Robust Control analysis and design

8/8/2019 Robust Control analysis and design

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ROBUST CONTROL:ANALYSIS AND DESIGN

GIPSAlab Control Systems department-

Olivier SENAME

ENSE3-BP 4638402 Saint Martin d'Hres Cedex, FRANCE

[email protected]

Why Robust H control

MIMO systems

Performance specifications linked to control design

Analysis of robustness properties

Design of robust controllers

Advanced optimisation tools for control synthesis

Extensions: Gain-scheduling, Linear Parameter Varyingsystems

Robust control : analysis and design Olivier Sename 20102


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S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: analysis

and design, John Wiley and Sons, 2005.www.nt.ntnu.no/users/skoge

K. Zhou, Essentials of Robust Control, Prentice Hall, New Jersey, 1998.

Bibliography

. . .

J.C. Doyle, B.A. Francis, and A.R. Tannenbaum, Feedback control theory,Macmillan Publishing Company, New York, 1992.www.control.utoronto.ca/~francis

G.C. Goodwin, S.F. Graebe, and M.E. Salgado, Control System Design,Prentice Hall, New Jersey, 2001.http://csd.newcastle.edu.au/


G. Duc et S. Font, Commande Hinf et -analyse: des outils pour la robustesse,Herms, France, 1999.

D. Alazard, C. Cumer, P. Apkarian, M. Gauvrit, et G. Ferreres, Robustesse etcommande optimale, Cpadues Editions, 1999.

OUTLINE

Motivation Industrial examplesIndustrial examples

H norm, stability

Performance analysis/specifications

H control design

Uncertainties and robustness

Performances quantifiersA first robustness criteria

Representing uncertainties

Mixed sensitivity problem


Performances limitations Bode and Poisson sensitivity integral

,Robust control design

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Modern industrial plants have sophisticated control systems crucial to their successful

operation: robotics aerospace semiconductor manufacturing industry

INTRODUCTION

Energy production and distribution ... automotive industry :

SI and Diesel engines suspension braking Global chassis control intelli ent hi hwa s


driver supervision .

5

AUTOMOTIVE CONTROL

Some actual important fields of investigation concern:

1. Environmental protection (Limiting of pollutant emissions Nox ; CO; CO2) Engine control

Automatic driving

Energy consumption optimisationElectrical and Hybrid vehicles

2. Road safety and monitoring (decrease the number ofaccidents)

Braking in dangerous situationsDetection of critical situations


Chassis controlTraffic controlDriver assistance (stop & start, anti-collision)

by wire technologyDiagnosis of embedded system

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injection control (CommonRail

ENGINE CONTROL

idle-speed control air to fuel ratio control cylinder balancing Torque control throttle control EGR + VGT


driveline control Post-treatment Energy recovery Downsizing

7

TopicsActive Control for safety and comfortMulti actuators (suspensions, braking, steering)

VEHICLE DYNAMICS CONTROL

Methodologies:

- Physical and behavioral modelling

- H control: LPV, fault-tolerant

- On line adaptation ofcomfort/handling criteria

-


,braking, steering)

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Analysis and robustness of frequency synthesizers

Modelling andoptimization offrequencysynthesizer

oops u y

integrated onchip

Analysis ofsemi-globalstability,


ro ustness,

observationand robust

control

9

CONTROL OF GLASS FIBER BUSHING

Process

Objective :enhance productquality i.e.,- avoid variations of fiberdiameter

-less production breaksaccounting for disturbances(air, input glass temp.)- Robustness requirementsas bushings are changed


- MIMO system

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DVD PLAYERS CONTROL

Disk

D=120mm

d=15mm

Control problem: minimizeposition error between thelaser spot and the real trackposition, both in the radial andin the vertical direction.

Focus movementup-down

Tracking movementin-out

pickup

otodiodes

SystemMeasurement unit and D/A conv.

Current

Amplifier

Pre-processing

unit

A/D

Converter

Digital

controller

Controller

Neither the track position northe true spot position can bemeasured.

Robustness pb: non idealconstruction of the device andnon perfect location of the hole


Motors

Ph

PWM

(PDM)

unit

Current

Amplifier

Actuation unit and D/A conv

pickup TDA...

6300 and 6301

at the center of the disc

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OUTLINE


H norm, stability


H control design







,Robust control design

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About H norm: MIMO GAIN

For a SISO system, y=Gd, the gain at a given frequency is simply

)()(

)()(

)(

)(

jG

d

djG

d

y==

The gain depends on the frequency, but since the system is linear it isindependent of the input magnitude

For a MIMO system we may select :


22

2

2

2 )()()(

jGdd

==

Which is independent of the input magnitude. But this is not a correctdefinition. Indeed the input direction is of great importance

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MIMO GAIN

d5 = 0.6000d4 = 0.7070d3 = 0.7070d 2 = 0d 1 = 1

Five different inputs

=

23

45G How to define and evaluate its gain ??

.-0.8000

.-0.7070

.0.7070

1

0

Input magnitude : norm2= 1

Norm(d1)=norm(d2) =norm(d3)=norm(d4)=norm(d5)=1

y5 = -0.20000.2000

y4 = 0.70700.7070

y3 = 6.36303.5350

y2 = 42

y1 = 53

Corresponding outputs


0.28280.99987.27904.47215.8310

and gains

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MIMO GAIN

6

7

8

MAXIMUM SINGULAR VALUE = 7.34

)(max 20

Gd

Gd

d=

1

2

3

4

5

||y

||2

/||d||

2

2

)(min

2

2

0G

d

Gd

d=


-5 -4 -3 -2 -1 0 1 2 3 4 5

0

d20 / d10

MINIMUM SINGULAR VALUE = 0.27

We see that, depending on the ratio d20/d10, the gain varies between0.27 and 7.34 .

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MIMO GAIN

Eigenvalues are a poor measure of gain. Let

=

00

1000G

Eigenvalues are 0 and 0

But an input vector leads to an output vector

Clearly the gain is not zero.

Now, the maximal singular value is = 100

It means that any signal can be amplified at most 100 times

1

0.

0

100


This is the good gain notion.

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MIMO GAIN

In the case of a transfer matrix G(s) : (m inputs, p outputs)u vector of inputs, y vector of outputs

)(2

y

)(2

u

Example ofA two-mass/spring/damper system


1 2

2 outputs: x1 and x2

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Singular Values

MIMO GAIN

A=[0 0 1 0;

0 0 0 1;

-k1/m1 k1/m1 -b1/m1 b1/m1;

k1/m2 -(k1+k2)/m2 b1/m2 -(b1+b2)/m2];

B=[0 0;0 0;1/m1 0;0 1/m2];

C=[1 0 0 0;0 1 0 0];

SingularValues(dB)

-20

-10

0

10

20

30

largest singular value

smallest singular value

Hinf norm: 21.1885 dBD=[0 0;0 0];

Control system toolboxG1=ss(A,B,C,D) : LTI systemTf(G1) : transfer functionnormhinf(G1)

sigma(G1)


Frequency (rad/sec)

10-1 100 101-50

-40

-30

Mu-analysis toolboxG2=pck(A,B,C,D)

hinfnorm(G2)

>> norm between 11.4704 and 11.4819

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MIMO GAIN

Mathematical backgrounds


PERFORMANCE ANALYSIS

Well-posedness

1,2

1=

+

= Ks

sGK(s) G(s)r(t) y(t)

di(t)dy(t)

+

+

+ +

+

u(t)e(t)

n(t)

-+ +

Therefore the control input is non proper: iy dsdnrsu3

1)(3

2 ++=

DEF: A closed-loop system is well-posed if all the transfer functions are proper

is invertible


In the example 1+1x(-1)=0Note that if G is strictly proper, this always holds.

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More important : Internal StabilityDEF: A system is internally stable if all the transfer functions of the closed-loop system are stable

K(s) G(s)r(t)

y(t)

di(t)

+

- +

+u(t)(t)

++

++=

id

r

GGKIKGKIK

GGKIGKGKI

u

y11

11

)()(

)()(

+

++=

=

=r

s

ss

s

sysKG

11)2)(1(

1

2

1

,1

,1

For instance :


There is one RHP pole (1), which means that this system is not internallystable.This is due here to the pole/zero cancellation (forbidden!!).

+

+

i

ss 22

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OUTLINE

Motivation

H norm, stability


Sensitivity functionsSome criteria and performances quantifiers

ExampleMIMO case

A first robustness criteria


Performances limitations


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Objectives of any control system :

shape the response of the system to a given reference andget (or keep) a stable system in closed-loop, with desiredperformances, while minimising the effects of disturbancesand measurement noises, and avoiding actuatorssaturation, this despite of modelling uncertainties,parameter changes or change of operating point.



Nominal stability (NS): The system is stable with the nominal model (no

Objectives of any control system

mo e uncer a n y

Nominal Performance (NP): The system satisfies the performance

specifications with the nominal model (no model uncertainty)

Robust stability (RS): The system is stable for all perturbed plants aboutthe nominal model, up to the worst-case model uncertaintyincludin the real lant


Robust performance (RP): The system satisfies the performancespecifications for all perturbed plants about the nominal model, up to theworst-case model uncertainty (including the real plant).

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FEEDBACK STRUCTURE

In the following:SISO (Single Input Single Output) and MIMO systems(Multi Input Multi Output) are considered

Classical one degree-of-freedom structure

SOME CONTROL STRUCTURES


Two degree-of-freedom structure

RST structure

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FEEDBACK STRUCTURE

Classical one degree-of-freedom structure

Input Outputdisturbance

K(s) G(s)r(t)y(t)

di(t) dy(t)

+

-+

+ +

+

+

re erenceOutput

uP(t)ControlInput

u(t)PlantInput


n(t)

PLANT = G(s)CONTROLLER = K(s) FEEDBACK

Measurementnoise

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FEEDBACK STRUCTURE

Two degree-of-freedom structure

di(t) dy(t)

K(s)

G(s)r(t) y(t)

n(t)

+

- +

+ +

+

++

Kp(s)


FEEDBACKImproves tracking performance

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FEEDBACK STRUCTURE

RST structure

di(t) dy(t)

R

Gr y(t)

n(t)

+

-+

+ +

+

++

1/STu(t)


POLYNOMIALSA two DOF structure with:Kp= T/S and K=R/S

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FEEDBACK STRUCTURE

w eDisturbance

Controlled Output

GENERAL CONTROL CONFIGURATION

P

K

u y

Control Input Measured output


P is the generalized plant (contains the plant, the weights,the uncertainties if any) ; K is the controller

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SENSITIVITY FUNCTIONS

Outputdisturbance

r(t)di(t)

dy(t)reference

Inputdisturbance

e(t)

The output & the control input satisfy the following equations :

Firstly, SISO case

K(s) G(s)

n(t)

+

-+ +

++

Measurementnoise


)()()(1

1)(

)()()(1

)(

iy

iy

KGdKnKdKrsGsK

su

GdGKndGKrsKsG

sy

+

=

+++

=

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)()()(1

1)( GKndGdGKr

sKsGsy yi +++

=

)()()(1

1)( KnKdKGdKr

sGsKsu yi +

=

)()(1

1)(

sKsGsS

+=Sensitivity

Let us define the well known sensitivity functions:


)()(1)()()(sKsG

sKsGsT+=ComplementarySensitivity

Loop transfer function )()()( sGsKsL =

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Input and Output Performance analysisusing the Sensitivity functions

Outputdisturbance

Input

K(s) G(s)r(t) y(t)

n(t)

di(t)dy(t)

+

-+

+ +

+

+ +

re erence

Output

Measurementnoise

u(t)e(t)


The output & the control input performances can bestudied through 4 sensitivity functions only.

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KS(s)r(t)

)()()(1

1)(

)()()(1

1

)(

iy

iy

KGdKnKdKrsGsK

su

GdGKndGKrsKsGsy

+

=

+++=

-T(s)

-KS(s)

u(t)

di(t)

dy(t)

T(s)

SG(s)

S(s)

y(t)

di(t)

dy(t)


-KS(s)

n(t)

Input performance Output performance

-T(s)n(t)

33


)()()(1

1)( KnKdKGdKr

sGsKsu yi +

=

The transfer functionKS(s)should be upper boundedso thatu t does not reach the h sical constraintsKS(s)

r(t)

The effect of the input disturbancedi(t)on the plantinputu(t)+ di(t)(actuator) can be made small by

even for a large referencer(t)

-T(s)

-KS(s)

u(t)

di(t)

dy(t)


ma ing e sensi ivi y unc ion s sma

The effect of the measurement noisen(t)on the plantinputu(t)can be made small bymakingthesensitivity functionKS(s)small (in High Frequencies)

-KS(s)n(t)

Input performance

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)()()(1

1

)( GKndGdGKrsKsGsy yi +++=The plant outputy(t)can track the referencer(t)bymaking the complementary sensitivity function T(s)equal to 1. (servo pb)

The effect of the output disturbancedy(t)(resp. inputdisturbancedi(t)) on the plant outputy(t)can be made small by making the sensitivity function S(s) (resp.SG(s)) small

The effect of the measurement noisen(t)on the plantoutputy(t)can be made small by making the

T(s)

SG(s)

S(s)

y(t)

di(t)

dy(t)


Sometrade-offsare to be looked for

BUT S(s) + T(s) = 1

Output performance

-T(s)

n(t)

35


These trade-offs can be reached if one aims : to reject the disturbance effects in low frequency to minimize the noise effects in high frequency

KS(s)

-T(s)

u(

r(t)

di(t)

dy(t)

We will require: S and SG to be small in low frequencies to reduce

the load (output and input) disturbance effects onthe controlled output

T and KS to be small in high frequencies toreduce the effects of measurement noises on the

-KS(s)

-KS(s)

n(t)

T(s)

SG(s)

r(t)

di(t)


contro e output an on t e contro nput actuatorefforts)

S(s)

-T(s)

y(t)

dy(t)

n(t)

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PERFORMANCE ANALYSIS: CRITERIA

Time domain performances

Classical performance indices

Rise time : the time, usually required to be small, that it takes for the output tofirst reach 90 % of its final value

Settling time : the time after which the output remains within 5 % of its final value,which is also usually required to be small.

Overshoot: the peak value divided by the final value : should typically be 1.2 (20%) or less

Decay ratio: the ratio between the second and first peaks, which should typicallybe 0.3 or less


Steady-state offset: the difference between the final value and the desired final value,this offset is usually required to be small.

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Time domain performances : other criteria

ISE (Integral Square Error) yredtteJISE ==

;)(2


0

A better and more advisable index should

ITAE (Integral Timeweighted Absolute Error)

yredttetJITAE ==

;)(0


include the control input effect

( )

+=0

22)()( dttuRteQJeu

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Frequency domain performances: criteria

GAIN, PHASE, DELAY and MODULE MARGINS

The gain marginindicates the additional gain that would Good value for


take the closed loop to the critical stability condition

Thephase marginquantifies the pure phase delay thatshould be added to achieve the same critical stabilitycondition

The delay marginquantifies the maximal delay that should

Gm : >6dB

Good value form : > 30/40

M=


conditionThe module marginquantifies the minimal distancebetween the curve and the critical point (-1,0j): this is arobustness margin

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M

Frequency domain performances: criteria

DELAY MARGIN :acce table ure time-


delay before instability

M

M

=



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MODULE MARGIN


m n

jGKM +=

Good value M > 0.5


Bandwidth

The concept of bandwidth is very important in understanding the benefits and

trade-offs involved when applying feedback control. Above we considered peaks


o c ose - oop trans er unct ons, w c are re ate to t e qua ty o t e response.

However, for performance we must also consider the speed of the response, and

this leads to considering the bandwidth frequency of the system.

In general, a large bandwidth corresponds to a faster rise time, since high

frequency signals are more easily passed on to the outputs. A high bandwidth also


indicates a system which is sensitive to noise and to parameter variations.

Conversely, if the bandwidth is small, the time response will generally be slow,

and the system will usually be more robust.

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Bandwith :Loosely speaking, bandwidthmay be defined as the frequency range [w1,

w2] over which control is effective. In most cases we require tight control


a s ea y-s a e so w1= , an we en s mp y ca w2 e an w .

The word effective may be interpreted in different ways : globally it

means benefitin terms of performance.

Definition1: The (closed-loop) bandwidth, wS, is the frequency where

|S(jw)| crosses 3dB (1/2) from below.


Remark: |S|


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The gain crossover frequency :

Definition3: The bandwidth (crossover frequency), wC, is the frequency

where |L(jw)| crosses 1 (0dB), for the first time, from above.


Remark:It is easy to compute and usually gives

wS< wC< wT

Note that the rise time can often be evaluated as :


Trt

3.2=

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PERFORMANCE ANALYSIS: example

Position control of a DC motor, using internal speed feedback

A 1/k v 1/(p+1) 1/(np) U0/(2)

1/kv1/kvRCp/(RCp+1)

Ve +

-

+

-

Vsx



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ar

Va

lues

(dB)

S

-20

0

20

ar

Va

lues

(dB)

T

-5

0

5

Good disturbance rejection


Frequency (rad/sec)

Singu

l

10-2 10-1 100 101 102-80

-60

-40

Frequency (rad/sec)

Singu

l

10-2 10-1 100 101 102-20

-15

-10

lues

(dB)

KS

-20

0

lues

(dB)

SG

0

5

10

Good noise rejectionWB=15.3 rad/s WBT=27.1 rad/s


Frequency (rad/sec)

Singu

lar

V

10-2 10-1 100 101 102-80

-60

-40

Frequency (rad/sec)

Singu

lar

V

10-2 10-1 100 101 102-15

-10

-5

Bad: control input sensitiveto noise

Bad: Input disturbance (di)are not rejected

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2

2.5

Output disturbance dy

Input disturbance di


1

1.5


0 1 2 3 4 5 6 7 8 9 100

.

reference step

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WS=15.3 rad/s

WT=27.1 rad/s

Bandwith :

Wc=21 rad/s

And it holds :

wS< wC< wTPhase margin= 72.4 degGain margin = infModule margin < 1.5 db, MT=0.5db

>>[Gm,Pm,W180,Wc]=margin(sys)

>> MT=hinfnorm(T)MS=hinfnorm(S);


eg);Magnitude(dB)

Bode Diagrams

-100

-50

0

50

100From: U(1)

1

-80

n-Loop

Ga

in(dB)

Nichols Charts

-50

0

50

100From: U(1)

To:

Y(1)


Frequency (rad/sec)

Phase(d

100 101 102-180

-160

-140

-120

-100

To:Y(1)

c=27 rad/s

Open-Loop Phase (deg)

Ope

-180 -170 -160 -150 -140 -130 -120 -110 -100 -90-150

-100

49

WS=15.3 rad/s

WT=27.1 rad/s

Bandwith :

1

1.2

1.4


Wc=21 rad/s

It holds :

0.2

0.4

0.6

0.8

rise timewS< wC< wT


mstT

r 853.2 ==

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.50

50


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PERFORMANCE ANALYSIS : MIMO case

Sensitivity functions: MIMO caseOutput

disturbance

r(t) y(t)

di(t)dy(t)

+ + +

reference

Inputdisturbance

p outputsu(t)(t)

The output & the control input satisfy the following equations :

n(t)

-+ +

+ + Measurementnoise

m controlinputs


)()())()(( iym

iyp

KGdKnKdKrsusGsKI

nrsyss

=+

=

BUT : K(s)G(s) G(s)K(s)

51

Sensitivity functions

)()())()((

)()())()((

iym

iyp

KGdKnKdKrsusGsKI

GdGKndGKrsysKsGI

=+

++=+


pyypypyITSGKGKITGKIS =++=+= ,)(,)( 11

Outputand Outputcomplementary sensitivity functions:

Inputand Inputcomplementary sensitivity functions:

muumumu ITSKGIKGTKGIS =++=+= ,)(,)( 11


Properties

yu

mu

py

KSKS

KGKGIT

GKIGKT

=

+=

+=

1

1

)(

)(

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Ty(s)

S G(s)

r(t)

di(t)

KSy(s)

-Tu(s)

r(t)

di(t)


Sy(s)

-Ty(s)

y

dy(t)

n(t)

-KSy(s)

-KSy(s)

dy(t)

n(t)


Output performanceInput performance

)()())()((

)()())()((

iym

iyp

KGdKnKdKrsusGsKI

GdGKndGKrsysKsGI

=+

++=+

53

As MIMO framework is concerned, MIMO gain of the sensitivityfunctions is considered through Frequency-domain plots of the singularvalues (sigma using the Control toolbox, and using the Mu toolbox:


The analysis of SISO systems can then be extended, except for


stability margins. 5 sensitivity functions have to be studied (Sy, Ty, Tu, Ksy, SyG) The robustness margins are the maximum peak of S and T. These

may not be sufficient to ensure robustness properties and should becompleted by a robust stability analysis

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A first approach to ROBUSTNESS ANALYSIS

A control system is robust if it is insensitive to differences between theactual system and the model of the system which was used to design thecontroller

Introduction: Skegestad & Postlewaite

A method: these differences are referred as model uncertainty.

How to take into account the difference between the actual system andthe model ?

A solution: using a model set BUT : very large problem and notexact yet


e approac

determine the uncertainty set: mathematical representation

checkRobust Stability

checkRobust Performance

55




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MODULE MARGIN/Maximum Peak criteria


m n

jGKM +=

=M

MS

1

Also :


==S max

Good value MS < 2 (6 dB)

57


The MODULE MARGIN is a robustness margin.

SMM 1=

Indeed, the sensitivity function allows to qualify the robustness of the

control system, as

)()(1

)()(

sGsK

sGsKTBF +

=Closed-loop transfer function


In uence o p ant mo e ing errors on t eCL transfer function GsGsKTBF

BF

+=

)()(1Sensitivity function

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Advantage: good module margin impliesgood gain and phase margins

SM 1


SSMM

1

For MS=2, then GM>2 and PM>30

Last one : )(max jTMT =


Good value MT < 1.5 (3.5 dB)

59

Robust Stability analysis: SISO case

I(s)

G(s)

+

+

uy

u z

Let us consider the case :


The loop transfer function is then: ;)( IIIIpp LwLwIGKKGL +=+==

Therefore RS System stable Lp. Lp should not encircle the point -1

+


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PERFORMANCE SPECIFICATION

Objective : good performance specifications areimportant to ensure better control system

mean : give some templates on the sensitivityfunctions

For simplicity, presentation for SISO systems first.



Templates on the sensitivity functions

Robustnessandperformancesin regulation can be specifiedby imposingfrequential templateson the sensitivity functions.

If the sensitivity functionsstay withinthese templates, the control

objectivesare met.

These templatescan be used for analysisand/or design. In the latter theyare considered asweightson the sensitivity functions


The shapesoftypical templateson the sensitivity functions are given in thefollowing slides

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Template on the sensitivity function - Weighted sensitivity

Typical specifications in terms ofSinclude:

1. Minimum bandwidth frequency wS(defined as the frequency where|S(jw)| crosses 0.707 from below).

2. Maximum tracking error at selected frequencies.

3. System type, or the maximum steady-state tracking error,

4. Shape of S over selected frequency ranges.

5. Maximum peak magnitude of S, ||S|| < MS.


The peak specification prevents amplification of noise at highfrequencies, and also introduces a margin of robustness; typically weselect MS=2.

63


Template on the sensitivity functionWeighted sensitivity

,upper bound, on the magnitude of S, given by another

transfer function :

1,)(

1)(

SW

jWjS e

e


where We(s) is a weight selected by the designer.

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Template on the sensitivity function )()(11

)( sGsKsS +=s b+=

1

e

Generally = 0 is considered,MS


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Template on the sensitivity function KS(s) )()(1)(

)( sGsK

sK

sKS +=

s BC1 1 +=uBCu

Muchosen according to LFbehavior of the process(actuator constraints:saturations)


wBC influences robustness : wBC better limitation of measurement noises

and roll-off starting from wBC to reduce modeling errors effects

67


Template on the sensitivity function SG(s))()(1

)()(

sGsK

sGsSG

+=

SGMs

MSG

SG

SGs

Limitation of input disturbance effectson the output by the choice ofwSGzero static error for constant inputdisturbance



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35


k

The templates can be defined more accurately by transferfunctions of order greater than 1, as

,)(b

es

ssW

bS

+

+=

if we require a roll-of of 20*k dB perdecade is required


In the MIMO case the simplest way is to defined the templates asdiagonal transfer matrices, i.e. using (Msi, bi, I, .)

69

THE MIXED SENSITIVITY PROBLEM

TWSW Te

In terms of control synthesis, all these specifications can be tackled inthe following problem: find K(s) s.t.

SGWKSW SGu

which is called a mixed sensitivity problem. Often, the simpler followingone is studied:

1KSW

SWe


The latter allows to consider the closed-loop output performance as wellas the actuator constraints.

70


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36

OUTLINE


H norm, stability


H control design







,

Robust control design

71

H CONTROL APPROACH

H control is devoted to SISO systems as well as MIMO ones

Main advantage: the plant modelling errors as well as

frequency domain.

H control is strongly linked to the weighted sensitivityfunctions.


er ormance spec ca on s en o grea mpor ance nH control approach.

72


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37

H CONTROL APPROACH

GENERAL CONTROL CONFIGURATION

This approach has been introduced by Doyle (1983). The formulation

makes use of the general control configuration.

K

u y

w eDisturbanceand reference

Control Input

Controlled Output

Measured output

P is the eneralized lant contains the lant the wei hts the

=

2221

1211

PP

PPP


uncertainties if any) ; K is the controller. The closed-loop transferfunction is:

21

1

221211 )(),()( PKPIKPPKPFsT lew+==

73

H CONTROL APPROACH

w eDisturbanceand reference

Controlled Output

based on the GENERAL CONTROL CONFIGURATION

=

2221

1211

PP

PPP

K

u yControl Input Measured output

P is the generalized plant (contains the plant, the weights, the uncertainties ifany) ; K is the controller. The closed-loop transfer function is:

21

1

221211 )()( PKPIKPPsTew+=


H suboptimal control problem : Given a pre-specified attenuation level, a Hsub-optimal control problem is to design a stabilizing controller that insures :

=

))((max)( jTsT ewew

74


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38


Robust Control toolbox (MATLAB R2009)

% Generalized plant P is found with function sysic%

systemnames = 'G We Wu';

inputvar = '[ r(1);u(1)]';

outputvar = '[We; Wu; r-G]';

input_to_G = '[u]';

input_to_We = '[r-G]';

input_to_Wu = '[u]';

sysoutname = 'P';

cleanupsysic = 'yes';

sysic;

-


nmeas=1; nu=1;[K,CL,GAM,INFO] = hinfsyn(P,nmeas,nu,'DISPLAY','ON');

gopt

75

H CONTROL APPROACH

The overall control objective is to minimize some norm of the transferfunction from wto e, for example, the H norm.

H suboptimal control problem : Given a pre-specified attenuation level,

H control problem: Find a controller K(s) which based on the informationin y, generates a control signal uwhich counteracts the influence ofwon e,

thereby minimizing the closed-loop norm from wto e.


a su -op ma con ro pro em s o es gn a s a z ng con ro er ainsures :

=

))((max)( jTsT ewew

76


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39


How to consider performance specification in H control ?In practice the performance specification concerns at least twosensitivity functions (Sand KS) in order to take into account thetracking objective as well as the actuator constraints.

A simple example :

K(s) G(s)r(t) y(t) +

-

(t) u(t)


KSruGurKyrKu

SryrGKGuy

===

===

:forceactuator)()(

:errortracking)(

77


1,1

KSWSW ueObjective w.r.t sensitivity functions:

SrWe e=1New controlled out uts :

The performance specifications on the tracking error & on the actuator canbe given as some weights on the controlled output as follows :

We(s)

W s

e1(t)

e2(t)

KSrWe u=2


K(s) G(s)r(t) y(t) +

-(t) u(t)

78


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40


K s G sr(t) y(t) +

We(s)Wu(s)

e1(t)

e2(t)

w=re=(e1, e2)

T

Externalinputs Controlled

The associated general control configuration is :

-(t) u(t)

W

GWW

u

ee

0


K

u r-y Control InputsMeasured outputs

GI

79


The corresponding H suboptimal control problem is therefore to find acontroller K(s) such that :

=SWe

+

+

=

+==

IGKIKGWW

PKPIKPPKPFsT

ee

lew

1

21

1

221211

)(

)(),()(where

KSWu

ew


=

KSW

SW

u

e

u

80


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41

H CONTROL APPROACH

The solution of the H control problem is based on a statespace representation ofP, the generalized plant, thatincludes the plant model and the performance weights.

&

++=

++=

=

uDwDxCy

uDwDxCeP

22212

12111

21

The calculation of the controller, solution of the Hcontrol problem , can then be done using the Riccati


approach or the LMI approach of the H

controlproblem (see Zhou 98, Skogestad&Postlewaite 96).

81

H CONTROL SOLUTION

=

++=

++=

++=

22212

12111

21

22212

12111

21

DDC

DDC

BBA

P

uDwDxCy

uDwDxCe

uBwBAxx

P

&

Assumptions:

, 2 s a za e an 2, e ec a eNecessary for the existence of stabilizing controllers

(A2) D12 and D21 have full rank (resp. m2 and p2)Sufficient to ensure the controllers are proper, hence realizable

(A3) [A-j I, B2; C1 D12] has full column rankn+m2 for all

(A4) [A-j I, B1; C2 D21] has full row rankn+p2 for all


o ensure a e op ma con ro er oes no ry o cancepoles or zeros on the imaginary axis which would result in CL instability

(A5)

not necessary but simplify the solution (can be relaxed)

[ ]

=

===

2

21

21

1

2121122211

0,0][,0,0

p

T

m

T

ID

D

BIDCDDD

82


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42

H CONTROL SOLUTION

Theorem1 : Under the previous assumptions, there exists acontroller K such that ||Tew||0, ( ) 01122112 =+++ CCXBBBBXXAAX TTTT

(iii)

imaginary axis TT ACC 11

has no eigenvalues on theimaginary axis


(iv) Y>0, ( ) 01122112 =+++

TTTT BBYCCCCYAYAY

( ) 2


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43

H CONTROL APPROACH

Advantage : if We and Wu are chosen according to the previoustemplates on the sensitivity functions, and if a solution does exist, thecontrol objectives are met.

This procedure can be easily performed by solving two Riccati equations ortwo LMIs.Using the Riccati formulation, the minimal value of can be approached byg-iteration (dichotomy).Using LMI formulation, the minimal value of is solved as an optimizationproblem


t s a so ava a e n c ass ca contro so tware, e.g.

MATLAB.

This can be completed by robust analysis and/or design according tomodel uncertainties.

85

H CONTROL APPROACH

LMI solution :

Linear Matrix Inequalities (LMIs) are powerful design tools in controlengineering, system identification and structural design.

Main advantages :

Many design specifications and constraints can be expressed as LMIs.

Once formulated in terms of LMIs, a problem can be solved exactlyby

efficient convex optimization algorithms.

While most problems with multiple constraints or objectives lack


analytical solutions in terms of matrix equations, they often remain

tractable in the LMI framework.

86


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44

H CONTROL APPROACH

0=+++ CCXXBBXAXA TTT

The Bounded Real Lemma (Scherer) :Let M(s)=C(sI-A)-1B with A stable. Then ||M||


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45

OUTLINE


H norm, stability


H control design







,

Robust control design

89

UNCERTAINTY AND ROBUSTNESS

A control system is robust if it is insensitive to differences between theactual system and the model of the system which was used to design thecontroller

Introduction: Skegestad & Postlewaite

A method: these differences are referred as model uncertainty.

How to take into account the difference between the actual system andthe model ?

A solution: using a model set BUT : very large problem and notexact yet


e approac

determine the uncertainty set: mathematical representation

checkRobust Stability

checkRobust Performance

90


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46


Lots of forms can be derived according to both our knowledge of thephysical mechanism that cause the uncertainties and our ability torepresent these mechanisms in a way that facilitates convenientmanipulation.

Several origins :

Approximate knowledge and variations of some parameters

Measruement imperfections (due to sensor)

At hign frequencies, even the structure and the model order isunknown (100% is possible)


Controller implementation

Two classes: parametric uncertainties / neglected or unmodelleddynamics

91


Example (Skogestad-Postlewaite, 96):

= k sh

,,,1+ sp

Nominal model: parameters h=k==2.5

25.05.3 +s


1+smmpMultiplicative uncertainties :

92


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47


Wm(s)

Relative uncertainties (Gp-G)/G

100

Magn

itu

de


10-2

10-1

100

101

10-1

Frequency

variations

93




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48

max0,

1

1)()(

+=

ssGsG


A simple example of unmodelled dynamcis

Then :

+ ;

11

)(

)(

max

max

0 j

j

sG

sG

1et1

avec)()(1)(

)(

max

max

0


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49

baabaas

sG +


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50

Now there is two ways to tackle the control problem underuncertainties:

UNSTRUCTURED UNCERTAINTIES: we i nore the structure of


considered as a full complex perturbation matrix, such that ||||


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51


Now the transfer matrix from w to zNow the transfer matrix from w to z

1

,),(

NNINNNFwhere

wNFz u+=

=

u

Objectives:Objectives:NSand;1,stableis),(RS

NSand;1NP

stableinternallyisNS

22


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52


ROBUST STABILITY :SMALL GAIN THEOREM

M=N11

uy

Small Gain Theorem: Su ose . Then the closed-loo s stem isRHM


well-posed and internally stable for all such that :

RH

/1)()

/1)()


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53


Robust Stability analysis for unstructured uncertainties

Application of the small gain theorem to the different uncertainty types.

1:1.. += KSwCNSswGG

1:;1..;)(

1:;1..;)(

1:;1..);(

1:;1..;)(

1

1

+=

+=

+=

+=

uiIiIiIiIiIp

yiOiOiOiOiOp

uIIIIIp

yOOOOOp

yp

SwCNStswIGG

SwCNStsGwIG

TwCNStswIGG

TwCNStsGwIG


This gives some robustness templates for the sensitivity functions.

However this may be conservative.

10 5


Robust Stability analysis: SISO case

I(s)

G(s)

+

+

uy

u z

Let us consider the case :

The loop transfer function is then: ;)( IIIIpp LwLwIGKKGL +=+==

Therefore RS System stable Lp. Lp should not encircle the point -1

+


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54


Robust Performance analysis for unstructured uncertainties

Robust control : analysis and design Olivier Sename 201010 7


Robust Performance analysis: SISO case

I(s)

G(s)

+

+

uy

u z

;)( IIIIpp LwLwIGKKGL +=+==

RP if NP is true for all plants:RP if NP is true for all plants:..

,,1

,,1

+


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55


Often, for SISO systems, when we have NP and RS we get RP. This is nota big issue for SISO systems.

For MIMO, systems, this approach may lead to very conservative results.,

needs to consider the structured singular value. It is defined as :

Find the smallest structured which makes det(I-M)=0.



Robust Stability analysis:

structured uncertainties case



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56

6. ROBUSTNESS ANALYSIS

uncertaintiesFictive uncertainties: full

complex matrix representingthe H norm specifications

r

uncertainties:block diagonal

matrix

Gw eDisturbances& references

Controlled out utsW i W o

f


uy

Control input Measured outputKP

N

11 1

r f

z Uncertainties outputsvUncertainties inputs

6. ROBUSTNESS ANALYSIS

1. Definition of the real uncertainties r and of the transfer templatese/w thanks to Wi et Wo

w eDisturbances& references

Controlled outputsN=

N N

zv zw

ev ew


2. Evaluation of

3. Computation of the admissible intervals for each parameter

(N ) , (N ) and (N)f rew zv

11 2


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57

ROBUSTNESS ANALYSIS

-analysis principle : Stable system for all uncertainties

6.

M

such that :

If and only if :

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Robust Control analysis and design