and gains are lumped in other TF.

MV

DV

CV

SPR

DV


• P control for set-point change (L=0)



– It is very unique that the integrating system will not have offset even with P control for the set point change.

– Even though there are other dynamics in sensor or actuator, the offset will not be shown with P control for integrating systems.

– Higher controller gain results faster closed-loop response: shorter time constant

( ) ( <0)c c cG s K K=/( )( ) 1

( ) (closed-loop TF)( ) 1 /( ) ( / ) 1

cCL

c c

K AsH sG s

R s K As A K s−

= = =+ − − +

1, /CL CL cK A Kτ= = −

no offset ev1 ( ( ) en wit( ), h p con ol)trCLK H s R s= =


• P control for load change (R=0)



– Higher controller gain results faster closed-loop response: shorter time constant

( ) ( <0)c c cG s K K=

1/( ) 1/( )( ) (closed-loop TF)

( ) 1 /( ) ( / ) 1c

CLc c

KH s AsG s

L s K As A K s−

= = =+ − − +

( 1/ ), /CL c CL cK K A Kτ= − = −

10, lim 0 (disturbance is compensated)

( ) cCL CLK

c

K GK →∞

= > =−


• PI control for set-point change (L=0)

– Closed-loop gain, time constant, damping coefficient


– As (-Kc) increases, closed-loop time constant gets smaller (faster response) and less oscillatory response can be achieved.

– As decreases, closed-loop time constant gets smaller (faster response) and more oscillatory response can be achieved.

– Partly anomalous results due to integrating nature– For integrating system, the effect of tuning parameters can be

different. Thus, rules of thumb cannot be applied blindly.

( ) ( 1) / ( <0)c c I I cG s K s s Kτ τ= +

2

( 1)/( ) / ( 1)( )

1 ( 1) /( ) / ( / ) 1c I I I

CLc I I I c I

K s As s sG s

K s As s A K s sτ τ τ

τ τ τ τ+ − +

= =+ + − − + +

11, ,

2I cI

CL CL CLc

KAK

K Aττ

τ ζ= = − = −

0lim ( ) 1 ( ( ) ( ), no offset)CL CLs

K G s H s R s→

= = =

Iτ

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CHE302 LECTURE VIII DYNAMIC BEHAVIORS OF … LECTURE VIII DYNAMIC BEHAVIORS OF CLOSED- ... control 1. Symbol size may vary according to the user's needs and ... Control Engineering