"Class-Type" Identification-Based Internal Models in Multivariable Nonlinear Output Regulation

This is the post peer -revie w accepted manuscrip t of: M. Bin and L. Marconi, ““Class-T ype” Identification - Based Internal Models in Multiv ariable Nonlinear Output Regulation, ” accep ted for publication in IEEE Transaction on Automatic Control. The published v ersion is a vailable online at: https://doi.org/10.11 09/T A C.2019.2955668 © 20 20 IEEE. P er sonal use of this ma terial is p ermitted. P ermission fr om IEEE must be obtaine d for all oth er uses, in any curr en t or futur e med ia, includin g r eprintin g/r ep ublishing this material for advertising or pr omotiona l purposes, cr ea tin g new collective works, for r esa le o r r edistribution to servers or lists, or r euse o f a ny cop yrighted compon e nt of this work in other works. 2 “Class-type” Identificati on-Based Internal Mo d els in Multi v ariable Nonlinear Output Re gulation Michelangelo Bin a n d Lo renzo Marco ni Abstract —The paper deals with the p roblem of output reg- ulation in a “non-equil ibrium” context f or a special class of multivar iable nonlinear systems stabilizable b y h igh-gain feed - back. A post-processing i nternal model design su itable for the multivar iable nature of the system, whi ch might hav e more inputs than regulation errors, i s proposed. Uncertainties in t h e system and exosystem are dealt with b y assuming t h at the id eal steady state input belongs to a certain “class of signals” by which an appropriate model set f or the i nternal model can b e deri ved. Th e adaptation mechanism for the internal model is th en cast as an identification problem and a least square solution is specifically deve loped. In line wit h rece nt developments in the field, the vision that emerges from the paper is that approximate, possibly asymptotic, r egulation is the ap p ropriate way of approaching the problem in a multivariable and u ncertain context. New insight s about the u se of id entification tools i n the d esign of adaptive internal models are also presented. I . I N T RO D U C T I O N W e co nsider non lin ear systems of the form ˙ x = f ( w, x , u ) , y = h ( w, x ) , e = h e ( w, x ) (1) with state x ∈ R n x , con tr ol input u ∈ R n u , measur ed outpu ts y ∈ R n y , “regulation error ” e ∈ R n e , and with w ∈ R n w an exogenou s signal generated b y the “exosystem” ˙ w = s ( w ) . (2) The p roblem of appr oxima te output regulation pe rtains the design o f an ou tput feedb a ck regulator of th e f o rm ˙ x c = f c ( x c , y ) , u = k c ( x c , y ) achieving the regulation ob jectiv e lim sup t →∞ | e ( t ) | ≤ ǫ , with ǫ ≥ 0 po ssibly a “small” numb er m e asuring the regulator’ s asymptotic performance. If ǫ = 0 , then the regulato r is said to achieve asymptotic r e g ulation . If ǫ can be re duced arbitrarily by o p portun ely tuning th e regu lator paramete r s, the r egulator is said to ach iev e practical re gulation . If the r egulation prop erties are obtained in spite of possible uncertainties in the system (1), the problem is referred to as r o bust outp ut re gulatio n [ 1], while the terminolog y ad aptive outpu t re gulation is typically used in pr esence of unce r tainties in the exosystem (2). An anchor p oint in the solu tio n of the prob lem is represen ted by the steady-state trajectories ( x ⋆ ( t ) , u ⋆ ( t )) solution of the so- called re gulator equation s ˙ w = s ( w ) , ˙ x ⋆ = f ( w, x ⋆ , u ⋆ ) , 0 = h e ( w, x ⋆ ) , (3) Michel angelo Bin (m.bin@impe rial.ac.uk) is with the De partment of Elec- trical and Elect ronic Engineering, Imperial Colle ge London, UK. L orenzo Marconi (lorenzo.marconi @unibo.it) is with the Depart ment of Electrica l, Electroni c, an d Information Engineering, Uni versity of Bologna, Italy . with x ⋆ representin g th e ideal state trajectory associated with a zero r egu lation error an d u ⋆ the associated inp ut (of ten referred to as “the friend” of x ⋆ ). As shown in [2], indeed, solv a b ility of (3) is a necessary condition f or the prob lem at hand. Regulator stru ctures propo sed in th e nonlinear co ntext are typically composed by two u nits, an in ternal model unit and a stabilizing unit , with a neat, alb eit limiting in many contexts, “role” conf erred on th e two at the design stage: the former is design ed to g e nerate the steady state input u ⋆ ( t ) required to keep the error at zero in stead y state, wh ile the latter is de sig n ed to steer the sy stem trajectories to x ⋆ ( t ) . What makes the desig n problem particularly challeng ing is, of course, the fact that ( x ⋆ , u ⋆ ) are unk n own as the in itial condition s of ( 3) and (2) are such and, in the robust/adaptive case, uncertainties in ( 1) and/or (2) stron gly af fe ct the solutio n of (3). The majo r ity of th e curre n t w orks o n the sub ject have some limiting aspects that is worth pointing out to better fram e the con tribution of this pape r . Non-equilib rium context. Cur r ent framew o rks typ ic a lly as- sume that the solution s of (3) depend on time throu g h w ( t ) , namely ( x ⋆ , u ⋆ ) = ( π ( w ) , c ( w )) fo r some π and c . Moreover , further restrictions are usually imp osed limiting the class of friends that can be dealt with, as for instance the so-called “immersion assumption” (the latter ev en m ore weakened over the year s, see [3], [4], [5], [6]). This assumption, far to b e necessary , leads to design princip le s of the interna l model unit just driven by th e exosy stem dyn amics and som e ap propr iate “distortions” tha t, howev er , do not co mpletely capture the full no nlinear context. A f o rmal framework to overcome this limitation w as giv en in [2], where a “non-equ ilib rium theory” for non linear output regulatio n was laid , by asserting that the internal model is in gener a l required to incor p orate a mixture of the residual p lant’ s and exosystem’ s dyna m ics, in this way making meaningless the d istinction between the plant and the exosystem from a design viewpoint (and thus between robust and ad aptive o utput regulation ). “F riend-centric” in ternal mod els. Many of the existing regulators are strongly “friend- c entric”, nam ely the design of the internal model unit is definitely tailored aroun d the specific u ⋆ resulting from the regulato r equations. Th is, in turn, leads to fragile designs in which unexpected variations of the system/exosystem easily lea d to in effecti ve regulators with unpred ictable asymptotic properties. Uncertainties in the system/exosystem are typ ica lly hand led b y parametrising th e internal mod el in terms of u ncertain parameter s and b y looking for “a daptive” mechanisms accord ing to the actual regulation error (see e.g . [5,7]). This way o f pro ceeding, howe ver, in- volves a “quan titati ve” information about how the u ncertainties reflect on the f riend that ar e hard to assum e, unless sub- stantially limiting the top ology describing sy stem/exosystem variations. Th ese difficulties p ushed the auth ors of [1] to conjecture that asy mptotic regulation in a gen e ral nonlinear and u ncertain context is un achiev able with finite dime n sional regulators and to promo te app roaches loo king for appro xi- mate regulato rs, which possibly b ecome asymp to tic if certain fortun a te conditions happen. In general, ho w a “q ualitative” informa tio n a b out the friend can be tran sferred into the design of an internal model that beha ves “well” fo r a “wide” r ange of system/exosystem v ariations is still an open point in literature. Pr e- versus post-pr o cessing schemes. A taxo nomy rec ently introdu c ed in th e literature regard s the distinction b e tween pr e- pr ocessing and post-pr ocessing internal models [8,9]. I n the latter , the intern al mo del un it directly processes the regulation error, while the stabilising unit stabilises the cascade of the system dri ving the intern al mo del unit. In the for mer , conv ersely , the two units are somehow “swapped”, with the internal mode l dire ctly g enerating the feedfo r ward input and the stabiliser stabilising the cascad e of the inte r nal model unit dr i ving the system. The regulato r structu res prop osed so far are definitely biased on pre-pro cessing solution s and, as such, limited to de al with sin g le input-sin gle erro r systems (i.e. n u = n e = 1 ) or some “squa r e” exten sions with n u = n e (see, e.g. , [ 1 0]). As ob served in [8,9], p ost-processing solutions seem more suited to handling gen e ral m ultiv ariab le contexts with po ssibly n u > n e . The latter , in turn, are also mor e promising to h andle contexts in which, besides the regulation er r ors, also extra measu rements are av ailab le that do not necessarily vanish at the steady state. Not surprising ly , the gen eral regulator structure for linear systems is post- processing [1 1]. The dr if t to wards post-pr ocessing solutions for non linear systems, howe ver, substantially co mplicates the design of the non linear regulator by raising an in tertwining in the d esign of the in ternal model and stab iliser (referr ed to as chic ken-egg dilemma in [12]) not p resent in pr e-processing approa c h es. T o the best knowledge of the authors, a general post-pro cessing n onlinear fram ework is still unavailable in literature with just some attemp ts don e in [13] and [14] f or simplified exosystems. In this paper we propo se a design technique b ased on the aforemen tioned non- equilibrium con text, in which the effects of the system and exo sy stem d y namics o n the stead y state are jointly considered in th e de sign of the internal mo del. The propo sed r egu la to r em beds a “po st-processing” internal model that applies to multiv ariab le systems not ne cessarily square, and wh ose construction is no t “friend -centric” but rathe r it is based on a “qualitative” info r mation o n the ideal error-zeroing steady state. I I . M A I N R E S U LT A. The c la ss of systems W e consider a subclass o f systems (1) with state x = col( x 0 , χ, ζ ) ∈ R n x satisfying th e following equatio ns ˙ x 0 = f 0 ( w, x ) + b ( w, x ) u (4a) ˙ χ = F χ + H ζ (4b) ˙ ζ = q ( w, x ) + Ω( w, x ) u (4c) e = C χ , y = col ( χ, ζ ) , (4d) in which x 0 ∈ R n 0 , y ∈ R n y , e ∈ R n e , ζ ∈ R n e , u ∈ R n u , with n u ≥ n e , χ = col( χ 1 , . . . , χ n e ) , with χ i ∈ R n i χ , i = 1 , . . . , n e , and n 1 χ + · · · + n n e χ =: n χ , C := blkdiag( C 1 , . . . , C n e ) , F := blkdiag( F 1 , . . . , F n e ) and H := blkdiag( H 1 , . . . , H n e ) , with C i :=  1 0 1 × ( n i χ − 1)  and F i :=  0 ( n i χ − 1) × 1 I n i χ − 1 0 0 1 × ( n i χ − 1)  , H i :=  0 ( n i χ − 1) × 1 1  . The χ sub system, in p articular, is described by n e chains of integrators with ζ enter ing a t the bottom an d the regulatio n error given by the first comp o nents χ i 1 of each chain χ i . Hence, χ and ζ are linear combinations of th e err or and its time der ivati ves. The functions f 0 , b , q an d Ω are sufficiently smooth functions, with Ω( w , x ) ∈ R n e × n u denoting the so- called “hig h-freq u ency matrix”. The fo rm (4) is representative of dif ferent f rameworks a d dressed in literature. F or instance, systems having a we ll-defined vector relative degree with re- spect to the inpu t-outpu t pair ( u, e ) and admitting a canonical normal form fit in the proposed framew o rk. In this case the x 0 dynamics in (4) d oes n ot depend on u and it represents the z er o dynamics o f the system re lati ve to the indicated in put-ou tput pair . On the o ther hand ( 4), with a sligh tly different structure of χ and of the m atrices F and H , is also representativ e o f systems that ar e “just” ( globally) stron gly invertible in the sense of [15,16] a nd feedback linear isable with respect to the in put-ou tput pa ir ( u, e ) and, as such, can b e tran sformed in partial normal form , see [ 17]. In this case the dy n amics (4b)-(4c) are th e partial normal form of the system and the subsystem ( 4 a) is indeed th e wh ole plan t (i.e. x = x 0 ). W e observe that th e mea su rable outpu ts y are assumed to be linear combinations o f the er r or and its time deriv ati ves, namely we look for a partia l state feedb ack solution. A pure error feedback regulator only processing e can be obtained by replacing the time der ivati ves with appropriate estimates via standard hig h -gain techniqu es (see [18 ]) whose details are not presented he re. In the re st of the p aper we assume the following. A1) Ther e exist β 0 ∈ KL , α 0 > 0 a nd, for e ach solution w of ( 2), each inp ut u , and each solution x of (4) corresponding to ( w , u ) , ther e exist x ⋆ 0 : R ≥ 0 → R n 0 and u ⋆ : R ≥ 0 → R n u fulfilling ˙ x ⋆ 0 = f 0 ( w, x ⋆ ) + b ( w , x ⋆ ) u ⋆ 0 = q ( w , x ⋆ ) + Ω( w, x ⋆ ) u ⋆ (5) in which x ⋆ := ( x ⋆ 0 , 0 , 0) , a nd | x 0 ( t ) − x ⋆ 0 ( t ) | ≤ β 0  | x 0 (0) − x ⋆ 0 (0) | , t  + α 0 | ( χ, ζ ) | [0 ,t ) for a ll t ≥ 0 . A2) Ther e e xists a full-r ank ma trix L ∈ R n u × n e such tha t the (square) matrix Ω( w, x ) L is bou nded, it sa tisfies L ⊤ Ω( w, x ) ⊤ + Ω( w , x ) L ≥ I n e for all ( w, x ) ∈ R n w × R n x , an d the map (Ω( · ) L ) − 1 q ( · ) is Lipschitz. Plant Int. Mo del Identifier Stabiliser e y η 1 u η θ Figure 1. Bloc k-diagra m of the re gulator . Equation s (5) are the specialisation of th e regulator equa - tions (3) in this no n-equilib rium context. The steady state ( x ⋆ , u ⋆ ) migh t be dependen t on the in itial con d itions of the system cohe r ently with [2]. Condition A1 ask s for unifo rm (in u ) detectability of the idea l steady state x ⋆ (see [19]). In case of systems with cano n ical normal form in which (4a) does not depend on u , th is assumption boils d own to a conventional minimum-ph ase requirement, far to be necessary although typically assumed in the pertinent literatu re. A2, instead, is a robust stabilisability requ ir ement and it imp lies that Ω( w , x ) is everywhere full rank. As a conseq uence, u ⋆ in (5) is given by u ⋆ = − Ω( w, x ⋆ ) ⊤  Ω( w, x ⋆ )Ω( w, x ⋆ ) ⊤  − 1 q ( w, x ⋆ ) . As clear fr om A1 an d A2, we deal with a simplified case in which a global result is sought u n der quite re strictive global Lipschitz an d bou ndedn e ss co nditions. Nevertheless, we rema rk th at the proposed result can be extended to a semiglobal setting by asking w to ev o lve in a comp act space and A1 and A2 to hold o nly locally (namely on e a ch comp act subset of R n x ). For reason of space and since the extensio n follows b y well-kn own argumen ts (see e .g. [18,20]) we omit this extension and we fo cus on the n ew adaptive frame work. B. The re gulato r structur e The pr oposed regulato r structure is depicte d in Figure 1. The p o st-processing intern a l model unit h as th e form ˙ η = Φ( η , θ ) + Ge, η ∈ R dn e (6) with d ∈ N , η = ( η 1 , . . . , η d ) , η i ∈ R n e , a n d Φ( η , θ ) =     η 2 · · · η d ψ ( η , θ )     , G =     g h 1 I n e g 2 h 2 I n e · · · g d h d I n e     , in wh ich h i , i = 1 , . . . , d , ar e fixed so that th e polyn omial s d + h 1 s d − 1 + · · · + h d − 1 s + h d is Hurwitz, g > 0 is a p arameter to be designed, ψ : R dn e × R n θ → R n e is a fun ction to be fixed, and θ ∈ R n θ , n θ ∈ N , is an “ad aptive” p arameter generated by th e id e n tifier subsystem, whose dy namics is describ e d by ˙ z = µ ( z , η , e ) , θ = ω ( z ) , (7) in which µ : Z × R dn e × R n e → Z and ω : Z → R n θ , with Z a normed vector space of finite d imension, ha ve to be fixed. Finally , the (static) stabiliser is taken as u = L  K χ χ + K ζ ζ + K η η 1 + K w ν ( x ⋆ , w )  , (8) in which the m atrices K χ , K ζ and K η are chosen as follows K χ ( ℓ, κ ) = ℓK ( κ ) , K ζ ( ℓ ) = − ℓI n e , K η ( ℓ, κ ) = ℓK ( κ ) C ⊤ with K ( κ ) = blkdiag( K 1 ( κ ) , . . . , K n e ( κ )) , where K i ( κ ) = −  c i 1 κ n i χ c i 2 κ n i χ − 1 . . . c i n i χ κ  (9) for i = 1 , . . . , n e , in which the coefficients c i j are ch osen so that the po lynomia ls s n i χ + c i n i χ s n i χ − 1 + · · · + c i 2 s + c i 1 , i = 1 , . . . , n e , ar e Hu rwitz, and ℓ, κ > 0 are design par ameters to be fix ed. The matrix K w and the function ν are introd uced for sake of g enerality a nd are possibly zero. These terms cou ld represent a “feed forward” contribution added by th e d esigner by employing possible knowledge of w and x ⋆ . Likewise, it could r epresent a term sho wing up in the norm al form (4) after a preliminary feedback of available mea surements that d o not vanish in steady state. Similarly to the other matrices in (8), the gain matrix K w can depend on κ and ℓ . The d egrees of freedom left to be fixed at this stage ar e the dimension d and function ψ of the in ternal model un it (6), the d ata ( Z , n θ , µ, ω ) of the ide ntifier (7), and th e con tr ol para meters g , ℓ and κ . C. De sign of the interna l model as pr ed iction model A ke y step in the regulator synthesis is the choice of the internal model (6) and of its adap ta tio n through the design of the identifier ( 7). Consistently with th e discussion in Section I, this mu st be done to achiev e a small, po ssibly zero, asym ptotic regulation erro r in spite of u ncertainties inv olving ( x ⋆ , u ⋆ ) and the underly ing dyn amics. W ith an eye to the last equation o f (6), we can write e ( t ) = ¯ c ( g )  ˙ η d ( t ) − ψ ( η ( t ) , θ ( t ))  (10) in wh ich ¯ c ( g ) := ( h d g d ) − 1 . Ou r design strategy to choo se ( d, ψ ) in (6) and the iden tifier (7) piv ots arou n d the idea that ˙ η d ( t ) − ψ ( η ( t ) , θ ( t )) can be interpr e te d as a “prediction error” attained b y th e “m odel” ψ in relating the “next de r iv ative” ˙ η d ( t ) to the “pre vious derivati ves” η ( t ) , and that, by minim is- ing this predictio n er ror, the actual regulation erro r is also minimised due to ( 1 0). This clearly suggests to lo ok at the problem of choosing d and ψ as an identifi cation p r oblem and, by bo rrowing the notation typ ic a lly adopted in that literatur e [21], to r efer to the map ψ ( · , θ ) as the prediction model relating th e “in put data” η to the “output” ˙ η d , an d to the set M := { ψ ( · , θ ) : θ ∈ R n θ } o f all th e possible candidate models as the co rrespon ding mode l set . Th e choice of d and of ψ thus must be done in such a way th a t the attainable prediction error is minim ised. Un less re lying on “un iversal” infinite-dimen sional models, h owe ver, this selection m ust be groun ded o n so me p reliminary k nowledge ab out the class of signals to which ˙ η d and η are expected to belong. In this context, the steady-state signals ( x ⋆ , u ⋆ ) resulting fr om the regulator equation s (5) are the anch or poin t on which that knowledge can be drawn. In particu lar , let η ⋆ 1 := Υ ( ℓ,κ ) ( w, x ⋆ ) , in which Υ ( ℓ,κ ) ( w, x ⋆ ) := −  Ω( w, x ⋆ ) L K η  − 1  q ( w, x ⋆ ) + Ω( w , x ⋆ ) L K ω ν ( w, x ⋆ )  , and d efine recu rsiv ely η ⋆ i , i = 2 , . . . , d + 1 , as 1 η ⋆ i := L i − 1 s ( w ) Υ ( ℓ,κ ) ( w, x ⋆ ) + L i − 1 f 0 ( w, x )+ b ( w ,x ) u ⋆ Υ ( ℓ,κ ) ( w, x ⋆ ) . Finally let ˙ η ⋆ d := η ⋆ d +1 . In view o f A2 a n d the defin ition of K η , th e m a trix Ω( w, x ) L K η is e very where inv ertible and, thus, a ll the pre- vious quantities are well-define d . M oreover , we observe that the quantities η ⋆ i , i = 1 , . . . , d + 1 , depend o n the design parameters κ and ℓ yet to b e fixed. The dimen sion d and the func tio n ψ shou ld be th en ideally chosen so that, with η ⋆ = col( η ⋆ 1 , . . . , η ⋆ d ) , th e f o llowing holds ˙ η ⋆ d ( t ) = ψ ( η ⋆ ( t ) , θ ⋆ ( t )) , (11) for some “ide a l” θ ⋆ ( t ) ∈ R n θ . This, in fact, would make ( x ⋆ , η ⋆ ) a trajectory of the clo sed -loop system in which the as- sociated regulation error is identically zero. Th e de sig n of the pair ( d, ψ ) so that (11) is fulfilled for all possible steady- state trajectories ( ˙ η ⋆ d , η ⋆ ) , however , is no t realistic unless lim itin g ev en further the class of treatable nonlin ear system s and of manageab le uncertainties on the solutio n of (5). Furthe r more, ev en in the fortu nate case in which the ideal re la tio n (11) cou ld be fu lfilled with a pe r fect pa rametrisation (may b e playing with large values of d ), this might requ ir e an unaccep table complexity of the inter nal mod el, and an app r oximated mo del with a possibly lower d would be p referable. Along this direction, we rather assume that the d esigner has a qu a li- tative knowledge ab out a “class” H ⋆ of signals 2 to which ( ˙ η ⋆ d , η ⋆ ) belong s in order to fix a model set M ne c essarily approx imated but optimised for the specific class. This is the “modelling part”, in which the “touch” o f th e designer and the knowledge on the stead y -state trajectories come into play . The class H ⋆ , in turn , is fixed on the basis o f the knowledge on the no minal solution ( x ⋆ , u ⋆ ) to (5), and after consid e r ing all the exp e cted system/exosystem uncertain ties that may affect it. The prob lem o f handling the overall u ncertainty on ( x ⋆ , u ⋆ ) is thus transferred to the adaptatio n side, and the idea of relying on system ide n tification techniq ues for it is furthe r motiv ated by the fact th at, typically , identification meth ods structurally manage large classes of signals [21]. From n ow on we sup p ose that the designer has fixed a class H ⋆ and, according ly , a model set M , so that the fo llowing assumption ho lds. A3) The map ψ is Lipschitz a nd differ entiab le with a locally Lipschitz derivative, and the Lip schitz constants d o not de- penden t on κ and ℓ . Moreo ver , th er e exists a compact set H ⋆ ⊂ R n e × R dn e , in depende nt on κ and ℓ , such tha t every ( ˙ η ⋆ d , η ⋆ ) ∈ H ⋆ satisfies ( ˙ η ⋆ d ( t ) , η ⋆ ( t )) ∈ H ⋆ for a ll t ∈ R ≥ 0 . The pr evious assumption for malizes th e “quan titati ve” prop - erties requ ir ed to the members of the class H ⋆ on which the design of the inter nal model and the id entifier is gro u nded. In particular, it is asked that th e elem ents of H ⋆ stay in a known co m pact set H ⋆ , and that the inferred prediction m odel ψ has som e stro ng regularity proper ties uniform in the con trol gains ( κ, ℓ ) . These req uirements, in principle no t needed in the 1 W e deno te by L g f the Lie deri vati ve of f along g . 2 Formall y , H ⋆ is a subset of the space of functions R ≥ 0 → R n e × R dn e . design of the identifier an d intern al m odel, a r e rather n e eded for the successiv e embedding o f the two units in the overall regulator, as they permit to brea k the “ch ic ken - egg dilemma” and sequ ence the d e sign of the remaining degrees of freedom. W e remark , m o reover , tha t in the “squa re” case, namely when n u = n e in (4), the m a trix Ω( w , x ) is square and κ and ℓ d o n o mix-up with Ω( w , x ) , q ( w , x ) an d ν ( w, x ) in the definition o f ˙ η ⋆ d and η ⋆ . Ther efore ( ˙ η ⋆ d , η ⋆ ) can b e always bound ed un iformly in κ and ℓ whenever they are taken lar ger than 1 and K ω / ( κℓ ) can b e bou nded unifor mly in κ and ℓ . D. The design of the iden tifier W ith d and ψ fixed, we shif t our attentio n to the design of the identifier . Th e fact that (11) is no t attainable exactly suggests to defin e a steady sta te p rediction err or as ε ⋆ ( t, θ ) := ˙ η ⋆ d ( t ) − ψ ( η ⋆ ( t ) , θ ) (12) and to look for a dynamical system wh ich is able to select the best p arameter, say θ ⋆ , wh ose co rrespond ing mod e l ψ ( · , θ ⋆ ( t )) is, at each t , the “best” mode l in M r elating ˙ η ⋆ ( t ) and η ⋆ ( t ) , minimising in some sense ε ⋆ . As customary in system identification, the m eaning of “best” in the mod el selection is b ased on the definition of a fitness criteria assigning to each model ψ ( · , θ ) ∈ M a su itable a n d com parable value. In particular, with C 0 ( R n θ , R ≥ 0 ) the sp ace of continuou s function s R n θ → R ≥ 0 , with each pair ( ˙ η ⋆ d , η ⋆ ) ∈ H ⋆ we associate the map J ( ˙ η ⋆ d ,η ⋆ ) : R ≥ 0 → C 0 ( R n θ , R ≥ 0 ) given by J ( ˙ η ⋆ d ,η ⋆ ) ( t )( θ ) := Z t 0 c ε  t, s, | ε ⋆ ( s, θ ) |  ds + c r ( θ ) , (13) with c ε : R ≥ 0 × R ≥ 0 × R ≥ 0 → R ≥ 0 and c r : R n θ → R ≥ 0 some user-defined positive fu n ctions characterising the par- ticular un derlying identificatio n pro blem. More precisely , the integral term of (13) measur es how well a giv en cho ic e of θ fits the histor ical data, while c r ( θ ) play s the role of a regularisation factor . W ith J ( ˙ η ⋆ d ,η ⋆ ) we associate the set-valued map ϑ ◦ ( ˙ η ⋆ d ,η ⋆ ) : R ≥ 0 ⇒ R n θ defined as ϑ ◦ ( ˙ η ⋆ d ,η ⋆ ) ( t ) := argmin θ ∈ R n θ J ( ˙ η ⋆ d ,η ⋆ ) ( t )( θ ) . Once a cost functio nal of the fo rm (1 3) is d e fined, th e ide ntifier subsystem ( 7 ) is constructed to guar antee the existence of an “optimal” steady state z ⋆ , which is robustly asymptotica lly stable for (7), and who se c o rrespon ding output θ ⋆ = ω ( z ⋆ ) is a pointwise m inimiser of J ( ˙ η ⋆ d ,η ⋆ ) ( t ) , i.e. satisfies θ ⋆ ( t ) ∈ ϑ ◦ ( ˙ η ⋆ d ,η ⋆ ) ( t ) for all t ≥ 0 . In particu lar , the iden tifier ( 7) is chosen as a system with state z = col( ξ , ς ) , ξ ∈ R 2 n e , ς ∈ Z ς , in which Z ς is a finite-dim ensional normed vector space, Z = R 2 n e × Z ς , and the pa ir ( µ, ω ) is chosen so that, with ξ 1 , ξ 2 ∈ R n e such that ξ = col( ξ 1 , ξ 2 ) , th e equatio ns (7) r ead as ˙ ξ 1 = ξ 2 − m 1 ρ ( ξ 1 − η d ) ˙ ξ 2 = ˙ ψ ( ξ 2 , η , ς ) − m 2 ρ 2 ( ξ 1 − η d ) ˙ ς = ϕ ( ς , ξ 2 , η ) θ = γ ( ς ) (14) where m 1 , m 2 > 0 are arb itrary , ρ > 0 is a de sig n p arameter, ˙ ψ : R n e × R dn e × Z ς → R n e is a function fixed below , and ( ϕ, γ ) is chosen to satisfy the following requirem ent. Requirement 1 (Identifier Requir ement) . Th e pa ir ( ϕ, γ ) is said to satisfy the id entifier r equ ir emen t r elative to a class H ⋆ and a co st functional (13) , if ϕ is loca lly Lipschitz, γ is Lipschitz and dif fer en tia ble with locally Lipschitz derivative, and ther e e xist β ς ∈ K L , a co mpact set S ⋆ ⊂ Z ς , α ς > 0 and, for each ( ˙ η ⋆ d , η ⋆ ) ∈ H ⋆ , a unique ς ⋆ : R ≥ 0 → S ⋆ , such that: • for every locally inte g rable δ ′ : R → R n e and δ ′′ : R → R dn e , all the ma ximal solutio ns to the system ˙ ς = ϕ ( ς , ˙ η ⋆ d + δ ′ , η ⋆ + δ ′′ ) are complete and satisfy | ς ( t ) − ς ⋆ ( t ) | ≤ β ς ( | ς (0) − ς ⋆ (0) | , t ) + α ς | ( δ ′ , δ ′′ ) | [0 ,t ) for a ll t ∈ R ≥ 0 ; • the signal θ ⋆ ( t ) := γ ( ς ⋆ ( t )) satisfies θ ⋆ ( t ) ∈ ϑ ◦ ( ˙ η ⋆ d ,η ⋆ ) ( t ) for a ll t ∈ R ≥ 0 . W ith H ⋆ and S ⋆ the compact sets intro d uced, respectiv ely , in A3 and in the identifier requiremen t, and with D ψ a n d D γ denoting th e Jacobian of ψ a n d γ respectively , we define ˙ ψ as any bou nded functio n satisfying ˙ ψ ( ξ 2 , η , ς ) = D ψ ( η , θ ) col  Φ( η , γ ( ς )) , D γ ( ς ) ϕ ( ς , ξ 2 , η )  (15) for all ( ξ 2 , η , ς ) ∈ H ⋆ × S ⋆ . With this construction, since und er A3 and the identifier requ irement, ψ , D ψ , ϕ , γ an d D γ are locally Lip schitz, and ˙ ψ is bo unded , there exists l ψ > 0 such that | ˙ ψ ( ξ 2 , η , ς ) − ˙ ψ ( ˙ η ⋆ d , η ⋆ , ς ⋆ ) | ≤ l ψ | ( ξ 2 − ˙ η ⋆ d , η − η ⋆ , ς − ς ⋆ ) | (16) for all ( ξ 2 , η , ς ) ∈ R n e × R dn e × Z ς and ( ˙ η ⋆ d , η ⋆ , ς ⋆ ) ∈ H ⋆ × S ⋆ . The identifier (14) is thu s com posed o f the two su bsystems ξ and ς . The dynamics and output maps ( ϕ , γ ) of ς are designed to fulfil the id entifier requiremen t. When dri ven by the “ideal” input pair ( ˙ η ⋆ d , η ⋆ ) , the subsystem ς is supposed to ha ve an attractive steady-state solution ς ⋆ along which its o utput θ ⋆ leads to the best m odel in the model set M accordin g to (13). In ad dition, a r obustness property , given in terms of in p ut-to- state stability with resp e ct to the ad ditiv e inputs ( δ ′ , δ ′′ ) , is required . T his ad ditional prop erty is needed since ( ˙ η ⋆ d , η ⋆ ) is not av ailab le for f eedback and in (14) th e system ς is instead driven by th e input ( ξ 2 , η ) , th e latter p laying th e role of a “proxy ” f or ( ˙ η ⋆ d , η ⋆ ) . While it is clear that η car ries some informa tio n on η ⋆ , the fact th a t ξ 2 acts as a proxy of ˙ η ⋆ d follows by the de fin ition of ξ , wh ich is ind eed designed as a derivative observer o f the deriv ative ˙ η d of η d , providing the missing informa tio n on ˙ η ⋆ d . W e stress that the ability to construct an identifier satisfy ing the requireme n t as indic a ted above hid es the need of qual- itati ve and qua ntitativ e k n owledge on the ideal steady-state signals ˙ η ⋆ d , η ⋆ and ς ⋆ , as e vid ent f or instance in the definition of S ⋆ and H ⋆ . W e r e mark, howe ver, that this inform ation concern s h igh-level proper ties o f the class H ⋆ , such as a unifor m bound on its elem e n ts, and not the precise kn owledge of th e actu al ( ˙ η ⋆ d , η ⋆ ) . In Section III, pair ( ϕ, γ ) fulfilling the identifier requirement wh en the m odel ψ ( · , θ ) is linearly parametrised and (13) is a least squ are functional is presented. E. The asympto tic stability r esu lt The overall regulator r eads as follows ˙ η = Φ( η , γ ( ς )) + Ge ˙ ς = ϕ ( ς , ξ 2 , η ) ˙ ξ 1 = ξ 2 − m 1 ρ ( ξ 1 − η d ) ˙ ξ 2 = ˙ ψ ( ξ 2 , η , ς ) − m 2 ρ 2 ( ξ 1 − η d ) u = L  K χ χ + K ζ ζ + K η η 1 + K w ν ( x ⋆ , w )  (17) W e finally show that the d esign par a meters ( g , ℓ, κ, ρ ) can be ch osen so that th e closed -loop system has an asymptotic regulation error that is bou nded by a fu nction of the best attainable p rediction error . The result is pr e c isely fo rmulated in the following theo r em. Theorem 1. Suppo se that A1 a nd A2 ho ld, and consider the r egulator ( 1 7) constructed in th e pr evious sections with H ⋆ and ψ satisfying A3 and ( ϕ, γ ) fulfilling the iden tifi er r equ ir emen t r ela tive to H ⋆ and a cost functional (13) . Suppose mor eover that ( ˙ η ⋆ d , η ⋆ ) ∈ H ⋆ for all κ > 1 and ℓ > 1 . Then ther e exist c, ρ ⋆ , g ⋆ ( ρ ) , κ ⋆ ( g ) , ℓ ⋆ ( g , κ ) > 0 such tha t, fo r all ρ ≥ ρ ⋆ , g ≥ g ⋆ ( ρ ) , κ ≥ κ ⋆ ( g ) an d ℓ ≥ ℓ ⋆ ( g , κ ) , every solution of the closed- loop system (4) , (1 7) satisfies lim sup t →∞ | e ( t ) | ≤ c g d lim sup t →∞ | ε ⋆ ( t, θ ⋆ ( t )) | , with c no t depen dent on the co ntr ol p arameters. Theorem 1 is proved in the Appendix . Its claim is an ap- pr oximate regulation re su lt, whic h beco mes asympto tic whe n - ev er ε ⋆ ( t, θ ⋆ ( t )) = 0 . This, in turn, hap pens when a “real” model exists and b elongs to the chosen model set M . As Assumption A3 and the iden tifier requir ement imply that ε ⋆ can be bounded uniformly in the con trol parameters, the c la im of the theor em is also a practical regu lation result, with the bound on the regu latio n err o r that ca n be r educed arbitrarily b y increasing g . Finally , we re mark that, if a “ saturated version” of ψ is implemented in th e internal model u nit (6) in place of ψ (for instan ce by satu rating ψ on H ⋆ × γ ( S ⋆ ) in th e same way as it is done in ( 15) for ˙ ψ ), and if ( ˙ η ⋆ d , η ⋆ ) is b ounde d unifor m ly in th e co ntrol p a rameters (wh ich is always true in the square case as remarked in Sectio n II-C), th en a practical regulation r esult is still preserved 3 also in the case in which ( ˙ η ⋆ d , η ⋆ ) / ∈ H ⋆ , thus paralleling the “can onical” pre-processing results (see e . g. [4,22]) . In this case, howev er , the asymptotic bound on e ( t ) cann ot be related to ε ⋆ any mor e . I I I . C O N T I N U O U S - T I M E L E A S T S Q U A R E S I D E N T I FI E R S W e dev elop here an examp le of a pair ( ϕ, γ ) that fulfils the identifier requ irement when the model ψ ( · , θ ) is a finite linea r combinatio n of kno wn functions of th e fo rm 4 ψ ( · , θ ) = n θ X i =1 θ i σ i ( · ) , (18) 3 This can be deduced by the proof of Theorem 1 by neglect ing the identi fier’ s dynamics and by noticing that, in (21), ˜ ψ ( ˜ η , ˜ ς , η ⋆ , ς ⋆ ) − ε ⋆ = ψ ( η , γ ( ς )) − ˙ η ⋆ d can be bounde d uniformly in ς . 4 For ease of exposition we present here the case in which n e = 1 , with the remark that an identifier of the same ki nd for n e > 1 can be always obtained as the compositi on of n e single-v ariable ide ntifiers. in which n θ ∈ N is arbitra ry a n d σ i : R d → R are known Lipschitz and bound ed fu nctions. In this case the mo del set M is the family of f unctions of th e form σ ( · ) ⊤ θ , having defined σ ( · ) := col( σ 1 ( · ) , . . . , σ n θ ( · )) an d θ := col( θ 1 , . . . , θ n θ ) . W e associate with M the following cost f unctional, obtained by letting in (13) c ε ( t, s, · ) := λ e x p( − λ ( t − s )) | · | 2 and c r ( θ ) := θ ⊤ Γ θ , with λ > 0 an d Γ ∈ R n θ × n θ symmetric and po sitive semi-definite J ( ˙ η ⋆ d ,η ⋆ ) ( t )( θ ) = λ Z t 0 e − λ ( t − s )   ε ⋆ ( s, θ )   2 ds + θ ⊤ Γ θ (19) in which th e pred iction error ( 12) a t time s reads as ε ⋆ ( s, θ ) := ˙ η ⋆ d ( s ) − σ ( η ⋆ ( s )) ⊤ θ . The optimisation proble m associated with (19) is recogn ised to b e a (weighted ) lea st squar e s problem with regularisation, in which λ and Γ play the role of the forgetting factor and th e regulariser respectively . Namely , except f or the regularisation term, minimising ( 1 9) means minim ising a weighted squared “norm” of th e prediction erro rs associated with all the past data. W ith S P n θ the spa c e of sym metric positi ve semi-definite matrices in R n θ × n θ , we let Z ς := S P n θ × R n θ and, by partitioning the state as ς = ( ς 1 , ς 2 ) , with ς 1 ∈ S P n θ and ς 2 ∈ R n θ , we equip Z ς with the norm | ς | := | ς 1 | + | ς 2 | . W e thus construct a p air ( ϕ, γ ) satisfying th e id e ntifier req u irement relativ e to (19) as fo llows ˙ ς 1 = − λς 1 + λσ ( η ) σ ( η ) ⊤ ˙ ς 2 = − λς 2 + λσ ( η ) ξ 2 θ = ( ς 1 + Γ) − 1 ς 2 , ς ∈ Z ς (20) The claim is form alized by the fo llowing proposition. Proposition 1. W ith c > 0 arbitrary , let H ⋆ be a class of locally integr able fu nctions ( ˙ η ⋆ d , η ⋆ ) : R ≥ 0 → R × R d satisfying | ( ˙ η ⋆ d , η ⋆ ) | ∞ ≤ c . Then, if Γ > 0 , the pair ( ϕ, γ ) constructed in (20) satisfies the identifier requir ement lo c a lly 5 r elative to H ⋆ and the least-sq uar es functional (18) with β ς ( s, t ) = s exp( − λt ) . Pr oof. As σ is Lipschitz an d bo unded , th e n ϕ ( ς , ξ 2 , η ) := ( − λς 1 + λσ ( η ) σ ( η ) ⊤ , − λς 2 + λσ ( η ) ξ 2 ) is locally Lipschitz. Pick an eigenvalue ǫ ( t ) of ς 1 ( t ) + Γ , an d let v ( t ) 6 = 0 be a correspo nding eigenvector . Then v ( t ) ⊤ ( ς 1 ( t ) + Γ) v ( t ) = ǫ ( t ) | v ( t ) | 2 , and since Γ > 0 a n d ς 1 ( t ) ∈ S P n θ , this implies ǫ ( t ) ≥ p , with p > 0 the smallest eigen value of Γ . Thu s ς 1 + Γ is in vertible an d the singular values of ( ς 1 + Γ) − 1 are boun d ed by p − 1 , which implies that γ ( ς ) := ( ς 1 + Γ) − 1 ς 2 is lo cally Lipschitz, smooth in ς an d, as a co nsequenc e , its deriv a tive is locally Lip schitz. Pick now ξ 2 = ˙ η ⋆ d + δ ′ and η = η ⋆ + δ ′′ , with ( ˙ η ⋆ d , η ⋆ ) ∈ H ⋆ and ( δ ′ , δ ′′ ) loca lly integrable. Forward com pleteness follows 5 The word ”locally” in the cla im of the propositi on refers to the fact that γ is only prov ed to be locally L ipschitz , and not globall y Lipschitz as requested by the identifie r require ment. Ne verthel ess, we remark that a glo bally L ipschit z γ can be simply obtained by saturating the expression in (20) on the compact set in which ς is supposed to range, the latter that can be inferred by the kno wledge of the bound c on ( ˙ η ⋆ d , η ⋆ ) , as specifie d in the proof. Detai ls are omitted for reason of space . by noticing that (2 0) is a stable linear system driven by the locally in tegrable input ( σ ( η ) σ ( η ) ⊤ , σ ( η ) ξ 2 ) an d that, as σ ( η ) σ ( η ) ⊤ ∈ S P n θ , th en S P n θ is forward inv ar iant for ς 1 . W ith Σ( η ⋆ , δ ′′ ) := σ ( η ⋆ + δ ′′ ) σ ( η ⋆ + δ ′′ ) ⊤ and π ( η ⋆ , ˙ η ⋆ d , δ ′ , δ ′′ ) := σ ( η ⋆ + δ ′′ )( ˙ η ⋆ d + δ ′ ) , define ς ⋆ 1 ( t ) := λ Z t 0 e − λ ( t − s ) Σ( η ⋆ ( s ) , 0) ds ς ⋆ 2 ( t ) := λ Z t 0 e − λ ( t − s ) π ( η ⋆ ( s ) , ˙ η ⋆ d ( s ) , 0 , 0) ds, and let ς ⋆ = ( ς ⋆ 1 , ς ⋆ 2 ) . If | ( ˙ η ⋆ d , η ⋆ ) | ≤ c f or some c > 0 , then clearly there exists c ′ > 0 such that ς ⋆ ( t ) ∈ S ⋆ := { ς ∈ Z ς : | ς | ≤ c ′ } . Furtherm ore, since σ is Lipschitz and bou nded, the r e exists l σ > 0 (possibly depending on c ) such that | Σ( η ⋆ , δ ′′ ) − Σ( η ⋆ , 0) | ≤ l σ | δ ′′ | and | π ( η ⋆ , ˙ η ⋆ d , δ ′ , δ ′′ ) − π ( η ⋆ , ˙ η ⋆ d , 0 , 0) | ≤ l σ | ( δ ′ , δ ′′ ) | for all ( δ ′ , δ ′′ ) ∈ R × R d . Hence, b y integration of (20), an d using ς ⋆ 1 (0) = 0 , we obtain | ς 1 ( t ) − ς ⋆ 1 ( t ) | ≤ e − λt | ς 1 (0) − ς ⋆ 1 (0) | + l σ | ( δ ′ , δ ′′ ) | [0 ,t ) , and a similar bou nd holds fo r | ς 2 ( t ) − ς ⋆ 2 ( t ) | , thus implying the first item of the iden tifier requiremen t with β ς ( s, t ) = s exp( − λt ) and with α ς = 2 l σ . For fixed t ∈ R ≥ 0 , differ- entiating (19) with respect to θ yields D θ J ( ˙ η ⋆ d ,η ⋆ ) ( t )( θ ) = 2(( ς ⋆ 1 ( t ) + Γ) θ − ς ⋆ 2 ( t )) . Since, ϑ ◦ ( ˙ η ⋆ d ,η ⋆ ) ( t ) := { θ ∈ R n θ : D θ J ( ˙ η ⋆ d ,η ⋆ ) ( t )( θ ) = 0 } , then θ ⋆ ( t ) = ( ς ⋆ 1 ( t ) + Γ) − 1 ς ⋆ 2 ( t ) ∈ ϑ ◦ ( ˙ η ⋆ d ,η ⋆ ) ( t ) , which is the seco nd item of the req uirement, thus conclud in g the proo f. W e observe that the r egularisation matrix Γ > 0 plays a fundam ental role in Proposition 1, as it en sures that ς 1 + Γ is un iformly non sin g ular . Howe ver, its presence fru strates the possibility of ha ving asymptotic regulatio n also when the “right” internal model belongs to the model set (18). As evident in (19), indeed , h aving Γ ≥ 0 mean s that, even if θ annihilates the pred iction err or ε ⋆ , and thu s the integral term of (19), it also produces a positive add end θ ⊤ Γ θ , thus possibly making such θ a non-station ary point o f J ( ˙ η ⋆ d ,η ⋆ ) ( t ) . In this case, θ app roaches a neigh bourh ood of θ ⋆ ( t ) of a size that depend s on th e m aximum eig en value of Γ that, howe ver, can be taken as small as d esired. Nevertheless, Γ can be chosen positive semi-d efinite (and possibly zero) . In this case, (7) can still be u sed by substitutin g the inv erse o perator with a pseu d o- in verse (indeed σ 1 + Γ needs not be inv ertible in this case), and the claim o f Propo sition 1 applies only if th e minimum non-ze r o singu la r value o f ς 1 + Γ is boun ded away from zero unifor m ly in t , w h ich can be seen as a pe rsistence of excitation condition . W e also remark that, in this case, the Lipschitz constant of γ and its deriv ative becomes d ependen t on ho w large is the minim um non-zero singu lar value of ς 1 + Γ , thus making the result o f Theorem 1 ob tained for a cer tain value of the gains ρ , g , κ and ℓ , app licable only to the solutions carrying sufficient excitation. I V . E X A M P L E : C O N T RO L O F T H E V T O L Consider th e lateral (p 1 , p 2 ) and angular (p 3 , p 4 ) dynamics of a VTOL airc raft descr ib ed by [23] ˙ p 1 = p 2 ˙ p 2 = d( w ) −  tan p 3 + v ˙ p 3 = p 4 ˙ p 4 = B u with  > 0 th e gravitational constant an d B = 2 LJ − 1 > 0 , with L > 0 th e leng th of the wings and J the m oment of inertia ( typically uncertain) . T he input u is the for ce on the wingtips, v is a vanishing in put taking into acco unt the (controlled) vertical dynam ic s (not co nsidered here) and d( w ) := M − 1 d 0 ( w ) , with d 0 ( w ) th e later al win d for ce disturbanc e , and M > 0 the VTOL m ass. Th e co ntrol goal is to eliminate the win d action f r om the later al p o sition dy namics, i.e. the regulation error is defined a s e ( t ) = p 1 ( t ) . W e also suppose to ha ve available for feedba ck th e entire s tate, namely y = p . Let w b e generated by an exosystem of the form (2) and chang e variables as p 7→ x := ( χ, ζ ) , with χ := (p 1 , p 2 , −  tan p 3 + d( w )) and ζ := L s d( w ) −  p 4 / (cos p 3 ) 2 . In the n ew coor dinates the f ollowing equations hold ˙ χ 1 = χ 2 ˙ χ 2 = χ 3 ˙ χ 3 = ζ ˙ ζ = q ( w, x ) + Ω( w , x ) u, in whic h 6 Ω( w, x ) := − B / (co s(tan − 1 (d( w ) − χ 3 ) / ))) 2 and q ( w , x ) pr o perly defin ed. Th is system is in the form (4), with A1 tr ivially fulfilled ( x 0 being absent) b y x ⋆ = 0 and u ⋆ = B − 1 ( L 2 s d( w ) − 2d(w) 2 L s d(w)) / (d(w) 2 +  2 ) , and A2 fulfilled o n each co mpact set with L a negative number 7 . W ith ( c 1 , c 2 , c 3 ) the coef ficients o f a Hurwitz po lynomial and κ, ℓ > 0 desig n parameter s, we fix the con trol law as u = −L  c 1 ℓκ 3 (p 1 + η 1 ) + c 2 ℓκ 2 p 2 + c 3 ℓκ ( −  tan p 3 ) + ℓ ( −  p 4 / cos 2 p 3 )  , with η 1 the first state o f the intern al mod el fixed later . In th e new coordinates ( χ, ζ ) , this co ntrol law is o f the fo rm (8), with K w = ℓ ( c 3 κ 1) and ν ( x ⋆ , w ) = c o l(d( w ) , L s d( w )) . Regarding the design of the in ternal model unit, we observe that, by following Section II -C, Υ ( ℓ,κ ) ( w ) = Q ( ℓ, κ ) D ( w ) , in which Q ( ℓ, κ ) := ( c 3 / ( c 1 κ 2 ) 1 / ( c 1 κ 3 ) 1 / ( c 1 ℓ L κ 3 )) and D ( w ) = co l(d( w ) , L s d( w ) , − Ω( w , 0) − 1 q ( w, 0)) . Thus, η ⋆ i = Q ( ℓ, κ ) L i − 1 s D ( w ) , i = 1 , . . . , d , and ˙ η ⋆ d = Q ( ℓ, κ ) L d s D ( w ) . The form of Q and the fact κ and ℓ have large values show that th e dominan t elements in η ⋆ i and ˙ η ⋆ d are L i − 1 s d( w ) and L d s d( w ) , regar d less the value of the dimension d of η . Now , suppose that d( w ) consists of a sing le harmo nic at a n u nknown frequen cy . The design o f ( d, ψ ) and th e iden tifier to reject d( w ) is then carr ied o ut by consider ing a single oscillator a s the model set, obta in ed with d = 2 , n θ = 2 , and ψ ( η , θ ) := θ ⊤ η . The adaptation phase, in turn, can be set up by using the least-squares iden tifier p r esented in Sec tio n I II with n θ = 2 and σ any bo unded func tio n satisfying σ ( η ) = η in the region where co l(d( w ) , L s d( w )) · c 3 /c 1 κ 2 is supp osed to rang e . V . C O N C L U S I O N S The pap er presented a post-processing design procedure for a class o f m u ltiv ariab le nonlinear systems stabilisable by hig h- gain fee d back h inging on a “no n-equ ilibrium” fram ew o rk. The internal mod el is adapti ve with the adaptation mech anisms c ast 6 Recal l that cos(tan − 1 ( s )) = 1 / √ s 2 + 1 . 7 In this respec t, we observ e that the ideal s teady- state v alue of the measurements (p 3 , p 4 ) is gi ven by (p ⋆ 3 , p ⋆ 4 ) := (tan − 1 (d( w ) / ) , L s d( w ) / (d( w ) 2 + 1)) , an d th us y is not in gen eral v anishing at the steady state . as an iden tification pr oblem and with the asym p totic regulation error that is directly r e lated to the identification erro r . The framework does not rely on an exact knowledge of the steady state frien d, n or on an exact parametrisation of it. Rather , it assumes the knowledge of some q u alitativ e \ quantitative informa tio n about the class of steady state signals used to choose the mod el set of the underlying identification pro b lem. The pap e r fits in the researc h direction of [1,14] in which approx imate, ra th er than asymptotic, regulation is envisioned as the rig ht perspective in presen ce of general u ncertainties. A P P E N D I X A P R O O F O F T H E O R E M 1 W ith κ > 1 and ℓ > 1 , pick a solution ( x, χ, ζ , η , ς , ξ ) to the closed-loo p system ( 4), (17) an d let ( x ⋆ , u ⋆ , η ⋆ , ˙ η ⋆ d ) be gi ven by A1 and Section II-C. Assume that ( ˙ η ⋆ d , η ⋆ ) ∈ H ⋆ , an d let ( ς ⋆ , θ ⋆ ) b e p r oduced by the id entifier requirement. Consider the fo llowing chan ge of variables η 7→ ˜ η := η − η ⋆ ς 7→ ˜ ς := ς − ς ⋆ χ 7→ ˜ χ := χ + C ⊤ ˜ η 1 ζ 7→ ˜ ζ := ζ − K ( κ ) ˜ χ ξ 7→ ˜ ξ := ξ −  η ⋆ d ψ ( η ⋆ , θ ⋆ )  e 7→ ˜ e := e + ˜ η 1 , where we recall that K ( κ ) is d efined in (8) and ˜ η 1 ∈ R n e represents th e first n e compon ents of ˜ η . By d efinition o f η ⋆ , ˙ η ⋆ i = η ⋆ i +1 , a n d in the new coordin ates we ob tain ˙ ˜ η i = ˜ η i +1 − h i g i ˜ η 1 + g i h i ˜ e, i = 1 , . . . , d − 1 ˙ ˜ η d = − h d g d ˜ η 1 + ˜ ψ ( ˜ η , ˜ ς , η ⋆ , ς ⋆ ) + h d g d ˜ e − ε ⋆ . (21) with ε ⋆ = ε ⋆ ( t, θ ⋆ ) given by (12) and with ˜ ψ ( ˜ η , ˜ ς , η ⋆ , ς ⋆ ) := ψ ( ˜ η + η ⋆ , γ ( ˜ ς + ς ⋆ )) − ψ ( η ⋆ , γ ( ς ⋆ )) that, since A3 and the identifier requ irement imply that ψ and γ are Lipschitz, fulfils | ˜ ψ ( ˜ η , ˜ ς , η ⋆ , ς ⋆ ) | ≤ c ψ ,γ | ( ˜ η , ˜ ς ) | for some c ψ ,γ > 0 ind e penden t on the co n trol parameters. Thu s, standard high-gain a rguments (see e . g. [24]) show that there exist a 0 , a 1 , a 2 , a 3 > 0 an d g ⋆ 0 > 0 such that for all g ≥ g ⋆ 0 the fo llowing bou nd hold s | ˜ η i ( t ) | ≤ a 0 g i − 1 | ˜ η (0 ) | e − a 1 gt + a 2 g i − d − 1 | ( ˜ ς , ε ⋆ ) | [0 ,t ) + a 3 g i − 1 | ˜ e | [0 ,t ) (22) for all t ∈ R ≥ 0 and ea c h i = 1 , . . . , d . Moreover, ˜ ξ satisfies ˙ ˜ ξ 1 = ˜ ξ 2 − m 1 ρ ˜ ξ 1 + m 1 ρ ˜ η d − ε ⋆ ˙ ˜ ξ 2 = − m 2 ρ 2 ˜ ξ 1 + ˜ µ ( ˜ η , ˜ ς , ˜ ξ 2 , η ⋆ , ς ⋆ ) + m 2 ρ 2 ˜ η d , in which, since by A3 ( ˙ η ⋆ d , η ⋆ ) ∈ H ⋆ implies ( ˙ η ⋆ d ( t ) , η ⋆ ( t )) ∈ H ⋆ , and by the identifier requirement we h av e ς ⋆ ( t ) ∈ S ⋆ , in view of (1 5) ˜ µ read s as ˜ µ ( ˜ η , ˜ ς , ˜ ξ 2 , η ⋆ , ς ⋆ ) := ˙ ψ ( ˜ ξ 2 + ψ ( η ⋆ , θ ⋆ ) , ˜ η + η ⋆ , ˜ ς + ς ⋆ ) − ˙ ψ ( ˙ η ⋆ d , η ⋆ , ς ⋆ ) . Mo reover , in view of (16), there exists l ψ > 0 such tha t | ˜ µ ( ˜ η , ˜ ς , ˜ ξ 2 , η ⋆ , ς ⋆ ) | ≤ l ψ  | ( ˜ η , ˜ ς , ˜ ξ 2 ) | + | ε ⋆ |  . He nce, cu stomary high -gain arguments show that there exist ρ ⋆ 0 > 1 and a 4 , a 5 , a 6 > 0 such th at, for all ρ ≥ ρ ⋆ 0 , the following hold s | ˜ ξ ( t ) | ≤ a 4 ρ | ˜ ξ (0) | e − a 5 ρt + a 6  ρ | ˜ η | [0 ,t ) + ρ − 1 | ˜ ς | [0 ,t ) + | ε ⋆ | [0 ,t )  . (23) W e can write ξ 2 = ˙ η ⋆ d + δ ′ and η = η ⋆ + δ ′′ , with δ ′ := ˜ ξ 2 − ε ⋆ and δ ′′ := ˜ η , so that the identifier require m ent yields | ˜ ς ( t ) | ≤ β ς ( | ˜ ς (0) | , t ) + α ς | ( ˜ η , ˜ ξ , ε ⋆ ) | [0 ,t ) . (24) In view of stan d ard small-gain argu m ents (see e.g. [25]), the bound s (22), (23), (24) yield the existence o f β 1 ∈ KL , a 7 > 0 , ρ ⋆ ≥ ρ ⋆ 0 and g ⋆ ( ρ ) ≥ g ⋆ 0 such that, f o r all ρ > ρ ⋆ and g ≥ g ⋆ ( ρ ) , we have | ( ˜ η ( t ) , ˜ ς ( t ) , ˜ ξ ( t )) | ≤ β 1 ( | ( ˜ η (0) , ˜ ς (0) , ˜ ξ (0)) | , t ) + a 7  g d − 1 | ˜ e | [0 ,t ) + | ε ⋆ | [0 ,t )  | ˜ η i ( t ) | ≤ β 1 ( | ( ˜ η (0) , ˜ ς (0) , ˜ ξ (0)) | , t ) + a 7  g i − 1 | ˜ e | [0 ,t ) + g i − 1 − d | ε ⋆ | [0 ,t )  . (25) By n oticing that ˜ e = C ˜ χ , d ifferentiating ˜ χ yield s ˙ ˜ χ = ( F + H K ( κ ) + g h 1 C ⊤ C ) ˜ χ + H ˜ ζ + C ⊤ ( ˜ η 2 − g h 1 ˜ η 1 ) , so that, in vie w of (9), quite standard hig h-gain argum ents (see e.g. [20]) sh ow that th ere exists κ ⋆ 0 ( g ) > 1 such that, for a ll κ > κ ⋆ 0 ( g ) the fo llowing hold | ˜ χ ( t ) | ≤ a 9 ( κ ) | ˜ χ (0) | e − a 10 κt + a 11 κ | ˜ ζ | [0 ,t ) + a 12 ( κ )  g | ˜ η 1 | [0 ,t ) + | ˜ η 2 | [0 ,t )  | ˜ e ( t ) | ≤ a 9 ( κ ) | ˜ χ (0) | e − a 10 κt + a 11 κ | ˜ ζ | [0 ,t ) + a 13 κ  g | ˜ η 1 | [0 ,t ) + | ˜ η 2 | [0 ,t )  (26) for some a 9 ( κ ) , a 10 , a 11 , a 12 ( κ ) , a 13 > 0 . Furthermore, in the new coordin ates, the control law (8) becomes u = − ℓ L ˜ ζ − L  Ω( w, x ⋆ ) L  − 1 q ( w, x ⋆ ) , an d differentiating ˜ ζ yields ˙ ˜ ζ = δ ( ˜ η , ˜ χ, ˜ ζ ) + ˜ φ ( w, x, x ⋆ ) − ℓ Ω( w , x ) L ˜ ζ (27) with δ ( ˜ η , ˜ χ, ˜ ζ ) := − K ( κ )(( F + H K ( κ ) + g h 1 C ⊤ C ) ˜ χ + H ˜ ζ + C ⊤ ( ˜ η 2 − h 1 g ˜ η 1 )) that satisfies | δ ( ˜ η , ˜ χ, ˜ ζ ) | ≤ a 14 ( κ, g ) | ( ˜ η , ˜ χ, ˜ ζ ) | , fo r some a 14 ( g , κ ) > 0 , and with ˜ φ ( w, x, x ⋆ ) := Ω( w , x ) L ((Ω( w, x ) L ) − 1 q ( w, x ) − (Ω( w, x ⋆ ) L ) − 1 q ( w, x ⋆ )) that, in view of A2 an d since | χ | ≤ | ˜ χ | + | ˜ η 1 | and | ζ | ≤ | ˜ ζ | + | K ( κ ) ˜ χ | , satisfies | ˜ φ ( w, x, x ⋆ ) | ≤ a 15 ( κ ) | ( ˜ x 0 , ˜ χ, ˜ ζ , ˜ η 1 ) | , fo r some a 15 ( κ ) > 0 and with ˜ x 0 := x 0 − x ⋆ 0 . Hence, usua l high- gain arguments show that, under A2, there exists an ℓ ⋆ 0 ( κ, g ) > 0 such th at, for all ℓ > ℓ ⋆ 0 ( κ, g ) the fo llowing bound ho lds | ˜ ζ ( t ) | ≤ a 16 | ˜ ζ (0) | e − a 17 ℓt + a 18 ( κ ) ℓ | ˜ χ | [0 ,t ) + a 19 ℓ | ˜ x 0 | [0 ,t ) + a 20 ( κ ) ℓ  g | ˜ η 1 | [0 ,t ) + | ˜ η 2 | [0 ,t )  (28) for som e a 16 , a 17 , a 18 ( κ ) , a 19 , a 20 ( κ ) > 0 . Furtherm ore, by noticing that | χ | ≤ | ˜ χ | + | ˜ η 1 | and | ζ | ≤ | ˜ ζ | + | K ( κ ) ˜ χ | , A1 yields the existence of b 2 , b 3 ( κ ) > 0 such th at | ˜ x 0 ( t ) | ≤ β 0 ( | ˜ x 0 (0) | , t ) + b 2 | ( ˜ η 1 , ˜ ζ ) | [0 ,t ) + b 3 ( κ ) | ˜ χ | [0 ,t ) . (29) Therefo re, in vie w of (25), (26), (2 8) and (29), re p eating the small-gain argu ments of [25] yields the existence of a κ ⋆ ( g ) ≥ κ ⋆ 0 ( g ) and o f an ℓ ⋆ ( κ, g ) ≥ ℓ ⋆ 0 ( κ, g ) such th at, f o r each ρ > ρ ⋆ , g ≥ g ⋆ ( ρ ) , κ ≥ κ ⋆ ( g ) , and ℓ ≥ ℓ ⋆ ( g , κ ) , it holds th at | ( ˜ x ( t ) , ˜ η ( t ) , ˜ ς ( t ) , ˜ ξ ( t )) | ≤ β ( | ( ˜ x (0 ) , ˜ η (0) , ˜ ς (0) , ˜ ξ (0)) | , t ) + p 1 ( κ, g ) | ε ⋆ | [0 ,t ) and lim sup t →∞ | ˜ e ( t ) | ≤ p 2 κ − 1 g 1 − d lim sup t →∞ | ε ⋆ ( t ) | lim sup t →∞ | ˜ η 1 ( t ) | ≤ p 3 g − d lim sup t →∞ | ε ⋆ ( t ) | for som e β ∈ K L a n d p 1 ( κ, g ) , p 2 , p 3 > 0 , and the result follows with c = p 2 + p 3 by noticing th at | e | ≤ | ˜ e | + | ˜ η 1 | and that, since κ ⋆ ( g ) c a n be taken to be larger than g , then κ − 1 ≤ g − 1 . R E F E R E N C E S [1] M. Bin, D. Astolfi, L. Marconi , and L. 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