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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 System s

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Dynamic Behavior of Closed-Loop Control Systems

Learning Objectives

• Become familiar with the major elements in the feedback control system

• Evaluate the dynamic behavior of processes operated under feedback control

• Develop closed-loop transfer functions

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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 difficult due to the presence of feedback.

Two useful tools:

1. Block diagram

2. Closed-loop transfer function

Block diagrams can provide quantitative information if each block is represented by a transfer function.

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Return to the previous example of stirred tank blending process (see Chapters 2 and 4, SEM).

Development of a Block Diagram

Control objective: Regulate tank composition xManipulated variable: Flow rate of pure A, w2

Primary disturbance: Inlet composition x1

Assume w1 constant

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Derive transfer functions for each component in the closed-loop system.

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Process

Approximate dynamic model of the stirred-tank blending system is available (see eq. 4-69, SEM):

( ) ( ) ( )1 21 2 (11-1)

τ 1 τ 1

K KX s X s W s

s s ′ ′ ′= + + +

11 2

ρ 1, , and (11-2)

wV xK K

w w wτ −= = =where

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Composition Sensor

Assume a first-order transfer function:

( )( ) (11-3)

τ 1m m

m

X s K

X s s

′=

′ +

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Usually τm << τ.

Compared to the (slow) process dynamics, sensor dynamics is considered as negligible (fast) dynamics; thus its transfer function can be further simplified asa steady-state gain Km.

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Controller

Suppose that a proportional plus integral (PI) controller is used. The controller transfer function is

( )( )

11 (11-4)τc

I

P sK

E s s

′ = +

, E(s) - Laplace transforms of the controller output ˙and the error signal e(t)

( )P s′ ( )p t′

p′ and e - electrical signals (units of mA)

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Set point expressed as an electrical current signal

Set point expressed as the actual physical variable

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

Transform

)()(~

)( '' sXsXsE msp −= (11-6)

)('~ tx sp is internal set-point related to the actual composition set-point by sensor gain Km :)(' tx sp

)()(~ '' txKtx spmsp =

msp

sp KsX

sX=

)(

)(~

'

'

(11-7)

(11-8)

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

Transform

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Current-to-Pressure (I/P) Transducer

Usually has linear characteristics and negligible (fast) dynamics; thus assume that the transfer function merely consists of a steady-state gain KIP.

( )( ) (11-9)t

IPP s

KP s

′=

′

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( )( )

2 (11-10)τ 1

v

t v

W s K

P s s

′=

′ +

Assume that the valve can be modeled as

Control Valve

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[psi]

The valve dynamics is generally nonlinear => approximated by linear (1st order) model in the vicinity of the nominal operating condition

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Combining the block diagrams for the individual components, we get the composite block diagram of the control system:10

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Block Diagram Simplification

Basic elements in a block diagram:

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

Circle represents algebraic relation of the input arrows, e.g. e = r – c.

Block represents the relevant dynamics (by transfer function model) between the input and output.

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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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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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G2 G1/G2A +

+Y

G2A G1A

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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 va riable

Gd – Effect of load variable on the controlled variable

Block Diagrams – General Treatment

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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

Ỹsp - internal set-point E - error

P - controller output U - manipulated variable

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Note• Each variable in the figure is the Laplace transform of a deviation

variable.

• For simplicity, the primes and “s” have been omitted; thus Y means Y’(s).

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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 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

GY

GGGG

KGGGY

mcvp

dsp

mcvp

mcvp

++

+=

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Effect of Ysp on Y Effect of D on Y

(11-30)

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

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Two Types of Control Problem

1. Servo problem (set-point change)

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

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

(closed-loop transfer function for load change)

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Remarks

1. Closed-loop transfer functions (eqs. 11-26 and 1 1-29) depend on dynamics of process, measurement device, controller and control valve.

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

4. (1+ GpGvGcGm) is often written as ( 1+ GOL) where GOL= GpGvGcGm is the open-loop transfer function. G OL relates Ym to ˙̇̇̇Ỹsp if the feedback loop is opened just before the comparator .

2. Denominator for both transfer functions, eqs. 11 -26 and 11-29, is the same => (1 + product of all the transfer functions in the loop), i.e. (1+GpGvGcGm).

5. For simultaneous changes in disturbance and set- point (i.e., ˙̇̇̇D ≠≠≠≠ 0 and Y sp ≠≠≠≠ 0), eq 11-30 holds => overall response is the ˙̇̇̇sum of the individual responses .

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1c v p m

sp c v p m

G G G KY

Y G G G G=

+

Negative feedback

1d

c v p m

GY

D G G G G=

+

Forward path from Y sp to Y

Forward path from D to Y

Product of all transfer functions in the loop

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Remark 3

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1c v p m

sp c v p m

G G G KY

Y G G G G=

+

Negative feedback

1p

c v p m

GY

D G G G G=

+

Forward path from Y sp to Y

Forward path from D to Y

Product of all transfer functions in the loop

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Consider this feedback system,

+

-Gc

Gm

Y+ Gp

D

+Gv

Ysp Km

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Feedback system – equivalent diagrams

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+

-Gc

Gm

Y+ Gp

D

+Gv

Ysp Km

spY~

Gc

+Y

D

+

Gp

Gv Gp

Gm

+

-Ysp Km

spY~

Ysp Y+

Gp

1+GcGvGpGmD

+GcGvGpKm

1+GcGvGpGm

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Analysis and Design Problems

Analysis Given particular Gp, Gv, Gc, Gm, ˙

- Is the closed-loop system stable? - Speed of response? Damping? …

Design Given particular Gp, Gv, Gm, and Gd, “design”Gc so that

- The closed-loop dynamics are stable- Y/Ysp has a gain of ?? and Y/D has a gain of ??

- The dynamics are sufficiently fast and smooth ˙ (without excessive oscillations)

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Effect of Proportional Control on Closed-Loop Response

Consider 1st-order process and P-controller.

Then 1+

=s

KG p

p τ 1+=

s

KG d

d τcc KG =

For simplicity, let and vv KG = mm KG =

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From eq. 11-30,

DKKKKs

KY

KKKKs

KKKKY

mpvc

dsp

mpvc

mpvc

+++

++=

11 ττ

Ds

KY

s

Ksp 11 1

2

1

1

++

+=

ττ

mpvc KKKK+=

11

ττ

mpvc

mpvc

KKKK

KKKKK

+=

11

mpvc

d

KKKK

KK

+=

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Decreases with increasing K cAlways < ττττ, i.e., CL response is faster than OL response

≠≠≠≠ 1 unless K c = ∞∞∞∞Always < 1

≠0 unless K c = ∞∞∞∞Always < Kd

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Note: CL system is 1 st order with time const τ1. For both TFs, τ 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

+=

τInverting, )1()( 1/

1τteMKty −−=

mpvc KKKK

M

+=

1

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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 never reaches new set-point. The discrepancy is called (steady-state) offset.

Offset decreases with increasing Kc. This is the characteristic of P-control.

Theoretically, offset → 0 when Kc→ ∞.

But, does it happen? If not, why?

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Regulator problem

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

KY

11

2

+=

τInverting, )1()( 1/

2τteMKty −−=

0

M D(t)

y(t)

no control (K c=0)KdM

Time0

K2M with controloffset

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

mpvc

d

KKKK

MKMK

+−=−=

10 2

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Kc offset

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Proportional-Integral Control for Disturbance Change

In this case, )1

1(s

KGI

cc τ+=

)1()1()

11(

11

1+++

=+

++

+=sKKKKss

sK

ss

KKKKs

K

D

Y

ImpvcI

Id

I

mpvc

d

ττττ

ττ

τ

Rearrange

12 332

3

3

++=

ss

sK

D

Y

τζτwhere

ττζττττ I

mpvc

mpvc

mpvc

I

mpvc

Id

KKKK

KKKK

KKKKKKKK

KK

)1(

2

1,, 333

+===

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Gain = ?

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For a unit step change in load, D = 1/s and hence

12)(

3322

3

3

++=

ss

KsY

τζτ

The output response for ζ3 < 1 is

−

−= −

3

23

/

233

3 1sin1

11)( 33

τζ

ζττζ t

eKty t

For this specific case, the responses for different Kc

and τI are given in the next slide.

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Remarks

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

ττττ = 1, Kp = 1, Kv = 1

τI = 0.25 Kc = 3.5y(t) y(t)

2. For Kc↑ or τI↓ � 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 valve and measurement dynamics. When these are included, the TF is no longer 2nd-order.

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4. For τI↓ � response more oscillatory (expected)

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Proportional-Integral Control for Set Point Change

)1/()/11(1

)1/()/11(

+++++

=ssKK

ssKK

Y

Y

Ipc

Ipc

sp ττττ

For this case,

Standard form

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1

3322

3 +++=

ss

s

Y

Y I

sp τζττ

τ3, ζζζζ3 – as defined earlier

Introduce unit step change in Ysp and inverse

3

2/

23

1sin1

)( 3

τζ

ζττ τζ t

ety tI −−

= −

−+−

−−+ −−

ζζ

τζ

ζτζ

21

3

2/

2

1tan1sin

1

11 3

te t

Gain = 1 always! No offset 2nd order dynamics

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y(t)Kc = 1, Kp = 1 ττττI = 1, ττττ = 1

Again, zero offset

Offset = ysp(∞) – y(∞) = 1 – 1 = 0

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Effect of Measurement Lag

As before let

1+=

s

KG p

p τ 1+=

s

KG d

d τ cc KG = 1== vv KG,,,

Assume significant measurement lag:1

1

+=

sG

mm τ

Resulting control system is

1+=

s

KG d

d τ

Kc 1+=

s

KG p

p τ

1

1

+=

sG

mm τ

+ ++

-

YspY

D

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Ym

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Consider set point change,

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)1(

)1)(1(1

1

5522

5

5

+++=

+++

+=ss

sK

ss

KKs

KK

Y

Y m

m

pc

pc

sp τζττ

ττ

τ

,15

cp

c

KK

KK

+= ,

15cp

m

KK+= τττ

cpm

m

KK++=

1

1

25 ττ

ττζ

where and

Produces 2nd-order system even for P-control. Response may be oscillatory depending on the choice of τ, τm, Kp

and Kc. One possibility is as follows:

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▪ Measurement lag produces poorer transients

Remarks

▪ Offset results from the use of P controller

Kp = 1

ττττ =1Kc = 8 Ysp = 1/s

y(t)

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Control of Pure Capacity (or integrating) Process

h controlled by manipulating q1.

Mass balance (in terms of deviation variables):

1qdt

dhA =

As

qh 1=or

(const)

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Let Gc = Kc, Gv = Gm = 1.

Kc 1/Ashspq1 h

spc

hKAs

h1)/(

1

+=

For set-point change hsp = 1/s, 1 1

( / ) 1c

hAs K s

= ⋅+

h(t→∞) = lim [s h(s)] = 1s→0000

Therefore offset = hsp – h (t→∞) = 1 – 1 = 0

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Alternatively, ..…

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When we place chemical process in a closed-loop wit h sensors, transducers, valves and controllers, we ha ve a more complicated system than the original process. Neve rtheless, 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 sys tems.

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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 11.1 and 11.3, SEM