Integral Action in Output Feedback for multi-input multi-output nonlinear systems

Transcription

1 1 Integra Action in Output Feedback for muti-input muti-output noninear systems Daniee Astofi and Laurent Pray arxiv: v1 [cs.sy] 3 Aug 015 Abstract We address a particuar probem of output reguation for muti-input muti-output noninear systems. Specificay we are interested in making the stabiity of an equiibrium point and the reguation to zero of an output robust to sma unmodeed discrepancies between design mode and actua system in particuar those introducing an offset. We propose a nove procedure which is intended to be reevant to rea ife systems as iustrated by a non academic exampe. Index Terms Robust reguation noninear contro output feedback semi-goba stabiization integra action observabiity high-gain observer forwarding non-minimum phase systems uncertain dynamic system. I. INTRODUCTION For a controed dynamica system it is of prime importance in rea word appications to be abe to design an output feedback contro aw which achieves asymptotic reguation of a given output whie keeping the soutions in some prescribed set in presence of constant uncertainties. We refer to this as the probem of robust output reguation by output feedback. The probem has been competey soved in the inear framework by Francis and Wonham in the 70 s see [44]. Important efforts have been done in order to extend this resut to the noninear case see for instance [9] [1] and many different soutions have been proposed see among others [10] [14] [3] [34] [4 Chapter 7.] [1] [18] [4] [1] [37] [8]. Nevertheess we are sti far from having a compete soution to the probem of output reguation in the noninear muti-input-muti-output framework simiar to what we have in the inear case. Indeed most of the works require a good knowedge of the effects of the disturbances on the system or they rey on structura properties as for exampe norma forms minimum phase assumption matched uncertainties or reative degree uniform in the disturbances. In particuar for singe-input singe-output minimum-phase noninear systems which possess a we defined reative degree preserved under the effect of disturbances a compete soution has been given in [4] further improved to the output feedback case in [36]. Under the same assumptions this work has been successfuy extended in [37] to square muti-input muti-output systems for which the notion of reative degree indices and observabiity indices coincides. Further with the technique of the auxiiary system introduced in [0] the minimum-phase assumption has been removed in [8] aowing the zero-dynamics to be unstabe. However as far as we know a genera soution is sti unknown when these structura properties do not hod. D. Astofi is with CASY-DEI University of Boogna Boogna 4013 Itay and with the MINES ParisTech PSL Research University CAS - Centre automatique et systémes Paris France e-mai: daniee.astofi@unibo.it. L. Pray is with the MINES ParisTech PSL Research University CAS - Centre automatique et systémes Paris France e-mai: Laurent.Pray@mines-paristech.fr. The approach to noninear output reguation foowed in this paper is motivated by the inear context deveoped in its fu generaity in the miestone paper [1] that we find usefu to briefy reca here. Consider the inear system ẋ = A 0 x+b 0 u y = C 0 x y = yr = y e C0r x C 0e where the state x is in R n the contro u is in R m and the measured output y is in R p. The output y is decomposed as y = y r y e where y r in R r r m is the output to be reguated to zero without oss of generaity. When the system above is supposed to be ony an approximation of a process given by ẋ = Ax+Bu+Pw y = Cx+Qw y = yr = y e Cr C e x+ Qr Q e w where w is an unknown constant signa to be either rejected or tracked the we posed reguator probem with interna stabiity addressed by Wonham for inear systems as shown for instance in [44 Chapter 8] is that of finding an output feedback aw based on the mode such that for a tripets {ABC} cose enough to {A 0 B 0 C 0 } and for a matrices pairs {P Q} the reguation-stabiization probem is soved i.e. the system admits a stabe equiibrium point on which the output to be reguated is equa to zero. According to [9 Proposition 1.6] for exampe this probem is sovabe if and ony if the foowing 3 conditions are satisfied: a the pair A 0 C 0 is detectabe; b the pair A 0 B 0 is stabiizabe; A0 B c the matrix 0 is right invertibe. C 0r 0 Precisey under the above 3 conditions it is aways possibe to design an output feedback aw of the form ż = y r η = Fη +Ly u = Kη +Mz +Ny which soves the reguation probem provided F L K M and N are chosen such that the foowing matrix A+BNC BK BM LC F 0 C r 0 0 is Hurwitz for a tripets {A B C} cose enough to {A 0 B 0 C 0 } and for a matrices pairs {PQ} Note that in this inear framework no structura properties are needed. Merging a the toos in iterature that are at our disposa we try to recover the same resut as in the inear case asking for possiby minima assumptions but at the same time paying particuar attention to proposing a design truy workabe in

2 appications. For exampe minimaity impies not to ask for any specific structura properties whereas appicabiity forbids noninear changes of coordinates when no expression is known for their inverse. Our answer to the probem uses bricks which can be found in other pubications as [39] [33] [5] that we gue together. But for making this gueing process efficient we have to address some new specific probems. As in the inear framework we extend the system with an integra action. Then as in [33] we rey on forwarding to design a stabiizing state feedback for the extended system. Next for transforming this state feedback into an ouput feedback it is sufficient to appy the techniques which have been proposed for asymptotic stabiization by output feedback. A ot of effort has been devoted to this question and many resuts have accumuated see for instance the survey []. In particuar the transformation is done by repacing the actua state by a state estimate provided by a tunabe observer i.e. an observer whose dynamics can be made arbitrariy fast. Stabiity of the overa cosed oop system is estabished via the common separation principe [39] [6] and output reguation foows from the integra action embedded in the contro aw. The tunabe observer we propose is as in [5] previousy inspired by [11] and [9] a high-gain observer written in the origina coordinates and appropriate for our muti-input mutioutput possiby non-square case. We propose a new set of sufficient conditions which guarantees the existence of such an observer. As opposed to what we have found in the iterature see for instance [8] [17] [15] our conditions can be verified in the origina coordinates and they do not need the expicit knowedge of the inverse of noninear change of coordinates which may be very hard to find. Aso in ooking for minima assumptions we do not ask for goba observabiity or goba uniformity with respect to the inputs. The atter impacts the state feedback design and we show how to address this point in [33] ony a goba soution is proposed. Finay we show that the proposed soution guarantees robust reguation. Robustness is here with respect to unmodeed effects not in the system state dimension but in the approximations of the functions which define its dynamics and measurements. This has been done aready in [33] but for the state feedback case and with an assumption on the cosed oop system. Here we show that if the mode is cose enough in a C 1 sense in open oop to the process then output reguation is achieved by our output feedback design. However as opposed to the inear case where the resut is goba with respect to the magnitude of the disturbances an unfortunate consequence of being in our ess restrictive context is that we need the perturbations to be sma enough. In this work to simpify we restrict our attention to systems affine in the input. The extension to the non affine case is made possibe by considering the system contros as state and their derivatives as fictitious contros. See [5] for exampe. The paper is organized as foows. Section II is devoted to show the main assumption and resuts of this work. In Section III and IV we present respectivey the state feedback design and the observer design. The proofs of the main propositions are given in Section V. Finay in Section VI we iustrate the proposed design with a non-academic exampe inspired from a concrete case study in aeronautics the reguation of the fight path ange of a simpified ongitudina mode of a pane. Notations For a set S S denotes its interior S denotes its boundary and dxs denote the distance function of a point x to the set S. When S is a subset of A B whose points are denoted abs a denotes the set{a A : b B : ab S}. For a functionhand a vector fied f L f denotes the Lie derivative of h aong f given coordinates x L f hx = h xfx. To x any stricty positive rea number v we associate a saturation function sat v defined as a C 1 function bounded by v and satisfying sat v s = s if s v 1+ς 1 where ς is a sma stricty positive rea number. II. ROBUST REGULATION BY OUTPUT FEEDBACK A. Probem Statement and Assumptions For a process we have at our disposa the foowing dynamica mode ẋ = fx+gxu y = hx = h r xh e x where the state x is in R n the contro u is in R m the measured output y is in R p and the functions f : R n R n g : R n R nm and h : R n R p are smooth enough and f and h are zero at the origin. We investigate the probem of reguating at zero the part y r of the output y decomposed as y = y r y e with y r R r and r m and this whie stabiizing an equiibrium for x. But being aware that the tripet f g h gives ony an approximation of the dynamics of the process we woud ike the above reguation-stabiization property to hod not ony for this particuar tripet but aso for any other one in a neighborhood. The rea process is described by equations of the form ẋ = ξxu y = ζxu 3 where the functions ξ : R n R m R n and ζ : R n R m R p are assumed continuousy differentiabe C 1. These functions are unknown but we assume that they are cose enough to f + gu and h respectivey in the sense that the discrepancies and ξ x ξxu fx gxu + ζxu hx xu f ζ x x xu h g x x xu ξ u xu gx x x ζ u xu are sma enough as made precise ater on. Mimicking the 3 necessary and sufficient conditions for the inear case given in the introduction we consider the foowing sufficient assumptions that we discuss after their forma statement. Assumption 1 There exists an open set O of R n containing the origin and an open star-shaped subset U of R m with the origin as star-center such that for any stricty positive rea number ū and any compact subset C of O there exist an integer d a compact subset Ĉ of O a rea number U and a cass-k functionαsuch that for each each integerκ we can findc 1 functionsϑ κ : R m R p O O U κ : O O R 0

3 3 a continuous function L ϑκ : O R 0 and a stricty positive rea number σ κ such that: 1. for any function t ut with vaues in Uū defined as Uū = {u U : u ū} 4 and any bounded function t yt the set Ĉ is forward invariant by the fow generated by the foowing observer ˆx κ = ϑ κ yˆx κ u ; 5. xˆx O O U κ xˆx = 0 x = ˆx ; 3. σ d κ α x ˆx U κxˆx σ d κ U 6 x C ˆx Ĉ ; 4. im σ κ = + ; 7 κ U κ 5. x xˆx κ[fx+gxu] 8 + U κ xˆx κ ϑ κ hxˆx κ u σ κ U κ xˆx κ ˆx κ u Uū xˆx κ C Ĉ. 6. For a y a y b ˆx κ u in R p O Uū ϑ κ y a ˆx κ u ϑ κ y b ˆx κ u L ϑκ ˆx κ y a y b 9 Assumption There exist an open subset S of R n and a continuous function β : S U which is zero at the origin and such that the origin of with u = βx is an asymptoticay and ocay exponentiay stabe equiibrium point with S as domain of attraction. Assumption 3 The matrix 0 g0 is right invertibe. f x h r x Assumption 1 is aimed at being a counter-part of the detectabiity condition a. But we have to face here probems specific to this noninear framework: In our construction we sha rey on the so caed separation principe. For noninear systems see [39] for exampe it asks for an observer with a tunabiity property i.e. an observer the speed of convergence of which can be made arbitrary fast see [7]. This property is provided here by the famiy of observers 5 satisfying 6 8 and 7. Observabiity may depend on the input. This expains why we impose the contro to beong to the set U. The tuning of observers for non inear systems may depend on the oca Lipschitz constant of the non inearities. This expains why the famiy of observers depends on the bound ū of the input. On the other hand to reduce the restrictiveness Assumption 1 is imposed ony for system states beonging to an open subset O of R n. In Section IV we sha see how the famiy of observers in this assumption can be designed as observers based on high-gain techniques. Assumption is the counter-part of the stabiizabiity condition b and caims the existence of a state feedback aw which asymptoticay stabiizes the system. Actuay it assumes that a preiminary design step can be done. For it any too Lyapunov design feedback partia inearization passivity use of structure of uncertainties in combination with gain assignment techniques etc. can be expoited. However because Assumption 1 imposes the contro be in U we propagate this restriction here asking the stabiizing contro β to take vaues in that set. On the other hand we can cope with having an arbitrary domain of attraction S no need for it to be the fu space or any arbitrariy arge compact set. Finay Assumption 3 corresponds to the non-resonance condition c and states that the first order approximation at the origin of the system does not have any zero at 0. B. Main resuts Assumptions 1 to 3 are sufficient to guarantee the existence of an output feedback aw soving the reguation-stabiization probem for the mode. Proposition 1 Suppose Assumptions 1 and 3 hod. There exists an open subset SO of S O R r such that for any of its compact set C xz there exist an integer κ a compact subset Cˆx of O a rea numberµ and C 1 functionsk : R n R r R r and ψ sat : R n R r Uµ such that the origin of the mode in cosed-oop with the dynamic output feedback ż = kˆxh r x ˆx = ϑκ yˆxu u = ψ sat ˆxz 11 with κ κ is asymptoticay stabe with a domain of attraction A containing the set Cˆx C xz. Proof: See Section V-A. In the case where S and O are the fu space R n this resut woud be a semi-goba reguation-stabiity resut. It caims the existence of a dynamic output feedback which asymptoticay stabiizes the origin of the mode. Such a resut is not new per se. It is in ine with many resuts reated to the separation principe as those in [39] [6] or [19 Chapter 1.3]. But as written in the introduction we do not state ony existence but instead we propose an expicit and workabe design. We refer the reader to Section III for the definition 1 of the set SO the rea number µ and the functions k and ψ sat and to Section V-A for the definition of the integer κ and the set Cˆx. In the foowing propositions under the Assumptions 1 and 3 and knowing the resut of Proposition 1 hods we study the process 3 in cosed-oop with the contro aw 11 designed for the mode. Proposition Let C be an arbitrary compact subset of the domain of attraction A given by Proposition 1 which admits the equiibrium as an interior point and is forward invariant for the cosed-oop system 11. For any open neighborhood N C of the boundary set C contained in A there exists a 1 See respectivey 33 and 56 for SO 36 for µ 30 and 38 for k and 37 for ψ sat. See successivey 6 64.

4 4 stricty positive rea number δ such that for any pair ξζ of C 1 functions which satisfies ξxu [fx+gxu] + ζxu hx δ xu N C Uµ 1 x the cosed-oop system 3 11 has equiibria and at any such point the output y r is zero. Proof: See Section V-B. If the domain of attraction were the fu space this resut woud foow from [38 Section 1]. It says that when the evauation on a spherica she -ike set of the mode and process functions are cose enough equiibria where output reguation occurs do exist. If this coseness is everywhere in the domain of attraction then we have even a soution to the the we posed reguator probem with interna stabiity. Proposition 3 For any compact sets C and C the atter being forward invariant for the cosed-oop system 11 which satisfy {0} C C A and for any open neighborhood N C of C contained in A there exists a stricty positive rea number δ such that to any pair ξζ of C 1 functions which satisfies ξxu [fx+gxu] + ζxu hx δ xu C x Uµ 13 and ξ x xu ξ u xu f g ζ x xu ζ x+ xu gx x x u xu h x x 0 δ xu C x Uµ 14 we can associate a point Xe = x e z e ˆx e which is an exponentiay stabe equiibrium point of 3 11 whose basin of attraction B contains C. Moreover any soution ˆXXtXXtZXt of 3 11 with initia condition X in B satisfies im t + ζ r XXtψ sat ˆXXtZXt = Proof: See Section V-C. This statement is of the same spirit as those caiming that under the action of sma perturbations asymptotic stabiity is transformed into semigoba practica stabiity. But here we have more since we have existence of a singe equiibrium for which the reguated output is zero. And for this no specific structure of the unmodeed effects is required. III. STATE FEEDBACK DESIGN A. Adding an integra action: design of the function k To sove the probem of reguating y r to 0 we foow the very cassica idea of adding an integra action. To do so we first seect a C 1 function k : R n R r R r satisfying 3 for a x in R n and a y a ry b r in R r kxy r = 0 y r = 0 16 kxy a r kxy a r L k x y a r y b r 17 where L k : R n R 0 is a continuous function. Of course the function k can be simpy h r. But in its choice we can take advantage of the properties of the physica system under consideration and it can simpify the feedback design or its impementation. An exampe is given in section VI. For the time being note that smoothness of k and 16 impies k x x0 = 0 x Rn 18 and the existence 4 of a continuous function η : R 0 R 0 satisfying η0 = 0 and y r [1+ x + y r ]η kxy r xy r R n R p. 19 Actuay the function k used in the output feedback 11 is the modified version given ater in 39. B. Design the function ψ via forwarding Let us consider the extended system ẋ = fx+gxu ż = kxh r x. 0 With Assumption we are eft with modifying the given state feedback β to obtain a state feedback stabiizing asymptoticay the origin for the extended system 0. Fortunatey it has the so-caed feedforward form which has been extensivey studied in the 90 s with in particuar the introduction of the forwarding techniques based on saturations as in [40] or on Lyapunov design with coordinate change as in [30] or couping term as in []. We reca briefy these techniques. They differ on the avaiabe knowedge they require. Specificay Assumption has two consequences : 1. With the converse Lyapunov theorem of [7] we know there exists a C 1 function V : S R 0 which is positive definite and proper on S and such that the function x x x fx + gxβx is negative definite on S and upperbounded by a negative definite quadratic form of x in a neighborhood of the origin.. Since the origin of the system in cosed-oop with βx is ocay exponentiay stabe there exists see [30 Lemma IV.] a C 1 function H : S R r satisfying H x xfx+gxβx = kxh rx H0 = 0. 1 Depending on whether or not we know the function V and/or the function H or ony its first order approximation at the origin eads to different designs. a Forwarding with V and H known 3 When L gl i fhrx = 0 for i in {0...ρ} 16 can be reaxed in { } kxh rx = 0 L f h rx =... = L ρ 1 f h rx = 0 h rx = 0. See [37] for exampe. 4 The function η is a smoothened version of s sup xyr: kxy r s y r 1+ x + y r.

5 5 When both V and H are known a stabiizer ψ for the system 0 is ψx z = βx J x [ L g Vx z Hx L g Hx ] with H defined by 1 and with J : R n R m R m any continuous function satisfying for a x R n J v Jxv > 0 v 0 det v x Foowing [30] this can be estabished under Assumptions and 3 with the function V e : S R r R 0 defined as V e xz = Vx+ 1 z Hx z Hx. 4 Remark 1 If V is known from the design of β it may not be proper on S. To make it proper we first define v S as v S = inf x S Vx Vx and we repace Vx by v S V x. See [3]. Unfortunatey in doing so the domain of definition of this new function V may be a strict subset of S. In the foowing we sti ca S this domain on which V is proper. b Forwarding with V unknown but H known When V is unknown but H is known there exists a function γ : S R 0 with stricty positive vaues such that a state feedback for the system 0 is ψxz = βx+γxl g Hx Jxz Hx 5 with H defined by 1 and J : R n R r R r bounded and satisfying 3. This can be estabished with the Lyapunov function 4. c Forwarding with V unknown and H approximated Instead of soving the partia differentia equation 1 for H and using 5 we pick ψxz = βx+γxgx H 0 Jxz H 0 x 6 where H 0 is obtained as H 0 = k 00 h [ ] 1 r f y r x 0 x 0+g0 β x 0. 7 The corresponding Lyapunov function is V e xz = dvx z H 0x z H 0 x 1 where d : R 0 R 0 is a C 1 function with stricty positive derivative to be chosen arge enough see [30]. In the case where the system ẋ = fx+gxβx +v 8 with v as input is input to state stabe with restriction i.e. provided v is bounded by some given stricty positive rea number then foowing [40] the state feedback can be chosen as ψxz = βx+ǫj x g0 H0 z H 0x 9 ǫ with J : R x R m R m bounded and satisfying 3 and ǫ is a sma enough stricty positive rea number. Whatever design route a b or c we foow we obtain the foowing emma. Lemma 1 Under Assumptions and 3 the function V e is positive definite and proper on S R r. Its derivative aong the extended system 0 in cosed-oop with u = ψx z is negative definite on S R r and upperbounded by a negative definite quadratic form of x z in a neighborhood of the origin. Consequenty for the corresponding cosed oop system the origin is asymptoticay stabe with S R r as domain of attraction without forgetting Remark 1 and ocay exponentiay stabe. Proof: Since V is positive definite and proper on S V e is positive definite and proper on S R r. Aso the derivative of V e aong the soutions of the cosed oop system is negative definite in xψxz and upperbounded by a negative definite quadratic form of xψxz in a neighborhood of the origin see [30] [40] for exampe. With this to compete the proof it is sufficient to show the existence of a rea number c such that z c ψ0z. Since we have ψ0z = J0L g H0z respectivey = J0H 0 g0z where the function J satisfies 3 the above inequaity hods if L g H0 respectivey H 0 g0 is right invertibe. But by differentiating 1 which hods at east in a neighborhood of the origin using 18 and 7 and since f and β are zero at the origin we have H x 0 = H 0. Assume the matrixh 0 g0 is not right invertibe i.e. the exists a vector v in R r such that Then we have v H 0 v H 0 g0 = 0. v k y r 00 which contradicts Assumption 3. f x 0 g0 = 0 h r x 0 0 Remark Because the set U in Assumption 1 is star-shaped whie satisfying 3 the function J can aways be chosen such that that the function ψ above defined takes vaues in U. A drawback of the integra action is the possibe wind-up. To prevent this phenomenon in a the above ż can be modified in ż = kxy r +ω[sat z z +Hx z +Hx] 30 with Hx repaced by H 0 x when needed and where the saturation function is defined in 1 ω is any stricty positive rea number and z shoud be chosen arge enough to aow the z-dynamics to converge to the right equiibrium point. This modification does not change anything to the asymptotic stabiity which can be estabished with the same Lyapunov functions.

6 6 C. Definitions of SO and µ and saturation of ψ to get the function ψ sat If we were to design a state feedback we coud stop here. But the output feedback we design is based on the previous state feedback and augmented with an observer. Since the estimated state may make no sense during some transient periods we need a mechanism to prevent any bad cosedoop effects during these periods. As proposed in [5] we use saturation. First we define the set SO where we woud ike the state to remain. For this et S be given by Assumption maybe modified as expained in Remark 1 above. Simiary et O be given by Assumption 1 maybe modified ater as in 54. Let aso the functionv e positive definite and proper on S R r be given by the above design of the state feedback or a converse Lyapunov theorem [7] satisfying V e xz = e x xz[fx+gxψxz]+e z xzkxh rx = W e xz 31 where the function W e defined here is positive definite on S R r. Then if S is not a subset of O we et v be the rea number defined as v = inf V exz. 3 xz S R r \O R r If not et formay v be infinity. We define the open set 5 SO = {xz S R r : V e xz < v }. 33 This set in non empty since it contains the origin. In the same way to each rea number v in [0 v we associate the set Ω v = {xz S R r : V e xz v}. 34 It is a compact subset of SO. Aso from Lemma 1 it is forward invariant for the extended system 0 in cosed-oop with u = ψxz. On the other hand for any C xz compact subset of SO we can find rea numbers v 1 < v satisfying C xz Ω v1 Ω v SO. 35 Then with µ the rea number defined as µ = 1+ς max xz Ω v ψxz 36 with ς a sma number as in 1 we consider the subset Uµ U see 4. As U in Assumption 1 it is star-shaped with the origin as a star-center. Let then the function ψ sat : R n R r Uµ be ψ sat xz = sat µ ψxz. 37 It is bounded and Lipschitz and as ψ it is C 1 on a neighborhood of the origin. Simiary we modify the function k defined in 16 by saturating its argument x. Namey we repace where kxhx by ksat x xh r x 38 5 See the further modification 56 x = 1+ς max xz Ω v x. 39 IV. OBSERVER DESIGN In the Assumption 1 we ask for the knowedge of the famiy of observers 5. Fortunatey it can be obtained as a high gain observer. A ot of attention has been devoted to this type of observers and many resuts are avaiabe at east for the singe output case. See for exampe the survey [6] and the references therein. We are interested here in some specific aspects as a the possibiity of writing the dynamics of the observer in the origina coordinates; b the muti-output case; as far as we know at the time we write this text the study of tunabe observers in the muti-output case is far from being concusive. Ony some sufficient conditions are known see for instance [7] [6] [41] [17] [8] [15] [6]; c the fact that observabiity hods ony on O a possiby strict subset of the fu space R n. To introduce them we find usefu to start with a very brief reminder on singe output high gain observers. A. Reminder on high gain observers in the singe output case It is known see [15 Theorem 3.4.1] for exampe that for a singe-input singe-output system of the form ẋ = fx+gxu y = hx x R n uy R 40 which is observabe uniformy with respect to the input and is differentiay observabe of order n o there exists an injective immersion Φ : R n R no obtained as φ = Φx = hx L f hx L no 1 f hx 41 which puts the system 40 into the so caed observabiity trianguar norma form φ = A no φ+b no bφ+d no φu y = C no φ 4 where A no = 0no 1 1 I no 1 n o 1 B 0 0 no = 1 no 1 C no = no 1 0no D no φ= d 1 φ 1...d i φ 1...φ i...d no φ 43 and where b d i are ocay Lipschitz function. An observer for the system 40 is ˆφ = A noˆφ+bno bˆφ+d no ˆφu +K no L no y C noˆφ ˆx = Φ -inv ˆφ 44 wherek no is such that A no K no C no is HurwitzL no = diag... no and Φ -inv is any ocay Lipschitz eft inverse function of Φ satisfying Φ -inv Φx = x x R n. In the φ-coordinates it is a standard high gain observer the dynamics of which can be made arbitrary fast by increasing the high-gain parameter see for instance [8].

7 7 B. On the possibiity of writing the dynamic of the observer in the origina coordinates As aready observed in [9] a main issue in impementing the observer 44 is about the function Φ -inv for which we have typicay no anaytica expression meaning that we have to sove on-ine a minimization probem as ˆx = argmin x φx ˆφ. Fortunatey as noticed in [11] and proposed aso in [9] this difficuty can be rounded when Φ is a diffeomorphism. Indeed in this case φ is simpy another set of coordinates for x and the observer 44 can be simpy rewritten in the origina x coordinates as 1 Φ ˆx = fˆx+gˆxu+ x ˆx K n L n y hˆx. 45 As a consequence there is no need to find the inverse mapping of the function Φ but infinitey more simpy ony to invert the matrix Φ ˆx. But for Φ to be a diffeomorphism we need x n o to be equa to n i.e. to have the fu order observer to have the smaest possibe dimension. C. High gain observer in the muti-output case As shown in [41] in the muti-input muti-output case a typica expression for Φ is Φx = Φ 1 x Φ p x Φ i x = h i x L f h i x L pi f ix h 46 where h i is the i-th component of h and p i are integers caed the observabiity indexes and p i=1 p i n. The dynamics of system expressed in these coordinates is where φ = Aφ+B bφ+dφu y = Cφ 47 A = bckdiag A p1...a pp B = bckco B p1...b pp C = bckrow C p1...c pp bφ = b1 φ...b p φ Dφ = bckco D p1 φ...d pp φ where bφ and Dφ are ocay Lipschitz functions. However even when the system is observabe uniformy in the input the functions b and D may not have the trianguar structure we need for the design of a high-gain observer. Conditions under which we do get trianguar dependence for bφ and Dφ have been studied for instance in [8] and [17]. Going on aong this route and imposing Φ to be a diffeomorphism in order to write the observer in the origina coordinates as done in 45 an aternative condition under which we do have an appropriate structure is given by the foowing technica assumption for which we do not need to know the inverse of Φ. Assumption 4 There exist i an open set O R n containing the origin and a starshaped set U with the origin as star-center ii a C 1 function Φ : O R n iii sequences of matrices L R n n M R n n and N R p p a matrix C R p n iv matrix functions u U Ku R n p and u U Au R n n v and for any positive rea number ū there exist a positive definite symmetric matrix P R n n and stricty positive rea numbers ν and d such that O1 the function Φ is a diffeomorphism on the set O and Φ0 = 0 O CΦx = hx O3 the matrices Au Ku P C satisfy for any u Uū PAu KuC+Au KuC P νp AuL = L M Au N CL = C O4 the matrix M is such that M P 1 is symmetric and satisfies im λ minm P 1 = + + O5 λ max L M P 1 L λmin M P 1 d 1 λ min L M P 1 L λmin M P 1 d. Moreover for any compact set C and Ĉ satisfying C Ĉ O there exists a sequence of positive rea numbers c such that O6 im + c = 0 O7 the function B : R n m R n defined as BΦxu = L f Φx+L g Φxu AuΦx 48 satisfies for a x a C x b Ĉ and u Uū P 1 M 1 L 1 [BΦx a u BΦx b u] P 1 c L 1 [Φx a Φx b ]. 49 Remark 3 As shown in the next Lemma the existence of a high-gain observer for the system is guaranteed if Assumption 4 hods. In particuar the properties O1 O O3 O6 and O7 guarantee the existence of a converging observer in the origina coordinates whereas properties O4 and O5 assure its tunabiity property. We remark that these conditions can be checked without need of finding formay the inverse mappingφ 1. In particuar given a system and a candidate diffeomorphism Φ property O1 one can immediatey check properties O inear dependence of the diffeomorphism on the output Then if this properties hods one can fix the degrees of freedom Ku M N L P which propery defines the high-gain observer as shown ater in Lemma see 50 and check aso the Lipschitz condition 49 in 07. Finay

8 8 property O3 guarantees the convergence of the observer see proof of Lemma. The conditions of Assumption 4 are satisfied in the singeoutput case considered in Section IV-A when n o = n by choosing Φ as in 41 and picking L = diag 1... n 1 M = N = 1 Au = A n BΦxu = B n bφx+d n Φxu and C = C n where the tripet A n B n C n and the functions b D are given in 43. In this case we set L n = L M N and Ku = K n in the observer 44. In this assumption A is aowed to be input-dependent to aow a broader cass of noninear systems. For instance it can be verified that the system ẋ 1 = x ẋ = u y = x 1 +x +x can not be transformed in the form 4 but it satisfies Assumption 4. In some cases the noninear terms 48 can be disregarded in the high gain observer design usuay aso caed dirty derivative observer. This is possibe for exampe when the notions of observabiity indexes and reative degree indexes coincide see [37] among others. In this case these noninear terms act through their bound and not their Lipschitzness. Unfortunatey then a very specific structure is needed because otherwise the gain between these noninear terms and some estimation error is increasing with the observer gain. Here we intend to consider a broader cass of systems and thus we do need to have these terms present in the observer. Lemma Under Assumption 4 for any compact set C and Ĉ satisfying C Ĉ O the famiy of systems 1 Φ ˆx = fˆx +gˆx u+ x ˆx [ L M KuN y hˆx ] 50 indexed by in R >0 satisfies points to 6 of Assumption 1. Proof: We et φ = Φx ˆφ = Φˆx φ = φ ˆφ. 51 With 48 and 50 systems and 50 are transformed in φ = Auφ+Bφu ˆφ = Auˆφ +Bˆφ u+l M KuN Cφ ˆφ With Assumption 4 and the notations 51 we define the Lyapunov Function U xˆx = 1 φ ˆφ [L M P 1 L ] 1 φ ˆφ. As Φ it is defined on O O and it takes vaues in R 0. Aso because the matrix L M P 1 L is positive definite we have xˆx O O U xˆx = 0 x = ˆx. So point of Assumption 1 hods. Aso we get U xˆx = φ ˆφ [ L PM 1 L 1 ] 5 [ Au L M KuN Cφ ˆφ ] +Bφu Bˆφ u which with using O3 and 49 gives for a xˆx in C Ĉ U ν P 1 L 1 φ +c P 1 L 1 φ 1+ u. Then with O6 for any u ū there exists a such that for any U xˆx ν φ L Since we have PL 1 φ xˆx C Ĉ. 53 P λ min Pλ min M P 1 PM 1 we obtain for a xˆx in C Ĉ U xˆx νλ minpλ min M P 1 U xˆx. So with O4 points 4 and 5 of Assumption 1 hod when we choose the integer κ as the integer part of the ratio / and with σ κ = νλ minpλ min M P 1. Next we have U xˆxλ min L M P 1 L = φ L M P 1 L 1 φ λ max L M P 1 L 1 φ ˆφ φ L M P 1 L 1 φ λ min L M P 1 L 1 So with O5 we get φ ˆφ U xˆx λ max L M P 1 L U xˆxλ min M P 1 d Φx Φˆx U xˆxλ min M P 1 d. But because Φ is a diffeomorphism defined on O for any compact subsets C and Ĉ of O there exist rea numbers Φ and L Φ 1 independent of such that for a x in C and ˆx in Ĉ we have x ˆx = Φ 1 Φx Φ 1 Φˆx This gives L Φ 1 Φx Φˆx Φ. x ˆx 1 L Φ 1 λ min M P 1 d U xˆx Φ λ min M P 1 d. So with O4 point 3 of Assumption 1 hods. Finay point 6 of Assumption 1 hods too. Indeed by definition of the set Uū the matrices KuM N L and the diffeomorphism Φ there exists a positive definite function L ϑ ˆx such that 1 Φ x ˆx L M KuN L ϑ ˆx for any > 0 u Uū and ˆx Ĉ.

9 9 D. Taking care of observabiity restricted to O by an observer modification In the above we are missing point 1 of Assumption 1 namey Ĉ may not be forward invariant. The probem is that the observer 57 does not guarantee that ˆx remains in O and therefore that Φ x ˆx is invertibe. To round this probem as in [9] we modify this observer here not by projection but by considering a dummy measured output extending the resuts in [5]. To make our point cear we introduce the foowing assumption. Assumption 5 Given the set O and the diffeomorphism Φ of Assumption 4 for any compact subset C of O we know of a C 1 function h : O R 0 such that: H1. the set {x R n : h x < 1} is a subset of O; H. the function x hx is continuous on O; h x x H3. for any rea number s in [01] and any x 1 and x in O satisfying h x 1 s h x s we have h x s for a x which satisfies for some λ in [01] Φx = λφx 1 +1 λφx. This means nothing but the fact that for any s in [01] the image by Φ of the set {x R n : h x s} is convex; H4. the set O mod defined as O mod = {x R n : h x 0} 54 contains C and has a non empty interior which contains the origin; H5. the set Ĉ = { x R n } : h x 1 is compact. Remark 4 There is a systematic way to define this function h when given the compact set C we know a positive definite symmetric matrix Q and a rea number R satisfying ΦC {φ R n : φ Qφ R} ΦO. Indeed in this case we et be the number defined as = sup R. R:{φ:φ Qφ R} ΦO Since O is a neighborhood of the origin is stricty positive. Then we seect a rea number ǫ in 01 and et { Φx QΦx h x = max ǫ 0}. 55 With this choice and since Φ is a diffeomorphism we can check that Properties H1 to H5 are satisfied. We may disike the convexity property mentioned in H3 of Assumption 5. Unfortunatey it is in some sense necessary. Indeed our objective with the modification E is to preserve the high-gain paradigm. This means in particuar that we choose to keep an Eucidean distance in the image by Φ as a Lyapunov function for studying the error dynamics. Aso we need an infinite gain margin as defined in Definition.8 in [35] since the correction term must dominate a the other ones in the expression of ˆx when h becomes too arge. Then as proved in Lemma.7 [35] with such constraints the convexity assumption is necessary. This impies that if we want to remove the convexity assumption we have to find another cass of observers. We are interested in the function h because it satisfies the property h x = 0 x O mod. This eads us to introduce a dummy measured output y = h x. Indeed y is zero when x is in O mod. But O mod being a strict subset of O we have here a stronger constraint. To dea with this restriction we need to reduce the set SO by modifying its definition given in 33 into v = inf Vxz xz S R r \O mod R r SO = { } 56 xz S R r : Vxz < v. With Assumption 5 point 1 of Assumption 1 can be estabished via a modification of the observer. Lemma 3 Assume Assumptions 5 hods. Let Φ : O R n be a diffeomorphism ū be a positive rea number and t ut be a continuous function with vaues in Uū and t yt be a continuous bounded function. The set Ĉ given in H5 is forward invariant for any system in the famiy indexed by in R >0 ˆx = fˆx +gˆx u 57 1 Φ + x ˆx [ L M KuN y hˆx ] +Eˆx uy where the term E is defined as Eˆx uy = τ ˆx uy Φ Φ x ˆx L M P 1 L x ˆx h x ˆx h ˆx where τ is a C 1 function to be chosen arge see59. If a conditions of Assumption 4 hod and the mode state x remains in O mod then a the points of Assumption 1 are satisfied. Proof: First we observe that h x ˆx ˆx = Rˆx uy+ 1 τ ˆx M P 1 1 L Φ x ˆx h x ˆx h ˆx where we have et Rˆx uy = h x ˆx

10 10 [ fˆx + gˆx u+ 1 Φ x ˆx L M KuN [y hˆx ]]. This motivates us for choosing τ satisfying 8h ˆx Rˆx uy τ ˆx uy 1 Φ M P 1 1 L x ˆx h x ˆx 59 which can be computed on-ine. Thanks to H the function x τ x defined this way is continuous on O. So we can use τ as ong as ˆx is in O. { { It impies that h x is non positive when h ˆx is stricty arger than 1. With uniqueness of soutions this impies that for each s in [ 1 1] the set {ˆx : h ˆx s} is forward invariant and so is the compact set Ĉ in particuar. This says that point 1 of Assumption 1 hod. On the other hand the modification E augments U in 5 with 1 τ ˆx [Φˆx Φx] Φ x ˆx h x ˆx h ˆx. But when h x is zero which is the case when the mode state x remains in O mod and when h ˆx is in [01] the convexity property of h in H3 gives 0 [Φˆx Φx] Φ x ˆx 1 h x ˆx h ˆx. We concude that when a conditions of Assumption 4 hod 53 hods even with the modification E. Hence from the proof of Lemma points to 5 of Assumption 1 hod. Finay with 58 and 59 the function defined by the right hand side of 57 satisfies aso the point 6 of the Assumption 1. Remark 5 An important feature is that thanks to the additiona term E no other modification as saturation is needed. This modification in fact guarantees that the estimate state ˆx remains in a compact subset of O which depends on the choice of the parameters. A. Proof of Proposition 1 V. PROOFS OF PROPOSITIONS This proof foows the same ines as in [5] inspired by [19 Chapter 1.3] with no meaningfu no originaity. We give it ony for the sake of competeness. We introduce the notations x e = x z u e = ψ e xz = u1 ψxz x u f e x e u e = ψ sate ˆxz = fx+gxu1 ku h r x ψsat ˆxz. sat x ˆx where the functions ψ ψ sat and sat x are defined in Sections III-B and III-C. The cosed-oop system 11 can be compacty written as ẋ e = f e x e u e u e = ψ e ˆxz. We have a function V e positive definite and proper on S R r which aows us to define the compact sets Ω v1 and Ω v as in 34 and satisfying 35. Aso with 31 and 37 the function W e xz = V e xz = e x xz[fx+gxψ satxz]+ e z xzkxh rx is continuous and negative definite on Ω v. These properties impy the foowing: 1. there exists a positive rea number W and a continuous function α : R 0 [0W] such that e x e x e [f e x e ψ sate ˆxz f e x e ψ e xz] α ˆx x xz Ω v ˆx R n ;. there exists a stricty positive rea number W such that e x e f e x e ψ sat e ˆxz W x e ˆxxz : xz Ω v \Ω v1 ˆx x δ xw where δ xw is the stricty positive rea number defined as 1 δ xw = α 1 min W e xz. xz Ω v \Ω v1 By coecting this we obtain e x e x e f e x e ψ sate ˆxz W e x e +α ˆx x W xz Ω v ˆx R n 60 W xz Ω v \Ω v1 ˆx : ˆx x δ xw. 61 On another hand et C = Ω v x. From 34 it is a compact subset of O. With this set and µ defined in 36 we can invoke Assumption 1. It gives us in particuar the sequence σ κ the integer d the rea number U and the function α. This aows us to define the integer κ as the smaest one satisfying σ d κ exp σ κ v v 1 W U αδ xw κ κ. 6 From now on we fix κ arbitrariy but arger than κ. Assumption 1 gives us aso the functions ϑ κ and U κ and the forward invariant compact subset Ĉ of O. Then because U κ is continuous and satisfies the point of Assumption 1 there exists 6 a cass-k function α κ satisfying U κ xˆx α κ x ˆx xˆx C Ĉ. 63 From Ĉ we define Cˆx and Γ as the sets Cˆx = Ĉ Γ = Ĉ Ω v. 64 Cˆx is a compact subset of O and Γ is a compact subset of O Ω v. Since a the functions are Lipschitz on Γ the soutions of the cosed-oop system 11 are we defined and unique as ong as they are in the interior set Γ of Γ. Moreover their vaues satisfy inequaities 60 and 6. Aso since Cˆx = Ĉ is forward invariant the ˆx-component of this soution cannot reach the boundary of this set in finite time. Stabiity 6 α κ can be constructed from s max xˆx C Ĉ: x ˆx s Uxˆx.

11 11 Let N κ contained in Γ be an open neighborhood of the origin whose points ˆx x z satisfy V e x e + α α 1 σ d κ α κ x ˆx < v 1. Consider a soution of the cosed-oop system starting from an arbitrary point ˆxxz in N κ. Let [0T[ be its right maxima interva of definition when it takes its vaue in the open set Γ. To simpify the notation we add t to denote those variabes which are evauated aong this soution. With 6 and 63 we have Uxtˆxt α κ x0 ˆx0 t [0T[ α xt ˆxt σ d κuxtˆxt t [0T[. This impies xt ˆxt α 1 σ d κ α κ x0 ˆx0 t [0T[. 65 This inequaity and 60 where W e is non negative give V e xtzt V e x0z0 + αα 1 σ d κ α κ x0 ˆx0 < v 1 t [0T[. 66 Thus if the initia condition ˆx0x0z0 is in N κ the soution remains inside a strict subset of Γ. Hence T is infinite and from 65 and 66 we can concude that the origin is stabe. Attractiveness Consider now a soution of the cosed-oop system with initia conditionˆxxz in Cˆx Ω v1 which according to 35 contains Cˆx C xz. Let [0T[ be the right maxima interva of definition of this soution when it takes its vaues in Γ. With 8 6 and 60 we have for a t in [0T[ Uxtˆxt exp σ d κ t Ux0ˆx0 exp σ d κ t σ d κu 67 V e xtzt V e x0z0 + W t v 1 + W t. Since the ˆx-component of the soution cannot reach the boundary of Cˆx in finite time and since V e xtzt is smaer than v we must have T v v 1 W and v v 1 v v 1 V e x z v + v 1 < v. W W 68 Then because κ satisfies 6 67 and 6 give for a t in [ v v1 T[ W and therefore α xt ˆxt σ d κuxt ˆxt αδ xw xt ˆxt δ xw t [ v v1 T[. 69 W Then with 61 and 68 we obtain } max {V e xtzt v 1 } v v 1 v v 1 max {V e x z v 1 < v W W 70 for a t in [ v v1 W T[. Since Cˆx is forward invariant this estabishes that the soution cannot reach the boundary of Γ on [0T[. This impies that T is infinite and that the soution remains in Γ for a t in R 0. So inequaities 69 and 70 and therefore inequaities 60 6 and 8 hod for a t arger. With LaSae invariance principe we concude than v v1 W im V extzt+uxtˆxt = 0. t + and thus that the soution of the cosed-oop system converges to the origin provided its initia condition ˆx0 x0 z0 is in Cˆx Ω v1 Cˆx C xz. B. Proof of Proposition We denote X = xzˆx ϕ m X = fx+gxψ satˆxz kˆxh r x ϑ κ hxˆxψ sat ˆxz ξxψ sat ˆxz ϕ p X = kˆxζ r xψ sat ˆxz. ϑ κ ζxψ sat ˆxzˆxψ sat ˆxz 71 A first eementary remark is that if Xe = x e z e ˆx e is an equiibrium point of ϕ p then we have in particuar 0 = ż X=Xe = kˆx eh r x e. With 16 this impies h r x e is zero. To prove the existence of Xe we use [16 Theorem 8.] which says that a forward invariant set which is homeomorphic to the cosed unit ba of R n contains an equiibrium. The Proposition 1 gives us a forward invariant set which may not be homeomorphic to the cosed unit ba. So our next task is to buid another set satisfying the required properties. The equiibrium of Ẋ = ϕ m X 7 being asymptoticay attractive and interior to C which is forward invariant C is attractive. It is aso stabe due to the continuity of soutions with respect to initia conditions uniformy on compact time subsets of the domain of definition. So it is asymptoticay stabe with the same domain of attraction A as the equiibrium. It foows from [43 Theorem 3.] that there exist C functions V : A R 0 and V C : A R 0 which are proper on A and a cass K function α satisfying α X VX V0 = 0 αdxc V C X V C X = 0 X C X Xϕ mx VX X A C X Xϕ mx V C X X A. 73

12 1 SinceCis compact andn C is a neighborhood of its boundary there exists a stricty positive rea number d such that the set {X A: dxc 0d]} is a subset of N C. Then with the notations v C = sup VX X A: dxc d = αd v C and since α is of cass K we obtain the impications V C X+ VX=αd αdxc V C X αd dxc d VX v C. With our definition of this yieds aso αd VX = V C X 0 < αd V C X 0 < dxc d X N C \C. 74 On the other hand with the compact notation we have VX = V C X+ VX X Xϕ mx < VX X A. 75 A this impies that V is a Lyapunov Function for 7 on A in the sense of [4 Page 34] and that the subeve set {X A : VX αd} is contained in N C C. It foows from [4 Coroary.3] 7 that the eve set {X A : VX = αd} is homeomorphic to the unit sphere. But with the fact that the origin is asymptoticay stabe and the arguments used in the proof of [4 Theorem 1.] this impies that the subeve set {X A : VX αd} is homeomorphic to the cosed unit ba. Then since the set C = {X N C : dxc [0d]} is a compact subset of N C A the rea number G = sup X X 76 X C is we defined and stricty positive. We get for a X in C Xϕ px = Xϕ mx+ X X So if ϕ p satisfies VX+G sup X C ϕ p X ϕ m X inf X C VX G we have for a X in {X A : VX = αd} X Xϕ px 1 VX. X[ϕ px ϕ m X] X ϕ p X ϕ m X. X N C 77 This impies the compact subeve set {X : VX αd} is homeomorphic to the cosed unit ba and forward invariant 7 Thanks to the contribution of Freedman [13] and Pereman [31] the restriction on the dimension is not needed. for the system 81. With [16 Theorem 8.] we concude that this subeve set contains an equiibrium of this system. Finay from points 1 and 6 of Assumption 1 we know that even when the observer in 11 is fed with y = ζxu and not with hx it admits a forward invariant compact subset Ĉ of O. So with L = sup{l ϑκ ˆxL k ˆx} X Ĉ with L ϑκ given by 9 and L k ˆx given by 17 we have for a xzˆxu in R n R r Ĉ U ϕ p X ϕ m X ξxu [fx+gxu] + Lζxu hx. Hence 77 hods when 1 is satisfied with δ = C. Proof of Proposition 3 1 inf X C VX 1+L sup X C X. X We start with the foowing Lemma which combines tota stabiity and hyperboicity and is a variation of [33 Theorem 6]. Lemma 4 Let a C 1 function ϕ m : R n R n be given such that the origin is an exponentiay stabe equiibrium point of: Ẋ = ϕ m X 78 with A as domain of attraction. For any compact sets C andc the atter being forward invariant for the above system which satisfy {0} C C A there exists a stricty positive rea number δ such that for any C 1 function ϕ p : R n R n which satisfies: ϕ p X ϕ m X δ X C 79 ϕ p X ϕ m X X X δ X C 80 there exists an exponentiay stabe equiibrium point X e of: Ẋ = ϕ p X 81 the basin of attraction of which contains the compact set C. Proof: Let Π be a positive definite symmetric matrix and a a stricty positive rea number satisfying Π ϕ m X 0+ ϕ m X 0 Π aπ λ min Π = 1 where λ max and λ min respectivey stand for max and min eigenvaues. By continuity there exists a stricty positive rea numberp 0 such that we have for a X satisfying X ΠX p 0 and Π ϕ m X X+ ϕ m X X Π a Π X Πϕ m X a 4 X ΠX.

13 13 Let ϕ p : R n R n be any C 1 function satisfying ϕ p X ϕ m X a p0 4 1λ max Π X : X ΠX = p We obtain X Πϕ p X = X Πϕ m X+X Π[ϕ p X ϕ m X] and therefore X Πϕ m X+ a 8 X ΠX + a [ϕ px ϕ m X] Π[ϕ p X ϕ m X] X Πϕ p X a 16 X ΠX X : X ΠX = p In this condition it foows from [16 Theorem 8.] that for each function ϕ p satisfying 8 there exits a point Xe satisfying ϕ p Xe = 0 Xe ΠXe p Assume further that ϕ p satisfies ϕ p X ϕ m X X X a 8λ max Π X : X ΠX p In this case we have for a X satisfying X ΠX p 0 Π ϕ p X+ ϕ [ p ϕp X X X Π = X ϕ ] m X Π X X +Π ϕ m X X+ ϕ m X X Π+Π Note aso that we have a 4 Π. [Xe +sx Xe] Π[Xe +sx Xe] p 0 [ ϕp X X ϕ m X X XXes : s [01] Xe ΠXe p 0 6 X ΠX p 0 3. Then with ϕ p X=ϕ p X ϕ p Xe= 1 0 ϕ p X X e+sx Xeds[X Xe] and 84 we get for a X satisfying X ΠX p0 3 [X Xe] Πϕ p X = 1 [X Xe] Π ϕ p 0 X X e +sx Xe[X Xe] ds a 4 [X X e] Π[X Xe]. Let { } a p0 δ 1 = min 4 1λ max Π a 8λ max Π and reduce p 0 if necessary to have that X satisfying Xe ΠXe p 0 is in C. Then 79 and 80 with δ = δ 1 impies 8 and therefore 84. We have estabished that the system 81 has an exponentiay stabe equiibrium with basin of attraction containing the compact set {X R n : X ΠX p 0 3 }. ] Now with d and V = V C + V as defined in the proof of Proposition we et v be a stricty positive rea number such that we have Let aso X ΠX p 0 3 X A : VX v 86 C = {X A : v VX dxc [0d]} It is a compact subset of N C A. By mimicking the same steps as in the proof of Proposition we can obtain that if ϕ p satisfies ϕ p X ϕ m X inf X C VX G X C 87 we have px 1 VX X X C. This impies the compact set {X A : VX v} is asymptoticay stabe for the system 81 with basin of attraction B containing the compact set {X A : VX αd} which contains C. Since with 86 we have { {X A : VX v} X R n : X ΠX p } 0. 3 with 8 85 and 87 we have estabished our resut with δ given as a δ = min 4 p0 1λ max Π a 8λ max Π inf X C VX sup X C X. X Proof of Proposition 3: In view of the above Lemma and 19 Proposition 3 hods if 13 and 14 impy 79 and 80. At the end of the proof of Proposition we have seen that 13 impies 79. So we are eft with proving that 13 and 14 impy 80. By using again the notations 71 and by dropping the arguments we see that ϕ p X ϕ m X X X kϑ p m u + + ϑ u y where kϑ = I 0 0 I 0 0 k ϑ κ 0 u = ψ sat ψ sat 0 y r y z ˆx ξ ξ f p = x u ζ ζ m = x + g x ψ sat g h 0 x u x k h r k 0 0 = yr 0 x ˆx ϑ κ h y x 0 ϑ κ 0 0 ϑ = ϑ κ 0 ˆx u y = ζxψ sat ˆxz hx. Reca that by construction the functions ψ sat k and ϑ are