Event triggered sliding mode control a new approach to control system design 1st edition bijnan band
Event Triggered Sliding Mode Control A
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Sliding Mode Control Methodology in the Applications of Industrial Power Systems Jianxing Liu
kmaxðminÞ f g Largest(smallest)eigenvalueofasquarematrix ‘ ’
rf ðxÞ Gradientofareal-valuedfunction f ðxÞ sup(inf)Supremumorleastupperbound(infi mumorgreatestlowerbound) x bc Floorfunctionthatreturnslargestintegerlessthanorequalto x ln Naturallogarithmwithbasee(¼ 2:71828)
K Setofstrictlyincreasingandcontinuousreal-valuedfunctions definedonthenonnegativeintervalwithzeroatzero
Event-triggeredcontrolisoneofsuchtechniquesthatgeneratesthesamplinginstant (alsocalledastriggeringinstant)forsamplingandupdatingthecontrolsignal.Toprovideapreliminaryideaonevent-triggeredcontrol,weconsideranonlineardynamical system
wherethefunction f ( , ) isLipschitzwithrespecttoboththearguments u ∈ Rm .Let thereexistsacontinuousfeedbackcontrollaw u (x ) = π(x ) suchthatthedynamics ˙ x = f (x ,π(x ))
isasymptoticallystable.Itisassumedthatthecontrolisimplementeddigitallyto theplant.So,thecontrolsignal π(x ) iscomputedforeverysamplinginstantandis appliedtotheplantatthesediscreteinstants.Then,thesystembecomesopenloop betweentwoconsecutivesamplinginstants.However,duetothis,thediscreteerror isintroducedintheplant,definedby e (t ) = x (ti ) x (t ) with e (ti ) = x (ti ) x (ti ) = 0 where t ∈[ti , ti +1 ).Thiserrorappearsintheplantduetodiscreteimplementationof continuous-timecontrol,butitsvalueiszeroifthecontroliscontinuouslyupdated asinanalogimplementation.
Further,weassumethatthesystem(1.1)isinput-to-statestable(ISS)withrespect totheerror e (t ).ThatmeansthereexistsacontinuouslydifferentiableLyapunov function V : Rn → R≥0 suchthat
forsomeclass-K∞ 1 functions a , a , a ,andclass-K function γ .Here,thenotation ‘ ·’denotesinner(scalar)product.Event-triggeringstrategyisdevelopedfordeterminingthesamplinginstantssuchthatdesiredstabilityisachieved.Inthiscase,the asymptoticstabilityofthesystemisdesiredwiththediscreteimplementationofthe controllaw.So,theobviousconditionforwhichthisholdsis γ( e )<σ a ( x ) for some σ ∈ (0, 1).Thiscanbesimplifiedfurther,byassuming a 1 and γ areLipschitz onsomecompacts,as σ x > Le e ,where Le isanappropriateconstant.Thus,the triggeringinstantmaybegeneratedbyexecutingthefollowing,
Thisisknownastriggeringruleforevent-triggeredcontrol π(x ).Itisseenthat thisensures Le e <σ x whichimpliesthat γ( e )<σ a ( x ) alsoholds.This impliesfrom(1.2)and(1.3)that
where K∞ a := a ◦ a 1 whichisthecompositionoftwofunctions a and a 1 . Thisshowsthatclosedloopsystemisasymptoticallystablewiththecontrolapplied atdiscreteinstants.Moreover,intheevent-triggeringtechniquetheinter-sampling behaviourisconsideredforstabilityoftheclosedloopsystem.
Let {ti }i ∈Z≥0 besequenceoftriggeringinstantsgeneratedbysomestabilizing triggeringrule.Wedefine Ti = ti +1 ti asthe inter-event/executiontime forany giventriggeringsequence {ti }i ∈Z≥0 .Forstabilityoftheevent-triggeredsystem,the inter-eventtimemustbestrictlygreaterthanzero,i.e. Ti > aT forall i ∈ Z≥0 and somepositiveconstant aT .Thisguaranteesthe Zeno-freeexecutionoftriggering sequence.Thepositiveinter-eventtimeensurescontrolisupdatedaftereveryfinitetimeintervalonly.Thisisessentialfortheprocessortoexecutethecontroltaskand updatethecontrolsignal.Inotherwords,itcanbesaidthat {ti }i ∈Z≥0 isanincreasing sequence,i.e. t0 < t1 < t2 < suchthat ti +1 > ti + aT .Suchatriggeringsequence isfeasibleforimplementingthecontrolpracticallytoensurethestabilityofclosed loopsystem.Thetriggeringinstantsgeneratedbysometriggeringrulethatisnot necessarilysatisfyingtheabovepropertywouldmaketheevent-triggeredsystem unstable.
where x ∈{[−c , c ]: c ∈ R>0 } whichisacompactset.Anystabilizingcontrollercan bedesignedfortheabovesystemtoensuretheasymptoticstability.Let u =−x 2 kx beafeedbackcontrolwhichensurestheasymptoticstabilityofthesystemwith k > 0. Thiscontrolisappliedtotheabovesystematdiscreteinstantsonlysuchthatclosed loopsystemisstable.So,thediscrete-timecontrolisgivenas
(t ) =−x 2 (
Itcanbeshownthattheclosedloopsystemwiththeabovediscretecontrolis ISSwithrespecttotheerror.Choose V (x ) = 1 2 x 2 .Then,withsomecalculation,we arriveat
Here,weusethefact |x |≤ c and x (ti ) = e (t ) + x (t ).NowapplyingYoung’s inequality2 tothefirstterm(ε = 2c+k k ),weobtain
where a (r ) and γ(r ) aregivenas a (r ) = k 2 r 2 and γ(r ) = (
Thevectorfield f : Rn × R → Rn givenin(1.5)isLipschitzwithrespectto itsfirstargument.Lettheslidingmanifoldbegivenforanycontinuousswitching function,denotedby s(x ) whichmaps Rn to R,as
Similarly,wealsodefine S + = {x ∈ Rn : s(x )> 0} and S = {x ∈ Rn : s(x )< 0}.Thecontrollawthatbringsslidingmodeinthesystem(1.5)isgiven by
Thecontrolfunctions u + and u arecontinuous,andalso u + = u .Clearly,this impliesthatthecontrol u isdiscontinuouson s = 0.Thecorrespondingresulting vectorfieldsare f + (x , u + ) and f (x , u ) duetothecontrolsignals u + and u , respectively.So,thevectorfield f ( , ) isalsodiscontinuouson S .Theexistenceof solutionsoftheclosedloopsystem(1.5)withcontrollaw(1.6)cannotbeexplained usingtheclassicalexistencetheorem.Indeed,inthiscase,theclosedloopsystem becomesdiscontinuousandisreferredassystemwithdiscontinuousright-handside. Thoughtherearemanytechniquesavailablefordefiningsolutionsofsuchsystems, inthisbookweunderstoodthesolutionsofthesysteminFilippov’ssense[1].
1.3.1DynamicsDuringSlidingMode
Theexistenceofsolutionsofthedynamicalsystem(1.5)isbrieflydiscussedin Filippov’ssense.Thedynamicalsystemisfirstreplacedbyaset-valuedfunction, calleddifferentialinclusion,onthezero(Lebesgue)measuresetwheresolutionisnot definedinclassicalsense.Wedenoteitby F (x , u ).Thus,thedifferentialinclusion forthesystem(1.5)iswrittenas
x ∈ F (x , u ). (1.7)
Theset-valuedmap F coincideswith f ;i.e., F containsonlyoneelement f , wheneverthelatteriscontinuousonitsrespectivedomain.Inotherwords,thesetvaluedmaprepresentsthevectorfieldsofthedynamicalsystematthepointsof discontinuity.Oneofthemainreasonstoreplacethedynamicalequation(1.5)by
Sincetheinclusion,givenin(1.7),capturesallthevectorfieldsinsomesufficiently small δ -neighbourhoodof s(x ) = 0,theset-valuedmapisconstructedbycollecting theconvexcombinationofthevectorfieldsinthatneighbourhood.Thus,thelimiting vectorfieldsinthesmallneighbourhoodofthedomains S + and S forany x ∈ S areobtainedas
Then,theset-valuedmap F (x , u ) isobtainedbycollectingallthevectorfields pointingthelinesegmentjoiningtheendpointsofthevectorfieldsin S + and S , i.e. f + and f .Notethateverytrajectoryin S + crosses S beforereaching S and viceversa.Thisimpliesthatthelinesegmentjoining f + and f alsointersects S . If Tx S representsthetangentvectorof S atthepoint x ,thenthelinesegmentalso intersects Tx S atsomepoint.Thus,thisintersectionpointgivestheendpointofthe vector f 0 (x , u ) suchthat
holdsforthepoint x ∈ S .Similarly,itcanbedefinedforotherpointson S .This givesthemotionofthesystemtrajectoryontheslidingmanifold.Inotherwords, thefunction x (t ) satisfies(1.8)isasolutionofthesystem(1.5).Thisisbecausethis solutionalsosatisfiestheinclusion(1.7)as f 0 iscontainedintheinclusion.Also, f = f 0 and f + = f 0 ;themotionofthesystemremainstangentto S andiscalled slidingmotion.Thus,duringslidingmode,thesystemtrajectoryisasolutionto(1.8) whichisalsoasolutionof(1.5)dueto(1.7).
Now,wediscusshowtofindthevelocityvector f 0 duringslidingmode.Define f + N and fN astheprojectionsofthevectors f + and f ,respectively,ontothenormal tothesurface S atthepoint x ∈ S .Then,thevectorfield f 0 iscalculatedusing (1.8)as
Thisshowsthevectorfieldduringslidingmodeisaconvexcombinationofthe vectorfields f + and f suchthatthetrajectoryistangenttosurface S .Thevalue of θ givenin(1.9)isobtainedfromtherelation
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N Y Evening Post p2 N 27 ’20 1350w
“To an American reader the chief criticism to be made of all these accounts of luncheons, dinners and concerts in the company of the rich and fashionable is that they are intolerably wearisome. Colonel Repington continually speaks of the play of wit in these high circles, but gives very few examples of it.”
+ − + + book. But if we do not get too weary of his ‘practically no English articles are read and discussed except mine,’ we may find illumination most of it unintentional in his accounts of his work running to and fro between the generals, the politicians and the press. ” F. H.
N Y Times p1 O 24 ’20 2400w
Reviewed by E. L. Pearson
Review 3:376 O 27 ’20 340w
“In short, his tendency to take his hostesses overseriously, apart from some waste of space, does little to impair the value of an
enlightening book. His taste may be a bit at fault but rarely his judgment.”
Review 3:559 D 8 ’20 800w
“This is the best book on the war that has appeared, and we hope it is the last. Everybody is sick of the war, its horrors and its squabbles, and wants to forget it. The excellence of the book consists in its twofold claim on our attention. There is the exhaustive criticism of the conduct of the war by a military expert of European reputation: and there is the picture of manners in that section of society ruled by American women, drawn by one who lived in its favour.”
Sat R 130:260 S 25 ’20 1450w
“Go into a shady part of the garden, or better still, into a damp shrubbery and lift up some big flat stone. Underneath you will find a quantity of crawling creatures, disturbed by the light so suddenly let in upon them.... Such a garden adventure recurs irresistibly to the mind as one reads Colonel Repington’s diary of the war years.... As to the enlightenment which his book should bring in regard to the way in which public affairs are too often handled, as to the advantages of the lessons to be learnt, and finally as to the value of this first step in the reform which comes with knowledge, we have no doubt whatever.”
Spec 125:434 O 2 ’20 2900w REPPLIER, AGNES. Points of friction. *$1.75 (4c) Houghton 814
