Guaranteed cost estimation and control for a class of nonlinear systems subject to actuator saturation

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European Journal of Control

journalhomepage:www.elsevier.com/locate/ejcon

Guaranteed cost estimation and control for a class of nonlinear systems subject to actuator saturation

Damiano Rotondo

^a,∗

, Mariusz Buciakowski

^b

aFaculty of Science and Technology, Department of Electrical Engineering and Computer Science (IDE), University of Stavanger, Kjell Arholmsgate 41, Stavanger 4036, Norway

bInstitute of Automation, Electronic and Electrical Engineering,University of Zielona Góra, ul. Podgórna 50, Zielona Góra 65–246, Poland

a rt i c l e i nf o

Article history:

Received 16 September 2020 Revised 28 June 2021 Accepted 7 July 2021 Available online 16 July 2021 Recommended by Prof. T Parisini Keywords:

Guaranteed cost estimation Guaranteed cost control Nonlinear systems Actuator saturation Linear matrix inequalities

a b s t r a c t

Theproblemsofguaranteedcostestimation(GCE)andguaranteedcostcontrol(GCC)concerndesigning astateobserveroracontroller,respectively,suchthatsomeperformanceismaintainedbelowanupper bound. Thispaperprovidesamatrixinequality-basedobserver/controller design procedureto perform GCEandGCCinaclassofnonlinearsystemsaffectedbyactuatorsaturation.Inparticular,thisclassof systemscorrespondstothoseforwhichtheoriginofthestatespaceisanequilibriumpointwhennull inputs areconsidered, and thenonlinearity isdifferentiablewith respecttothe stateand linearwith respecttothesaturatedinput.Simulationresultsobtainedusinganumericalexampleandarotational single-arminvertedpendulumareusedtoillustratetheeffectivenessoftheproposeddesignprocedure.

© 2021TheAuthor(s).PublishedbyElsevierLtdonbehalfofEuropeanControlAssociation.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Theproblemofoptimalcontrolconsistsinfindingacontrollaw for a given system so that some performance criterion is mini- mized.Foranonlineargenericsystem,thisproblemcanbesolved using Ponytryagin’s maximum principle or solving the Hamilton- Jacobi-Bellman (HJB) equation [8,13,49]. However, in most ofthe cases,obtainingtheanalyticalsolutionisahardtask,whichmoti- vated thesearchforalternative approachestoperformtheabove- mentioned minimization. A quite successful approach is the so- calledguaranteedcostcontrol(GCC), whichwasproposedfirstby ChangandPeng[3]asawaytoguaranteetheperformanceforun- certain systems by requiring it to be below some upper bound.

Early results on GCC were obtained in the 1990s by Ian R. Pe- tersen andhiscolleagues,whostudiedthedesignofrobuststate- feedbackcontrollersthatminimizeanupperboundonaquadratic cost function [25,26]. A computationally efficient framework for findingtheoptimalguaranteedcostcontrollerwasprovidedbylin- earmatrixinequalities(LMIs:see[31]foratutorial),suchthatsev- eral approacheswere developed, e.g. [6,43]. LMIs were alsoused in[19]toachieveGCCinbilateralteleoperationsystemswhichin- cluded time-varyingdelays and model uncertainty. Although the initial attention of the research community was devoted to lin-

∗Corresponding author.

E-mail address: [email protected] (D. Rotondo).

eartimeinvariant (LTI)systems,soonitwasdriventowardsother classes of systems which could take into account variability in time, for example linear systems with varying parameters (LPV) [28,30] or fuzzy Takagi-Sugeno (TS) systems [9,41,42,46]. GCC is still a very active field ofresearch, withseveralworks appearing everyyearintheliterature,mainlydealingwithnonlinearsystems, e.g.[11,15,33,39,45].

Strictly related to GCC is the problem of guaranteed cost es- timation (GCE) in which instead of designing a controller, one wishes to design a state observer that minimizes some upper boundonaperformancecriterion,whichisafunctionoftheesti- mationerrorandthemeasurementnoise.[25,26]showedthatGCE extendedthemuchcelebratedKalmanfiltertouncertainsystems.

A GCE-based approach was later developed by Petersen [24] us- ing a classof state estimators that includecopiesof theglobally Lipschitz system nonlinearities within them. More recently, Ishi- hara et al. [12] developed an approach based on regularization and penalty functions to solve the optimal filtering problem for discrete-time systemswith norm–bounded parametric uncertain- ties.

Anothertopicthathasattractedmuchattentionby thecontrol theory research community is how to deal withsaturation non- linearities.Saturation canbe found everywhere inphysicalappli- cations,since real-worldactuators areconstrainedin thenumber of deliverablecontrol actions. The control techniquesthat ignore theseactuatorlimitscan beaffectedby degradedperformance or

https://doi.org/10.1016/j.ejcon.2021.07.002

0947-3580/© 2021 The Author(s). Published by Elsevier Ltd on behalf of European Control Association. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

instability oftheclosed-loopsystem.Hence,theanalysisandsyn- thesis of control systems with saturating actuators has been in- vestigatedbyseveralworks,seee.g.[4,20,27,37,38].Thedeveloped approachescanbedividedintotwomaincathegories:thetwo-step paradigm(alsoreferredtoasanti-windupcompensation[14,50])ig- noresthesaturationatthecontrollerdesignstep,andhandlesitby adding acompensator; onthe other hand,theone-step paradigm (alsocalleddirectcontroldesign[7,36])takesintoaccountthesatu- rationduringthecontrollerdesignphase.Amongthemostrecent results concerning this topic, one may mention [48], where the problemofinputsaturationwassolvedbyintroducinganauxiliary design system, Shahri et al. [32], which employed the Lyapunov direct method forthe stability analysis of fractional order linear systems subject to input saturation, and [29], which proposed a virtual actuator-basedfault tolerant control strategy to deal with actuatorsaturationsinunstablelinearsystems.

Thiswork ismotivatedbythe bigimportanceheld bythe op- timaldesignofobserverandcontrollergainsinautomaticcontrol systems.Theliterature reviewhasshownthat, althoughthereex- ist a few results on GCC for systems with input saturation, e.g.

[17,44,47],theproblemofGCEforthesesystemshasnotbeencon- sideredyet.Hence,thispaperaimsatdevelopinga designproce- durethataddressesboththeGCEandtheGCCforaclassofnonlin- ear saturatingsystems,whileatthesametimeanalysingthecase inwhichthecontrollerusestheestimateproducedbytheobserver inordertoupdatethecontrollaw.

Morespecifically,thispaperproposesamatrixinequality-based guaranteed cost estimation and control design procedure for a classofdiscrete-timenonlinearsystemssubjecttoactuatorsatura- tion.Thisclassofsystemscorrespondstothoseforwhichtheori- ginofthestatespaceisanequilibriumpointwhennullinputsare considered, and the nonlinearity isdifferentiable withrespect to thestateandlinearwithrespecttothesaturatedinput.Itisworth highlightingthatthesenonlinearities,whichhavebeenconsidered previouslyinthecontextoffaultestimationbyZhu[2,40],encom- pass cases which cannot be dealtusing the traditional bounding boxmethod[35].Hence,analternativeapproachbasedontheap- plication ofthemeanvalue theorem,asdescribedby Lewisetal.

[1,23],mustbeobtained.

Thecontributionsofthepapercanberesumedasfollows:

1.a polytopicapproach basedontheapplicationofthe mean valuetheoremisdescribedforthecharacterizationofaclass ofdiscrete-timenonlinearsystemssubjecttoactuatorsatu- ration;

2.sufficient conditions for the synthesis of a state observer that achieves GCE and a state-feedback controller that achievesGCCforthe above-mentionedclass ofsystemsare providedintheformofanLMI-basedfeasibilityoroptimiza- tionproblem;

3.it isshown that fortheabove-mentioned class of systems, the celebrated separation principle holds only one-way in the sensethat theobservercan bedesignedindependently from the controller, but the converse is not true. Hence, sufficient conditionsforthedesignofan estimate-feedback guaranteedcostcontrollerareobtainedintheformofbilin- earmatrixinequalities(BMIs).

The paperisstructured asfollows.Section 2describestheno- tationandsome lemmaswhichareusedintheproofsofthethe- oreticalresults.InSection3,theclassofconsiderednonlinearsys- tems is defined, andthe differentdesign problemsconsidered in thispaperareformulated.Section4provides LMI-basedsufficient conditions for the design of the state observer. Section 5 is de- voted to providingLMI-based sufficient conditionsfor thedesign of the state-feedback controller. In Section 6, the state-feedback control lawisreplacedbyan estimate-feedbackcontrol,andBMI-

basedconditionsforthedesignofthecontrollergainareobtained.

Section7summarizesthefinalprocedurefordesigningandimple- menting the componentsof thecontrol systemthat provide GCC andGCE.Thetheoreticalresultsareillustrated bymeansofan il- lustrativeexampleinSection8,whereasanapplicationtoanonlin- earrotationalsingle-arminvertedpendulumisgiveninSection9. Finally,themainconclusionsareoutlinedinSection10.

2. Notationandpreliminaries

ForarealsymmetricmatrixA∈R^n×n,thenotationA0(A≺0) standsfora positive(negative) definite matrix andindicatesthat alltheeigenvaluesofAarepositive(negative).GivenamatrixA∈ R^n×nwithA0,thesymbolEA denotestheellipsoid:

EA=

x∈Rⁿ:x^TAx≤1

.

ThesymbolCo

{

^Ai,i=1,...,N

}

^{denotestheconvexhullofafinite}

numberofN vertexmatrices:

Co

{

^Ai,i=1,...,N

}

=

N

i=1

μ

iAi

|

^N

i=1

μ

i=1,

μ

i≥0

∀

ⁱ=1,...,N

.

Given a vector u∈R^m, the symbol

σ

(·) ^{denotes the standard} saturation function, such that

σ

(^u)=[

σ

(^u1),

σ

(^u2),. . .,

σ

(^um)^]T, where

σ

(^ui)=sgn(^ui)^min

{

¹,

|

^ui

|}

^{. Given a matrix A}∈R^m×n, the symbolLσ(^A)^denotes:

L_σ

(

^A

)

=

{

^x∈Rn:

σ (

^Ax

)

=Ax

}

^.

Thefollowinglemmaareusedthroughoutthepaper.

Lemma1. GiventwomatricesK,H∈R^m×nandavectorx∈Rⁿ,sup- posethat

|

^Hjx

|

≤1for j=1,2,...,m,whereH_jdenotesthe jthrow ofH.Then:

σ (

^Kx

)

∈Co

DiKx+D⁻_iHx,i=1,...,2^m

, (1)

where thematrices D_i areall thepossiblem×m diagonalmatrices whosediagonalelementsareeither1or0,andD⁻_i =I−D_i.

Proof. See[10].

Lemma 2. Given the matrix H∈R^m×n, if the following inequality holdsfor j=1,...,m:

P PH^T_j H_jP 1

0, (2)

then

|

^Hjx

|

≤1

∀

^x∈EP⁻¹. Proof. See[21]. 3. Problemformulation

Considerthefollowingdiscrete-timenonlinearsystem:

xk+1=Axk+g

(

^xk,

σ (

^uk

) )

^{, (3)}

y_k=Cx_k, (4)

wherex_k∈Rⁿ isthestate,u_k∈R^misthecontrolinput, y_k∈R^pis the output,A andC are constant matricesofappropriate dimen- sions,andthenonlinear functiong:Rⁿ×R^m→Rⁿ isassumedto satisfythefollowingassumptions:

1. g(^x,

σ

(^u))^isaffinein

σ

(^u)^{,sothatitcanberewrittenas:}

g

(

x,

σ (

^u

))

= f

(

^x

)

+F

(

^x

) σ (

^u

)

, (5) with f:Rⁿ→Rⁿ and F:Rⁿ→R^n×m appropriate functions suchthat f(⁰)=0andF(⁰)=0.Notethataconsequenceof thisfactisthatg(⁰,0)=0;

2.g(^x,

σ

(^u))^is differentiable withrespectto xwith bounded partialderivatives:

a_i,j≤

∂

^gi

(

^x,

σ (

^u

))

∂

^xj

≤a_i,j, i=1,...,n, j=1,...,n. (6)

Byapplyingthemeanvaluetheorem[5],thefollowingrelation holds:

g

(

^a,

σ (

^u

) )

^−g

(

^b,

σ (

^u

) )

^=M

(

a,b,

σ (

^u

))(

^a−b

)

, (7) forsomematrixM(·)^{obtainedasfollows:}

M

(

a,b,

σ (

^u

))

=

⎡

⎢ ⎢

⎢ ⎣

∂

^g1

∂

^x

⁽

^c1,

σ (

^u

))

..

∂

^gn .

∂

^x

⁽

^cn,

σ (

^u

))

⎤

⎥ ⎥

⎥ ⎦

^{, (8)}

with:

c1,...,cn∈

{

^c∈Rn:c=

α

^a+

β

b,

α

+

β

=1,

αβ

≥0

}

^{. (9)}

Takingintoaccountlowerandupperboundsa_i,j,a_i,jandeachpos- siblepermutationofthesebounds,matricesM_i∈R^n×n,i=1,...,N, canbeobtained¹suchthat:

M

(

a,b,

σ (

^u

))

∈MCo

{

^Mi,i=1,...,N

}

^{, (10)}

sothat:

g

(

^a,

σ (

^u

) )

^−g

(

^b,

σ (

^u

) )

^∈M

(

^a−b

)

. (11) Moreover,takingintoaccount(5),thefollowingholds:

g

(

^x,

σ (

^a

) )

^−g

(

^x,

σ (

^b

) )

^=F

(

^x

)( σ (

^a

)

−

σ (

^b

))

. (12) Hereafter, theproblemsofobserverandcontrollerdesignare for- mulated.

3.1. Stateobserverdesign

Letusconsideranonlineardiscrete-timeobserveroftheform:

xˆk+1=Axˆk+g

xˆk,

σ (

^uk

)

+Ko

(

^yk−Cxˆk

)

, (13) wherexˆ_k denotestheestimate ofthestate x_kandKodenotesthe observergaintobedesigned.Then,thedynamicsoftheestimation errore_k=x_k−xˆ_kisgivenby:

e_k+1=x_k+1−xˆ_k+1=A˜e_k+s_k, (14) whereA˜=A−KoCands_k=g(^xk,

σ

(^uk))−g

ˆ x_k,

σ

(^uk)

.Apartfrom the asymptotical convergence to zero of the estimation error, a boundonthefollowingcostfunction:

Jo= ∞ k=0

e^T_kQee_k, (15)

withgivenQe0,isconsidered asobjectiveforthedesignofthe observer.Hence,theGCEdesignproblemcanbeformulatedasfol- lows.

Problem 1. (State observer design problem) Given Q_e0 and a scalar

γ

^o>1,designKo suchthat thedynamicsofthe estimation error(14)isasymptoticallystablewith:

Jo<

γ

^oeT0Qee0. (16)

1In general, N = 2 ⁿ² matrices are obtained using this approach, which is com- monly named bounding box method , since it generates a hyperbox of matrices [35] . However, if the matrix function M x,ucontains h constant elements which arise from the linearity of some function g i(x, σ(u )) with respect to some state variable x j, then a reduced number of matrices N = 2 ^(n−h)2is obtained.

3.2. State-feedbackcontrollerdesign

Todesignarobustcontroller,letusconsiderthefollowingcon- trollaw:

u_k=Kcx_k, (17)

where Kc is the controller gain to be designed. Substituting the aboveequationinto(3)givesthefollowingclosed-loopsystem:

xk+1=Axk+g

(

^xk,

σ (

^Kcxk

) )

. (18) Let us note that, since g(⁰,0)=0, then g(^xk,

σ

(^Kcx_k)) ^{can be} rewrittenas:

g

(

^xk,

σ (

^Kcxk

) )

^=g

(

^xk,

σ (

^Kcxk

) )

^−g

(

^0,

σ (

^Kcxk

) )

+g

(

^0,

σ (

^Kcx_k

) )

^−g

(

⁰,0

)

. (19) Takingintoaccount(11)and(12),thefollowingisobtained:

g

(

^xk,

σ (

^Kcxk

) )

^−g

(

^0,

σ (

^Kcxk

) )

^∈Mxk, (20)

g

(

^0,

σ (

^Kcx_k

) )

^−g

(

⁰,0

)

=F

(

⁰

) σ (

^Kcx_k

)

, (21) sothat:

g

(

^xk,

σ (

^Kcxk

) )

^∈Mxk+F

(

⁰

) σ (

^Kcxk

)

^{. (22)}

Thefollowing objectivesare takeninto accountforthe design ofthecontroller:(i)asymptoticalconvergencetozeroofx_k when x₀belongstotheellipsoidEQ definedbyagivenmatrixQ0;and (ii)boundonthefollowingcostfunction:

Jc= ∞ k=0

x^T_kQxxk+

σ (

^uk

)

^TQu

σ (

^uk

)

, (23)

withgivenQx0andQu0.Hence,theGCCdesignproblemcan beformulatedasfollows.

Problem2. (State-feedbackcontrollerdesign problem)Givenma- trices Q0,Qx0,Qu0andthe scalar

γ

^c>1, designKc such that(18)isasymptoticallystableand:

Jc<

γ

^cxT0Qxx0, (24)

whenx₀∈EQ.

3.3. Estimate-feedbackcontrollerdesign

Thecontroller designproblempreviously formulated (Problem 2)assumesthattherealstatex_kisavailableforfeedback.However, amore realisticsituationis theone inwhichtheestimatedstate shouldbeusedinstead,i.e.(17)changesinto:

u_k=Kcxˆ_k, (25)

wherexˆ_k istheestimatedstategivenby theobserver(13).Inthis case,thequestionaboutwhetheritispossibleornottodesignthe observerandthecontrollerseparatelyarises.

Letusconsider theinterconnection ofthe system(3)and(4), the observer (13) andthe control law (25) such that the overall systemobeys(seeFig.1forablockdiagramdepictingtheirstruc- turesandinterconnections):

e_k+1=A˜e_k+g

x_k,

σ (

^Kcxˆk

)

−g

ˆ

x_k,

σ (

^Kcxˆk

)

, (26)

x_k+1=Ax_k+g

x_k,

σ (

^Kcxˆ_k

)

. (27)

Letusnotethat:

g

x_k,

σ (

^Kcxˆk

)

=g

x_k,

σ (

^Kcxˆk

)

−g

0,

σ (

^Kcxˆk

)

+g

0,

σ (

^Kcxˆk

)

−g

(

⁰,0

)

. (28) Takingintoaccount(11):

Fig. 1. Block diagram of the proposed control and estimation strategy.

g

x_k,

σ (

^Kcxˆ_k

)

−g

0,

σ (

^Kcxˆ_k

)

∈Mx_k, (29) g

x_k,

σ (

^Kcxˆk

)

−g

ˆ

x_k,

σ (

^Kcxˆk

)

∈M

x_k−xˆ_k

=Me_k. (30) Moreover,dueto(12):

g

0,

σ (

^Kcxˆk

)

−g

(

⁰,0

)

=F

(

⁰

) σ (

^Kcxˆk

)

. (31) Hence:

g

x_k,

σ (

^Kcxˆ_k

)

∈Mx_k+F

(

⁰

) σ

Kcxˆ_k

, (32)

whichmeansthattheoverallsystemcanbeputintheequivalent form:

xk+1∈

(

^A+M

)

^xk+F

(

⁰

) σ (

^Kcxk−Kcek

)

^{, (33)}

e_k+1∈

(

^A˜+M

)

^ek, (34)

wherexˆ_k=x_k−e_khasbeenused.

From(33)to(34)itcan beseen that,due tothe nonlinearterm

σ

(^Kcxk−K_ce_k)^{,theseparationprincipleholdsonlyone-way,inthe} sensethat,whiletheobservercanbedesignedindependentlyfrom thecontroller,thisisnottrueforthecontroller,whosedesignpro- cedure shouldbe modified totake into account the effectofthe evolution ofe_k,drivenby thespecific choiceoftheobservergain Ko,on the nonlinearity

σ

(·)^{.In order todeal with thissituation,} the requirementsofProblem 2are changedby requiringthemto holdfor[x^T₀,e^T₀]^T∈E[QS;S^TR],wheretheellipsoidE[QS;S^TR]isde- finedbyagivenmatrix:

Q S S^T R

0, (35)

and that J_c<

γ

c

x^T₀Q_xx₀+e^T₀Q_xe₀

. Note that with this choice, in casex0=0,thenJc<

γ

cxˆ^T₀Qxxˆ0isobtained,whereasife0=0,then thecaseJc<

γ

^cxT0Qxx₀isrecovered.Hence,theGCCdesignproblem canbemodifiedasfollows:

Problem 3. (Estimate-feedback controller design problem) Given matricesQ0,S,R0,Qx0,Qu0,theobservergain Ko and thescalar

γ

^c>1,designthecontrollergainKc such that(33)and (34)isasymptoticallystableand:

Jc<

γ

c

x^T₀Qxx0+e^T₀Qxe0

, (36)

when[x^T₀,e^T₀]^T∈E[QS;S^TR].

4. Designofthestateobserver

TheobjectiveofthissectionistosolveProblem1byobtaining sufficient conditions for the synthesis ofthe state observer (13), whicharegivenbythefollowingtheorem.

Theorem1. LetP0,

γ

o>1andU besuchthatthefollowingholds:

P−

γ

^{oQe≺0, (37)}

Qe−P ∗ P

(

^A+M_i

)

−UC −P

≺0, i=1,...,N. (38)

Then, the nonlineardiscrete–time observergiven by (13), with gain calculatedasKo=P⁻¹U,issuchthat(14)isasymptoticallystableand Jo<

γ

oe^T₀Qee0.

Proof. Letusconsiderthefollowinginequality[30]:

^Vk+e^T_kQeek<0 for k=0,...,∞, (39) where ^Vk=V_k+1−V_k, with V_k=e^T_kPe_k, P0. Since e^T_kQee_k>

0

∀

^ek,ifinequality (39)holdsthen^Vk<0,whichcorrespondsto the Lyapunov conditionfor asymptoticstability of (14).Then, by summing(39)from0to∞,thefollowingisobtained:

∞ k=0

e^T_kQee_k=Jo<V0=e^T₀Pe0, (40)

which,dueto(37)ensuringthatV₀<

γ

^oeT0Qee₀ foranyinitialcon- ditione₀,provesthatJo<

γ

^oeT0Qee₀ [22].

The remaining ofthe proof showsthatinequality (39) follows from(38).In fact,takingintoaccount (13),(39)can be rewritten as:

^Vk+e^T_kQee_k=e^T_k

_˜

A^TPA˜−P+Qe

e_k+e^T_kA˜^TPs_k

+s^T_kPA˜e_k+s^T_kPs_k<0. (41) From(11),itfollowsthat:

s_k=g

(

^xk,

σ (

^uk

))

−g

(

^xˆk,

σ (

^uk

))

∈M

(

^xk−xˆ_k

)

=Me_k, (42)

sothat(41)issatisfiedif:

A˜^TPA−P+Qe+A˜^TPM+M^TPA˜+M^TPM≺0,

∀

^M∈M, (43) which,usingSchurcomplements,leadsto:

Qe−P

_˜

A^T+M^T

P P

_˜

A+M

−P

≺0,

∀

^M∈M. (44)

Replacing:

PA˜=PA−PKoC=PA−UC, (45) into(44),whereU=PKo,leadsto:

Qe−P P

_˜

A+M

−UC −P

≺0,

∀

^M∈M, (46)

whichissatisfiedif(38)holds,thuscompletingtheproof.

Remark 1. The problemofdetermining theobserver gainmatrix describedbyTheorem1canbetreatedasanoptimizationproblem inwhichthecostperformanceindex

γ

^{oisminimized.}

5. Designofthestate-feedbackcontroller

TheobjectiveofthissectionistosolveProblem2byobtaining sufficientconditionsforthesynthesisofthecontroller(17)forthe system(3)-(4),whicharegivenbythefollowingtheorem.

Theorem 2. Let P0,

γ

^c>1 and ,Z be such that the following holds:

Q I I P

0, (47)

P Z^T_j Zj 1

0, j=1,. . .,m, (48)

⎡

⎢ ⎣

−P ∗ ∗ ∗

(

^A+Ml

)

^P+F

(

⁰

) φ

i −P ∗ ∗ Q_x^1/2P 0 −I ∗

φ

^{i 0 0 −Q}u⁻¹

⎤

⎥ ⎦

^{≺0, i}_l⁼₌¹₁^,_,^._.^._.^._.^,_,²_N^m,

(49)

γ

^{cQx I}

I P

0, (50)

with

φ

i=D_i +D⁻_iZ,where Z_jdenotes the jthrowofthematrixZ, the matricesD_i areall thepossiblem×m diagonalmatrices whose diagonalelementsareeither1or0,andD⁻_i =I−D_i.Thenthestate- feedbackcontrollaw(17),withgaincalculatedasKc= P⁻¹,issuch that(18)isasymptoticallystableandJc<

γ

^cxT0Qxx₀ whenx₀∈EQ. Proof. InordertoensurethatProblem2issolved,letusdefinethe function V_k=x^T_kP⁻¹x_k, P0, andlet usrequirethat the ellipsoid EQ iscontained inE_P−1,which isequivalent toQ−P⁻¹0 and,by Schur complements,to (47). Then, to ensure asymptotic stability

forx₀∈EQ,itissufficienttoensureitforx₀∈E_P−1,whichtogether withtheconstraintonJc,leadstothefollowingconstraintsonV_k:

^Vk+x^T_kQxx_k+

σ (

^Kcx_k

)

^TQu

σ (

^Kcx_k

)

<0, (51)

P⁻¹−

γ

cQx≺0, (52)

where^Vk=V_k+1−V_k. Bydefining:

¯

x^σ_k =[x^T_k,

σ (

^Kcx_k

)

^T]T, (53) andtakingintoaccount(22),theinequality(51)issatisfiedif:

(

^x¯σk

)

^T

^σ11

^σ12

∗

^σ22

¯

x^σ_k =

(

^x¯σk

)

^T

^σx¯σk <0,

∀

^{M∈M, (54)}

where:

^σ11=

(

^A+M

)

^TP−1

(

^A+M

)

−P⁻¹+Qx,

^σ12=

(

^A+M

)

^TP−1F

(

⁰

)

^,

^σ22=F

(

⁰

)

^TP−1F

(

⁰

)

+Qu.

According to Lemmas 1–2, by introducing an auxiliary feedback matrix Hc andthe constraint (48),which enforces EP−1⊆Lσ(^Hc) andthat isobtainedfrom(2)throughthe changeofvariableZ= HcP,then

σ

(^Kcxk)^{canbeplacedintotheconvexhullofagroupof} linearfeedbacks:

σ (

^Kcx_k

)

∈Co

{

^DiKcxk+D⁻_iHcx_k,i=1...,2^m

}

^{. (55)}

Hence,from(55)andbyconvexityofthefunctionV_k,itfollows:

(

^x¯σk

)

^T

^σx¯σk ≤ max

i=1,...,2^mx^T_k

^ixk, (56)

where:

ⁱ=

^TiP⁻¹

i−P⁻¹+Qx+

ψ

i^TQu

ψ

i, (57) with:

i=A+M+F

(

⁰

) ψ

^{i. (58)}

By requiringthat ⁱ≺0for i=1,...,2^m, andapplying Schur complements,withanappropriatecongruencetransformation,the followingisobtained:

⎡

⎢ ⎣

−P P

^Ti PQ_x^1/2 P

ψ

i^T

iP −P 0 0

Q_x^1/2P 0 −I 0

ψ

iP 0 0 −Qu⁻¹

⎤

⎥ ⎦

^≺0,

∀

^M∈M, (59)

which,byreplacing:

iP=

(

^A+M

)

^P+F

(

⁰

) ψ

iP

=

(

^A+M

)

^P+F

(

⁰

)

^DiKcP+F

(

⁰

)

^D−iHCP

=

(

^A+M

)

^P+F

(

⁰

)

^Di +F

(

⁰

)

^D−_iZ, (60) where =KcP,leadsto:

⎡

⎢ ⎣

−P ∗ ∗ ∗

(

^A+M

)

^P+F

(

⁰

) φ

i −P ∗ ∗ Qx^1/2P 0 −I ∗

φ

i 0 0 −Q_u⁻¹

⎤

⎥ ⎦

^{≺0, i=}

∀

^1M,.∈^..M^,2m,

(61) whichissatisfiedif(49)holds.ByapplyingSchurcomplementsto (52),(50)isobtained,whichcompletestheproof.

Remark2. Alsointhiscase, theproblemofdeterminingthecon- troller gain matrix describedby Theorem 2can be treatedasan optimizationproblem in whichthe cost performance index

γ

^{c is}

minimized.

6. Designoftheestimate-feedbackcontroller

TheobjectiveofthissectionistosolveProblem3byobtaining sufficientconditionsforthesynthesisofthecontroller(25)forthe system(3)and(4)withstateobserver(13),whicharegivenbythe followingtheorem.

Theorem3. GiventheobservergainK_o(hence,thematrixA˜),letP 0,

γ

^c>1andthematricesKc,Hcbesuchthat:

Q S S^T R

−P0, (62)

P−

γ

^c

Qx 0 0 Qx

≺0, (63)

⎡

⎣

^P

H_c^T_,j

−H_c,^T_j

H_c,j −Hc,j

1

⎤

⎦

0, j=1,...,m (64)

⎡

⎢ ⎢

⎣

P P

A+M_l1+F

(

⁰

) φ

i −F

(

⁰

) φ

i

0 A˜+Ml2

P−

Qx+

φ

i^TQu

φ

i −

φ

^TiQu

φ

i

−

φ

^TiQu

φ

i

φ

i^TQu

φ

i

⎤

⎥ ⎥

⎦

^0,

i=1,...,2^m l₁=1,...,N l2=1,...,N ,

(65) hold,where

φ

i=D_iKc+D⁻_iHc,H_c,jdenotesthe j-throwofthematrix H_c,andthematricesD_iareallthepossiblem×mdiagonalmatrices whosediagonalelementsareeither1or0,andD⁻_i =I−D_i.Thenthe estimate-feedbackcontrollaw(25)ensuresthat(18)isasymptotically stablewhen[x^T₀,e^T₀]^T∈E[QS;S^TR]andJ_c<

γ

c

x^T₀Q_xx₀+e^T₀Q_xe₀

. Proof. Letusconsiderthefunction:

V_k=

x_k ek

T

P

x_k ek

, (66)

withP0,andletusrequirethatE[QS;S^TR]⊆EP,whichisequiv- alent to (62). Then, to ensure asymptotic stability for [x^T₀,e^T₀]^T∈ E[QS;S^TR],it issufficient to ensure itfor[x^T₀,e^T₀]^T∈EP,which to- getherwiththeconstraintonJc,leadsto(63)and:

^Vk+x^T_kQxx_k+

σ (

^Kcxˆ_k

)

^TQu

σ (

^Kcxˆ_k

)

<0. (67) ByperformingareasoningsimilartotheoneinTheorem2,us- ingLemma2theconstraint(64)ensuresthat:

EP⊆L

Hc −Hc

, (68) and,accordingtoLemma1:

σ (

^Kcxˆ_k

)

∈Co

ψ

˜i

x_k e_k

=D_i

Kc −Kc

x_k e_k

+D⁻_i

Hc −Hc

x_k e_k

. (69)

sothatthefollowingholds:

x_k+1 e_k+1

∈

A+M+F

(

⁰

) φ

i −F

(

⁰

) φ

i

0 A˜+M

x_k e_k

, (70)

whichmeansthat

∃

^M1,M₂∈Msuchthat:

xk+1

e_k+1

=

A+M1+F

(

⁰

) φ

^{i −F}

(

⁰

) φ

ⁱ

0 A˜+M2

xk

e_k

=A¯

xk

e_k

. (71)

Then,(67)leadsto:

A¯^TPA¯−P+

Qx 0

0 0

+

φ

i^T

−

φ

i^T

Qu

φ

i −

φ

i

≺0,

∀

^M1,M2∈M, (72)

which,by meansofSchur complements,andanappropriate con- gruencetransformation,leadsto:

⎡

⎢ ⎢

⎣

P P

A+M1+F

(

⁰

) φ

i −F

(

⁰

) φ

i

0 A˜+M2

P−

Qx+

φ

i^TQu

φ

^{i −}

φ

i^TQu

φ

ⁱ

−

φ

i^TQu

φ

i

φ

i^TQu

φ

i

⎤

⎥ ⎥

⎦

^{0, i=1,...,2}

m

∀

^M1,M₂∈M,

(73) whichissatisfiedif(65)holds,thuscompletingtheproof.

Note that when applyingTheorem 6, the matrix A˜ is consid- ered to be known, since the observer gain Ko is assumed to be designed beforehandusing Theorem 1.Taking the above consid- erationsintoaccount,similarly toprevious cases,an optimization problem concerning minimization of the cost performance index

γ

^{ccanbedefined.Itisworth}highlightingthattheconditionspro- videdby Theorem6arebilinear matrixinequalities(BMIs)dueto theproductbetweentheunknownvariablesPand

φ

i,hencetheir resolutionsuffersfrombeinganon-convexproblem.

7. Designandimplementationprocedurefortheguaranteed costestimationandcontrol

The problemof determining the state observerand controller gainmatricesissolvedusingtheresultsgivenbyTheorems1–3.It isdonebyanoptimizationproblemsubjecttominimizationofthe costperformance indexes

γ

^oand

γ

^{c.Thedesignand}implementa- tionprocedurecanbesummarizedasfollows:Off-linecomputation:

1. Obtain a representation of the system of interest as in (3)and(4);

2. Calculatethe Jacobiansofthe functiong(·)^{withrespect to} stateandinput;

3. Compute lower andupper boundsfor the elementsof the Jacobiansand usethem to obtain (10) usingthe bounding boxapproach;

4. (Observerdesign)Obtaintheobservergain Kobysolvingthe optimizationproblem:

minimize

γ

^o

subjectto subjectto(37)-(38)

5. (Controller design) Obtain the controller gain Kc by solving theoptimizationproblem:

minimize

γ

^c

subjectto subjectto(47)-(50) On-linecomputation:

1. Computethestateestimateusing(13); 2. Computethecontrolactionusing(25).

Remark 3. The above proceduresummarizes the necessarysteps forthedesignofthestateobserverorthestate-feedbackcontroller.

Itcouldbeappliedtothecaseoftheestimate-feedbackcontroller, albeitsomeminorchanges.

8. Numericalexample

Letusconsiderthefollowingsystem:

x1

(

^k+1

)

=a11x1

(

^k

)

−0.5x2

(

^k

)

+0.1x3

(

^k

)

+ cos

(

^x1

(

^k

) ) σ

1

(

^u1

(

^k

) )

3x²₁

(

^k

)

+2 ,

x₂

(

^k+1

)

=−0.2x₁

(

^k

)

+a₂₂x₂

(

^k

)

+0.1x₃

(

^k

)

+

σ

2

(

^u2

(

^k

) )

2+x²₂

(

^k

)

^,