10. closed-loop dynamics - 2013

8/10/2019 10. Closed-loop Dynamics - 2013

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Min-Sen Chiu

Department of Chemical and Biomolecular Engineering

National University of Singapore

CN3121 Process Dynamics and Control

10. Dynamic Behavior of Closed-loop Control Systems


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Dynamic Behavior of Closed-Loop

Control Systems

Learning Objectives

Become familiar with the major elements in thefeedback control system

Evaluate the dynamic behavior of processesoperated under feedback control

Develop closed-loop transfer functions

10

DynamicsofClosed

-LoopControlSystems


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10

DynamicsofClosed

-LoopControlSystems

The combination of the process and the feedback

controller is called the closed-loop system.

Variables of a closed-loop system:1. Inputs set-point and disturbance variables.

2. Output - controlled variable.

The analysis of closed-loop systems can be difficultdue to the presence of feedback.

Two useful tools:

1. Block diagram2. Closed-loop transfer function

Block diagrams can provide quantitative information if

each block is represented by a transfer function.


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Process

Approximate dynamic model of the stirred-tank blending

system is available:

1 21 2 (11-1) 1 1

K KX s X s W s

s s

11 2

1

, , and (11-2)wV x

K Kw w w

where

10

DynamicsofClosed

-LoopControlSystems


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Composition Sensor

Assume a first-order transfer function:

(11-3)

1

m m

m

X s K

X s s

10

DynamicsofClosed

-LoopControlSystems

Usually m

.

Compared to the (slow) process dynamics, sensor

dynamics is considered as negligible (fast) dynamics;

thus its transfer function can be further simplified as

a steady-state gain Km.


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Controller

Suppose that a PI controller is used, its

transfer function is

1

1 (11-4)

cI

P sK

E s s

, E(s) - Laplace transforms of the controller outputand the error signal e(t) P s p t

and e(t) - electrical signals (units of mA)

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DynamicsofClosed

-LoopControlSystems

Set point expressed as an

electrical current signal

Set point expressed as the

actual physical variable

p t


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The error signal is )()(~)( '' txtxte msp (11-5)

Transform

)()(~)( '' sXsXsE msp (11-6)

)('~ txsp is internal set-point related to the actual set-point

by sensor gain Km :)(' tx sp

)()(~ '' txKtx spmsp

m

sp

spK

sXsX

)()(

~

'

'

(11-7)

(11-8)

Eqs. 11-4, 11-6 and 11-8 are shown in the controller

block diagram.

Transform

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D

ynamicsofClosed

-LoopControlSystems


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Current-to-Pressure (I/P) Transducer

Usually has linear characteristics and negligible (fast)dynamics, thus the transfer function merely consists of

a steady-state gain KIP.

(11-9)t

IP

P s

KP s

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D

ynamicsofClosed

-LoopControlSystems


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2

(11-10) 1

v

t v

W s K

P s s

Assume that the valve can be modeled as

Control Valve

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D

ynamicsofClosed

-LoopControlSystems

[psi]

The valve dynamics is generally nonlinear =>

approximated by a linear 1st-order model in the vicinity

of the nominal operating condition


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Combining the block diagrams for individual components

obtains the overall block diagram of the feedback control

system as follows:10

D

ynamicsofClosed

-LoopControlSystems


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Block Diagram

Basic elements in a block diagram:

Arrowindicates flow of information, e.g. p = Ge = G(r - c).

Circlerepresents algebraic relation of the input arrows,

e.g. e = r c.

Blockrepresents the relevant dynamics (by transfer

function model) between the input and output.

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D

ynamicsofClosed

-LoopControlSystems

Gr +

-

p

p

e

c


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Some basic rules

1. Y = A B C

2. Y = G1G2A G1 G2A Y

G2 G1A Y

G1G2A Y

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D

ynamicsofClosed

-LoopControlSystems

A +

-

+

-

B C

YA - B

B -

-

+

+

C A

Y- B - C


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3. Y = G1(A B)

G1A

B

Y+-

G1

G1

YA

B

+

-

4. Y = (G1+G2)A

G1

G2

A Y+

+

G1+G2A Y

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D

ynamicsofClosed

-LoopControlSystems

G2 G1/G2A+

+

Y

G2A G1A


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D

ynamicsofClosed

-LoopControlSystems

This is a general diagram that can be used to represent

a wide variety of practical control problems.

Gp Effect of manipulated variable on the controlled variable

Gd Effect of load variable on the controlled variable

Block Diagram for Closed-loop System


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Standard symbols

G = transfer function

(subscripts c, v, p, d, m = controller, valve, process,

disturbance, and measurement respectively)

Y - process output D - disturbance or load variable

Ym - measured output Ysp - set-point

sp - internal set-point E - error

P - controller output U - manipulated variable

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D

ynamicsofClosed-LoopControlSystems

Note

Each variable in the figure is the Laplace transform of a deviationvariable.

For simplicity, the primes and s have been omitted; thus Ymeans Y(s).


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10

D


Closed-Loop Transfer Functions

The objective is to find the transfer functions between

the inputs (Ysp and D) and the output (Y) of the closed-loop or feedback control system.

For the process input,

)()~

( YGYKGGYYGGEGGPGU mspmcvmspcvcvv

Process output is obtained as

DGUGY dp

From the above two equations,

DGYGYKGGGY dmspmcvp )(


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Rearranging,

DGGGG

GYGGGG

KGGG

Ymcvp

dsp

mcvp

mcvp

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Effect of Yspon Y Effect o f D on Y

(11-30)

Eq. 11-30 illustrates the important role of Laplace Transformin the analysis of feedback control system.

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D


Ysp Y+

Gd

1+GcGvGpGmD

+

GcGvGpKm

1+GcG

vG

pG

m

Closed-loop dynamics can now be

given in an open-loop manner


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Two Control Problems

1. Servo problem (set-point change)

This is equivalent to Ysp 0 (set-point change) and D = 0(disturbance change is zero), from the last equation

mcvp

mcvp

sp GGGG

KGGG

Y

Y

1

(11-26)

(closed-loop transfer function for set point change)

2. Regulator problem (disturbance change)

mcvp

d

GGGG

G

D

Y

1(11-29)

In this case, D 0 and Ysp = 0 (constant set-point):

(closed-loop transfer function for load change)

10

D



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Remarks

1. Closed-loop transfer functions (eqs. 11-26 and 11-29)

depend on dynamics of process, measurement device,

controller and control valve.

3. Overall transfer function = (product of transfer functions inthe forward path)/(1 + product of all transfer functions in the

feedback loop).

2. Denominator for both transfer functions, eqs. 11-26 and 11-29,

is the same => (1 + product of all the transfer functions in the

feedback loop), i.e. (1+GpGvGcGm).

4. For simultaneous changes in disturbance and set-point (i.e.,

D 0 and Ysp 0), eq 11-30 holds => overall response is the

sum of the individual response.

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D


Ysp

Y+

Gd

1+GcGvGpGmD

+GcGvGpKm

1+GcGvGpGm


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1

c v p m

sp c v p m

G G G K Y

Y G G G G

Negative feedback

1d

c v p m

GY

D G G G G

Forward path from Ysp to Y

Forward path from D to Y

Product of all transfer

functions in the loop

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D


Remark 3


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1

c v p m

sp c v p m

G G G K Y

Y G G G G

Negative feedback

1

p

c v p m

GY

D G G G G

Forward path from Ysp to Y

Forward path from D to Y

Product of all transfer

functions in the loop

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D


Consider this feedback system,

+

-

Gc

Gm

Y+ Gp

D

+

GvYsp Km


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Analysis and Design Problems

Analysis Given particular Gp, Gv, Gc, Gm,

- Is the closed-loop system stable?

- Speed of response? oscillatory

response ?

Design Given Gp, Gv, Gm, and Gd, design Gc toachieve requirements like:

- Closed-loop stability;

- Closed-loop dynamics are sufficiently fast andsmooth (without excessive oscillations);

- Y/Ysp has an (ideal) gain of?? and Y/D has an

(ideal) gain of??

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D

ynamicsofClosed-LoopControlSy

stems


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Effect of Proportional Control on Closed-Loop Response

Consider 1st-order process and P-controller.

Then1

s

KG

p

p 1

s

KG dd

cc KG

For simplicity, let andvv KG mm KG

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stems


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From eq. 11-30,

DKKKKs

KY

KKKKs

KKKKY

mpvc

dsp

mpvc

mpvc

11

Ds

KY

s

Ksp

11 1

2

1

1

mpvc KKKK

11

mpvc

mpvc

KKKK

KKKKK

11

mpvc

d

KKKK

KK

12

Decreases with increasing KcAlways < , i.e., CL response is

faster than OL response

1 unless Kc =

Always < 1

0 unless Kc =

Always < Kd

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D


stems

Note:

CL system is 1st order with time

constant 1. For both TFs, time

constant is the same, but gain is

different


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Servo problem

Step change of magnitude M in set-point, i.e., Ysp = M/s

and D = 0.

Then from the last eq.s

M

s

KY

11

1

Inverting, )1()( 1/

1

teMKty

mpvc KKKK

M

1

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D


stems

Less than the desired

value M.


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Offset = current set-point (final value of the response)

mpvcmpvc

mpvc

KKKK

M

KKKK

KKKMKMMKM

111=

As t , output response neverreaches new set-point.The discrepancy is called (steady-state) offset.

Offset decreases with increasing Kc.

This is the characteristic of P-control.

Theoretically, offset when Kc .

Can it happen in practice? If not, why?

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D


stems


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Regulator problem

In this case, D = M/s Ysp = 0 and Ds

KY

11

2

Inverting, )1()( 1/

2

t

eMKty

0

M D(t)

y(t)

no control (Kc=0)KdM

Time0

K2Mwith control

offset

Offset = current set-point (final value of the response)

mpvc

d

KKKK

MKMK

10 2

10

D


stems

Kc offset


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Proportional-Integral Control for Disturbance Change

In this case, )1

1(

s

KG

I

cc

)1()1()

11(

11

1

sKKKKss

sK

ss

KKKKs

K

D

Y

ImpvcI

Id

I

mpvc

d

Rearrange

12 332

3

3

ss

sK

D

Y

where

I

mpvc

mpvc

mpvc

I

mpvc

Id

KKKK

KKKK

KKKKKKKK

KK

15.0,,

333

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D


stems

Gain = ?


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Remarks

1. Integral action eliminates offset, ysp() y() = 0

= 1, Kp = 1, Kv= 1

I

= 0.25 Kc= 3.5

y(t) y(t)

2. For Kc or I response speeds up

3. For Kc response more oscillatory (unexpected)

5. Note In general, closed loop response becomes more oscillatory

as Kc. The anomalous result above is due to neglected valveand measurement dynamics. When these two dynamics are

included, the TF is no longer 2nd-order.

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D


stems

4. ForI response more oscillatory (expected)


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Proportional-Integral Control for Set Point Change

(1 1 / ) / ( 1)

1 (1 1 / ) / ( 1)

c p v m I

sp c p v m I

K K K K s sY

Y K K K K s s

For this case,

Standard form

12

1

33

22

3

ss

s

Y

Y I

sp

3, 3 as defined earlier

For a unit step change in Ysp

Gain = ?

Offset = ?

2nd order dynamics

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D


stems

Kc= 1, Kp= 1

Kv= 1, Km= 1

I= 1, = 1


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Effect of Measurement Lag

As before let

1

sKG pp 1

sKG dd

cc KG 1 vv KG,,,

Assume significant measurement lag:1

1

sG

m

m

Resulting feedback control system:

1

s

KG dd

Kc1

sKG

p

p

1

1

sG

m

m

+ +

+

-

Ysp Y

D

10

D


stems

Ym


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Consider set point change,

12

)1(

)1)(1(1

1

55

22

5

5

ss

sK

ss

KKs

KK

Y

Y m

m

pc

pc

sp

5 ,1

c p

p c

K KK

K K ,15

cp

m

KK

cpm

m

KK

1

1

2

5

where and

Response may be oscillatory depending on the choice of

, m, Kp and Kc. One possibilityis as follows:

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D


stems


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Measurement lag produces poorer transients

Remarks

Offset results from the use of P controller

Kp = 1

=1Kc = 8

Ysp = 1/sy(t)

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D


stems

S


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When we place chemical process in a closed-loop with

sensors, transducers, valves and controllers, we have a more

complicated system than the original process. Nevertheless,

our analysis shows that a closed-loop system can be written

as one single transfer function.

Thus we recognize that a block diagram provides a

convenient representation for analyzing control systems.

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D


stems

Summary

We understand the key features of P and PI control using

simple first-order processes.

We notice that measurement dynamics can cause

deterioration in the control system performance.

Further reading: Chapter 10.1 to 10.3, SEM

10. closed-loop dynamics - 2013

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