1

Robust Control Systems (Chapter 12)Feedback control systems are widely used in manufacturing,

mining, automobile and other hardware applications. In response to increased demands for increased efficiency and reliability, these control systems are being required to delivermore accurate and better overall performance in the face of

difficult and changing operating conditions. In order to design control systems to meet the needs of improved performance and robustness when controlling

complicated processes, control engineers will require new design tools and better control theory. A standard technique of improving the performance of a control system is to add extra sensors and actuators. This necessarily leads to a multi-input

multi-output (MIMO) control system. Accordingly, it is a requirement for any modern feedback control system design

methodology that it be able to handle the case of multiple actuators and sensors.

Robust means durable, hardy, and resilient

2

Why Robust?• When we design a control system, our ultimate goal is to control a

particular system in a real environment.

• When we design the control system we make numerous assumptions about the system and then we describe the system with some sort of mathematical model.

• Using a mathematical model permits us to make predictions about how the system will behave, and we can use any number of simulation tools and analytical techniques to make those predictions.

• Any model incorporates two important problems that are often encountered: a disturbance signal is added to the control input to the plant. That can account for wind gusts in airplanes, changes in ambient temperature in ovens, etc., and noise that is added to the sensor output.

3

A robust control system exhibits the desired performance despitethe presence of significant plant (process) uncertainty

The goal of robust design is to retain assurance of system performance in spite of model inaccuracies and changes. A system is robust when it has acceptable

changes in performance due to model changes or inaccuracies.

D(s) Disturbance

R(s) PrefilterGP(s)

ControllerGC(s)

PlantG(s)

Sensor1

+

-

+

+

++

Y(s)Output

N(s)Noise

4

Why Feedback Control Systems?

• Decrease in the sensitivity of the system to variation in the parameters of the process G(s).

• Ease of control and adjustment of the transient response of the system.

• Improvement in the rejection of the disturbance and noise signals within the system.

• Improvement in the reduction of the steady-state errorof the system

5

Sensitivity of Control Systems to Parameter Variations

• A process, represented by G(s), whatever its nature, is subject to a changing environment, aging, ignorance of the exact values of the process parameters, and the natural factors that affect a control process.

• The sensitivity of a control system to parameter variations is very important. A main advantage of a closed-loop feedback system is its ability to reduce the system’s sensitivity.

• The system sensitivity is defined as the ratio of the percentage change in the system transfer function to the percentage change of the process transfer function.

( ) ( )

( )GHSsHsG

S

GHGG

GHTG

GTS

sGHsGsT

GGTT

sGsGsTsTS

sRsYsT

GT

GT

larger for Less)()(1

1

1/.

11. ;

)(1)()(

//

)(/)()(/)(ty)(sensitivi ;

)()()(

2

+=

++=

∂∂

=+

=

∂∂

=∆∆

==

6

The sensitivity of the feedback system to changes in the feedback element H(s) is

( )

y)sensitivit(Root /

y)Sensitivit (System //

rulechain the Use.block a offunction transfer within the a is where, determine toneed Often we

11/.

1.

2

αα

αα

α

αα

∂∂

=

∂∂

==

+−

=+−

+

=∂∂

=

ir

GTG

T

Tα

TH

rS

TTSSS

GαS

GHGH

GHGH

GHG

TH

HTS

i

parameter

7

Robust Control Systems and System SensitivityA control system is robust when: it has low sensitivities, (2) it is stable

over the range of parameter variations, and (3) the performance continues to meet the specifications in the presence of a set of changes

in the system parameters.

( )

( )( )

( )11 ;

1 isroot The1

1.

parameter theoft independen are )( of zeros/

:isy SensitivitRoot

function) tranfer theis parameter; theis ( // :isy sensitivit System

1

1

++−=−−=−

++=++

−=

+−=

=

=

∑=

αα

ααα

ααα

ααα

ααα

αα

α

α

sSSS

rsrs

SS

sTd

drS

Td

TdTS

i

i

i

rTT

i

n

i

rT

ir

T

1/s+α+

-R(s) Y(s)

8

Let us examine the sensitivity of the following second-order system

( )Kss

sssGH

S

kssKsT

TK

+++

=+

=

++=

2

2

1)(1

1 that(4.12) Eq. from know We

)(

K/s(s+1)+

-R(s) Y(s)

9

Example 12.1: Sensitivity of a Controlled System

1.get to2Then . and 1condition normal heConsider t

)(

)(11

n21n

122

12

122

2

====

++

+=

++=

+=

ξbb

bsbsbsbsT

bsbss

sGGS

C

TG

ωωξ

diagram Bode aon Tlog 20 and Slog 20Plot

Controllerb1+b2s

Plant1/s2

R(s) +

-

Y(s)G(s)GC(s)

controller (PD) derivative-alproportion a is )(sGC

10

Bode PlotFrequency response plots of linear systems are often displayed in the form of logarithmic plots, called Bode plots, where the horizontal axis represents the

frequency on a logarithmic scale (base 10) and the vertical axis represents the amplitude ratio or phase of the frequency response function.

11

Disturbance Signals in a Feedback Control System

• Another important effect of feedback in a control system is the control and partial elimination of the effect of disturbance signal.

• A disturbance signal is an unwanted input signal that affects the system output signal. Electronic amplifiers have inherent noise generated within the integrated circuits or transistors; radar systems are subjected to wind gusts; and many systems generate all kinds of unwanted signals due to nonlinear elements.

• Feedback systems have the beneficial aspects that the effect of distortion, noise, and unwanted disturbances can be effectively reduced.

12

The Steady-State Error of a Unity Feedback Control System (5.7)

• One of the advantages of the feedback system is the reduction of the steady-state error of the system.

• The steady-state error of the closed loop system is usually several orders of magnitude smaller than the error of the open-loop system.

• The system actuating signal, which is a measure of the system error, is denoted as Ea(s).

G(s)

H(s)

Y(s)R(s)Ea(s)

1)( When )()(1

1)(1

)()()()()( =+

=+

−=−= sHsRsGsGH

sGsRsYsRsE

13

Compensator• A feedback control system that provides an optimum performance

without any necessary adjustments is rare. Usually it is important to compromise among the many conflicting and demanding specifications and to adjust the system parameters to provide suitable and acceptable performance when it is not possible to obtain all the desired specifications.

• The alteration or adjustments of a control system in order to provide a suitable performance is called compensation.

• A compensator is an additional component or circuit that is inserted into control system to compensate for a deficient performance.

• The transfer function of a compensator is designated as GC(s) and the compensator may be placed in a suitable location within the structure of the system.

14

Root Locus Method

• The root locus is a powerful tool for designing and analyzing feedback control systems.

• It is possible to use root locus methods for design when two or three parameters vary. This provides us with the opportunity to designfeedback systems with two or three adjustable parameters. For example the PID controller has three adjustable parameters.

• The root locus is the path of the roots of the characteristic equation traced out in the s-plane as a system parameter is changed.

• Read Table 7.2 to understand steps of the root locus procedure.• The design by the root locus method is based on reshaping the root

locus of the system by adding poles and zeros to the system openloop transfer function and forcing the root loci to pass throughdesired closed-loop poles in the s-plane.

15

The root Locus Procedure

( )( )

zero. aat arrival of angle theand pole a from locus theof departure of angle theDetermine :10 Stepaxis. real on thepoint breakway theDetermine :9 Step

axis.imaginary thecrosses locus heat which tpoint theDetermine :8 Step. angle with and at centered asymptotes alonginfinity at zeros the toproceed loci The :7 Step

axis. real horizontal therespect to with lsymmetrica bemust lociroot The :6 Steppoles). ofnumber the to(equal loci separate ofnumber theFind :5 Step

zeros. at the ends and poles at the begins locus Thedegrees. zero is 4- sat pole and zero thefrom angle theand ,180 isorigin at the

p1 angle thebecause 2,- and 0 points ebetween th axis real on the satisfied iscriterion angle The :4 Stepaxis. real on the zeros and poles thelocate we

(Step3) 0gain for the roots of locus thedetermine To .2 isparameter gain tivemultiplica The

04221 :zeros and poles of in terms written is )(function transfer The :2 Step

0141

121

1)(1equation sticcharacteri The :1 Step

AA

2o

φσ

=

∞≤≤

=++

+

=

+

+

+=+

p

KKsssKsGH

ss

sKsGH

16

Example

( )( )321)(

++=

sssG

ControllerGC(s)

PlantG(s)

R(s) Y(s)

-+

z1=-3+j1

-1-2

-z1

-z1

( )( )( )( )( )112

11ˆ

ˆ3)()(1

)()()(

rsrsrszszsK

sGsGsGsG

STC

C

+++++

=+

=

j1

j2

17

Analysis of Robustness

PrefilterGP(s)

ControllerGC(s)

PlantG(s)

Sensor1

+-

+

+

++

Y(s)Output

N(s)Noise

small. )(Better ;1)()( then 1,)( When ;)()(1

)()()(

function transfer loop-closed The .)]()(1[)(

r so )( tosmall bemust )( noiseSensor

).( edisturbanc afor small )(output thekeep and )(input an for )]()()([error trackingSmall

1

sSsTsSsGsGsG

sGsGsT

sGsGsS

ntrtn

tdtytrtytrte

PC

C

C

=+=+

=

+=

⟩⟩

−=

−

:goals System

18

The Design of Robust Control Systems• The design of robust control systems is based on two tasks:

determining the structure of the controller and adjusting the controller’s parameters to give an optimal system performance. This design process is done with complete knowledge of the plant. Thestructure of the controller is chosen such that the system’s response can meet certain performance criteria.

• One possible objective in the design of a control system is that the controlled system’s output should exactly reproduce its input. That is the system’s transfer function should be unity. It means the system should be presentable on a Bode gain versus frequency diagram with a 0-dB gain of infinite bandwidth and zero phase shift. Practically, this is not possible!

• Setting the design of robust system requires us to find a propercompensator, GC(s) such that the closed-loop sensitivity is less than some tolerance value.

19

PID ControllersPID stands for Proportional, Integral, Derivative. One form of controller

widely used in industrial process is called a three term, or PID controller. This controller has a transfer function:

A proportional controller (Kp) will have the effect of reducing the rise time and will reduce, but never eliminate, the steady state error. An integral

control (KI) will have the effect of eliminating the steady-state error, but it may make the transient response worse. A derivative control (KD) will have the effect of increasing the stability of the system, reducing the

overshoot, and improving the transient response.

∫ ++=

++=

dttdeKdtteKteKtu

sKs

KKtG

DIp

DI

pC

)()()()(

termderivative a and n term,integratioan term,alproportion a provides controller The

)(

20

Proportional-Integral-Derivative (PID) Controller

kp

ki/s

kis+

+

+

e(t) u(t)

sKsKsK

sEsUsG

sEsKs

KKsU

KKKtytrte

tseKsteKteKtu

IPDPID

DI

p

DIP

DIp

++==

++=

−=

++=

2

)()()(

)()()(

ly.respective gains,feedback derivative and integral,al,proportion theare and ,, output; system theand

signal reference ebetween therror theis )()()(

)()()()(

21

Time- and s-domain block diagram of closed loop system

PIDController System

r(t) e(t) u(t)

-+R(s) E(s) U(s)

y(t)

Y(s)

)()(1)()(

)()()(

sGsGsGsG

sRsYsG

PIDsys

PIDsys

+==

22

PID and Operational AmplifiersA large number of transfer functions may be implemented using

operational amplifiers and passive elements in the input and feedback paths. Operational amplifiers are widely used in control systems to

implement PID-type control algorithms needed.

23

Figure 8.5

Inverting amplifier

1

2

)()(

RR

tVtV

S

o −=

24

Figure 8.30

Op-amp Integrator

sCRsZsZ

sVsV

sGFSs

out 1)()(

)()(

)(1

2 −=−==

)(1 sZ

)(2 sZ

25

Figure 8.35

Op-amp DifferentiatorThe operational differentiator performs the differentiation of the input signal.

The current through the input capacitor is CS dvs(t)/dt. That is the output voltage is proportional to the derivative of the input voltage with respect to time, and

Vo(t) = _RFCS dvs(t)/dt

sCRsZsZ

sVsV

sG SFS

o _)()(

)()(

)(1

2 =−==

)(2 sZ

)(1 sZ

26

Linear PID Controller

vo(t)

R1

R2

vs(t)

C1 C2

Z2(s)

Z1(s)

( )( )

122121

22112

2121

2211212

21

2211

;1;;)(

111

)()(

)(

CRKCR

KCR

CRCRKPs

KsKsKsG

sCR

sCR

CRCRsCR

sCRsCRsCR

sVsV

sG

DIIPD

PID

S

o

−=−=+

−=++

=

++

+=

++−==

27

Tips for Designing a PID ControllerWhen you are designing a PID controller for a given system, follow the

following steps in order to obtain a desired response.

• Obtain an open-loop response and determine what needs to be improved

• Add a proportional control to improve the rise time• Add a derivative control to improve the overshoot• Add an integral control to eliminate the steady-state error• Adjust each of Kp, KI, and KD until you obtain a desired overall

response.

• It is not necessary to implement all three controllers (proportional, derivative, and integral) into a single system, if not needed. For example, if a PI controller gives a good enough response, then you do not need to implement derivative controller to the system.

28

The popularity of PID controllers may be attributed partly to their robust performance in a wide range of operation conditions and partly to their

functional simplicity, which allows engineers to operate them in a simple manner.

( ) ( )( )

plane- hand-left in the anywhere locatedbecan that zeros twoandorigin at the pole oneith function w transfer a

introduces PID y theAccordingl ./ and ;/ Where

)(

3231

2132

3

212

33

21

s

KKbKKas

zszsKs

bassKs

KsKsKsK

sKKtGC

==

++=

++=

++=++=

29

Root LocusRoot locus begins at the poles and ends at the zeros.

( )( )

( )( )( )

error. statesteady zero and responsefast a have willsystem thelarge, is If ;)(

have we1,)( Setting equal.ely approximat are rootscomplex theand zeros theBecause)()(

)()()(

ˆ)()(1

)()()()(

isfunction transfer loop closed The zero. theapproaches rootscomplex theincreases, controller theof K3 As locus.root plot thecan wezeros,complex with controller PID a use weAssume

521)(

33

3

2

3

2

3

112

113

KKs

Krs

KsT

sGrs

sGKsG

rsrsrszszsK

sGsGsGsGsG

sT

sssG

P

PP

C

PC

+≈

+=

=+

=+++

++=

+=

++=

-2

r2

j 2

j 4K3 increasing

z1

z1

r1

r1

30

Design of Robust PID-Controlled SystemsThe selection of the three coefficients of PID controllers is basically a search

problem in a three-dimensional space. Points in the search space correspond to different selections of a PID controller’s three parameters. By choosing different points of the parameter space, we can produce different step responses for a

step input.The first design method uses the (integral of time multiplied by absolute error

(ITAE) performance index in Section 5.9 and the optimum coefficients of Table 5.6 for a step input or Table 5.7 for a ramp input. Hence we select the three PID

coefficients to minimize the ITAE performance index, which produces an excellent transient response to a step (see Figure 5.30c). The design

procedure consists of the following three steps.

dttetT

∫=0

)(ITAE

31

The Three Design Steps of Robust PID-Controlled System

• Step 1: Select the ωn of the closed-loop system by specifying the settling time.

• Step 2: Determine the three coefficients using the appropriate optimum equation (Table 5.6) and the ωn of step 1 to obtain GC(s).

• Step 3: Determine a prefilter GP(s) so that the closed-loop system transfer function, T(s), does not have any zero, as required by Eq. (5.47)

on

nn

o

bsbsbsb

sRsYsT

++++==

−− 1

11 ...)(

)()(

32

Input Signals; Overshoot; Rise Time; Settling Time• Step: r(t) = A R(s) = A/s• Ramp: r(t) = At R(s) = A/s2

• The performance of a system is measured usually in terms of step response. The swiftness of the response is measured by the rise time, Tr, and the peak time, Tp.

• The settling time, Ts, is defined as the time required for the system to settle within a certain percentage of the input amplitude.

• For a second-order system with a closed-loop damping constant, we seek to determine the time, Ts, for which the response remains within 2% of the final value. This occurs approximately when

2

2

1/

1/

2

100 (PO)Overshoot Percentage

Response)(Peak 1 Time);(Peak 1

ratio) damping : frequency; natural undamped :(44 ;4 ;02.0

ξξπ

ξξπ

ξω

ξω

πξω

τξω

−−

−−

−

=

+=−

=

==≅⟨

e

eMT

ξωTTe

pn

p

nn

ssnTsn

33

Example 12.8: Robust Control of Temperature Using PID Controller employing ITAE performance for a step input and a settling time of less than 0.5 seconds.

( )

( ) ( )( )3223

212

33

212

31

212

3

02222

15271 :5.6) (Table ITAEfor equation sticcharacteri theof tscoefficien optimum The

12)(1)(

)()()(

is 1)(ith function w transfer loop-closed The;)( :controller PID a Using

seconds. 0.5 than less of timesettling afor input step afor eperformanc ITAE optimiman obtain todesire Weinput. step afor seconds 3.2 is criterion) (2% timesettling theand 50%, iserror state-steady the1,)( If

122%;50

21;1)( lim;

1 ;

21

121

11)(

nnn

C

C

PC

C

sss

pp

ssnn

ωsω.sω.s

KsKsKsKsKsK

sGGsGG

sRsYsT

sGs

KsKsKsG

sG

esGKKAe

ssssssG

+++

+++++

++=

+==

=++

=

=

======+

=++

=++

=+

=→

ξωξω

R(s)GP(s) GC(s) G(s) Y(s)

D(s)

+++

-

U(s)E(s)

34

Cont. 12.8

( ) ( )

( )1000. tonumerator overall thebring andequation previous in the zeros

theeliminate order toin 5.648.13

5.64)( require weTherefore

10002155.171000

)(1)()()(

response ITAE desired theachieve that toso )(prefilter aselect We10002155.17

1.49.6 1.49.65.1510002155.17

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2

23

2323

223

1

++=

+++=

+=

+++

−+++=

+++

++=

===

==

sssG

ssssGGsGGsGsT

sGsss

jsjssss

sssT

KKK

T

P

c

Pc

P

n

nsnωξ

ξωω

35

Results for Example 12.8

0.4%0.4%52%y(t)/d(t)maximum

0.0%0.0%50.1%Steady-state error

0.450.203.2Settling time (s)

1.9%31.7%0Percent overshoot

PID with GP(s) Prefilter

PID & GP(s)=1GC(s)=1Controller

36

E12.1: Using the ITAE performance method for step input, determine the required GC(t). Assume ωn = 20 for Table 5.6. Determine the step response with and without a prefilter GP(s)

prefilter with theand without response step theDraw ;8.14

8.14)( Where

40028400

)(1)()(

)()( isgain loop-closed theprefilter, aWith

4002840027

)()( is system loop-closed theprefilter, aWithout

400 and 27 have then we20When 4.1 :isequation sticcharacteri ITAE The

)1(it Find )(1

)()(

by given controller PI a Use;1

1)(

2

2

21

22

212

21

+=

++=

+=

++

+=

===++

+++==+

=

+=+

=

ssG

sssGGsGGpsG

sRsY

sss

sRsY

KKss

KsKssGG

sGGsT

sKKG

ssG

P

C

C

n

nn

C

C

C

ωωω

GP(s) GC(s) G(s) Y(s)

D(s)

+++

-

U(s)E(s)R(s)

37

E12.3 A closed-loop unity feedback system has

( )

iprelationsh plot theThen 9

by determined is in changes to ofy sensitivit The9

)( isfunction y sensitivit The

99)( isfunction transfer loop-closed The

plot Bode aon )S( and )T(plot and Find ;3;9)(

2

2

2

2

++−==

++

+=

++=

=+

=

pssps

Tp

dpdTS

pTpss

psssS

psssT

jωjωSppss

sG

TP

TP

38

E12.5

( )( )

( )( )

( )

( )10000012.0)( controller The ms; 40100

44 is timesettling The

40000K and 200 200;2 20%. P.O.for 0.5Let 200

)( isfunction transfer loop-closed The

200)(

100/Select .1018.3 where200100

200100

10215900 isfunction transfer loop-open The

ms. 60 than less is timesettling theand 20% than less is step a toovershoot the that so )(Design .)(r compensato PDith feedback w

unity negative a and 1

2001

100

900,15)(plant a has systemA

2nnn

2

21282

1

4

2

12

21

+====

====⟨=

++=

+=

=×=++

+

=

++

×

+

=

+=

+

+

=

ssGT

kssKsT

ssKsGG

KKKKsssKKsK

sssKKsK

GG

sGsKKsG

ssssG

cn

S

c

c

CC

ξω

ωωξωξ

39

DP12.10

6.1862.706.186)( ;/6.1862.70)(

6.186;2.70;7.5 :ITAE Use;10

)(

)( :controller PI the )c(

1004100)(;1004)(

4;100 :method ITAE Use;)10(

)(

)( :controller PD (b)73.50)( ;73.50 ;702.0 4.5%; For 10)(1

)()( ;)( (a)

function)sfer robot tran space (The)10(

1)(

2121

2321

2121

2112

221

21

2

+=+=

===+++

+=

+=+=

+=+=

==+++

+=

+====⟨

++=

+==

+=

ssGssG

KKKsKss

KsKsT

sKsK

sKKsG

ssGssG

KKKsKs

sKKsT

sKKsGsGcKPO

KssK

sGGcsGGcsTKsG

sssG

pc

n

c

pc

c

c

ω

ξ

Consider

Consider

Consider

40

5.14) (Eq. 1

;5.15) (Eq 100

peak time and time,settling overshoot, means ePerformancPID and PI, PD, K, :casesfour for the eperformanc theofsummary a Find

10002155.71000)(

10002155.7)( ;1000 ;215 ;5.7

10 with ITAE Use;10

)(

)( :controller PID (d)

21

2

2321

322

123

322

1

322

1

2

ξω

π

ω

ξξπ

−==

++=

++====

=++++

++=

++=

−−

np

l

c

n

c

tePO

sssGp

ssssGKKK

KsKsKssKsKsKsT

sKsKsKsGConsider