Closed-Loop Control

Systems

Chapter 10

Control Diagram of a Typical

Control Loop

F1

T1

T

F2

T2

TCController

Actuator

System

TT

Sensor

System

Components and Signals of a

Typical Control Loop

T

F2

T2

Thermocouple

millivolt signal

Transmitter4-20 ma

DCS

Control

Computer

A/D

3-15 psig

D/A4-20 ma

Operator

Console

Tsp

I/PAir

F1

T1

Thermowell

• Example: Two Control Loop For Steam Heated Stirred Tank

Feedback control system: Valve is manipulated to increase flow

of steam to control tank temperature

Closed-loop process: Controller and process are interconnected

TT TC

IP

Ps

Condensate

Steam

Fin,Tin

F,T IP

LT

LC

Le

ve

l Co

ntro

l

Temperature Control

Feedback Control Control Objective:

– maintain a certain outlet temperature and tank level

Feedback Control:

– temperature is measured using a thermocouple

– level is measured using differential pressure probes

– undesirable temperature triggers a change in supply steam pressure

– fluctuations in level trigger a change in outlet flow

Note:

– level and temperature information is measured at outlet of process/

changes result from inlet flow or temperature disturbances

– inlet flow changes MUST affect process before an adjustment is

made

Feedback Control: Basic components of control systems : – requires sensors, actuators and controller: e.g. Temperature Control Loop

Controller:

– software component implements math

– hardware component provides calibrated signal for actuator

Actuator:

– physical (with dynamics) process triggered by controller

– directly affects process

Sensor:

– monitors some property of system and transmits signal back to controller

Tin, F

Tout e A C P

M

Controller

Process -

+

Actuator (Valve)

Measurement (Thermocouple)

TR

Feedback Control

• Study of process dynamics focused on uncontrolled or Open-loop processes

– Observe process behavior as a result of specific input signals

• In process control, we are concerned with the dynamic behavior of a controlled or

Closed-loop process

– Controller is dynamic system that interacts with the process and the process

hardware to yield a specific behaviour

Gp Y(s) U(s)

Y(s) YSP(s) or R(s)

+

-

Process Final Element Controller

D(s) Gd

Gc Gv Gp

Gm

+

+ U(s) V(s)

Ym(s)

E(s)

Gp(s) - Process Transfer Function

Gc(s) - Controller Transfer Function

Gm(s) - Sensor Transfer Function

Gv(s) - Actuator Transfer Function

Gd(s) -Load Transfer Function

D Disturbance or load variable

Y controlled variable

Ysp setpoint or desired or reference variable

Ym measured variable

E error

V controller output variable

U manipulated variable

Closed-Loop Transfer Function For control, we need to identify closed-loop dynamics due to:

- Setpoint changes Servo

- Disturbances Regulatory

1. Closed-Loop Servo Response

– transfer function relating Y(s) and R(s) when D(s)=0

Isolate Y(s)

Y s G s V s

Y s G s G s U s

Y s G s G s G s E s

Y s G s G s G R s Y s

Y S G s G s G s R s G s Y s

p

p v

p v c

p v c s m

p v c m

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

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

( )

Y sG s G s G s

G s G s G s G sR s

p v c

p v c m

( )( ) ( ) ( )

( ) ( ) ( ) ( )( )

1

2. Closed-loop Regulatory Response with no disturbance dynamics (ie Gd=0)

– Transfer Function relating D(s) to Y(s) at R(s)=0

– Isolating Y(s)

Y s D s G s V s

Y s D s G s G s U s

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

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

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

p

p v

p v c

p v c m

p v c m

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

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

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

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

0

0

Y sG s G s G s G s

D sp v c m

( )( ) ( ) ( ) ( )

( )

1

1

Closed-loop Transfer Function

2. Regulatory Response with Disturbance Dynamics

3. Overall Closed-Loop Transfer Function

Y sG s

G s G s G s G sD sd

p v c m

( )( )

( ) ( ) ( ) ( )( )

1

Regulatory

Servo

Y sG s G s G s

G s G s G s G sR s

G s

G s G s G s G sD s

p v c

p v c m

d

p v c m

( )( ) ( ) ( )

( ) ( ) ( ) ( )( )

( )

( ) ( ) ( ) ( )( )

1

1

Example: Blending Process

Transfer function for each of the four 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.

Process

Dynamic model of a stirred-tank blending system

1 21 2 (11-1)

τ 1 τ 1

K KX s X s W s

s s

where

11 2

ρ 1, , and (11-2)

wV xK K

w w w

(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

Current-to-Pressure (I/P) Transducer

Because 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 Valve 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 K

P s s

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 K

X s s

Controller

Suppose that an electronic proportional plus integral controller is

used. As will be shown later, the controller transfer function is

1

1 (11-4)τ

cI

P sK

E s s

where and E(s) are the Laplace transforms of the controller

output and the error signal e(t).

P s

p t

1. Summer

2. Comparator

3. Block

•Blocks in Series

are equivalent to...

G(s)X(s)Y(s)

Y

In general, the standard closed loop block

diagram is

“Closed-Loop” Transfer Functions

•Indicate dynamic behavior of the controlled process

(i.e., process plus controller, transmitter, valve etc.)

•Set-point Changes (“Servo Problem”)

Assume Ysp 0 and D = 0 (set-point change while disturbance

change is zero)

•Disturbance Changes (“Regulator Problem”)

Assume D 0 and Ysp = 0 (constant set-point)

*Note same denominator for Y/D, Y/Ysp.

MPVC

PVCM

sp GGGG

GGGK

sY

sY

1)(

)(

MPVC

L

GGGG

G

sD

sY

1)(

)(

Example: Draw block diagram for this Process

(PI control of liquid level – regulatory loop)

In general

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

0KAs

1)s(G

As

1)s(G

K)s(G

K)s(Gs

11K)s(G

CL

P

VV

MM

I

CC

For these assumptions, the transfer functions are:

MPVC

L

GGGG

G

Q

H

D

Y

11

MV

I

C KAs

Ks

K

As

D

Y

11

11

1

MvCIMVCI

I

KKKsKKKsA

s

D

Y

2

02 MvCIMVCI KKKsKKKsA

1s2s

K)s(G

22

The closed-loop transfer function is:

Substitute,

Simplify,

Characteristic Equation:

Recall the standard 2nd Order Transfer Function:

(2)

(3)

(4)

(5)

For 0 < < 1 , closed-loop response is oscillatory. Thus

decreased degree of oscillation by increasing Kc or I (for constant

Kv, 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

A

KKK

2

1 IMVC

10

Comparing coefficients (5) and (6) gives:

Substitute,

22

KKK

A

KKK

A

II

MVC

I

MVC

I2