Maximum Output Amplitude of Linear Systems for certain Input Constraints

for certain Input Constraints

WolfgangReinelt

DepartmentofElectricalEngineering

Linkoping University,S-58183Linkoping,Sweden

WWW:

http://www.control.isy.liu.se/~wolle/

Email:

wolle@isy.liu.se

June 1999

REG

LERTEKNIK

AUTO

_{MATIC CONTR}

OL

LINKÖPING

Reportno.: LiTH-ISY-R-2165

Technical reports from the Automatic Control group in Linkoping are available by

anony-mous ftp at the address

ftp.control.isy.liu.se

. This report is contained inthe portable

for certain Input Constraints

WolfgangReinelt

June 1999

Abstract

Wedeterminethemaximumoutputamplitudeofasystem,whenitsinputfullscertain

constraints. Inparticular,theamplitudeandtherateofchange(i.e.therstderivative)

havetobebounded. Weshowpropertiesoftheworstcaseinputandpresentanalgorithm

thatallowstoconstructthisinputandcalculatesthemaximumamplitudeoftheoutput.

Thisproblemhasapplicationswithinthenon-conservativedesignofcontrollersforplants

withboundedinputs. Nevertheless,itisinterestingasasystemtheoretictaskitselfand

thereforestatedseperately.

Keywords: LinearSystemTheory,Maximum OutputAmplitude,HardConstraints.

1 Introduction and Motivation

Linearsystemswithboundedinputsproduceboundedoutputs,stabilityprovided. Giventhe

maximumamplitudeoftheinputandcalculatingthemaximumamplitudeoftheoutputisthe

problemofdeterminingthe

1

-normofthesystem.Amorechallengingtaskistocalculatethe

maximumoutputamplitudeforasystemwithinputsthatfullcertainadditionalconstraints.

Formotivationwelookatprocessengineeringforexample,andinparticularatthe

liquid-levelina tank,interpretedas atimesignal. Notonly the liquid-level inatankis bounded

(bythetank'sheight),additionallytheliquidcannotchangeitslevelarbitrarilyfast. Sothis

signalisboundedinamplitude,aswellasitsrstderivativedue tothetime. Regardingthe

liquid-levelasareferencesignalforacontrolsystemwithcontrolsignallet'ssaytheopening

angleofsomevalve(feedingthetank)itisquitereasonabletodesignacontrollerthattakes

careforthepossiblereferencesignalsfortheliquidlevelaswellasfortheamplitudebound

forthecontrolsignal. Theboundforthecontrolsignalis,looslyspeaking,therangebetween

fullyopenandfullyclosed.

Interpretingthisinsystem-theoreticterms,thetransferfunctionfromreferencesignalto

controlsignalisasystemwithaninputthatfullscertainconstraints(bounds foramplitude

and rst derivative) and a bounded output signal. Task of the controller design is now to

designthecontrollersothatthemaximumamplitudeofthecontrolsignalisnothigherthan

"fullyopen" inorder toprevent wind-up and stability problems. Therefore itis obviously

helpfultocalculatethemaximumoutputamplitudeforagivensystemwithprescribedinput

constraints onamplitudeandrstderivative.

At a rst glance it might be suÆcient to neglecet the constraint on the reference

resultingcontroller willbetooconservative. Duringrun-time,theamplitudeofthecontrol

signalwillbemuchsmallerthantheallowedone;thiseectwillobviouslyreducethecontrol

systems'sspeedandperformance.

However, the solution of the technicalproblem has many applications in the design of

controllers thatguaranteeboundedcontrolsignals[1]. In particular,optimalcontrol

prob-lems can be solved [4, 5], as well as robust control problems [6, 7]. The term optimal

refers to the fact, that the constraint

|

r(t)

_

| ≤

_

R

allows a minimization of the error signal

|e(t)| = |r(t) − y(t)|

. An extension for the discrete time case is possible [3, 2, 8]. These

controlapplicationswillbepresentedseperately.

Thisworkisorganizedasfollows: section2denestheproblem,whichwillthenbesolved

in section 3. Section 4 sketches shortly the numerical realisation. The ideasare extended

to the multivariable case in section 5, suitedfor multivariable control systems. Section 6

illustrates the theory with anexample. Conclusions aredrawnin section 7, related works

as well as thedirectionof future research are also given. Problem statementand solution

aredue to[1,4], anumericalimplementationwasoutlinedin[5]and theextensiontothe

multivariablecasewasgivenin[6, 7].

2 Problem Statement

We examinea linear and time invariantstable system, which is represented by its transfer

function

G(s)

resp.itsimpulseresponse

g(t)

. Wepostponetheextensiontothemultivariable

case tosection 5and concentrate on the SISO case. The input is denoted by

r

, the output

by

x

. The followingconstraints hold forthe continuousand piecewise 1 dierentiable input signal

r

:

|r(t)| ≤ R

(1)

|

r(t)

_

| ≤

_

R

(2) for

t > 0

,where

R,

_

R > 0

aregivenconstantvaluesand

r(t) = 0,

t

≤ 0.

(3)

Wecallthosereferencesignals,whichfulleqns.(1-3)

(R,

_

R)

-admissible,orshort

r

∈ A(R,

_

R)

. Wearelookingforthemaximumamplitude

X

m

(t)

oftheoutput

x

(uptoacertaintime)with

(R,

R)

_

-admissibleinputs,i.e.

X

m

(t) :=

sup

r

∈A(R,

R)

_ sup

0<τ

≤t

|x(τ)| =

sup

r

∈A(R,

R)

_ sup

0<τ

≤t

|g(τ) ? r(τ)|.

(4)

where"

?

"is the convolution:

g(t) ? r(t) =

R

_t

0

g(τ)r(t − τ)dτ

. Quiteclear fromlinear system

theoryis,thatforsystemswiththeonlyinputconstraint(1),themaximumoutputamplitude

isgivenbymax

t

≥0

|x(t)| = R

R

_∞

0

|g(τ)|dτ

,producedbythebang-banginput:

r(t − τ) = R

·

sign

(g(τ)).

(5)

Thustheproblemistrivialunlesstheadditionalconstraint(2)isimposed.

1

We now turn to the construction of the maximum output amplitude as stated in eqn.(4)

according to [4]. Basic idea is rst to show some properties of the input signal

r

, which

produces the output with the maximum output amplitude. In the following, we call this

inputsignaltheworstcaseinput. Thisstrategyismotivatedbytheexistenceofaworstcase

inputinthesimplecaseineqn.(5). Westartwiththecalculationoftheoutputbyconvolution

foracertaintimestamp

t

:

x(t) =

Z

t

0

g(τ)r(t − τ)dτ =:

Z

t

0

g(τ)r

t

(τ)dτ

(6)

andabbreviatethe"timeinverted"inputsignalby

r

t

(τ) := r(t − τ)

withthefollowing conse-quencesfortheconstraints(1-3):

|r

t

(τ)

| ≤ R, τ < t

(7)

|

r

_

t

(τ)

| ≤

_

R,

τ < t

(8)

r

t

(τ)

=

0,

τ

≥ t

(9)

3.1LemmaThefunction

X

m

(t)

ismonotoneincreasing(i.e.notdecreasing)intime

t

.

There-fore, the maximum amplitude as dened in eqn.(4) appears for sure for

t

→ ∞

, so

X

m

=

lim

t→∞

X

m

(t)

isthemaximumoutputamplitude.

Proof. Let

t

0

> 0

and

r

t

0

∈ A(R,

_

R)

theinput 2

thatproducesthemaximumamplitude

X

m

(t

0

)

.

For

t

1

> t

0

dene

r

t

1

(τ) = r

t

0

(τ), 0

≤ τ ≤ t

0

resp.

r

t

1

(τ) = 0, t

0

< τ

≤ t

1

. Clearly

r

t

1

∈ A(R,

_

R)

andfromeqn.(6) followssup

0<τ

≤t1

|x(τ)| =

sup

0<τ

≤t0

|x(τ)|

andthus

X

m

(t

1

)

≥ X

m

(t

0

)

.

2

3.2DenitionSupposethereexists aninput

r

∞,o

=: r

o

withmaximumoutputamplitude

X

m

. Then

−r

o

producedobviouslythemaximumoutputamplitude

X

m

,too. Foroneofthem,say

r

o

,holds

X

m

=

Z

_∞

0

g(τ)r

o

(τ)dτ

≥ 0,

(10)

i.e., ineqn.(4) the absolute valueis super uous. In thefollowing, we constructthisworst

caseinput 3

thatproduces thismaximumoutputamplitudeaccordingtoeqn.(10).

3.3Algorithm (Construction ofanauxiliaryinput)Let

r

∈ A(R,

_

R)

beanarbitraryadmissible input. Weconstructan auxiliaryinput

r

H

for

r

uniquelyby the steps givenbelow (gure 1 illustratestheconstruction). Thesetofallpossibleauxiliaryinputs (i.e.allsignalswiththe

sameproperties)isdenotedby

A

H

(R,

_

R)

.

1. Let

t

i

, i = 1, . . . , N

thezerosof

g

. Dene

r

H

(t

i

) = r(t

i

)

.

2. If

g(t) > 0

in

(t

i

, t

i+1

)

, let

r

_

H

(t) = +

_

R

in the neighbourhood of

t

i

and

r

_

H

(t) = −

_

R

in

the neighbourhood of

t

i+1

. In the case that this denition leads to the non-unique situationthat the two'slopes' intersect insome

t

∗

∈ (t

i

, t

i+1

)

, let

r

_

H

(t) = +

_

R

in

[t

i

, t

∗

]

resp.

r

_

H

(t) = −

_

R

in

[t

∗

, t

i+1

]

. Denenally

r

H

=

min

{r

H

, +R

}

. 2

uniquenessisnotnecessaryforthisargumentation.

3

0

0

0

g(t)

_

r

H

(t)

+R

−R

+

R

_

−

R

_

r

H

(t)

r(t)

t

t

t

t =

∞

t =

∞

t =

∞

t

1

t

2

t

3

Figure1: Construction ofauxiliaryinput

r

H

forgiveninput

r

.

3. If

g(t) < 0

in

(t

i

, t

i+1

)

, do as in step 2. but with changed signs for

r

_

H

and resulting

obviousmodications.

4. Finally,choose

|

r

_

H

(t)

| =

_

R

forlarge times

t

inordertoensurelim

t

→∞

r

H

(t) = 0

, arising fromconstraint(9).

3.4Corollary thefollowingpropertiesoftheauxiliaryinputareclearbyconstruction:

1.

A

H

(R,

_

R)

⊂ A(R,

R)

_

, i.e.anauxiliaryinputisalsoadmissible.

2. Twodierentinputs

r

1

, r

2

∈ A(R,

_

R)

with

r

1

(t

i

) = r

2

(t

i

)

havethesameauxiliaryinput.

3.

r

H

(t)

≥ r(t)

for

g(t)

≥ 0

and

r

H

(t)

≤ r(t)

for

g(t)

≤ 0

.

4. Fixanadmissibleinput

r

andsupposeanarbitraryadmissiblesignal

r

∗

_{6= r}

H

with prop-erty3.,then

R

_∞

0

g(τ)r

H

(τ)dτ >

R

_∞

0

g(τ)r

∗

(τ)dτ

.

5. Themaximumwidthofthepulsesof

r

_

H

inAlgorithm3.3is givenby

T = 2

R

_

R

.

3.5TheoremThe worstcaseinputisauxiliaryinput:

r

o

∈ A

H

(R,

_

Proof. Forall

r

∈ A(R,

_

R)

thefollowingholds byconstructionof

r

H

,seeCorollary3.4(3.):

Z

_∞

0

g(τ)r(τ)dτ

≤

Z

_∞

0

g(τ)r

H

(τ)dτ

(11)

and"

=

"holdsonlyfor

r

≡ r

H

(exceptthecase

g

≡ 0

). Assume

r

o

∈ A(R,

_

R)\A

H

(R,

R)

_

,thenthe

constructionofanauxiliaryinput

r

oH

is possible(because

r

o

is admissibleinput). Applying eqn.(11)to

r = r

o

:

X

m

=

Z

_∞

0

g(τ)r

o

(τ)dτ <

Z

_∞

0

g(τ)r

oH

(τ)dτ

(12)

which contradicts the denition of

X

m

as the maximum output amplitude. Consequently

r

o

∈ A

H

(R,

R)

_

.

2

Until now, we did not construct a unique worst case input thatleads to the maximum

outputamplitude,butweshowedsomenecessarypropertieswhicharelistedinthefollowing

3.6LemmaThefollowingnecessarypropertiesoftheworstcaseinput

r

o

hold:

1. The(derivativeofthe)worstcaseinputhasapulse-shape:

r

_

o

(t)

∈ {±

_

R, 0}

,where_

r

o

(t) =

0

implies

|r

o

(t)

| = R

.

2. Thewidthofthesinglepulsesisconstrainedby

T = 2

R

_

R

. 3. Twoadjacentpulseshavedierentsigns.

4. lim

t

→∞

r

o

(t) = 0

and

|

lim

t

→∞

r

_

o

(t)

| =

_

R

.

The previous Lemmastates mostly properties of

r

_

o

. Partial integration of eqn.(10)

to-getherwithLemma3.6(4.) givesanexpressionforthemaximumoutputamplitudeinthese

terms,where

s

isthestepresponseofthesystem(i.e.

s = g

_ ):

X

m

= −

Z

_∞

0

s(τ)

_

r

o

(τ)dτ

(13)

Despitetheminusineqn.(13),

X

m

≥ 0

holdsbyDenition3.2. Itappearsonlyduetothe partialintegration.

Lookingontoeqn.(13)andknowingtheshapeoftheworstcaseinputasstatedinLemma3.6,

thesolutionisquiteclear: inordertomaketheintegralmaximal,putsomepulses(of

maxi-mumwidth,seeCorollary3.4(4.)) inthenearofextremaofthestepreponse: positiveones

inthenearoftheminimaandnegativeonesinthenearofthemaxima. Inthefollowing,we

willprove this. Inorder nottooverloadthediscussion withtechnicaldetails,wemakethe

1. Lettheimpulseresponse

g

haveonlyanitenumber

N

ofzeros,i.e.thestepresponse

onlyanitenumberofextrema.

2. Lettherstextremumof

s

tobea(local)maximum.

Undertheseassumptions,themaximumamplitude

X

m

is givenby:

X

m

=

R

_

N

X

i=1

(−1)

i+1+k

Z

t

_i

00

t

_i

0

s(t)dt + R(−1)

N+k

lim

t

→∞

s(t).

(14)

The lastpart of the sum exists becausethe system is stable. The pairs

(t

0

i

, t

i

00

)

refer to the unknownpositionsofthepulses. Additionally,thesignofthepulsesisstillunknown,

there-fore we added

k

∈ {0, 1}

in eqn.(14). Obviously the problem is solved, when weknow the exactlocationofthepulsesandtheirsign. Wemakeonemore

3.8TemporaryAssumptionLettheextremaofthestepresponsehaveadistance

> 2T

,which

ensuresthatallpulseshavemaximumwidth

T

,i.e.

t

00

i

= t

i

0

+ T

.

3.9TheoremNecessaryconditionforamaximum

s(t

0

i

) = s(t

i

0

+ T )

,suÆcientconditionis

k = 0

ineqn.(14).

Proof. Necessaryforamaximumis

∂X

m

∂t

_i

0

= 0

forall

i

,whichleadsimmediateleyto

s(t

0

i

) = s(t

i

0

+

T )

. SuÆcientis thattheHessianis negativedenite,whichleadsdirectlyto

_

R(−1)

i+k

_(g(t

0

i

+

T ) − g(t

_i

0

)) < 0

thus

k = 0

, becauseofassumingarstlocalmaximumof

s

inAssumption3.7.

2

ApplyingTheorem3.9toeqn.(14),wegetthemaximumoutputamplitudeby

X

m

=

R

_

N

X

i=1

(−1)

i+1

Z

t

_i

0

+T

t

_i

0

s(t)dt + R(−1)

N

lim

t

→∞

s(t).

(15)

whereweplacethepulses

(t

0

i

, t

i

0

+ T )

,sothat

s(t

0

i

) = s(t

i

0

+ T )

holdswithanegativesignunder

themaximaof

s

andwithapositivesignundertheminimaof

s

.

3.10Remarkontemporaryassumptions3.7and3.8:

1. Thestabilityofthesystemensures

g(t)

→ 0

forlarge

t

,i.e.

s(t) =

const. Inthecase,that

g

has a innite numberof zeros the maximum output amplitude is givenby eqn.(15) withanarbitraryprecisionwhen usingarbitrarymanyzeros(i.e.

N

→ ∞

)Foranexact

proof,wereferto[4 ,AppendixA2.1].

2. Supposearstlocalminimumfor

s

. Inthiscase,thesuÆcientconditionwouldleadto

k = 1

, i.e.toachangeofsignsforallpulses. We cansimplytreatthesedierentcases

bytaking theabsolutevalueofthesumineqn.(15).

3. The"well-distinctness"oftheextrema,assumedinAssumption3.8ensuresthatpulses

withwidth

T

posed aroundtheextremaof

s

willnotintersecteachother. Technically,

this assumption simplies the proof of Theorem 3.9. In general, we can prove that

R

t

_i

00

t

_i

0

g(t)dt = 0

is necessary and suÆcient for a maximum. The proof is of technical

T

,isviewedasonemaximumresp.minimum. Fortheentirediscussionofallcases,we referagainto[4, AppendixA2.2].

3.11 Remark For _

R

→ ∞

(i.e. no restriction on the input's speed), the worst case input convergestothebang-banginput,asgivenineqn.(5).

4 Numerical Algorithm

Theorem 3.9 and Remark 3.10 give us necessary and suÆcient conditions to construct the

worstcaseinput. Themaximumoutputamplitude canbecalculatedwithhelp oftheworst

caseinputandeqn.(15). Werestrictourselvesonanotveryformal descriptionofthe

algo-rithmthatconstructsthe worstcaseinputnumerically. Numericaldetailsand thehandling

ofspecialcasesareoutlinedin[5].

1. Determineextremaofthestepresponse.

2. Putinitialpulseswithcorrectandalternatingsignsundereachextremumof

s

.

3. Makethemwider,ensuring

s(t

0

i

) = s(t

i

00

)

,aslongastheydonotintersect.

4. Inthecaseofintersections,handle"not welldistinct"pulsesseparately.

5. Calculatethemaximumoutputamplitudefromeqn.(15).

The problem canalso be viewed as a nonlinear optimisation problem in the pulse

po-sitions: determine

t

0

i

, t

i

00

, so that rhs of eqn.(15) becomesmaximal. Indeed, a solution by nonlinearoptimisationispossible,havingagoodinitialvaluegivenbystep(2.) ofthe

pro-cedureabove;thissolution isoutlinedin[6].

5 Multivariable Case

Weextendthepreviousresulttothecaseofmultivariablesystems,i.e.

r

and

x

arenow

vec-tor signals[6 ]. What wehaveinmind withthisextensionis thetreatmentof multivariable

control systems with constraint control signals, i.e.weregard

x = u

and

r

is the (external)

referencesignalof thecontrol system. Therefore, itis usefultorestrict theinput

r

compo-nentwise:

|r(t)|  R, t > 0

(16)

|

r(t)

_

| 

_

R, t > 0

(17)

r(t)

=

0, t

≤ 0.

(18)

inanalogy toeqns.(1-3). Read



asacomponentwise

≤

and evaluate

| · |

therefore compo-nentwisely. Consequently, wecall the set of allsignals

r

fullling these constraints (

R,

_

R

)-admissible, with

R,

_

R

arenow beingvectors with positive entries. Furthermore, we dene themaximumoutputamplitudecomponentwiseby

wherethe

X

i,m

aredenedasinLemma3.1.

Letusrstlookontoasystemwithone output

x

and

k

inputs

r = (r

1

, . . . , r

k

)

T

_{∈ A(R,}

R)

_ .

Then

x(s)

isgivenby

x(s) = G

1

(s)

· r

1

(s) +

· · · + G

k

(s)

· r

k

(s)

(20)

Weabbreviate

x

~

i

(s) = G

i

(s)

· r

i

(s)

. Now wearelookingforthemaximumoutputamplitude

X

m

. Usingeqn.(20),themaximumoutputamplitudeisgivenby

X

m

=

k

X

i=1

~

X

i,m

.

(21)

Itfollowsdirectly,that

X

m

isachievedforacertainvector

r = (r

1

, . . . , r

k

)

T

_{∈ A(R,}

R)

_ .

Inthemultivariablecasewith

n

outputs,wesimplyapplytherststep: accordingtothe

denition,thecomponents

X

i,m

of

X

m

canbecalculatedasinequation(21).

6 Illustrative Example

Weexaminethesystemrepresentedbythetransferfunction

G(s) =

s

2

_{+ 0.4}

s

2

_{+ 1.4s + 1.0}

,

with input constraints

R = 1.0

resp. _

R = 0.8

. According to Lemma 3.6 (2.), the maximum

pulsewidthis

T = 2.5

. Numericalsolution byconstructionoftheworstcaseinputyields

X

m

= 0.76.

(22)

Fig. 2 shows the construction of the worst case input with pulses of

r

o

located at

[0, 1.20]

,

[1.20, 2.39]

,

[5.24, 7.74]

and at of width

T/2

at innity (in reversed time) with signs "- +

-+" following max-min-max sequence of the step response. For simulation of this worst

case input we need to reverse this reversed time, therefore we choose the "innite" time

to

t

∞

= 120

. With this transformation, we receive the depicted worst case output with

a maximum amplitude of

x(t

m

) = 0.73

for

t

m

= 108.80

as a good approximation for the

maximumamplitudeascalculatedineqn.(22).

7 Conclusions and Related Works

Weshowedconditionsoftheworstcaseinputwithboundedamplitudeandspeed,that

pro-duces thatmaximumoutputamplitudeforagiven(stable)multivariablesystem. A

numeri-calalgorithmwassketchedtoconstructthisworstcaseinputandtocalculatethemaximum

outputamplitude. Thisis,aswebelieve,animportantstepwithinthenon-conservative

con-trollerdesignforsystemswithhardboundsonthecontrolsignal. Minimisationoftheerror

signalinacontrolsystemispossiblewiththeseassumptionsonthereferencesignal.

More-over itenables us to check the maximumamplitude anarbitrary signal within the control

0

1

2

3

4

5

6

7

8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

s(t)

(a)Step response

s

with calculated posi-tionofthepulsesfortheworstcaseinput.

0

20

40

60

80

100

120

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t

r_o(t)

(b)Worstcaseinput.

0

20

40

60

80

100

120

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t

y(t)

(c)Worstcaseoutput.

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