The development of mathematical systems theory is

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1 LECURE NOES Lear Operator Equatos wth pplcatos Cotrol ad Sgal Processg By Radal W Beard he developet o atheatcal systes theory s arguably oe o the greatest acheveets o the th cetury [] ew key cocepts have eabled ths developet, oe o whch s lear operator equatos Probles that gve rse to lear operator equatos clude lear regresso, optal resource allocato, optal lterg, optal cotrol, ad solutos to tegral ad partal deretal equatos, to ae but a ew Gve the wde varety o probles that ca be posed as lear operator equatos, a ablty to pose ad solve the s oe o the ost portat tools that systes egeers ca have ther bag o trcks he objectve o ths artcle s to preset the essetal deas behd the soluto to lear operator equatos he teded audece s graduate studets specalzg cotrol ad sgal processg lthough lear operator equatos are preseted ad studed uerous textbooks, they are usually studed the cotext o specc applcatos ere we preset a ued raework or lear operator equatos a way that we hope s sghtul I addto, we oer several coo exaples to llustrate soe potetal applcatos We wll study equatos o the or x = b, where s a lear operator ad b s gve he objectve s to d a x that satses the equato he theory dscussed ths ote s subject to the ollowg assuptos: he operator s lear ad aps oe lbert space to aother lbert space hs assupto lts the scope o applcato to spaces wth a well-deed er product, whch excludes, or exaple, the space o bouded sgals (L he operator ust be bouded hs assupto lts the scope o applcato to cotuous operators, whch excludes, or exaple, partal deretal equatos he rage space o the operator ust be closed hs s a ld assupto that s satsed by alost all terestg bouded lear operators ro oe lbert space to aother lthough these three assuptos ay appear ltg, they clude a large class o portat applcatos Probably the ost portat o these s lear atrx equatos o the or x = b Ideed, ay approxate solutos to partal deretal equatos ca be reduced to ths or Sce ost sgal processg ad cotrol applcatos are pleeted o dgtal coputers, they ust evetually be reduced to equatos that ca be solved uercally coo approach to solvg lear operator equatos, whe the rage ad/or doa o the operator has te deso, s to approxate the operator equato by a atrx equato ece, eve whe the above assuptos are ot satsed, the techques troduced ths ote ca ote be used to approxate ther solutos he reader o ths colu s orgazed as ollows I the ext secto, we gve several detos, cludg the deto o lear vector spaces, er products, ad lbert spaces Next we dee lear operators ad the lbert adjot operator ad gve several llustratve exaples he ollowg secto cotas the heart o these otes Fg s troduced as the key to uderstadg lear operator equatos We beleve that ths gure s a pedagogcally portat tool or uderstadg lear operators ad thereore sped sgcat te dscussg ts detals Whe atteto s restrcted to lear atrx equatos, the sgular-value decoposto copletely characterzes the udaetal subspaces o the operator, as s also dscussed he ext secto presets several applcatos o the theory, cludg least squares, u-or solutos, cotrollablty ad observablty o lear systes, optal cotrol, optal estato, ad odelg echacal systes he exaples were chose to llustrate the wde varety o probles that ca be solved usg the theory preseted the prevous sectos Basc Detos hs secto dees soe basc atheatcal cocepts that wll be used throughout ths ote I partcular, we dee vector spaces, er product spaces, lbert spaces, ad closed subspaces ad gve exaples o each he ost udaetal cocept s that o a vector space Deto : vector spacev = ( X, Sover the set o scalars S s a set o vectors X together wth two operatos, addto ad scalar ultplcato, such that the ollowg propertes hold [], [3]: xy, Xad α, β S ples that αx + βy X x + y = y + x 3 ( x + y + z = x + ( y + z 4 here exsts a zero vector X such that x + = xor all x X 5 α( x + y = αx + αy he author (beard@eebyuedu s wth the Departet o Electrcal ad Coputer Egeerg, Brgha Youg Uversty, Provo, Utah 846, US prl IEEE Cotrol Systes Magaze 69

2 N ( R ( 6 ( α + β x = αx + βx 7 ( αβ x = α( βx 8 here exst scalars adsuch thatx = adx = x Exaple ( / : he ost coo vector space s, the set o real -vectors over the reals, or ore geerally, the set o coplex -vectors over the coplex eld Exaple ( l : he set o real (respectvely, coplex te -tuple sequeces over the real (respectvely, coplex eld ors a vector space Exaple 3 ( L : he set o cotuous uctos : ors a vector space I act, s ot requred to be cotuous, oly easurable [4] hs class o uctos cludes pecewse cotuous uctos Essetally, L cludes the set o all physcally realzable sgals, addto to a large uber o ophyscal sgals he deso o a vector space s deed through the oto o a bass set set P = { p, p, K }, s called a ael bass [5] or a vector space V = ( X, S the vectors P are learly depedet, ad every vector X ca be represeted by a te lear cobato o eleets ro P udaetal theore o uctoal aalyss s that every vector space has a ael bass [5, p ] he oto o a ael bass s prarly used aalyss (see the proo o heore I applcatos, t s ore coveet to work wth a coplete bass, where te sus are allowed exaple s the set o coplex expoetals o L (,, whch are a coplete bass but ot a ael bass (sce ay dscotuous ucto L (, caot be represeted by a te lear cobato o coplex expoetals Deto : he deso o a vector spacev s the cardalty o the sallest ael bass or V Exaple 4: he deso o ad s he deso o l s he deso o L s er product ro oe vector space to aother vector space s deed by the ollowg propertes Let V = ( X, Sbe a vector space, let xyz,, be eleets o X, ad let λ be a eleet o S; the, : X X s a er product x, y = y, x, d ( R ( = d ( R ( N ( R ( Fgure Fudaetal relatoshp betwee a operator ad ts adjot hs dagra was orgally troduced to the author by Pro Joh We at Resselaer Polytechc Isttute, who attrbutes t to the late George Zaes x + y, z = x, z + y, z, 3 λx, y = λ x, y, 4 x, x wth equalty x =, where z s the coplex cojugate o z Every er product o V duces a or o V : x = x, x er product space s a vector space wth a er product Exaple 5( / : he stadard er product o s deed as x, y = y x = x y, = where y deotes the coplex cojugate traspose o yiw s a postve dete erta atrx, the aother vald er product o s x, y = y Wx = w x y = j= j j Exaple 6 ( : vald er product o te -tuple sequeces s a sple exteso o the er product o Let x = ( x, x, K ad y = ( y, y, K be te sequeces wth x, y Vald er products clude the ollowg: l x, y = y x x, y = y Wx, W erta x, y l l l = = y xw w = =, > he set o te -tuple sequeces such that x l < s deoted as l or sply as l whe s uderstood Exaple 7 ( L : he er product or uctos s aga aalogous to the er product or l ad I ths case, the su s replaced by a tegral he ollowg are vald er products or uctos: x, y = y ( t x( t dt L ( Ω x, y = y ( t Wx( t dt, W erta L ( Ω Ω Ω x, y = y ( t x( t w( t dt, w( t > ad easurable, L ( Ω Ω ( where the tegral s a Lebesgue tegral [4] he set o easurable uctos o such that x L < s a vald vector space ad s deoted as L ( Ω or sply L whe adω ( Ω are uderstood ( 7 IEEE Cotrol Systes Magaze prl

3 vector space V s sad to be coplete all Cauchy sequeces (e, coverget sequeces V coverge to a eleet o V lbert space s a er product space that s coplete he classc exaple o a er product space that s ot coplete s the set o cotuous uctos wth the er product deed ( For exaple, the Fourer seres o a perodc square wave cossts o a te su o cotuous uctos Let s deote the rst ters ths seres; the the sequece { } s a Cauchy sequece that coverges, s the ea squared sese, to a dscotuous ucto hereore, the set o cotuous uctos s ot a lbert space; however, the set o te eergy easurable uctos L s a lbert space he ost cooly used lbert spaces are, C, l, ad L Deto 3: Let V ad W be subspaces o a lbert space ; the V s orthogoal to W, wrtte V W, or all v V ad all w W, vw, = Deto 4: I V s a subspace o a lbert space, the the orthogoal copleet o V s the set { x : v V, x, v } V = = [ αu + βu ] = α[ u ] + β[ u ] ece, s a lear operator over the reals Exaple (State rasto Matrx: he lear deretal equato &x = x y = Cx wth tal codto x( = x dees a lear operator ro = R to = L [, ]or every < < Let t [ x ]( t = Ce x he y( t = [ x ] s L[, ], ad sce he objectve o ths artcle s to preset the essetal deas behd the soluto to lear operator equatos [ αx + βx] = α[ x] + β[ x ], s a lear operator over the reals Exaple (Zero-State Deretal Equatos: he zero-tal-state lear deretal equato Deto 5: I V ad W are orthogoal subspaces o a lbert space, the the orthogoal su o V ad W s V W = { x : x = v+ w, v V, w W} he ollowg lea s proved [3, p 8] Lea : Let V be a closed subspace o a lbert space he = V V Operators ad djots hs ote s lted to lear operators that ap oe lbert space to aother operator : s sad to be lear or all x, x ad α, β S, where S s the eld assocated wth, ( αx + βx = αx + βx Exaple 8 (Matrces: Let be a coplex atrx he aps to ad s a lear operator over the coplex eld Exaple 9 (Covoluto: Covoluto s a lear operator ro = L (, to = L (, Letu L (, ad dee t []( u t = u( τ h( t τ dτ or t [, ] Iy( t = [ u]( t, the y L (, u L ad h s absolutely tegrable [4, p 8] Clearly, &x = x + Bu y = Cx wth tal state x( = dees a lear operator ro = L [, ]to = p he soluto to ths equato at te s gve by Let ( τ y ( = Ce Bu( τ dτ ( τ [(] u t = Ce Bu( τ dτ Sce [ αu( t + βu( t] = α[ u( t] + β[ u( t], s a lear operator over the reals Exaple (Fourer rasor: he Fourer trasor jωt [ x( t] = x( t e dt π dees a lear operator ro = L (, to = L (, over the coplex eld he lbert djot Operator udaetal cocept lear algebra ad uctoal aalyss s the lbert adjot operator Gve a lear operator (3 prl IEEE Cotrol Systes Magaze 7

4 :, the adjot o, deoted, s a operator ro to that satses x, y = x, y (4 or all x ad all y Exaple 3 (Matrces: Let be a coplex atrx he adjot o s oud by applyg the deto (4: x, y = y x = x, y = ( y x, ad y Fro ths equa- whch ust hold or all x to, we see that y = ( y = y = Exaple 4 (Covoluto: he adjot o the covoluto operator deed Exaple 9 s deed by where ad [], x y = x, [] y, L (, L (, t L (, t= τ = [], x y = x ( τ h ( t τ d τ y ( t dt = τ = x( τ ht ( τ yt ( dt dτ t= τ (5 x, [] y = x( τ []( y τ dτ τ = By equatg (5 ad (6, we get that [ y]( t = h( σt y( σ dσ σ = t Exaple 5 (State rasto Matrx: Let be the state trasto ap deed Exaple ; the the adjot o s deed by where ad =, L [, t] x, y x, y x, y = y ( τ Ce x dτ L [, ] τ = = y ( Ce d τ τ τ x τ = τ = e τ C y( τ dτ x τ = (6 (7 x, y = ( y x (8 Settg (7 equal to (8 gves τ = τ y = e C y( τ dτ Exaple 6 (Zero-State Deretal Equatos: Let be the lear operator assocated wth the zero-state deretal equato Exaple ; the the adjot o s deed by where ad x, y = p x, y, L [, ] x, y = ( p x ( B e C d y τ τ τ = x ( τ( B e ( τ C y dτ L [, ] x, y = x ( τ [ y]( τ dτ Settg ( equal to ( gves ( τ (9 ( ( [ y]( t = B e C y ( Exaple 7 (Fourer rasor: Let be the Fourer trasor operator deed Exaple ; the the adjot o s deed by where [ x]( jω, Y( jω = x( t, [ Y]( t, L [, ] L [, ] j t [[ x], Y = ω x( t e dt Y( L [, ] jω dω π ω= t= jωt = x( t e Y( jω dω dt π t= ω= (3 ad x( t, [ Y]( t = x( t [ Y]( j d dt L (, ω ω t= (4 Settg (3 equal to (4 gves jωt [ Y]( t = e Y( jω dω, π ω= the verse Fourer trasor oy( jω Fudaetal Subspaces o a Lear Operator he portace o the adjot o a lear operator coes ro the udaetal relatoshp betwee a operator, ts adjot, ad ther assocated rage ad ull spaces he rage space o a operator : s deed as 7 IEEE Cotrol Systes Magaze prl

5 R( = { y : x such that y = x} d( R( = d( R( he ull space o : s deed as N ( = { x : x = } he objectve o ths secto s to show that the udaetal relatoshp betwee a operator, ts adjot, ad ther assocated rage ad ull spaces s that llustrated Fg, aely: aps eleets o tor( ad aps eleets o to R( ; s the orthogoal su o N ( adr( ad s the orthogoal su o N ( ad R( ; 3 he deso o the rage space o equals the deso o the rage space o he proos o these stateets are cluded the ollowg three results Lea : Let : be a bouded lear operator, where ad are lbert spaces Furtherore, let R( ad R( be closed he [ R( ] = N ( ad [ R( ] = N ( Proo: o show that [ R( ] = N (, we eed to show that N ( [ R ( ] ad that [ R( ] N ( o show that N ( [ R( ], let y be ay eleet N ( ad let $y be ay eleet R( he there exsts a $x such that y$ = x$, ad y$, y = x$, y $ = x, y = x$, =, whch ples that y [ R( ] o show that [ R( ] N (, let y be ay eleet [ R( ] ad let $x be ay eleet he y$ = x$ s a eleet o R(,so y$, y = x$, y = By deto o the adjot, ths ples that x$, y = Sce ths s true or every $x, we ust have that y =, whch ples that y N ( he proo that [ R( ] = N ( s show by slar arguets + Fro Lea, we thereore have the ollowg theore heore : Uder the hypothess o Lea : = R( N ( = R( N ( 3 d( = d( R( + d( N ( 4 d( = d( R( + d( N ( heore : Uder the hypothess o Lea, Proo: We eed to show that a d( R( d( R( ad b d( R( d( R( o prove part a, let P = { p, p, K} be a ael bass or R( so d( R( equals the cardalty o P For each p P, there exsts a $q such that p = q$ Sce = R( N (, there s a uque decoposto $q = q + where q R( ad N ( hereore, p = q Let Q = { q, q, K } We wll show that Q s a learly depedet set, whch ples that ay ael bass o R( cotas Q, whch ples that d( R( d( R( P s a ael bass or R(, he portace o the adjot o a lear operator coes ro the udaetal relatoshp betwee a operator, ts adjot, ad ther assocated rage ad ull spaces whch ples that all te subsets o P are learly depedet, e, cp = c =, I, I where I s a te dex set But cp = cq = I I cq = I I cq =, where the last plcato ollows ro the act that cq s a eleet o R(, whch s orthogoal to the N ( hereore, Q s learly depedet he proo o part b ollows by substtutg or ad or the above arguet + By spectg Fg, t ca be see that ay eleet orgatg aps to R(, cludg eleets R( Sce ay eleet s apped by tor(, t s also apped by to R( hereore, the ollowg lea holds Lea 3: Uder the hypothess o Lea : R( = R( ; R( = R( other portat result that ollows drectly ro Fg s Fredhol s alteratve [3], whch s stated as Lea 4 I prl IEEE Cotrol Systes Magaze 73

6 Lea 4: Uder the hypothess o Lea, the operator equato x = bhas a soluto b, ν = or every vector ν N (, e, b N ( P = P hereore, the operator ( s vertble, the operator I ( projects oto N ( Mappg ro to R( : Clearly, aps to R( wth o addtoal assuptos Mappgs ad Projectos I applcatos, t s desrable to be able to project eleets o ad to the our udaetal subspaces o Whe Mappg ro to N ( : I s te desoal wth deso ad d( R( = p <, the a appg ro to N ( ca be costructed by dg orthooral s apped to N ( or R(, we would lke these vectors up+ u that are orthogoal to R( ad lettg appgs to be projecto operators ssug that the hypotheses U = ( up+, up+, K, u he the atrx U aps ( to o Lea are satsed, the ollowg paragraphs deostrate how these operators ca be costructed N ( Let Z be ay oto operator ro to ( ; the U Z aps to N ( Projecto ro tor( : Usg arguets slar to costructg the udaetal cocept lear algebra projecto ro to R(, t ca be ad uctoal aalyss s the lbert show that the operator : s vertble, the the adjot operator operator ( projects otor( Whe the operator s Projecto ro to R( : I x, the x ca be ot vertble, the S aps to R(, where S s ay uquely decoposed to a copoet R( ad a copoet N ( Let x = xr + x, where x r R( ad x N ( Sce x r R(, there exsts a eleet y vertble operator o Projecto ro to N ( : Slarly, s vertble, the the operator I ( projects oto such that xr = y, so N (, where I s the detty operator ro to Mappg ro to R( : Clearly, aps tor( x = y + x = y wth o addtoal assuptos Mappg ro to N ( : Usg arguets slar to costructg the appg ro I the operator : s vertble, the to N (, we get that y = ( x, whch ples that xr = ( d( x = <, ad the colus o V spa the ull space Note that P = ( o, thev s a projecto operator sce Z aps to N (, where Z s ay oto operator ro P = P hereore, the operator( s vertble, the operator to ( ( projects oto R( Whe the op- Characterzato va erator s ot vertble, the S aps tor(, Sgular-Value Decoposto where S s ay vertble operator o I the operator s a coplex atrx, the each o Projecto ro to N ( : Fro the prevous paragraph, ote that characterzed by the sgular-value decoposto o [3] the udaetal spaces show Fg ca be copletely I, wth rak( = p, the the sgular-value decoposto x = x xr o s gve by = x ( x = ( I ( x, V = U U Σ U V V = Σ, (5 where I s the detty operator ro to Note that P = I ( s also a projecto operator sce where Σ=dag( σ,, σp L σ p >, ad U = ( U U ad V = ( V V are utary atrces, e, C V V = I ad U U = I he odcato C o Fg or the atrx case s show Fg Lea 5: I ad the sgular-value decoposto o s gve by N( = spa( V N( = spa( U R( = spa( V R( = spa( U (5, the R( = spa( U N ( = spa( U 3 R( = spa( V Fgure Fudaetal relatoshp betwee a atrx ad ts erta 4 N ( = spa( V 74 IEEE Cotrol Systes Magaze prl

7 Proo: Let y be ay eleet o R( ; the there exsts a x such that y = U Σ V x Let z =ΣV x; the y = U z, whch shows that y spa( U Coversely, let y be ay eleet the spa o U ; the there exsts soe z such p that y = U z Let x = VΣ z; the z =ΣV x ad y = x,so y R( Sce the colus o U are orthooral to the colus o U, ad spa( U = ad R( N ( =, we ust have that N ( = spa( U Slar arguets show that R( = spa( V ad N ( = spa( V + he atrx U s a ull-rak p atrx ad thereore dees a appg ro p to, where p Slarly, p p U :,V :, adv p : coplete characterzato o the udaetal subspaces ters o the sgular-value decoposto s depcted Fg 3 Fro Fg 3, t s clear how to costruct operators ro ad that ap to each o the udaetal subspaces o he operators are deed as ollows: VSV : N (, where S s ay vertble ( p ( p atrx I partcular, S V = ( V, the the operator s a projecto VSV : R (, where S s ay vertble p p atrx I partcular, S V = ( V, the the operator s a projecto USU : N (, where S s ay vertble ( p ( p atrx I partcular, S U = ( U, the the operator s a projecto USU : R (, where S s ay vertble p p atrx I partcular, S U = ( U, the the operator s a projecto USV : R (, where S s ay vertble p p atrx I partcular, ote that S =Σ, the the appg s sply VSU : R (, where S s ay vertble p p atrx U Z: N (, where Z s ay ull-rak ( p atrx VZ : N (, where Z s ay ull-rak( p atrx pplcatos he objectve o ths secto s to llustrate, through a varety o applcatos, the power o the cocepts preseted the prevous secto I partcular, we wll show applcatos to solvg lear systes o equatos, cotrollablty ad observablty, optal cotrol, Kala lterg, ad odelg dyac systes Lear Systes o Equatos Our rst applcato s the soluto o the lear syste o equatos x = b, (6 C p V N( C where, x, ad b I= = rak(, the N ( = N ( = { } ad (6 has the uque soluto x = b O the other had, s ot square or ot ull rak, the as show Fg, ether N ( or N (, or both, wll be otrval Least-Squares Solutos: Cosder rst the case whe s tall ad ull rak (e, > ad rak( = I ths case, Fg reduces to Fg 4 I b R(, the x = b has a soluto Sce rak( =, verso ca oly take place hereore, appgb to va gves b = x Notg that s vertble, we ca solve or x to obta x = ( b (7 I b R(, the o soluto exsts he least-squares soluto s deed to be $x = arg x b C N( R( R( V V V We ca decopose b as b = br + b, where br R( ad b N ( hereore, $x = arg x br b Sce b R( ad br R(, usg (7, we ca ze x br b by settg $ x = ( br to obta a u value o b ( ( br + b = ( br ples that the least- squares soluto s gve by (7 + he atrx = ( s called the Moore-Perose verse o [6] ad s a exaple o a pseudo-verse o I C p U U U U C p Fgure 3 Fudaetal subspaces o ters o the sgular-value decoposto R( C rak( = N Fgure 4 Least-squares lear equatos C N( R( prl IEEE Cotrol Systes Magaze 75

8 N( R( C C R( to wll have a zero copoet ro N ( ; thereore, xr = V ζ, where ζ p ccordgly, we seek the least-squares soluto to the equato V ζ= b Sce b ay ot be R( V = R(, we projectb otor( vau (see Fg 3 to get U Vζ= U b Sce = U V Σ, we get Σζ = U b, whch has a soluto ζ= Σ U b, plyg that xr = VΣ U b he soluto s thereore aga gve by the Moore-Perose pseudo-verse o : rak( = + V Σ U = Fgure 5 Mu-or lear equatos the sgular-value decoposto o s = U V Σ, the + = VΣ U Note that R( + = R( ad N ( + = N ( hereore, Fg 4 could have bee draw wth + stead o Mu-Nor Solutos: Now cosder the case whe s at ad ull rak (e, < ad rak( = I ths case, Fg reduces to Fg 5 I ths case, the equato x = b wll have a te uber o solutos sce, $x s a soluto, the $x + x wll also be a soluto, where x s ay vector N ( Suppose that we would lke the soluto that has the u Eucldea or; e, we would lke to solve the optzato proble x subject to: x = b he u-or soluto s oud by orcg the copoet o the soluto ro N ( to be zero hereore, we seek xr R( such that x r = b xr R( ples that there exsts a ζ such that xr = ζ; thereore, ( ζ= b Note ro Fg 5 that ( aps oto tsel ad s vertble; thereore, ζ= ( b, whch ples that x = ( b he soluto r x = ( b s the u-or soluto o the equato x = b he atrx = ( s also called the Moore-Perose verse + Further sght about the structure o + ca be gaed ro the sgular-value decoposto o Let = UΣV ; the + V = U Σ Note or coparso that = V Σ U ; thereore,r( = R( + ad N ( = N ( +, whch eas that Fg 5 could have bee draw wth replaced by + Mu-Nor Least-Squares Solutos: he al case we wll cosder s the geeral proble show Fg, where rak( = p < (, I ths case, the soluto s ost easly derved usg Fg 3, where t ca be see that the soluto ca be decoposed to two probles: a least-squares soluto ro p to ad a u-or soluto ro to p Fro Fg 3, we see that the u-or solu- Note that the stadard deto o the Moore-Perose pseudo-verse s gve as ollows [6, p 43] I, the the Moore-Perose pseudo-verse o s the uque atrx that satses =, =, ( =, ad 4 ( = It s straghtorward to show that = U V Σ s the sgular-value decoposto o, the + V = U Σ satses the our equatos lsted above (use the act that V V = I ad U U = I Moreover, t ca be show that t s the oly atrx that satses these equatos Cotrollablty Let &x = x + Bu be a lear syste wth x ad u he syste s sad to be cotrollable gve ay tal codto x, a arbtrary desred al posto x, ad a te te >, there exsts a cotrol u( t L [, ] that drves the syste ro x to x he soluto to the deretal equato at te gve the tal codto x s Rearragg we get ( τ x ( = e x + e Bu( τ dτ ( τ x ( e x = e Bu( τ dτ = [], u where s deed (3 ad aps = L [, ]to = Clearly, to allow x = x( to be arbtrarly assged or ay tal codto x requres that R( =, whch case Fg reduces to Fg 6 Sce : L [, ], the syste wll be cotrollable ad oly rak( = rak( = o derve the stadard cotrollablty crtera, we solve or the u-or soluto to ths operator equato ccordgly, let u R( ; the u = ζ, where ζ ad s gve ( hereore, ζ satses ( ζ= x e x, where the atrx ( has deso By Sylvester s rak equalty [3, p 5], rak( =, the ( has rak equal to ad s hece vertble hereore, there exsts a cotrol sgal L [, ]that drves the syste ro ay tal codto x to ay al cogurato x ad oly the rak o the atrx 76 IEEE Cotrol Systes Magaze prl

9 ( τ ( ( = e BB e τ dτ (8 L [, ] s equal to he atrx (8 s called the cotrollablty Graa Usg the Cayley-alto theore [6], t s straghtorward to show that the cotrollablty Graa has rak ad oly the atrx ( B B B B has rak, whch s the stadard cotrollablty crtero [7] N ( R ( Fgure 6 Fudaetal subspaces or cotrollablty R R ( Optal Cotrol Cosder the optal cotrol proble L p [, ] u subject to: x& = x + Bu L x( = x x( t = x R R ( N ( R ( he costrat could also be wrtte as x = e x + τ e Bu( τ dτ t t ( t τ t ( t Let [] u = e Bu( τ dτbe the lear operator deed Exaple ; the the objectve s to d the - u-or soluto to the lear operator equato [] u = x e x (9 I the prevous secto, we showed that the u-or soluto s gve by u = ζ, where ζ satses t ζ= x e x t ad where s the cotrollablty Graa gve (8 I the syste s cotrollable, the s vertble, whch ples that the u-or cotrol s gve by t ( ( ( ( τ ( t τ t C dτ ( x e x u( t = x e x t t t t = e C Ce e Observablty Let &x = x y = Cx ( be a lear syste wth x, y, ad tal codto x( = x he syste s sad to be observable the tal codto x s uquely deted by the output y( tover a p Fgure 7 Fudaetal subspaces or observablty te perod o te [, ] s dscussed Exaple, the output y( ts gve by the equato t y( t = Ce x = [ x ]( t, where : L [, ] o allow x to be uquely detered requres that the deso or( be equal to, whch case Fg reduces to Fg 7, where s gve by (9 I ths case, y( t R(, ad thereore o orato s lost by appg y to R va to obta [ y] = [ x ] he atrx P = e τ τ C Ce dτ τ = s called the observablty Graa o syste ( By Lea 3,R( = R(, whch ples that syste ( s observable ad oly rak( = Usg the Cayley-alto theore [6], t s straghtorward to show that the observablty Graa has rak ad oly the atrx ( ( C C C has rak, whch s the stadard observablty crtero [7] Optal, Ubased Estato Cosder the dyac syste x& = x, x( t = x y = Cx + ξ, ( prl IEEE Cotrol Systes Magaze 77

10 y x x$ = e C Ce d e C y( d τ τ τ τ τ τ τ = τ = Sce the covarace o the ose s detty, ths s also the u varace estate [3] g Fgure 8 Pot-ass suspeded by assless rod where ξ s a zero ea Gaussa process wth covarace equal to the p p detty atrx he objectve s to develop a ubased estate o x ro easureets o y( t over the terval[ t, t ] he soluto to the deretal equato ( s t y( t = Ce x + ξ( t = [ x ] + ξ( t, where : Lp[ t, t] s gve Exaple s show the prevous exaple, the syste s observable, the rak( = rak( = he objectve s to d a lear operator K : Lp[ t, t] such that x$ = K[] y s a ubased estate o x ; e, hereore, we have { } E x$ = x { $ } = { K[] } = E{ K [ [ x] + ξ] } = E{ ( K [ x] + Kξ} = ( K [ x] + E{ Kξ} E x E y = ( K [ ] Note that K : s a atrx I K = I, the Ex { $ } = x, whch ples that $x s a ubased estate hereore, K s a let verse operator o FroFgts clear that such a operator s x K = ( hereore, a ubased estate o x s L Dyacs o Costraed Mechacal Systes Mechacal systes are ote odeled usg Lagraga or altoa orulatos alteratve ethod, preseted [8], uses Gauss s prcple ad the techques developed these otes he techque s best llustrated va a exaple Followg [8], let x = ( xy, be the coordates o a pot-ass wth ass that s subjected to a orce F = ( Fx, Fy he ucostraed accelerato o the pot x s gve by a = M where M = I ad a s the ucostraed accelerato lteratvely, the pot-ass s costraed, the the actual accelerato wll der ro the ucostraed accelerato Suppose, or exaple, that the pot-ass s costraed to be at the ed o a assless rod o legth L rgdly suspeded ro the pot(,, as Fg 8 I ths case, the pot-ass satses the costrat φ( x = x + y L = Deretatg ϕ twce, we coclude that the actual accelerato o the pot-ass ust satsy the costrat F ( x&& x = b( x,& x, where ( x = ( x, y ad b(,& x& xx = y& Gauss s prcple states that the accelerato o the pot-ass wll satsy the ollowg zato proble: M ( x&& a (&& x a subject to: x&& = b lteratvely, we ca wrte ( / && / M M ( M / && M / x a x a / / subject to: M ( M x && = b Lettg ξ= M &&x M / / a gves ξ subject to: M ξ = b a, / ( whch s the stadard u-or least-squares proble dscussed earler ths secto he soluto s thereore gve by / ( ξ= M ( b a, + 78 IEEE Cotrol Systes Magaze prl

11 or alteratvely, ( && / / x = a + M M ( b a For the pot-ass show Fg 8, we get + ( x y gy x && = & & + x + g L y Coclusos hs lecture ote has preseted a ued approach to bouded lear operator equatos o lbert spaces We have show that lear operators ca be copletely uderstood va the dagra show Fg, whch represets the udaetal subspaces o a bouded lear operator We have also show a uber o applcatos o the uderlyg theory he objectve o these otes has bee to preset the udaetals a pedagogcally clear aer, the hope that they wll eable readers to successully apply the cocepts to ther ow probles Reereces [] GC Goodw, SF Graebe, ad ME Salgado, Cotrol Syste Desg Eglewood Cls, NJ: Pretce all, [] DG Lueberger, Optzato by Vector Space Methods New York: Wley, 969 [3] K Moo ad WC Strlg, Matheatcal Methods ad lgorths Eglewood Cls, NJ: Pretce all, [4] F Joes, Lebesgue Itegrato o Eucldea Space Bosto, M: Joes ad Bartlett, 993 [5] E Kreyszg, Itroductory Fuctoal alyss New York: Wley, 978 [6] P Lacaster ad M seetsky, he heory o Matrces, d ed Sa Dego, C: cadec, 985 [7] WJ Rugh, Lear Syste heory, d ed Eglewood Cls, NJ: Pretce all, 996 [8] FE Udwada ad RE Kalaba, alytcal Dyacs: New pproach New York: Cabrdge Uv Press, 996 Radal W Beard receved a BS electrcal egeerg ro the Uversty o Utah 99 ad a MS electrcal egeerg 993, a MS atheatcs 994, ad a PhD electrcal egeerg 995, all ro Resselaer Polytechc Isttute, roy, NY Sce 996, he has bee a ssstat Proessor the Electrcal ad Coputer Egeerg Departet at Brgha Youg Uversty, Provo, U I 997 ad 998, he was a suer aculty ellow at the Jet Propulso Laboratory, Calora Isttute o echology s research terests clude coordated cotrol o ultple-vehcle systes ad olear cotrol e s a eber o the IEEE, I, ad au Beta P prl IEEE Cotrol Systes Magaze 79