Dynamic Behavior of Closed-Loop Control Systems

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Next, we develop a transfer function for each of the five elements in the feedback control loop. For the sake of simplicity, flow rate w1 is assumed to be constant, and the system is initially operating at the nominal steady rate.

ProcessIn section 4.3 the approximate dynamic model of a stirred-tank blending system was developed:

1 21 2 (11-1)

τ 1 τ 1K KX s X s W ss s

where

11 2

ρ 1, , and (11-2)wV xK Kw w w

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(11-5)sp me t x t x t

or after taking Laplace transforms,

(11-6)sp mE s X s X s

The symbol denotes the internal set-point composition expressed as an equivalent electrical current signal. This signal is used internally by the controller. is related to the actual composition set point by the composition sensor-transmitter gain Km:

spx t

spx t spx t

(11-7)sp m spx t K x t

Thus

(11-8)spm

sp

X sK

X s

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Current-to-Pressure (I/P) TransducerBecause transducers are usually designed to have linear characteristics and negligible (fast) dynamics, we assume that the transducer transfer function merely consists of a steady-state gain KIP:

(11-9)tIP

P sK

P s

Control ValveAs discussed in Section 9.2, control valves are usually designed so that the flow rate through the valve is a nearly linear function of the signal to the valve actuator. Therefore, a first-order transfer function usually provides an adequate model for operation of an installed valve in the vicinity of a nominal steady state. Thus, we assume that the control valve can be modeled as

2 (11-10)τ 1

v

t v

W s KP s s

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Composition Sensor-Transmitter (Analyzer)We assume that the dynamic behavior of the composition sensor-transmitter can be approximated by a first-order transfer function:

(11-3)τ 1

m m

m

X s KX s s

ControllerSuppose that an electronic proportional plus integral controller is used. From Chapter 8, the controller transfer function is

11 (11-4)τc

I

P sK

E s s

where and E(s) are the Laplace transforms of the controller output and the error signal e(t). Note that and e are electrical signals that have units of mA, while Kc is dimensionless. The error signal is expressed as

P s p t p

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Transfer Functions of Control Systems

1

Set-Point Change: ?

Load Change: ?

sp

X sX s

X sX s

1. Summer

2. Comparator

3. Block

• Blocks in Series

are equivalent to...

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Closed-Loop Transfer Function for Set-Point Change (Servo

Problem)

2

0

1

p v p

v p c

v p c sp

v p c m sp m

m c v p

sp c v p m

D s

Y s X s G s U s G s G s P s

G s G s G s E s

G s G s G s Y s B s

G s G s G s K Y s G Y s

K G G GY sY s G G G G

Closed-Loop Transfer Function for Load Change (Regulator

Problem)

1 2

0

1

sp

d p

d v p c m

d v p c

m s

m

d

c v m

p

p

Y s

Y s X s X s

G s D s G s U s

G s D s G s G s G s GK Y Y s

G s D s G s G s G s G Y s

Y s GD s G G

s

G G

General Expression of Closed-Loop Transfer Function

1where

, : any two variables in block diagram: product of the transfer functions in the

path from to : product of the transfer functions in the

feedback loop.

f

e

f

e

YX

X Y

X Y

Servo Problem

open-loop transfer function

sp

f m c v p

e c v p m OL

X Y

Y YK G G G

G G G G G

Regulator Problem

open-loop transfer function

f d

e c v p m OL

X DY Y

G

G G G G G

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I O

Example2 1

42 1 2

1 1 2 3

1 2 3 1

1 1 2 3

1 2 3 1

1 1 2 1 2 3

2 1 2 1 2 1

4

4

2 3 1

2 1

2 1 2

2 1

2 1 2

1

1

1

11

1

c

c m

m c

sp c m

m c

c m

c

c m

c

c

m c c

c m c c m

m

GG

G GG G

G GO GI G G G

K G G GCY G

G

G G G

K G G G

G G G G

K G G G G GG G G G G G

GG G

G

G

G

G

G

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Closed-Loop Response with P Control (Set-Point Change)

1

1

1

1

111

1where

and open-loop gain = 01

1

pm c v

pspc v m

OLOL m c v p

OL

OL

KK K KH s Ks

KH s sK K Ks

KK K K K K KK

K

Closed-Loop Response with P Control (Set-Point Change)

1/1

1

1

1

offset1

offset and

sp sp

t

spOL

OL

Mh t M S t H ss

h t K M e

Mh h M K MK

K

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Closed-Loop Response with P Control (Load Change)

1

2

1 1

2

/1 1 2

2

1

11

11

where 1

1

offset 01

offset and

p

pc v m

p

OL

t

psp

OL

OL

KH s Ks

KQ s sK K K

sK

KK

Mq t M S t Q s h t K M es

K Mh h K M

K

K

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Closed-Loop Response with PI Control (Set-Point Change)

11

11

1 1

1

111

1

1

c cI

ppm vm v

p pspv mc m

ccI

cI

v

G

K

s Ks

KK K KK KH s ssK KH s

G s

K K K Ks

Ks s

G

Closed-Loop Response with PI Control (Load Change)

2 23

1

3

2 3 3

3 3 3

1 1

1 11 1

11

where

1

11

2 1

1, , 2

p p

p pv m v m

I

c v m

OLII

OL OL

OLI I I

c v m O

c

O

I

L

c

L

K KH s s s

K KQ sK K K K

s s

sK K K K s

Ks s

K K

KK

K

G

K K K K

Ks

s s

Closed-Loop Response with PI Control (Load Change)

3

3

1 1

32 23 3 3

233

233 3

1

2 1

1sin

1

Note that offset is eliminated!

t

q t S t Q ss

KH ss s

Kh t e t

Closed-Loop Response with PI Control (Load Change)

3 3

3 3

faster response , less oscillatory

faster response , more oscillatory

c

I

K

EXAMPLE 1: P.I. control of liquid level

Block Diagram:

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Assumptions

1. q1, varies with time; q2 is constant.

2. Constant density and x-sectional area of tank, A.

3. (for uncontrolled process)

4. The transmitter and control valve have negligible dynamics (compared with dynamics of tank).

5. Ideal PI controller is used (direct-acting).

)h(fq3

0KAs1)s(G

As1)s(G

K)s(G

K)s(Gs

11K)s(G

CL

P

VV

MMI

CC

For these assumptions, the transfer functions are:

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MPVC

L

GGGGG

QH

DY

11

MVI

C KAs

Ks

K

AsDY

1111

1

MPCIMVCI

I

KKKsKKKsAs

DY

2

02 MPCIMVCI KKKsKKKsA

1s2sK)s(G 22

The closed-loop transfer function is:

Substitute,

Simplify,

Characteristic Equation:

Recall the standard 2nd Order Transfer Function:

(11-68)

(2)

(3)

(4)

(5)

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For 0 < < 1 , closed-loop response is oscillatory. Thus decreased degree of oscillation by increasing Kc or I (for constantKv, KM, and A).

To place Eqn. (4) in the same form as the denominator of the T.F. in Eqn. (5), divide by Kc, KV, KM :

01ssKKK

AI

2

MVC

I

AKKK

21 IMVC

10

Comparing coefficients (5) and (6) gives:

Substitute,

22

KKKA

KKKA

II

MVC

I

MVC

I2

• unusual property of PI control of integrating system• better to use P only

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