U.P.B. Sci. Bull., Series A, Vol 74. Iss 4, 2012 ISSN

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1 UPB Sc Bull, Seres A, Vol 74 Iss 4, 202 ISSN ON THE NONLINEAR OUTPUT REGULATION PROBLEM PART MIMO NONLINEAR SYSTEMS NORMAL FORMS AND A DISCUSSION ON THE NECESSARY CONDITIONS FOR SOLVING THE CONTROL PROBLEM Andreea UDREA, Alexandru TICLEA 2,Crstan FLUTUR 3, Valentn TANASA 4 In aceasta lucrare este prezentată problema reglar eşr dacă se consderă ssteme nelnare MIMO In acest context formele normale pentru reprezentarea sstemelor nelnare sunt de mare nteres Două forme normale pentru ssteme nelnare MIMO sunt prezentate Prma releva vectorul de grade relatve s poate f aplcată pentru ssteme pătrate nversable (forma normală clască), a doua poate f utlzată ş pentru ssteme nepatrate ş dă detal/nformaţ despre propretatle de nversabltate ale sstemulu scrs în această formă (forma normala recentă) Condţle necesare pentru rezolvarea probleme reglar eşr pentru ssteme scrse în forma clască ş cea recentă sunt prezentate Dn dscuţa prezentată rezultă că problema pusă se poate rezolva sub presupuner ma puţn restrctve dacă sstemul este scrs în forma normală obtnută prn algortmul structur zerourlor la nfnt (forma recenta) Această abordare permte de asemenea dscutarea probleme de reglare ş în cazul sstemelor MIMO nepătrate In ths paper the general problem of output regulaton when dealng wth nonlnear MIMO systems s presented In ths context, normal forms for nonlnear systems are of great mportance Two normal forms for MIMO nonlnear systems are presented The frst one reveals the relatve degrees vector and can be appled for square nvertble systems (classc normal form), whle the second can be used for non square systems and gves detals on the system nvertblty propertes (recent normal form) The necessary condtons for solvng the output regulaton problem for the classcal normal form and, respectvely, the recent normal form are presented From ths dscusson t results that the problem can be solved under weaker assumptons f consderng the later form (obtaned by usng the nfnte zero structure algorthm) Ths approach also permts dscussng the regulaton problem for non square MIMO systems Keywords: output regulaton, MIMO nonlnear systems, normal forms Assstant professor, Department of Automatc Control and Systems Engneerng, Unversty POLITEHNICA of Bucharest, Romana, e-mal: andreeaudrea@acsepubro 2 Assstant professor, Department of Automatc Control and Computers, Unversty POLITEHNICA of Bucharest, Romana, e-mal: alexandrutclea@acsepubro 3 Teachng assstant, Department of Automatc Control and Systems Engneerng, Unversty POLITEHNICA of Bucharest, Romana, e-mal: crstanflutur@acsepubro 4 Teachng assstant, Department of Automatc Control and Systems Engneerng, Unversty POLITEHNICA of Bucharest, Romana, e-mal: valentntanasa@acsepubro

2 4 Andreea Udrea, Alexandru Tclea, Valentn Tanasa, Crstan Flutur Introducton A defnng problem n control theory s the desgn of feedback controllers so as to have certan outputs of a plant to track partcular reference trajectores An appealng dea s dynamc nverson, but ths can rarely be carred on n an exact manner through open loop control In fact, closed-loop control (whch acheves an approxmate dynamcal nverson) s almost always the soluton of choce, snce, n any realstc scenaro, the control goal has to be acheved n spte of a good number of phenomena whch would cause unexpected system behavor (for nstance: parameter varatons, addtonal undesred nputs) One partcular (determnstc) form of ths problem s to consder that the dynamcs of the system that generates the references and the dsturbances (the exosystem) are known and consequently desgn a controller that steers to zero certan outputs of the augmented system plant-plus-exosystem, thus achevng what s called the property of output regulaton Problems of ths knd have been extensvely studed n the 970s for lnear MIMO systems; the works of Francs and Wonham for nstance, provde an exhaustve presentaton of the theory [, 2] The results culmnated wth the Internal Model Prncple (IMP), whch states that a structurally stable soluton (e robust to plant parameter varatons) necessarly has to use feedback of the regulated varables and ncorporate n the feedback path a (possbly redundant) model of the exosystem A nonlnear enhancement of ths theory was ntated at the begnnng of the 990s [3, 4] The semnal paper of Isdor and Byrnes [3], although lmted n scope (t only secured local, nonrobust, regulaton about an equlbrum pont), hghlghted fundamental deas whch shaped all subsequent developments n ths area of research For nstance, t ponts out the basc challenges n solvng the output regulaton problem n a nonlnear settng, namely to create an nvarant set on whch the desred regulated varable vanshes, and to render ths set asymptotcally attractve It also hghlghts the fundamental lnk between the problem n queston and the noton of zero dynamcs (a concept ntroduced and studed earler by the same authors) In the past 20 years, the desgn phlosophy ntroduced n the paper above was extended n several drectons One goal was to move from local to nonlocal convergence, for whch several approaches at ncreasng level of generalty have been proposed [5, 6, 7] An mportant advance of [7] was to gve a general (nonequlbrum) defnton of the problem, through a convenent defnton of the noton of steady state for nonlnear systems Another concern was to obtan desgn methods whch are nsenstve, or even robust, wth respect to model uncertantes (ether n the plant or n the exosystem) [8, 9]

3 On the nonlnear output regulaton problem - Part - MIMO nonlnear systems normal forms 5 A general framework n whch the output regulaton problem s solved fnally emerged The basc deas were captured wthn two fundamental propertes, the nternal model property and the stablzablty property Once the frst proprety s acheved, the output regulaton problem can be smply solved through hgh-gan stablzaton technques [0] A crucal observaton was that the problem of achevng the asymptotc IMP s closely related to, and actually can be cast as, the problem of desgnng a nonlnear observer By usng avalable observer desgns [, 2, 3], ths approach has lead to effectve desgn methods that fall n two classes: based on mmerson (they mply rather strong assumptons) [4, 5] and newest results droppng the mmerson/observablty condton [6, 7] The results referred to above are by no means general; they can be appled to partcular classes of systems, under specfc hypotheses These deas were pursued manly for SISO nonlnear systems, leadng to some effectve desgns The fact that so far there have been lmted attempts to solve the problem n the MIMO case (eg[8]) s not entrely surprsng, and for varous reasons Frst, seeng how the desgn of observers s nstrumental n the desgn of controllers that solve the output regulaton problem, the desgn of observers n the multple-output case s known to pose serous techncal dffcultes, especally n obtanng the rght canoncal forms that allow a meanngful (constructve) characterzaton of the observablty propertes [9] Second, MIMO normal forms are not smple extensons of SISO normal forms For nstance, whle SISO normal forms lend themselves naturally to the defnton of nonlnear equvalents for the lnear fnte and nfnte zero structures and nvertblty propertes, the extenson of these notons to MIMO systems s nowhere near as straghtforward On the other hand, a normal form of some knd represents the only tool to (robustly) handle nonlnear dynamcs for control purposes, whle normal forms that seem to be adequate for the MIMO nonlnear output regulaton problem have been ntroduced just recently Last, but not the least, whle there s a wealth of stablzaton tools for SISO nonlnear systems, there are not so many avalable for MIMO systems In ths paper we are gong to present the assumptons under whch the problem of output regulaton can be solved n the case of nonlnear MIMO systems Ths problem s strongly lnked to the normal forms of nonlnear MIMO systems The normal forms evolved n close relatonshp wth the control technques The paper s structured as follows: n chapter 2 the classc and recent results on normal forms used for MIMO nonlnear control problems are ntroduced and the zeros dynamcs of the system s dscussed, n chapter 3 the general problem of output regulaton for nonlnear MIMO systems s presented, n chapter 4 the necessary condtons under whch the output regulaton problem

4 6 Andreea Udrea, Alexandru Tclea, Valentn Tanasa, Crstan Flutur can be solved are dscussed and the assumptons consdered n [8] are relaxed and chapter 5 deals wth conclusons and further developments 2 Normal forms and zero dynamcs for MIMO nonlnear systems Normal forms for nonlnear MIMO systems are meant to reveal key structural propertes of the analyzed/consdered system: relatve degrees, zero dynamcs and nvertblty propertes The relatve degrees expose the nfnte zero structure, whle the zero dynamcs characterze the fnte zero structure of the system These notons are mportant (as n the case of lnear systems) when control problems are of nterest Consderng the followng MIMO system: x = f( x, u) () y = h( x) n m p wth: x R the state, u R the nput, y R the output, f the system s nput affne t can be represented as: x = f( x) + g( x) u (2) y = h( x) where: u = col( u,, u m ), y = col( y,, y p ), g( x) = [ g( x),, gm( x)], hx ( ) = ( h ( x),, hp ( x)) and g(x) s a n m matrx and u(x), y(x), h(x) are vectors and system (2) can be rewrtten as (3): m x = f( x) + g ( x) u y = h( x) y = h ( x) p p In the case of square systems, m=p> the system can be wrtten n the normal form ntroduced by Isdor n [20] Ths s based on the exstence of some vector relatve degrees {r,,r m } at a pont x 0 : k L L h( x ) = 0; j=,m; =,m; k<r -; n a neghborhood of x 0 2 gj f r r Lg L f h( x) Lg L ( ) m f h x M = rm rm Lg L ( ) f hm x Lg L ( ) m f hm x j= j nonsngular n x 0 (3)

5 On the nonlnear output regulaton problem - Part - MIMO nonlnear systems normal forms 7 Each relatve degree r s assocated to the th system output The sum of the relatve degrees s at most n By applyng the followng transform Θ: x > ( ze, ): r [ h ( x) Lf h( x)] Θ ( x) =, rm [ hm( x) Lf hm( x)] j (consderng that dlfh( x ), j=0,r -, =,m are lnearly ndependent) the system n (2) can be wrtten n normal form as: = f (,) z e + P(,) z e u 0, = e,2 e r, = er, m r, = (,) + j(,) j j= e e q z e m z e u y = e,, =,m (4) where Pj (, ez) = Lg z (, ) j Θ ez, and the functon Θ : x > ( ez, ) s a dffeomorphsm from x to (e,z) Observaton : In the case of SISO systems there s always possble to fnd a set m of functons such that Lg z ( ) 0, j x j=,p; =,n- r For the MIMO case, ths s possble only f the dstrbuton spanned by the column vectors { g, g2, g m } s nvolutve n a neghborhood of x 0 Observaton 2: If the matrx M(e,z) s sngular and rankm<m s constant (the system can not be wrtten n the normal form wth m relatve degrees), a dynamc extenson algorthm proposed n [20] can be used n order to extend the system by addng ntegrators on the nput channels such that the system mght be wrtten n normal form The zero dynamcs of (4) Consderng the system (4), the zero dynamcs s gven by the followng expresson: = f (,0) 0 z (5) =

6 8 Andreea Udrea, Alexandru Tclea, Valentn Tanasa, Crstan Flutur A generalzaton of the above algorthm was made n [99] The normal form of system (2) s: = f ( x) + g ( x) u 0 0 e = e + δ ( x) v + σ ( x) u, j =, n, j, j+, j, l l, j l= e n, y = e ; =, m = v, wth n n2 n m v = a( x) + b( x) u; =, m a( x) = Lge, n ( x) b( x) = L e ( x) f, n and the matrx { b( x), b2( x),, bm ( x )} s smooth and nonsngular Consderng δ, j, l(x)=0 the vector { n, n2,, n m } of system (6) represents exactly the relatve degrees vector (whch, n the case of lnear system gves the nfnte zero structure) If δ, j, l( x) 0 the vector { n, n 2,, n m } s not lnked to the nfnte zero structure of a lnear system [22] Under the stronger assumptons that some matrx ranks are constant and the dstrbuton spanned by the column vectors { g, g2, g m } s nvolutve, system (6) takes the form: = f ( x) 0 e = e + δ ( x) v, j =, r, j, j+, j, l l l = e r, y = e ; =, m = v, The zero dynamcs of (6) Usng these generalzatons of the normal form - (6) and (7), the zero dynamcs s gven by: z = f ( x) 0 (8) The assumptons needed for the elaboraton of the presented normal forms are rather strong In a recent publcaton [22] these assumptons are substantally weakened Moreover, the nfnte zero dynamcs algorthm [22], [23] allows the representaton of MIMO nonlnear systems that are not necessary square and nvertble under the followng form: (6) (7)

7 On the nonlnear output regulaton problem - Part - MIMO nonlnear systems normal forms 9 md = f ( x) + g ( x) u + ϕ ( x) v e e e l d, l = ξ = ξ + δ ( xv ), j=, q, j, j+, j, l d, l l= ξq, = v d, yδ = hδ( z, zd ) (9) yd, = ξ,; =, md wth q q2 q md, ξ = { ξ,, ξ,, ξ, q}, =, m d vd, = a( x) + b( x) u yd = col( ξ,, ξ2,, ξm d,) and col{ b( x), b2( x),, bm d ( x)} s nonsngular From the nfnte zero structure algorthm, the dynamcs ξ, j does not depend on v l, l>j and t follows that : δ, j, l(x)=0, j<q l, =,m d (0) The relatons (9), (0) hold under Assumpton B[22] In (9) m d represents the largest number for whch the system can be transformed n the normal form (ths value results by applyng the nfnte zero structure algorthm) The algorthm also dentfes a vector of nteger values { q, q2,, q m } that represents the nfnte zero structure In addton, f Assumpton C [22] holds there exst a coordnate transform that puts the system n the form : = f ( x) + g ( x) u ξ = ξ + δ ( xv ), j=, q, j, j+, j, l d, l l= q, d, e e e ξ = v () ye = he( x) y = ξ ; =, m d,, d and δ, j, l(x)=0, j<q l, =,m d Moreover, the form (9) gves an nsght on the system nvertblty propertes If the term u e s absent, the system s left nvertble; f the term y e s absent, the

8 0 Andreea Udrea, Alexandru Tclea, Valentn Tanasa, Crstan Flutur system s rght nvertble; f the terms ue and nvertble and f the terms ue and e ye are both absent, the system s y exst, the system s degenerate [23] In the case of square nvertble systems wth m=p= m d, the terms u e and y e do not exst and the system takes the followng normal form [22]: = f ( z, ξ) ξ = ξ + δ ( z, ξ) v, j =, q ξ, j, j+, j, l l l= q, e = v y = ξ ; =, m, d (2) wth q q2 qm and δ, j, l(x)=0, j<q l, =,m (3) It can be observed that the system n (2) form has a trangular structure [22] between ξ, j and the nputs Relaton (3) reveals the fact that there s also a trangular dependency of δ, j, lon the state varables [23] The zero dynamcs of (2) The system s (2) zero dynamcs s gven by: = f (,0) e z Ths type of normal form wth a structure n whch the nputs are enterng the system n a trangular fashon and the δ, j, l functons have trangular dependences on the systems state presents valuable propertes (asde weaker assumptons and the fact that t shows the nvertblty propertes of the system) from the control pont of vew [23] Usng t to solve the output regulaton problem hasn t been yet pursut In what follows we are gong to gve the general context of output regulaton problem n the case of MIMO systems and to compare the condtons for the problem solvablty n case of usng the two normal forms presented above n terms of assumptons restrctvty 3 The general output regulaton problem for MIMO nonlnear systems The problem of output regulaton consders that the models for the process to be control and the exosystem are known The latter s supposed to contan the reference and/or the perturbatons A regulator solvng the problem

9 On the nonlnear output regulaton problem - Part - MIMO nonlnear systems normal forms n closed loop must assure: the boundedness of the state trajectory and unform convergence to 0 of the error We consder a multvarable system gven by the followng expresson: x = f( w, x, u) y = k( w, x) (4) e= h( w, x) n m p wth: x R the state, u R the control nput, e R the regulated output, p r y R the measured output, w R the exosystem s state The exosystem s an autonomous system: w= s( w) (5) The functons f(w,x,u), h(w,x), k(w,x) and s(w) are consdered to be of class C k (suffcently large) n ther arguments The ntal condtons for the system vary on a fxed closed set x(0) X 0 and for the exosystem - vary on a nvarant compact set w(0) W We further consder that the system s of fnte dmenson, tme nvarant, and can be put n a normal form: such that t has a well defned relatve degrees vector and the zero dynamcs s stable (the system s of mnmum phase) Isdor approach n [8] or 2 as descrbed n (9) The regulator s supposed to be of the form: ψ = ϕψ (, y) (6) u = γψ (, y) v wth: ψ R the regulator state and the functons ϕ( ψ, y) and γ ( ψ, y) of class C k The ntal condtons for the regulator can vary on a compact set ψ (0) Ξ The system (4), (5) and (6) n closed loop form s: w= s( w) x = f( w, x, γψ (, k( w, x))) (7) ψ = ϕψ (, kwx (, )) e= h( w, x) Consder that X s a compact subset of X 0, the regulator (6) solves the output regulaton problem f the postve trajectory on W X Ξ s bounded and lm et ( ) = 0, unformly on W X Ξ (when the system s n steady state) t

10 2 Andreea Udrea, Alexandru Tclea, Valentn Tanasa, Crstan Flutur The form of the regulator, ts ntal condtons set and ts propretes are to be determned In the context of the output regulaton problem as presented n [7], [4] wth the notatons and lemmas of [7] the trajectores of the system n closed loop are supposed to be bounded Ths leads to the concluson that the ω lmt set ω ( W X Ξ ) s not empty, compact and nvarant, and unformly attracts the trajectores of the system n closed loop and the steady state error s 0 f and only f: ω( W X Ξ) ( w, x, ψ): h( w, x) = 0 { } 4 Necessary condtons for solvng the output regulaton problem MIMO nonlnear systems 4 The system can be wrtten n the normal form (4) In ths case the system wth the exosystem (5) has the form: = f0(, z w) + f(,, z e w) e, =,m (8) e= q(,, z e w) + M(, z w) u wth: ntal condtons n the set Z E W where Z s fxed and compact and E s bounded and the functons f 0, f, g, s, M are smooth enough The couplng matrx M(z,w) s consdered nvertble (n the SISO systems case the condton s that bwze (,,,, er ) 0) Ths means that system (8) has a vector of relatve degrees: {,,} between the control nput u and the regulated output e A system wth the relatve degrees vector {r, r m } can always be transformed nto the form (8) Consderng that a controller of the form (6) solves the problem of output regulaton and applyng lemma 2 [7] the steady state locus of the system n closed loop (7) must be a subset of the set for whch the error s zero (e=0) If system (8) s n steady state the followng condtons are fulflled: - the steady state locus of the system n closed loop ω ( W Z E Ξ ) s a s n m v subset of: R R {0} R - the restrcton of the system n closed loop to the steady state locus ω ( W Z E Ξ ) s (wth the zero dynamcs gven by the frst two relatons) w= s( w) z = f ( w, z) ; cu e=0 0 ψ = ϕψ (,0) - ( wz,,0,,0, ψ ) ω( W Z E Ξ) 0 = qz (,0, w) + M( zwγ, ) ( ψ,0) <=> 0 = qz (,0, w) + M( zwu, )

11 On the nonlnear output regulaton problem - Part - MIMO nonlnear systems normal forms 3 It was consdered that the postve trajectory of the exosystem (5) on W s bounded f the trajectores asymptotcally approach ω ( W ) Ths assumpton does not dmnsh the generalty of the problem because t can be consdered that W= ω ( W ) whch means that the exosystem s n steady state Assumpton MIMO n The set W R s compact and nvarant under (5) If the postve orbt of the set W Z E Ξ under (4), (5) and (6), then the system dynamcs n closed loop s the graph of a functon defned on the whole of W parts Notng A {(, ) (,,0,,0, ) ( ), v ss = wz wz ψ ω W Z E Ξ ψ R} (9) and consderng the functon m uss : Ass R ( wz, ) M ( wzqwz, ) (,,0) by constructon the set descrbed by (9) s the codoman of a functon defned on the whole of W, whch s nvarant under the zero dynamcs of the system n the normal form (4) and the exosystem: w= s( w) (20) z = f0( w, z) The functon u ss s the control law that forces the system to evolve on A ss In concluson, f the controller (6) solves the output regulaton problem for the system n normal form wth the exosystem, then there s a functon defned on the whole of W whch has the codomane A ss and A ss s nvarant under (20) v Moreover, for each ( w0, z0) Ass, ψ 0 R such that the ntegral curve of (20) s exactly the ntegral curve of ψ = ϕψ (,0) startng n ψ 0 and satsfyng uss ( w(), t z()) t = γ ( ψ (),0), t t R In other words, one can buld a controller that reproduces the nput for steady stare such that the regulated error s 0 (nternal model for nonlnear system) Consderng ths approach t can be observed that Assumpton I MIMO can be consdered only f the couplng matrx M s nvertble The nvertblty mples two aspects: the system s square

12 4 Andreea Udrea, Alexandru Tclea, Valentn Tanasa, Crstan Flutur 2 the nverse of M must be formally computed whch represents a major drawback for real mplementaton cases Ths s probably why no publcaton follows the artcle gvng ths soluton n [8] 42 Recent normal form Followng the above reasonng, we consder that the system s wrtten n the normal form (2) wth the exosystem (5): w = s( w) = f (, z w) ξ = ξ + δ (, zw, ξ) v, j=, q ξ, j, j+, j, l l l= q, e = v y = ξ ; =, m, d Assumpton I MIMO takes the followng form: Assumpton MIMO N n The set W R s compact and nvarant under (5) If the postve orbt of the set W Z E Ξ under (2), (5) and (6), the system dynamcs n closed loop s the graph of a functon defned on the whole of W parts If we note: A' {(, ) (,,0,,0, ) ( ), v ss = wz wz ψ ω W Z E Ξ ψ R} (2) and f n relaton (3) ql we can consder the functon: m u' ss : A' ss R a( w, z,0)/ b( w, z,0) ( wz, ) fp( u',, u' p, a( wz,,0),, ap ( w, z,0), b( w, z,0),, bp ( w, z,0)) By constructon, the set descrbed by (2) s the codoman of a functon defned on the whole of W, whch s nvarant under the zero dynamcs of the system n the normal form (2) and the exosystem (5): w = s( w) = f (,0) e z The functon u ss s the control law that forces the system to evolve on A ss

13 On the nonlnear output regulaton problem - Part - MIMO nonlnear systems normal forms 5 The trangular propertes of the new normal form (2) lead to a feasble soluton for wrtng the necessary condtons for the output regulaton problem n a real case, wthout formally nvertng the couplng matrx 5 Conclusons In the lght of recent advancements on normal forms we consder that a more sutable approach for solvng the nonlnear output regulaton problem n the case of square MIMO systems s to use the normal form proposed n [22] In ths case the condtons under whch the problem of output regulaton s solvable are more relaxed from the followng ponts of vew: the relatve degrees vector s not requred; there s no assumpton that mples the fact that the couplng M matrx s to be nverted explctly, whch means that nonsquare MIMO systems could be consdered for control too; the assumptons needed for the normal form of [22] are less restrctve than the ones for the normal form of [2] Acknowledgement Ths work was fnanced from the project TE_232 Nonlnear solutons for the nonlnear output problem, Human Resources Program, UEFISCDI B I B L I O G R A P H Y [] BA Francs, WM Wonham (976) The nternal model prncple of control theory, Automatca, vol 2, pp [2] BA Francs (977) The lnear multvarable regulator problem, SIAM Journal on Control and Optmzaton, vol 4, pp [3] A Isdor, CI Byrnes (990) Output regulaton of nonlnear systems, IEEE Transactons on Automatc Control, vol 25, pp 3-40 [4] J Huang, WJ Rugh (990) On a nonlnear multvarable servomechansm problem, Automatca, vol 26, pp [5] K Khall (994) Robust servomechansm output feedback controllers for feedback lnearzable systems, Automatca, vol 30, pp [6] A Serran, A Isdor, L Marcon (2000) Semglobal nonlnear output regulaton wth adaptve nternal model, Internatonal J of Robust and Nonlnear Control, vol 0, pp [7] CI Byrnes, A Isdor (2003) Lmt sets, zero dynamcs and nternal models n the problem of nonlnear output regulaton, IEEE Transactons on Automatc Control, vol 48, pp [8] J Huang, CF Ln (994) On a robust nonlnear multvarable servomechansm problem, IEEE Transactons on Automatc Control, vol 39, pp [9] A Serran, A Isdor, L Marcon (200) Semglobal nonlnear output regulaton wth adaptve nternal model, IEEE Transactons on Automatc Control, vol 46, pp 78-94

14 6 Andreea Udrea, Alexandru Tclea, Valentn Tanasa, Crstan Flutur [0] L Marcon, A Isdor (2006) A unfyng approach to the desgn of nonlnear output regulators, n Bonveto et al, eds, Advances n Control Theory and Applcatons, LNCIS vol 353, pp , Sprnger Verlag [] JP Gauther, I Kupka (200) Determnstc observaton theory and applcatons, Cambrdge Unversty Press [2] R Marno, P Tome (992) Global adaptve observers for nonlnear systems va fltered transformatons, IEEE Transactons on Automatc Control, vol 37, pp [3] V Andreu, L Praly (2006) On the exstence of a Kazants-Kravars/Luenberger observer, SIAM Journal on Control and Optmzaton, vol 45, pp [4] CI Byrnes, A Isdor (2004) Nonlnear nternal models for output regulaton, IEEE Transactons on Automatc Control, vol 49, pp [5] F Dell Prscol, L Marcon, A Isdor (2006) A new approach to adaptve nonlnear regulaton, SIAM Journal on Control and Optmzaton, vol 45, pp [6] LMarcon, L Praly, A Isdor (2006) Output stablzaton va nonlnear Luenberger observers, SIAM Journal on Control and Optmzaton, vol 45, pp [7] L Marcon, L Praly (2008) Unform practcal nonlnear output regulaton, IEEE Transactons on Automatc Control, vol 53, pp [8] NK McGregor, CI Byrnes, A Isdor (2006) Results on Nonlnear Output Regulaton for MIMO Systems, Proceedngs of the 2006 Amercan Control Conference Mnneapols, Mnnesota, USA [9] G Besancon, ed (2007) Nonlnear observers and applcatons, LNCIS vol 363, Sprnger Verlag [20] A Isdor (995) Nonlnear Control Systems - 3 rd edton, Sprnger [2] A Isdor, (999) Global normal forms for MIMO nonlnear systems wth applcatons to Stablzaton and dsturbance attenuaton, Math Control Sgnals Systems, Sprnger- Verlag [22] X Lu, Z Ln (20) On normal forms of nonlnear systems affne n control, IEEE Transactons on Automatc Control, vol 56, no 2 [23] X Lu, Z Ln (2007) On Normal Forms of Nonlnear Systems Affne n Control, Unv of Vrgna, Charlottesvlle, Amercan Control Conference