European Journal of Control (1009)2:143-151; " 2009 EUCA DOl: [(), 3166lEJC. J 5. J.t 3- J 56

European Journal of

Control

A Discrete-time Observer Based on the Polynomial Approximation of the Inverse Observability Map

Alfredo Germaniu ,- and Costanzo Manes 1.3,"

'Department of Electrical and Information Engineering. Uni ver.;ity of L 'Aquila, Poggio di Roio, 670.tO L' Aquila. Italy: "Uni\'er,itil Campus Bio·Medim of ROUIe, Via Al vaTU del Portillo. 11. 00 12H Room Italy: 'ht ituto di Anali,i d~i Si,tcmi ed Infonnatica dd CNR "A RUOerti", Viale Manmni 30, ()(llll5 Roma. Italy

This paper inrestigates the prohlem (~f asymptutic slate ohserl'alion .lin mwfylic nontillear discrete-lime

.ITslems. A fI('\\" leclllliquefor Ihe ohserl"er cOllstfllctioll. hased Oil Ihe Taylor polYllumial approximalio/l of Ihe i!lrefSe of Ihe ohserrahililY map, is presented alld disclI.n-ed. The degree dlOsell for Ihe approx imalioll i,l'

the most important desiKn parameler oflhe ohsen'er. ill Ihat il call he (,/IOSt'lI ,\"[(ch 10 ells lire the ('olll"ergence of Ihe ohserratioll error at allY desired expollential rate, ill liny prescrihNI convergellce region (semiKlohal exponcllt ia! COli rergellce ) .

Keywords: Observer design, discrete-time nonlinear systems, polynomial approximation.

I. Introduction

The importance of the problem of asymptotic state observation of nonlinear dynamical systems is well known, and a large amount of scientific literature on this subject is available. Although the case of con­tinuous time systems has been more intensively stu­died in the past. an increasing number of authors is devoting the attention to the state observation prob­lem in the discrete-time framework. Many papers discuss the extension to discrete-time systems of observer design techniques devised and developed for

• Email: alfredo.germani(li univaq .it ••

Corfl'V)()"'/l'na /0: C. Mancs, Email: costanzo.mancs (,! univaq,it

continuous time systems. An approach widely inves­tigated is 10 find a nonlinear change of coordinates and, if necessary, an output transformation, that transform the system into some canonical form suit­able for the observer design using linear methodolo­gies. First papers on the subject are [10] , [23],[24]. where autonomous systems are considered. and quite restrictive conditions are given for the existence of the coordinate transformation that allows the observer design with linearizable error dynamics. Other papers concerned with the problem of finding conditions for the existence of the change of coordinates are [301 and [31] . for autonomous systems, and [3J and P] for non autonomous systems. The drawback of this approach is thaI in general the computation of the coordinate transformation, when existing, is a very difficult task. Also the technique proposed in [22] requires to solve functiona l equations that in general do not admil a closed for11l solution. An interesting approach for the construction of observers with linear error dynamics for systems admitting a differential / difference representation is reported in [26] . Another approach consists in designing observers in the original coordinates, finding iterative algorithms. typically inspired by the Ne\vton method. that asymptotically solve suitably defined oh,l"i'n'ahifity maps (see e.g. [12], [13] and [27]) . Sufficient conditions of local convergence are provided, in general, under

R,'cn"'l'{/IO Apri! ]()Ol!. Acup/"d II! D"c<'I"ha ]()()8

Recommend"d h)" D. NormmU!·Cyrol "nd S. Monaco.

144

the assumption of Lipschitz nonlinearities . The use of the Extended Kalman Filter as a local observer for noise-free systems has been investigated in [29J, [4J. [5] and l28j. In [28]. sufficient conditions of convergence are given in term of existence of a solution for some Linear Matrix Inequalities (LMI ). Some authors pro­pose observers for particular classes of systems. In [32] an observer with standard Luenberger form and constant gain is proposed for autonomous systems with the state transition function characterized by Lipschitz non­linearities and with linear output transformation. Suffi­cient convergenL'C conditions are given in term of LM [. Nonlinear systems with linear measurements are also considered in llJ and [6]. The problem of asymptotic state reconstruction for systems described by bilinear state-transition functions and rational output functions is considered in [17]. where. using the tool of the Kronecker algebra . the problem has been solved through the immersion of the nonlinear system into a linear one of larger dimension . In [19] the use of polynomial approximations of nonlinear discrete-time systems for solving the state observation problem has been investi­gated, and conditions for the exponential convergence of the polynomial extended Kalman filter lIS], when used as an observer, are studied.

T his paper presents a ne\v method for the con­struction of an observer for nonlinear discrete-time systems of the type

X(I + 1) = '/lx(I)). I E 2, ( 1 )

y(l) = "(X(I )),

where X(I) E IRn is the system state, Y(I) E III is the measured output. I: R." ......... lRn is the one-step state transition map, and II: IRn ......... 1R is the output function. System (I) is assumed to be Klohally mw/rlic. that means thai both f and II are analytic functions in all compact sets ofIR". The approach employed here can be considered the discrete time equivalent of the approach followed in [II] and [14]. The method is based on the Taylor polynomial approximation of the inverse of the observability map (the function that maps states into output sequences) up to a given degree v > I. and constitute a nontrivial extension of the one presented in [12]. where only the first degree Taylor approximation has been considered, and only local convergence is guaranteed . T he main novelty of the contribution here presented is that. thanks to the analyticity assumption. the degree of the approxima­tion can be chosen such to guarantee the exponential convergence of the observation error to zero when the initial observation error is inside a ball of pre­scribed radius. centered at the origin. It follows that

A. Gemumi and C. MIlIlCI

the proposed observer has the important property to be a sl'lIJiKlohal (,xpol/I'ntial uhsl'fl"£'r, in that both the convergence rate and the radius of the convergence region can be arbitrarily assigned.

T he paper is organized as follows. In section II some preliminary definitions and notations are given, and previous work of the authors on the subject is discussed . In section III the new observation algo­ri thm is presented and its convergence properties are discussed . Simulation results and conclusions follows. An extended Appendix is also present. where the theory of the Taylor approximation of the inverse of a nonlinear vector function is explained in some detail.

2. Preliminaries

In this section some definitions. notations and con­cepts that will be used throughout the paper are presented. and relations with the authors' previous work are discussed. T he sYlllboIF(.r ) , with r EN. is recursively defined as:

f o(x) = X ,

J'+'(x) = ((of')( x) = ((f ' (!')), I > 1. (2)

(the symbol 0 denotes the composition of functions). and is used to denote the r-steps stale transition , thai allows the computation of future system states as a function of past states:

X(I + I ) = ('(X(I)). (3)

The output at time f + r can be written as a function of x ( t) as follows

y( t + r) = hoF(x(t)). (4)

Definition 2.1: The ohsl'rl"Uhilify map fur Ihl' sl"sfl'm

( J) is a I'eclor jimclion :: + 1> ( x ) . Ifi!1! 1>: !lll.n ......... R.n.

defined as:

Lei Y, denote Ihe slacked Olllplil sequence y( f) inll'rl"U/ /1,/ + n)

Y, =

)"(1 + ;1 - 1)1.

Y(I+ 1) j Y( I)

(5)

in Ihl'

(6)

Poll'",,,,,;,,1 Appwx;mm;OI' <If [/ ... 1m.." .... Oh. ... ,.."f,iIi,y ;\I"p

By Ill(' definilion of Ihl' ohserrahililY mllp. Ihe following hold.,:

Y, = "' (X( I)) . (7)

Definition 2.2: Tile oh,lw)'ahiliIY /I/(/Irix for Ihc syslem t l ) i,I','

d Q(x) = - >!>(x ).

(Ix (8)

Followiflg Ihe 1I0WliOI/.I· illfl'Odlln!lJ ill 111l' Appelldix , Q t x ) will he also )1'f i I /('11 a,~ \7 , (I)(.\') or v\ z. 4>(.\') (,n'e

f 1(0) ill Ihc ApPl'm/ix ) ,

Remark 1: ,Vole Ihal for lim'af syslems Ihe aho!'e definilio/l of ohser)'{/hililY I/!(Ilrix ('(Iindt/I',I' lI'ilh Ille dassico/linear onc ,

Dl' finilion 2.3: The nOllfint'(Jr .I'plel// ( / ) is ,mid /(J he observable ill all opell sllh.~el 1! c R:~ if il.l· ohsenahifitll map (5) i,~ im'ertihlt, ill 0, i,t' .. Ihl'n' '·Xi.~/,I· ~ - I f::) ,~lIch Ihat (1)-1 (~(x)) = x,for (III x E ~t I/o. = R~ , Ihell Illl' sJ.l'tem ( / ) is saill /0 he globally o bserva ble.

The Dljinitioll 2,3 of I!h,~Cf\'(/hiliIY ,~ ltlle.\· Ihl' IlIeO(­etiml po.l'.I'ihility (~l thl' recOIHtnlNiOIl of rhe .~Iale of a lIon/iliear ,\,),slelll, al a gil'('11 lillie I, exploitillg Ihe k/lowledge (~f Ihe Oil/pili wi/II/:'/ice ill Ihe illlenal /1.1 + II), as

(9)

Such a po.uihili(v ill 1II0,~1 C(W!,~ relllaills Iheoretical m'('(III,~l'. ill gelleral. closcd furms fur Ihl' illra,n' of 'he ohsf'(mhilily map lire /101 ami/ahle, Note fhat the oh,I'('f1'ahilil), properly in a .1'('1 n implies Ihal Ih(' oh.l'erl'ahilil), malrix Q fx ) i.l' IWI/.I'ingular ill 0,

Definition 2.4: The 1I01llillear s),stem ( I ) i,l' said 10 he uniform ly Lipschitz observable ill (II! opell suh.l'eT n C R IO flTit is oh,lwwlhle. alld ifhoth lIIaps 4' ( x ) ami (I) - 1(:), (lrt' IIlIiformly Up.l'dtit : ill n (lml in (1)(0 ). r/:,,~pel'fil'(")', Thi,~ mCIIIH f/rm there e ,\'I:I'1 111'0 r('(ll ItIll/lh('f,~ "'I'" , and 1 <1> ,nid, Ihat

11 4>(x r) - (1) (.\: ])11 < 1<1> 11,\'1 - Xl II . V_\:I, X! E n, 11 "'- ' (0, ) - " - '(0,)11 < 1. '110, - 0,11·

'''':: 1, =2 E <I> (n ),

( ' 0)

If n = an , Ihen lhe s,nlem ( I ) i.l' .I'aid 10 he globally uniformly Lipschitz observable,

Dl' finilion 2.5: Co/Hider a '~l'lIIl1/lh ' ,1'p'le", of Ih,' form

((t + ' ) = g«( I), ' ·(1). 0). I E Z. ( " )

« I) = m«(I ). ye ,l. 0).

145

II'/r('f(' « I) E IR' i.~ the ,~y_\' I('m ,I'I(l/l', 0 E lR'" i~· II .~t'I of ('0/1.1'((1111 paraml'lers. allll y (t) E R i,l' lire 01111'111 of .\)'Stel/l ( I ; ,

• System ( II) . with a given cho ice o f O. is said to be a lo('(J! (uymptolic oh.l'en'('f for (I) if there exist an open set n C IRn and Ii E R+ such that. for any pair X(IO ),€(/o) E O.

IIX(lo) - ~( lo)1I < 8 lim IIx(l) - W) II = o. ,- 0 (12)

• The system ( I I) is sa id to be a .I'1'II1I)dohal asymplolic ob.l'er rer if fo r any H C IR" and b E IR+ there exists a choice 0 s tich to guarantee Ihe implication (1 2) fo r :lily pair x( ro ) ,~(l() E n,

• T he system (II). with a given choice of 0. is s.1id \0 be a glohal lI.~ympfOli{ · oh.I'('fI'('r fo r (I) if for .IIlY pair x ( tO).t,;( /O) E Rn it is limt_o ll x(l)­,(1) 11 = O.

Remark 2: A ssllme Ihm a sy,l'lem of Ihe Iypl' ( 11 ) "rmides a sequcnce U l j .filch tllm for (Ill pairs x(to).Wo) E R" il is

';m II Y, - <1> « (1))11 = O. ,_0 ( ' J)

If thl! ,~y,l'Ielll ( I ) i,~ IIl1iformly U{1sc/lil:: ohserrahlt,. IhclI ( II ) i,~ (I globa l asym ptotic observer. T"i,~ IIlIPPl' IIS heall/.\l' tht, Upschit:: propert), implies Ihe follOirillg

Il x (l) - (1)11 = 11 "'- ' ('Nx(I ))) - " - '("'«0)))11

= 11"'- '( Y,) - .,,- ' ("'(((1)))11 < 0.· ,II Y, - .,,(,(1)) (1.

(1 4 )

(111(/ thm:fore II x (/ ) - !;'(/) II ...... 0, COl/sideI' II glohlll~r oh.~{'r1'(/h/(' ,~y,~ tl'lII of Ilu' type

( J) , Thanks /0 Ihe anumplioll (!{ illlwrihili/r of lhe oh.l'('f\'(/hility map. Ihis CII/I he IIs('d to defille a /lCII' .I'l'l of coordinates for syslem ( 1) :

::(1):::: <l1(X(/ - 1i + I )). ( IS)

/'11'01(' lila t :: ( I ) i.\· lI,wd h(,ft, 10 rel1f,',\'ellf the .I')"I'telli s I (I{ C

al lillie t - n + I . RemlliliK fhe i(/emiIY (7). il foll01l's I hili

:(/) = Yt-~+I ' ( '6)

140

Let (A",B",C,,) he a BfllflOlfsk), Iriple oI dimensioll II,

defilled as:

0 0 0 0 0

I ]] 0 ]] 0

A h= 0 I 0 0 0 ElRnx" .

000 I 0

( ' _ [ 0 0 0 ... " -

I ] ER" .

o

Using IllesI' malrices Ihefollolrillg equalions holds

Y I - n_2 = A h Y I - n_ 1 + Bby( 1 + I) ,

Y( I - /I + I) = eh Y,- 1H I .

( 17)

( ")

~Vilh Ihe suhslilulioll :: ( 1) = Y, _ JI + I, Ihese ("an he rewrillen (IS

Z(I + I ) = Ab::(I) + Bh)'(1 + I ) .

y( l - n+ I) = Chz(t) ( 19)

If Ihe im·ene lIlap 1> - 1(.) Ifere availahie, Ihen Ihe oriKinal.I·lale X(I) could he ('{lsil}" mmpllledfrolll z(t) in IU·(I sleps:firsl X(I - 11 + I) is wmpuled as

.\"(1 - n + I) = 1> -1('::(1)), (20)

am/lhen X(I) is compliled as

.\"(1) = fn-l(x(l _ n+ I)) =/,,-1(1)-1(::(1)) .

(21 )

II {t)l/mfs Ih(l[, IISillK Ihe ::'-c(lordinales. syslem ( I ) ("{III

he re1frilten a.I·:

::'(1 + I) = Ab'::(I) + Bhh 0.f(1) -1(::'(1))).

Y( I) ~ ;' 0/"- ' (<1>- ' ('(1))) . (22)

The prohlelll is Ihal a closed form for Ihe in rase of Ihe ohsl'rmhilily map (If a giren syslem is 11M aWlilahle, in Keneral. Numerical ileralil'e //Ielhods call he Ilsed{tJr Ihe COlllplllalion of X(I - n + I) , hr findillg Ille (unique) rOOf of

'(I) - "'(x) ~ II, (23)

A. Germani /lnt! C. MOII<·.\"

/11 [/2]theIol/owinK ohserl'er Slrllclllre was proposed

V'( I+I) ~j(W))

+ (Q(f("o( I)) lr' ( B; (y( I + I ) - 1>0 i"(0( I)) )

+ K(Y(I - II+ I) - h(0( t )) ))

i (I)=/,,-I(!t'(I)) . I>IO = II - I ,

(24)

Ifhere Q( . ) is the ohserrahility matrix (8 ) (If .ITslem ( I ) . For I his ohserrer, I he fvllml"inK local COli l"ergC!1("e

reslllllw!tl:; lrue ( Theorem 2.1 (?llI2Jj

Theorem I: For a s .... slelll of Ihe Iype ( I ) Ihere I'xi.I"ls a jinite Kalil )'ector K E ]R" and a slIitahfe b > () SlIch Ihal

Ihe system (24) is a local ohserl'a, prol"ided Il1al:

• Systelll ( J) is glohally Lipschil::' ohsen·ahle definition 2.4 );

• hand fare IIl1i/armly Lipschil::' ill ]RI!: • IV"-' (",(10)) - x(lo)11 > O.

(see

The proolof this Iheorem ill [12] exploils Ihe rep­resell/alivn (22) vfs)"siem (I). Note Ihatlhe thcorem is here slaled IIsing Ihe nola!ions ill/ro(/ueed in Ihis paper.

and Ihal Ih£' Lipschil:: ohs£'n'ahilily condilioll implies Ihe./illl rank properlr of Ihe ohserl"ahilily //latrix Q.

3. The Polynomial Observer

The ohserver proposed in this paper is based on the idea of using the Taylor approximation of a chosen degree of the inverse map as a numerical method for solving (23) . This approach has strong connections with the iterative technique presented in [16]. which generalizes the Newton- Rapson approach. The observation algorithm derived is characterized by the classical prediction-correction structure, but in this observer the correction term is a polynomial function of the output prediction error. whereas standard algorithms have a linear correction term.

T he main assumption needed for the construction of the proposed observer is that both functionsfand h in (I) are analytic in all ]Rn. Well known results of real analysis establish that the composition of analytic functions gives back an analytic function. and that if an analytic function is invertible. then the inverse is analytic too . It follows that the observability map ::. = 1>(x) defined in (5) is analytic in all ]Rn. and that globally observable analytic systems admit an analytic inverse x = 1> - 1(:) in all ]RI!.

Polynomial Approxinwrjm[ of rh(' h 1l'<'rs(' Oh"w m hi!iry !'.lap

It is important to stress that the Taylor polynomial approximation of the inverse of the observability map ~ = 1>(x) can be computed without the explicit knowledge of the inverse function x = <P - t(~) . by using only the derivatives of q)(x). The mathematic<ll tools needed for the construction of the polynomial Taylor approximation of inverse functions are described in the Appendix . Using the notations stated in the Appendix. the following derivatives can be defined

where GO<~) = 1> - 1(:) and G I(:) is the Jacobian V,, {J- 1(: ) . Let

r, ~ "'P II Gd' )II, "ER'

The Taylor theorem states that

" , " -'( -) - " C (0)( _ ')'" + , (- ' ) '" - - ~k! ' k - - - .:; ,,+ 1 - . .:; .

,."

(26)

(27)

where the remainder of the series is infinitesimal of order I) + I W.f.t. (z - z):

II II r ,,+, II "11 '+' r ,,+1 « v + I )! z - z . (28 )

The exponents /kj in the Taylor formula (27) denote the Kronecker powers of matrices and vectors (see Appendix).

Proposition 2: If 'i/ p > () , Ihe sequence r k dej;lIed ill (26) is slIch Ihal

lim rl.l (29) /..-o( k + I )! - 0,

then 1> - I (~ ) is IlIw/ )l lic ill all RI!.

Proof The proof is a straighlforward consequence of the inequality (28). •

Ld

Ck(x) = Gk 0 <:/)(x) = (v1k. <5 1> -1(:)) "~ "'(.t I

(30)

and let:: = 1> (s') and x = <:p - I (=) . Observing tha t V 10]'3 (Vi ( ~) = 1> -1 (.:) = i , i.e. Go(X) = .r, the Taylor formula (27) becomes

" , - - , ,, c- ( ,)(- "( ")) '" + (- "'( - )1 .\ - .\ T Lk! ' A X _ - "¥ X r ,,_ 1 _ ,'I' x ., Ie_ I .

(31 )

147

where

(l2)

Note Ihat rk = SUP: ER' II Gd:)11 = SUP'ER' IIGk(x)ll· The main resu ll proved in the Appendix is that the

sequence of derivatives (;~ (x ) of the in verse funct ion 1> - 1(.) can be computed as functions of the matrix coefficients F".d x) recursively defined as

Fu = ( v.< 1» I~1 = Qlkl(x), k = 1, 2, . . .

F h.J. = v x ® FH _ I + (v ,.<I» 0 Fh_IJ.._I ,

h = I , · · · ,k - 1,

Fo.1. = O. (33 )

Lemma 9 of Ihe Appendix proves Ihat the sequence of matrices Cdx) needed in the Taylor formula (31) can be computed as

Go = X.

ti l = (V, <:/) - I .

k = 2.3 , ···.

(l4)

Note that

- . -I _ I G I = (v .\"1I) = Q (x ), (l5 )

What is proposed in this paper is a family of observers, whose construction is based on the truncation of the Taylor series (31). Each observer in the family is characterized by the degree of the pol y­nomial approximation , and recursively provides at time I an estimate of the past state x(/ - 1/ + I), denoted x(/) . From this, an estimate _\-(t ) of the cur­rent state is obtained as _\"( 1) =f'- I (X(I»). The obser­ver starts the operations at a given time 10 , while the output measurements are available starting from time 10 _ n + t· T he observer state is X(t), whose initial va lue X(to) is an a priori estimate of X( lo - II + I). The recursive equat ions of the proposed observer a re the following:

Polynomial Observer of degree v

(l6)

(l7J

k = 2, ... , I)

148

~ \ .. , ( I + I) = K(y( 1 - 1/ + I ) - h(A (I )

+ S,(),(, + I ) - 1. 0/' (,(,))).

" 1 xl' + I) ~ flxi')) + (; k! 9,( , )~ ( i , i' + i ),

.,(1) ~ r' (X(, )) ,

(38)

(39)

(40)

Equa tio n (40) p rovides the observed state _;:-(/ ). In the observer equations. the o bscTvability matrix Q(x ) defined in (2 .2) and the matrix functions F;,;. (x) defi ned in (33) are needed. NOle that FI.I(x) = Q(x). The gai n vector K EIRn must he chose n sllch th:H all eigenva lues or AI> - KC" are in the open unit circle in the com plex p lane (A" a nd C" arc the Brunowsky pair defined ill (17)).

Remark 3: COli/par/ilK ('qllaliolls (3t'i) - (J7) )f;III equa­

tiulI.~ (34) it i.~ clear thaI Ihl' m(l/riCl'.~ 91 (I) ill 'he summalioll (39) are the COI!jJki/!lIIs GA (~r fhe Tarlur approximafioll of the jlll'erst! 111(11' e)'a/lIl11ed at z = /(X( /)) . llio/e a/so lha/, II'h/'/l11 = J fhe a/gurilll/ll

(36) - (40) ('oil/ddes Irith t/J {' ohWI'rer (24) presented ill [12J,

Rema rk 4: ,Vote that ill Ihe ob,lwlw NIl/til ions (36)­(4U), lIlt' 011/.1' matrix im'a,n' /() ht, /IImn'rit-ally ("om­PII/('il m clIl'h I ime .~/ep i .l· Ifl(' il/1'Cr,l'l' of I he ob.~erl'llhili I J' lIIa1rix Q( . ). n(lllIllled (II J{ \,(1 )). ill (36), illdepelldemly

of Ihl' orilc,. II of I"l' oh.\·en'('f. The IIImrix /IIIIClioll.l· Fh .

A{ .\' ) at eadl lime slep /1111.1'1 he (' I'lIll1(1/et! til /( X ( l jj . COl/siderillK Iht' dimellsioll of Ihe m(Jlri{ ·l' .~ ill I},e sum­

lIIaliollS (37) (lnd (39)

(4i )

it is easy to see how the compiexity increases by inc reasing tbe o bserver degrce . Ho wever, the only opcr'Hio ns in volved. after the eva luatio n o f the coef­ficient s Flo.'. a re simple prod ucts .. nd sums of ma trices.

The following theorem provides conditio ns fo r scmiglo bal ex ponen tial convergence of the obsen 'er (36) (40). For convenience. in the proof the gain vector Kin (3R) is chosen such th.1l nil eigenval ues of A/, - Kef, are real and d istinct.

T heorem 3: Consider IIII' .1')'.\'11'111 (I ) and il.l· ohsf'n'lI­hilil)' 111111' ,Ie filled ill (5). COII.l'idl!r .~.r.\·lem (36) - (40). Il'here K i.~ .W l'h that all l'igenl'O/ue.l· 0/ A" - KC" are

relll amI dislillct ill llEe illlen'lIl ( - I , I ). An I/me Iflm : 11" ,: Ihe syHe/ll (I) i.~ IIIli!,ormlJ' glohilll.!' Lip.n·hil:

ob,l'er I'ohle ..

.of . v <,,,,wni "",I ('. ,If"",.,

i'I".: The fiIl/CliulI.I· / alit!" are allalYlie ill all R" {lml II/I [formly Lipsdril: ill IRI!, i.e.:

'hI> (] , III(x') - j(x,) ll < O}lI x, - .<,[1-(42)

\I'-"I, x1 E [Rn:

i'I,,) Ifll' l'UII.Wmls r \;. defim'd ill eqlllll iofl (26) {lrt' .nldl that: r~ < 00 . \l'k E N, amI

\1'(1 ) 0 , . ,f

lrm r A t. , = O. A_ ·.... ,.

(43)

Theil . .for allY pair of real /llIlI/hi' I'S o· lIlId 0, Ififlr o E (O./) lind Ii> O. there exi.\·1 (/II illlegcr 1.1, Ihe

,k'gfl'f' o/liIe Oh.IHI'('f (36) - (40). c/lld 1I flosilire rC(I! It ,I'urll IIJ(H

11 .\' (/) - .,"(1) 11 < /IO I-I"U .\' ( IO - 11 + 1) - x(to) 11, (44)

fi" all fJlIir.~ \, (10) . .\'(/0 - 11 + l )eRn such that II x(lo - /I + I ) - X(to )[[ < O.

Pruof" By ass umption Hpo, bo th /and II are analytic functio ns in all R". and therefore the o bservahility l11:rp 4' is analytic in all [R". Thanks to the global observab ility a ss umption fip " the function {I is invertible and analytic in all RIO. and the refore acts a s a global cha nge of coord inates (note tha t global analy­ticity o f (I ) - 1 is also ensu red by 11",. see Propositio n 2), Let (I ) be the {I-transformed sta te of the o bserver (36) (40), defined by:

(45 )

wh ich is a n estimate of :(/) = .1'(1 _ " + I ) (recall that = (I) = 'Nxu - II + I))). and defi ne the error in : ·coo rdinates as e (t ) = ::(I) - :(1), From the observer equ,:lI io ns (36)- (40). considering the eq uivalence C'A( /) = GA(!{X( t))) . the upda te la w of : (I ) takes the fo rm

, (H I)

( " 1 ) = (1) f(x(t» + 6k!GAV(X( I » A~~\( t + 1) .

(46)

Consider now a sequence :(/) defi ned by the update law

'I' + I) ~ "' (f(X(' ))) + 6 ",.(' + i ), (47)

where A \, (1+ 1 )= K(y( 1- 11+ I) - It( \ (/ )))+ Bh(I'( t+ I) - hOj' {\ (f)) . as defined in (3X). Note that. by de r­ini tion of the map <P. il is

Pohllom i,,1 Approx imlil ioll ,,/,111l' l" I'<'r ,,',- Ob.'nmhilil )' ,Hat!

q> (f{X(I) )) = A/I <!> (X(t )) + B/I !! Of"( X(I»,

and h(X( I)) = CIJ ': (I). (4H)

T hus. recalling (45 ), it follows

5(1 + I ) = A/l 5(1) + Bn IT of" (X(t») + ~\ . , ( I + I)

~ (A" - KC,,)'(I ) + Kh (X(I»

+ B,.. II c f "(x( I)} + ~\.y( 1 + I )

(49)

and, by the definition of ~\ . .{I + I), it is

5( I + I ) = (A/I - KCn ).:( I) + Bhy( t + I)

+Ky(I - II + I). (50)

Now, consider the T aylor expansion of <1> -1(5(1 + I )) around <1>(f{X(I)) (see eq.(31)):

"> - ' (' (1+ I» ~ /(X( I » , I

+ Lk

,G, (f(X (I)))1> t', (1 + I)

k _ I .

+r~+ d t + I)

= X(I + I) + r ,,+ I(1 + I),

( 5 I )

where the remainder r,, + 1(1 + I) is such Ihat

11" , ,( 1+ 1) 11 < (: :+: ), 11'(1+ I ) - <I> (f(x(t))) II '+ '

< , r~+l 11 1> (1+ 1)11,,+1. - (v+l) l \ .,'

(52)

Rewrite equation (51) as

Then

where

e( I + I ) ~ , ( I + I ) - '(I + I )

~ ' (I + I ) - <I>( X( I + I )) ~ ' (I + I ) - <1>(<1>- ' ('( 1 + I ))

- r ,,+ I(1 + I)) ~ '( I + 1) - '(1+ I)

(53 )

+ (1 1

(Yx (I>(X»)~( .\is)r"+ I (I + I) (54)

(55 )

Now, rewrite (19) for ':;(1 + I) as follows

::(1 + I) = (An - KC,,)':; (I) + B,..y(1 + I)

+ Ky( I - I1 + I).

Using this and (50) yields

149

(56 )

' (I + I) - ' (I + I) ~ (A" - KC,,)(,( I) - ' (I)),

(57 )

and using (54)

e(1 + I ) = (AI> - Keh )e( t ) + ; (t + I ); (58 )

where

r(1 + I) = (1 1 (Yx1> (x )),_((,) d.1)r"_ 1( I + I ).

(59 )

By the Uniform Lipschitz Observability assumplion II V x 1>(x) 11 < 1<1>, \Ix E IR" , and therefore

(60)

Recalling the hound (52) on f " + 1(1 + I). and observing that, from definition (38),

11 1>, ,( I + 1)11 < o,( II KII + oj) Ilx(l - n + I) - x (I) 11

< o,(11K11 +07h.-, 11 '(1) - '(1)11

(61 )

it follows

II il l + 1)11 < li (v )

where

11 ' (1) 11 ' +' (v + I)! '

(62)

(63)

By assumption, K is such that all eigenvalues of Ah - KC" are real and dist inc\. Lei K >. denote the gain tha t assigns a set of eigenvalues.A = {.A l ..... .A,,} , and let .AM u = maxj_ L .. ,, {II.Aj ll} · Choose .A sllch that .A M ax

< n . Let VA denote the Vandermonde matrix

I -' ,

VA =

I .A"

),'1-1 ,

),'1-1 "

(64)

which is nonsingular, being .Ai -I- .Ai, i -I- j, and such that

(65)

150

It is easy to verify that

K>. = _ V);" I >''' ,

>''' , (66)

Note that. being >"Max < l. it is .fii < IIV>.II < nand II K>. II < II V);" ' II.fii· Consider the following trans­formation of the error:

T hen

1/(1 + I) = Ar}(I) + ~~'>. i'(1 + I)

which implies that :

11"(1 + 1)11 < >'"" II "It)11 + IIV, IIII ' (I + 1)11 < '",,11"(1)11 + II V, II il( v ) 11,, ',,(1)11"+'

(1/+ I )!

< A"d,,(I)11

+11 V, II I1 V-' r - '3(p) 11,,(1)11"+' >. (1/+ I)!

< (>'M" + IIV,II IIV, ' r +'

d(v ) 11 ,,(1)11 ")11"(1)11 . (1/+ I )!

(67 )

(68)

(69)

Now. choose the degree of the observer v such that. for the given n E (0.1) and b > O. it is

(: ~- : )' II IV,1I11 V,' II (IIK, II + o;bn. ·, )"- '

< b(n - >"~1a ') .

(70)

This inequality can be satisfied for a sufficiently large v. thanks to the assumption HI', From this. thanks to the definition of :3(v) given in (63), it follows

A + II V illi V-' 11 "- ' l(v) (II V, lh"bj" < n. M~, " >. f (1/+ I )! -

(71 )

It is easy to see that if at a given time I It ]s Ilx(l) - X ( I - n + 1)11 < b. then

11 ,,(1)11 ~ II V,v(I )1I ~ II V' (" (X(I - n + I) )

- "' (x (I )) 1I < II V, lh"b. (72)

A. (iermani and C. Mal!{,.\

so that

1I ,,(I)r +' ~ 11,,(1)11"11"(1)11 < (IIV,1I 7. WII,,(I)II· (73 )

From this. thanks to inequality (69) it follows

11 ,,(1 + 1)11 < (>."", + (IIVxIIIIV,' 111"+'

il(p) (,;':'!;,) 11,,(1)11 . (74)

and thanks to (71)

(75 )

Th us. being n: E (0.1). if at time I It ]s II '}(I)II < II V>. lh4.E. then also at time I + I it is 1111(1 + 1) 11 < II V>. lh4.E. Therefore. ifat 10 Ilx(lo) - X(lo - n + 1) 11 < 8. then II r}(lo) 11 < II V>. II : ¢b. and thanks to (75)

(76)

This implies

II V(I)II < "H' IIV,II IIV;' ll ll v(lo) ll. (77)

and. being e(l ) = 1I <I> (x(l - n + I)) - (l> (x(I)) II, II follows

IIX(I - n + I ) - x( I)11 < [( /-1°,1" ... _1 II V >.1 111 ~":\ ' 1I IIx( 10 - II + I ) - X( 10) II·

(78 )

and finally

Ilx(l) - .\"(/) 11 < d-'":j'-1:4.: 4.·, It V>. II II V);" ' II Ilx(lo - n + I) - x(lo)ll. (79)

which is the thesis (44).

IIVxIIIIV;' II·

4. Simulation Results

with "

- _.,,- 1- _. ,, - Ir I¢ 14.- 1

•

The polynomial observer (36) - (40) has been tested on many systems. [n general. the theoretical convergence results have been confirmed by the simulation tests. Here. some results concerning a chaos synchroniza­tion problem have been presented.

Consider the well known HhlOlI map [20J

(SO)

Polynomial Approxim"lioll ol'llll' b'l'as" Ohsa rahililV ;\fap

which, for some values of the parameters a and h, del1nes a discrete time chaotic system. The canonical values of the parameters of the Henon map are (f = 1.4 and h = 0.3, which produce a chaotic beha­vior of the system.

The polynomial observer (36) - (40) has been applied to the problem of estimating the state of the Henon system when the constant parameter h is un­known. Only the I1rst state variable XI(I) is assumed to be measured . This problem is known as a chao.\"

s),llclironi::alio!l prohlem wilh ullcertain parmllelef.\" (see '.g. [lSI. [25]).

The unknown parameter h is modeled as a constant state variable. i.e. x.,(1 + I) = x://) = h. T he system eq ua tions considered for the observer construction are the following

XI ( I + I )

.\":>(1 + I)

X3 (1+ 1)

I - 1.4xi(l ) + X:>(I)

XI(I).'J(t)

,;(t)

r(t) = xl.

(Xl )

The polynomial observers of orders // = I. 2. 3 and 4 have been applied. The matrix functions Fh .k(X). del1ned in (33). where k = 1, ... , 4 and II = I, ... k, have been computed using the symbolic toolbox of MATLAB ' · . by reproducing exactly the recursive computations in (33).

The map (I:> is as follows:

, , 1 - 1.4(1 - l.4xl + x2t + .\"3XI

1> (x) = I - 1.4xi + X2

XI

(Hl)

F,,(x) ~ Q(x) ~ 7.84(1 - 1.4xf + X2 ).\·1 + X3

- 2.8x l I

- 32.928xf + 7.84 + 7.84x2 Fu(x) = - 2.800

o

7.84.\'1 o o

In matrix Fu(x) the only nonzero elements are [FuJI.1 = - 65.856xl and [FuJ1.2 = [FI..'lt.4 = [F1..111.1o = 7.84. Matrices Fh. l for II > I are rather big sparse matrices, and can not be easily reported. Note that matrices Fu can be computed as ptll (see (33».

The gain matrix used in the simulations here reported is the following

K = [- 0.0017 0.0428 - 0.3600 j, (85)

I 5 I

u ,----------------,

-u.~

.,

-1.1,L,-------,------m------,c-----~ ~ l IU 15 , 20

Fig. I. True and observed variahle x ,_

and is such to assign eigenvalues AI = 0.1, A2 = 0.12 and A3 = 0.14 to the Brunowsky pair (A h. en) of dimension n = 3. The Figs 1- 3 report simulation results where all observers for v = 1,2.3 and 4 pro­vide convergent state estimates. The true and observed initial states are

x (O) ~ [ - 0.6

;(0) ~ [-0.3

_ 0.IO.3j T

- 0.2 O.2]T. (8 6)

It can be seen that higher order observers produce faster transients.

In many cases, the first order observer does not converge. while higher order ones are con vergen!. For instance. when the initial states are as follows

x(O) ~ [-0.6 - 0.1 0.31'

; (0) ~ [0 0 0.21'. (87)

- 2.8 + 3.92xf - 2.8x2 XI

10 (83)

I o o

o 0

7.840_\'1 o o

- 2.800 o o

o I () 0 o 0

o o o

o o o

the first order observer does not converge.

5. Conclusions

(84)

A new technique for the observer construction for analytic discrete-time nonlinear systems has been pre­sented in this paper. T he method is based on the Taylor

"2

----~, ----~-----, ." I).!

, , ., " / •• ., ..

11.6 • • • .-, 1).4

11.2

" -11.2

In!< . .... - 1).4

0 , '" HI: . 2. True and obscrved \'ariablc -'~ .

0.,

0.'

0.' I

" t -(u I

• (rue . [ale ,

•

'\ , , ,-,

!

1 ,

1

i ,

j

" 2()

--{)-41 ~' ~~-~-~-~ o 5 10 •

HI: . 3. True and obscrved variable- -'.1 (the constant parameter).

polynomial approximation of the inverse of the observability map. The degree I) chosen for the approximation is the most important design parameter of the ob~rver. Thanks 10 the analyticity assumption. it is shown that the degree v can be suitably chosen to ensure the error convergence al any desired exponential ratc. when an upper bound for the norm of the initial observation error is assigned (semiglobal exponential convergence of the observation error). The resulting observation algorithm has a prediction-correction structure. where the correction term is a polynomial function of the output prediction error. It must be noted that when !J = I the proposed observer coincides with the one presented in [12] . Computer simulations support the theoretical convergence results.

Acknowledgment

T his work is supported by AfiUR (It alian Ministry of University and Research) and by C-VR (National Research Council of Italy) .

A. Camalli ""d C. MalJl'.\

References

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IJ. Ciccarella G . Dalla Mora M, Germani A. A robust observer for discrete time nonlinear systems. 5y.~1 COllirol Le/l 1995; 24: 291 - 300.

14. Dalla Mora M. Germani A, Manes C. Design of state observers from a drift-observability property. IEEE Trwl'\" Aulum COII/rol 2000; 45: 1536- 1540.

IS . De Angeli A. Genesio R. Tesi A. Dead-Beat chaos synchronization in discrete-time systems. IEEE Tram Circuil.l· Syst/ , Filmlalll Thcory Appl6: 1995: 42: 54 .

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17 . Germani A. Manes C. An exponential observer for discrete-time systems witb bilinear drift and rational outplll functions. Proceedings of the European Control Conference. 2007. 5777 - 57R2. Kos. G reece. 2007.

IR . Germani A, Manes C, Palumbo P. Polynomial extended Kalman filter. It:!:;!:: Trans AlIlom Conlrol 2005: 50: 2059-- 2064.

19 . Gennani A. Manes C. The polynomial extended Kalman niter as an exponential observer for nonlinear discrete~time systems. Proceedings of the 47th IEEE Conference on Decision and Control (C DCOR). Cancun. Mexico. 200R.

P()/.,-"mui,,1 Appr().\inlllli,,~ ()llh" /""('TJi' OhUTWhililY M ap

20. Henan M. A two-dimensional mapping with a strange aUractor. emll/IIIIII M mh Ph)" 1976: 50, 69-77.

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u '1I2001 : 42. 8 1- 94. 23. Lee W. Nam K. Observer design for autonomous discrete­

time nonline:lr systems. 5)".1"1 COJlfrOIUII 1991: 17: 49-58. 24. Lin W. Bymes C I. Remarks on linearization of discrete·

lillie autolWI!lOUS s),s tems and nonlinear observer deSign. Sy.w C(O/tIm{ Lt'll 1995; 25: 31 - 40.

25. Millcrioux G. Daafnuz J. Glnbal chaos synchronization and mbust lillering in noi~)' context. I t:t:t: TriUl.I· lirerlil.l· SYSI I , Film/alii Tlwory Appl 2001; 41\(10): 1170- 1176.

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Appendix

Pol)'nomial Approximalionof Inverse Mappings

T his appendix describes the mathematical tools of the Kronecker algebra. used in this paper to represen t the rormu las or thl! Taylor seril!s or vector funct ions or severa l variables and or thei r inverse funct ions, Here, the symbol I" denotes the identity mat rix in IRn>cn,

The Kronecker product ortwo matrices AI and Nof d imensions p x q and r x ,~respl.,'c t ively. is the (p . r) x (q , .\.) matrix

m1l fli ... ",l qN

,H 81 N = (X8 )

II1rlN . . . /II pqN

153

where the 1111) arc the entries of /t. / . The Kronecker power or iI matr i.\( M is recu rsively defi ned as

Note thaI if .H E IRI'X'I, then M!o1 E Rl"x( Some useful propcrlies of the Kronecker prOOucl are the following:

(A + 0 ) ® (C + D) ~ A '" C + A '" D + B ", C

+B 0 D

(90 )

A ® (B ® C) ~ ( A ® O) ® C (9 1)

(AC) "' (B· f) ) ~ ( A ® O)·(C® D) (92)

dOI (A ® 0 ) ~ del (B ® A ) (93)

Repeated applicalio n of the properties (9 1) and (92) provides the iden tities

{AB)I" = AI' Ji~ and (A [t)- I = (A-t )IAI.

(94)

Using (92). given II E IR""" and B E R",xm, the fol­lowing properties can he e:l sily derived

(A ,g, B)- t = A - I ® trl, (95)

A ® B ~ (A 0 ' .. )(1, ® 0 ). (96)

From the las\, thank s to (93), it follows del (A g . 8 l = del (A ® I",) dct(/" Z. B). Using (93) it is del(/" 0 B) = det( B 0 " ')' Recognizing the block diagonal structure of /'" 0 A and or I" ,g, B it follows

del(A ® 0) ~ (del (A )t(del( B ))". (97 )

A quick survey on the Kronecker algebra can be found in the Appendix of [91. See (211 for more properties.

The Kronecker formalism is a lso usefu l fo r repre­senting deriva tives of malrix functions of several variables. :lIld provides :. compact nOlation for the Taylor pol ynomia 1 ,Ipproxima tion of non linear vector functions. Let /I1 (x ) be a smooth ma trix function of x elRn. I.e.: M :Rn_R'''~ , and lei 'V'.r 0 ,H : IRn ...... R,>c"r denote the following ma lrix of deri vatives:

'V'x 0 M = [DM (hi

, .. {)M 1 ih:"

(98)

154

T his ma tri x Jacobian is a fomHiI Kronttker product of the ve<: tor V ... = [aj ax 1 ". aj ax,,1 with the matrix M (x). Repeated deriv<t tives C<1 n be recursively defined as

VIO]~' .lH = M . . '

'\71:+1) ® M = V_, 'z. ('\7 ~1 ® M ), i > I.

(99)

Let:F: !R" - Rm be a smooth non linear map. In this C:l SC V .• :g F is the standard Jacobian of :F. and therefore throughout the paper the following symbols will be indiITerently used

(1 00)

The las\ symbol provides the most compact notation. Note thaI for matrix functions the use of V_,0 is Ilttess.'l ry.

A usefu l property of the oper:llor V < is the fol­lowing:

v .• ® (A lx) · Blx)) ~ IV., e A )(/" '" B)

+ A i"., '" B).

where A and B are generic matrix functions.

(1 01 )

Let .1'{x) :md H ( : ) be functions :F: R"-[R" and H : !R" - R'x<". The composition H 0 F : R" _ lR'x,' is defined as

H 0 F (x ) ~ H(F(x )). ( 102)

Proposition 4: If H ( ::) and :F (x) (Ire dil ferelll iahfe.

Ihell

Vx ", (H 0 F) ~ « V , '" 'I) 0 F )( VxF) ", /,.). (1 03)

where Ihe rerlll

(V , '" H) 0 F(x). ( 104)

del/OIl!S Ihe derimtire (~l H 11".r.l. z compll led al

Fix ).

Proof Ohserve that

V., ® (H o F)~ [ J(H OF ) Ox,

J(~ ;"F) 1 (105)

and th.1I

a.1';(x) nXi

O:FI I,. Ox/

~ « V , '" H) 01')

( 106)

Substi tutio n of this into the right ha nd side of (105) provides eq uation (103).

Throughout the paper. the sy mbol S,.(p) will denote the open ball of rad ius II centered at Xo E R" . i.e. S .• u(p) = (x E R" : II x - xo ll > pl. The symbol 5.'Q(/I) wi ll denote the closu re of S~~(fJ)·

Using the Krone<:ker nOlation. the T:lylor Theorem fo r vttlor func tions of several Vi' ria hies ca n be written as follows:

Th(>orem 5: Lei F : R" ---4 IR'" he (/ /r1ll(,lioll defined all

Ihe dosed hall . . <:;_~O (p) J()r sOllie Xo E E" (llId p > O. If,for (I girel/ inlcger v. all deril"{Jlirf!.f V'~ I:>!) .1', k = I , .... /1 + I . f!Xisl lI11d arc cOllfill/w/iS (1/ nay poinl of

S.,) f' ) . fhell fhe jollOlring kiellfilY holfl.~Jor all x E S 'o(p):

F (x ) ~ t kl, (V~I 0 F (x )) (x - x,) ··1 0,-0 . TO

+ R,,_I (x. xo)(x - xo )l,,+ II.

( 107)

where lilt' remainder slIlisfies II\(, inequalily

II R"+,(x )1I < ( ~ I)' '"P II vl~'10 F(x)lI · v . . ' E.~~.lp)

( 108)

TIll' fW!(,fioll .1' i.~ .mki Iu he al/a/r/ it" in Ihe .\'('1 S~,.(p) if Iltt' Taylor s('ri('.~ is nm\"('/"gen/ for all x E S~Jp).

Defining

~dxO, fJ) = s_up II v ~q :>!) :F(x) ll, ( 100) _'ES",(Vi

(/ .I'liffidelll ("(Jlldilion for Ihe l/Iw/Ylil"if)' ofF in S~./ p)

i.f lite following

I· 'Pk(XO. p) - 0 lin k' - .

~ -.'- . ( 110)

Polylwmilll Approximmiol1 of rhe IlI l'a,\',· Ob,,<'fwhilily Map

Theorem 6: Let F: En ....... Rfl he a smoolh limelio/l defined on all R". If. for an)' choice of XI) E IR" ami of p E IR + the sequence ;;d Xo. (J ) defined ill ( J(9 ) i.l'such Ihal

1 I II )

then F is analy tic in aff ]R".

Proof The proof is a direct consequence of the remainder expression (108).

Let F - I: )R" __ X" denote the in verse map of the function F , i.e.

(112)

so that F - 10 F is the identity map in IR": F - I(F (x» = x. Let us define the following notation for the derivatives of the inverse function F - I:

A useful equi valent recursive definition is

90(' ) ~ r' lc), 9k+1 (::) = 'V, @9d :: ), k = 0, L ... .

(114)

Lemma 7: The deriJ'alires of the im'er.I'e mapping F - I. i.e. the fIIalrix lilllc/ions 9d :: j . k = i.2 ..... defined in eqllation (J 13), sati.lfy the foll()1 ring elwin of idenlities:

where

, L (9h 0 F )Fn.1 = O. k = 2, 3 ..... .-,

FI._k = (v_,F )lk], k = 1.2, . ..

Fhk = 'V , (9 Fhk _ 1 + ('VxF ) 0 Fh_ Lk _I,

h = I , .... k - 1.

Fiu = o.

1115)

(116)

(11 7)

Proof The property (91 0 F )FI.I = in. with Ff.! = 'V, F, it will be proved in Lemma X. From this

(118)

The computation of the derivative. using (10 1), gives

v , &1(9, o f )Fu ) ~ 19,oF )lv ,F &I")(I"&Fu )

+ ( 0 I o f ) ("V\ .@ FI .I) = O.

(119)

155

From the property (A ,g, 8)(CO D) = (AC}g{CD) it follows

('V_,F 3 I")( J,, @FI.I) = v ,F 3 F I.I = ('V_,F )-<]

= F2_2 ·

(120)

Moreover. from (117). it is Fu = 'V ,-rx· FI I. T here­fore equa tion (119) can be written as

(91 o f )h2 + (91 of)F u = 0. (121 )

so that (116) is proved for k = 2. The identity (116) for k > 2 will be proved by induction. T hen. ass ume that (116) holds for some k > 2, and prove that it holds also for k + I. If ( 116) is true, then

( 122)

Differentiating, and using (101), yields

, I: (19.-, 0 F )((v ,F ) @ I" )(1" C0 F,, ) n_1 ( 123)

+ (9h 0 F )( 'V , @ Fhd ) = O.

Reorganizing the subscripts one gets

(9k+ l 0 F ) ((v _, F ) :?:' i".) (I" (0 Fu) , + L (Oh 0 F )(V t .g, Fn.1. .-, + (lv,.:F) & 1", ~, )l I" '" F._, , I) ~ O~

that can be rewritten as

,+, L (9h 0 F )h_l.+l = O. .. ,

where

Fk+l.k+ 1 = ('V_,.F ) O l n. )(I" (2;: FI.-I.-),

( 124)

( 125)

F/,-,:_l = 'Vx (2;: Fu + «'VxF ) 0 J ", - I ) (I" .g Fh _ u:),

I < h < k.

( 126)

Exploiting the property (A 3 8)(C3 D) = (Aqg, (BD)

Fk+ IJ<+ l = ('V_<F) S, h .h

hk+1 = 'Vx 3 F h,1.- + ('V \'F ) ':29 F/,_ l.k, (127 )

I < Ii < k.

"6

Consideri ng tha t Fl.l ;: 'V •. F, it fo llows F~+l.~-I- I = C'V.f F )iA+1J. The comparison with (117) shows that Fu = Fu , for all pairs h,k, a nd therefore

,., L Wh of)hk+ 1 = O. ( 128) h.,

This equa tion completes the induction and concl udes the proor.

lemma 8: The deriralire.f of Ihe im"('f.fC mflppillK :F - 1(:: ) m/1/pllled ill :: = :F ( X ) J{/fi.~f)" the f()//Oll'jIlK

dwill of ielell!ilie.I'

(1 29)

9~ + 1 0 F = (v\ g. ({h 0 .1"))(91 0 F ) -g, In'-) '

( 130)

Proof: Since F - I 0 F is the identity map, then '\7, (F - 1 o f } = In. From Ihis. using the chain rule

'l.,.(r ' of) ~ (('l,r' )o F )('l.,F )

~ (9, of) ('l,F) ~ I,,, ( 1 3 1 )

and therefore equation ( 129) is proved. Since for any differentiable matrix function H (:: ) .

H : IR" _IR'~' il is

'l., ,. (H 0 F ) ~ (('l, 6'0 H) 0 F )(('l ,.1") 6'0 q . (132)

see Proposit ion 4. it foltows

'l., " (9, o f ) ~ (('l , 0 9.) of)(('l ,F ) '" 1,,)

= (9k+1 0 .1')((91 oFr t 0 In.J ·

( 133)

From thi s

~h -l- I o :F = (\7 , g (9, 0 F )){(9. 0 .1")- 1 ® 1n')~ I.

(134)

Th:lllks to property (95) it is

(1 35 )

and the theorem is proved.

Let

g~(x) = ~h 0 F (x) = (\7141 g: F~ I (=»):_F(<l .

( 136)

Note tlmt go(=) -= .1"-1 (=) and tha t 90(x) = x. Denoting : -= F(.\" ) and : = F (.i ), the Taylor series exp;.lnsion of F - I (=) around the point = can be written as

.4. Cia",,,,,; lI"d C. ,\I",,,,,

x ~ x+ t ,',g, (.,)(o- o)'·' A_ II\· ( 137)

+ R,_, (0 , ')(0 - 0) "+ '.

It is kn own that if F is anal ytic in ,t set S'o(p) and admits the inverse function .1" - I, then F - I is ana­lyt ic too. The coelTicients ~,(xo) of the Taylor series e.xpa nsion of F - 1(=) aro und ;\ IlOin t =0 = F(xo) can he recursi vely computed usi ng the algorithm of Lemmil K withou t the explicit knowledge of a closed form of F - I ( :). However. the computations in volved in the iterations require the deriva tives of inverse matrices (consider for instance G2 = (v .• ·g: (V.\ F)~I) «'\7 ,.F f l Z, l ,,» . The followi ng Theo rem shows that the coemcients 91 can be computed wit hout the dif­ferentiation of inverse matrix fu nct ions.

Lemma 9: The matrix JUllcliom· g~ : R" 0----. R:n ~ '" defined ill ( 136) C(l1l rn.' reclIrxil'e1y compllle(1 {{.\Iullo,...~:

go= x. - -, 9, ~ ('l,F) ,

( 138)

k = 2. 3 ... .

Irllere the m(({rix c(leificiel1lJ comp/(Ied liS

/·i,.1i are re("ur.l·ire/\"

"1 Fu. = (V.,F ) . k = I. 2 .. ..

Fh).: -= 'il , €I FIrJ<_ 1 + (v .TF) ~ Fit_lA_ I,

h -= I ..... k-I , . .

FOA -= O. ( 139)

Proof The proof is a direct conseq uence of Lemma 7. Equations (138) are simply deri ved froll1 the identities (11 5) - ( 116). proved in LemnHl 7. In particular, rewrite ( 116) as

~~ I

9,Fu + L 9,FH -= O. ( 140) h_'

Solving this equation for 9h(lr~ and consideri ng that. thanks to (94)

(F ... )- ' ~ (( 'l,F)"' ) - , ~ (( 'l"F) - , )"! ~ 9 \';. ( 14 1 )

gives back the equations (138).

Remark 5: Theorem 9 Xh01t"J 111m. gin'lI (111 {lIIai)"lic jimCli(J// F : Rn

0--- IR~. Ifitll /wlIsillKlilar Jacohiull 'V ~F (1/ (I gil·('11 X" E 1R". Ihe t"lH.f{idelll of Ihe Tll_l"Ior .l"eril'.~

('xp(lll.~ioll of Ihe ill rene fll/lClioll F - f lifO/ifill

=0 -= F (.\·o) exi.fI.~ lIlId CllII he f('cllr.~i l-eJ)" compllle(I tI.~ fllll( ·'iUlH of Ihe (ieriratirex oj IIIl' clirt' c/ map F m/ll·

Jlllled (1/ Xli. ((.1' long a.l· 'V ~F(xu) is nOllsillgular.