Time scaling for observer design with linearizable error dynamics

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1 Available online at wwwsciencedirectcom Automatica 40 (2004) wwwelseviercom/locate/automatica Brie paper Time scaling or observer design with linearizable error dynamics W Respondek a;, A Pogromsky b, H Nimeier b a Institut National des Sciences Appliquees de Rouen, Laboratoire de Mathematiques de l INSA, Place Emile Blondel, Mont Saint Aignan, Cedex, France b Department o Mechanical Engineering, Eindhoven University o Technology, PO Box 513, MB 5600, Eindhoven, The Netherls Received 12 July 2002; received in revised orm 24 July 2003; accepted 19 September 2003 Abstract In this paper, we consider the problem o observer design or dynamical systems with scalar output by linearization o the error dynamics via coordinate change, output inection, time scaling We present necessary sucient conditions which guarantee the existence o a coordinate change output-dependent time scaling, such that in the new coordinates with respect to the new time the system has linear error dynamics? 2003 Elsevier Ltd All rights reserved Keywords: Observers; Time scaling; Output inection; Linearization 1 Introduction An important control problem studied extensively is that o observer design or a dynamical system with output A typical approach to this problem is to nd a dynamical system (observer) coupled with the observed system by output inection in such a way that the overall system possesses an invariant asymptotically stable set o a specic structure Although a solution to the observer design problem in its ull generality is not known yet, it is clear that the problem statement is coordinate independent invariant with respect to time scaling thereore it is natural to seek conditions, ensuring the existence o an observer, that would also be coordinate independent invariant with respect to time scaling, similar to the corresponding properties o asymptotic stability The problem o transorming a system to observer orm via change o state output coordinates has been intensively studied during the last 20 years (see eg Bestle & Zeitz, 1983; Gauthier, Hammouri, & Othman, 1992; Gauthier & Kupka, 2001; Hammouri & Gauthier, 1988; Krener & Isidori, 1983; Krener & Respondek, 1985; A preliminary version o this paper was presented at the IFAC NOLCOS Symposium, St Petersburg, 2001 This paper was recommended or publication in revised orm by Associate Editor Jessy W Grizzle under the direction o Editor Hassan Khalil Corresponding author address: wresp@lmiinsa-rouenr (W Respondek) Plestan & Glumineau, 1997; Xia & Gao, 1989) A recent paper by Moya, Ortega, Netto, Praly, Pico (2001) suggests enlarging the class o systems admitting an observer exploiting the additional reedom o introducing possible time scaling In this paper we are going to address this problem A dual problem o linearization, using time scaling, o dynamics with inputs has been considered by Sampei Furuta (1986), Respondek (1998), Guay (1999) Expressing a system in physical coordinates is very natural but sometimes it may complicate the observer design based on the second Lyapunov method At the same time or the system written in other coordinates that design can be much easier to perorm To be more precise, consider dynamics with output given by the ollowing equations: ż = s(y)az; y = Cz; (1) where z =(z 1 ;:::;z n ) is the state, y is the scalar output, s(y) is some nonvanishing positive real-valued unction In this case, it is possible to linearize the dynamics (1) via time scaling o the orm d = s(y(t)) For the linear system written with respect to the new time, the observer design problem can be easily solved using linear techniques Thereore, it is interesting to nd conditions which guarantee that a nonlinear system with scalar output ẋ = (x); y= h(x) can be put in orm (1) /$ - see ront matter? 2003 Elsevier Ltd All rights reserved doi:101016/automatica

2 278 W Respondek et al / Automatica 40 (2004) The problem which we are going to address in this paper is to nd a coordinate change, which transorms the dynamics with a scalar output either to orm (1) or to orm (1) up to an output inection Equivalently, we are looking or a coordinate change a time scaling that either linearize the observed dynamics or linearize it up to an output inection For the related problem o bilinearization up to output inection we reer the reader to Hammouri Gauthier (1988) It is worth mentioning that in a recent paper (Moya et al, 2001) it was reported that time scaling can signicantly simpliy the controller design or the so-called reaction systems In particular, in Moya et al (2001) it was shown that the model o a reaction system can be written in a coordinate system suitable or time scaling that signicantly simplies the controller/observer design A preliminary version o our results is presented in Respondek, Pogromsky, Nimeier (2001) An alternative approach to the problem o transorming an observed dynamics to the observer orm via a change o state-space coordinates a time rescaling has been independently proposed by Guay (2001) His method, based on the language o dierential orms thus dual to ours, also leads to solving (a series o) ordinary dierential equations on the output space In this paper, we consider only the case o dynamical systems with scalar output The more general case o dynamical systems with multiple outputs ( inputs) will be reported elsewhere The paper is organized as ollows The problem is ormulated in Section 3 In Section 2, we describe a model o a batch reactor whose analysis can be simplied by a natural time rescaling Section 4 contains the main results In Section 5, we compare the systems that can be brought to observer orm under a state-space dieomorphism a time rescaling with those that can be transormed into observer orm under state- output-dieomorphism Finally, in Section 6, we come back to the model o a batch reactor o Section 2 illustrate our results with it the state model or the reactor appears as ollows: dc A = (k 1 e E=RT + k 2 e E=RT )c A ; dc B = k 1 e E=RT c A k 3 e E=RT c B ; dc C = k 3 e E=RT c B ; dc D = k 2 e E=RT c A ; dt = J 1k 1 e E=RT c A + J 2 k 2 e E=RT c A + J 3 k 3 e E=RT c B + QH v =%c % V; where c A;B;C;D are the concentrations, J i = H ri =%c %, i = 1; 2; 3, QH v =UA t (T st T), V is the reaction volume, c % is the heat capacity o the reaction mixture, % is the density o the reaction mixture, H v is the enthalpy o the vaporization, U is the heat transer coecient, A t is the area o the heat exchange surace, T st is the steam temperature the concentrations (c A ;c B ;c C ;c D ) satisy the conservation law c A + c B + c C + c D = const which ollows rom the system equations Now consider the ollowing problem: to estimate the concentrations c A ;c B ;c C c D provided the temperature T is measurable First, we rescale the independent variable: d =e E=RT : Then introducing the state vector x =(c A ;c B ;c C ;c D ;T) t output y = T the system equations can be rewritten in the orm dx = Fx + (y); (3) d y = Gx; (4) 2 Motivating example Consider a batch reactor where the ollowing reactions between chemicals A, B, C, D take place: A B k1 C; k3 A D; k2 (2) where k 1 ;k 2 ;k 3 are the reaction rates Suppose that all reactions are endothermic have rst-order kinetics, the reacting mixture is heated by steam, which ollows through a acket around the reactor with a rate Q Suppose additionally that the activation energies E i o the three reactions are equal, that is, E 1 = E 2 = E 3 = E Under these assumptions where k 1 k k 1 k F = 0 k k J 1 k 1 + J 2 k 2 J 3 k G =(00001), (y)=(0000ua t (T st y)e E=RT =V%c % ) t : It is now clear that system (3), (4) admits an observer, with linear error dynamics, or the largest observable subsystem (which is the (c A ;c B ;T)-subsystem; see also Section 6)

3 W Respondek et al / Automatica 40 (2004) Problem statement Consider nonlinear observed dynamics o the orm :ẋ = dx = (x); y= h(x); where x( ) R n, y( ) R is the measurement In this paper, we will deal with the ollowing two questions Question 1: when does there exist a (local) dieomorphism z = (x) a time scaling o the orm d = s(h(x(t))), where s is a nonvanishing real-valued unction, such that becomes d = Az; y = Cz; where the pair (A; C) is observable? Question 2: when does there exist a (local) dieomorphism z = (x) a time scaling o the orm d = s(h(x(t))), where s is a nonvanishing real-valued unction, such that becomes = Az + (y); d y= Cz; where the pair (A; C) is observable is a vector eld whose components depend on y = Cz only? To answer the rst question means to characterize nonlinear dynamics that are linearizable via a dieomorphism an output-dependent time scaling, while to answer the second question means to characterize dynamics that are linearizable via a dieomorphism, an output-dependent time scaling, an output inection It is obvious that the rst question is equivalent to the ollowing one: when is (locally) equivalent under a dieomorphism z = (x) to = s(y)(az + Ky); y= Cz = z 1; (5) where k 1 A = ; K = ; C =(1; 0;:::;0) s(y) is a nonvanishing real-valued unction? The second question is obviously equivalent to the ollowing one: when is (locally) equivalent under a dieomorphism z = (x) to = s(y)(az + (y)); y= z 1 = Cz; (6) k n where (z 1 ) A = ; (z 1 )= ; n (z 1 ) C =(1; 0;:::;0) s(y) is a nonvanishing real-valued unction? A motivation or our study is the ollowing immediate observation I is equivalent, via a change o coordinates, to (6), then we can construct the ollowing observer (compare eg Krener & Isidori, 1983; Krener & Respondek, 1985): ẑ = s(y)(aẑ + (y)) + L(y Cẑ) yielding the error e =ẑ z that satises ė = s(y)(a + LC)e thus gives the linear equation de =(A + LC)e; d with respect to the new time We will work around a xed initial condition x 0 R n we will assume that the dieomorphism satises (x 0 )=0 Clearly, a necessary condition or to be equivalent to one o the above-discussed orms is the ollowing local observability rank condition (see eg Isidori, 1989; Nimeier & Van der Schat, 1990): dim span{dh(x 0 ); dl h(x 0 );:::;dl n 1 h(x 0)} = n we will assume this throughout the paper 4 Main results Following Krener Isidori (1983) (see also Krener & Respondek, 1985) dene a vector eld g by { 0 i 06 6 n 2; L g L h = (7) 1 i = n 1: For 2weput ( 1) l = +1: 2 In order to avoid the trivial case, we will assume throughout that n 2 We have the ollowing results Theorem 1 is, locally around x 0, equivalent under a dieomorphism z = (x) to system (5) i only i in a neighborhood o x 0 it satises (i) dl g L n h = l ndl h mod span{dh}, or some smooth unction ;

4 280 W Respondek et al / Automatica 40 (2004) (ii) [ad i g; ad g] =0,or 0 6 i 6n, where = 1 s, g = s n 1 g, s = exp, with being a solution o { 0 i n 2; g = ( 1) n 1 i = n 1: Theorem 2 is, locally around x 0, equivalent under a dieomorphism z = (x) to system (6) i only i in a neighborhood o x 0 it satises conditions (i) (ii) o Theorem 1 with i in item (ii) satisying 0 6 i 6 n 1 Remark 1 Notice that the system { 0 i 06 6 n 2; g = ( 1) n 1 i = n 1 o rst-order partial dierential equations on the state space R n is actually a rst-order ordinary dierential equation on the output space R For instance, bring the system, via the (local) dieomorphism x i = L i 1 h, or 1 6 i 6 n, toits observability normal orm ẋ 1 = x 2 ; y= x 1 ; ẋ n 1 = x n ; ẋ n = n (x 1 ;:::;x n ): (8) Then the above system o rst-order partial dierential equations becomes the ordinary dierential equation d(y) dy = (y) on the output space R Remark 2 In order to avoid redundancy we can take i =0 =1; 3;:::;2n 1 in (ii) o Theorem 1 i = 0 =1; 3;:::;2n 3 in (ii) o Theorem 2 (compare Krener & Isidori, 1983) In small dimensions we can, instead o checking the commutativity o the rame {ad g}, bring the system into the observability normal orm (8) veriy whether the component n satises the conditions o the ollowing proposition, which can be proved by tedious but straightorward computations Proposition 1 is, locally around x 0, equivalent under a dieomorphism z = (x) to the system (6) i only i in a neighborhood o x 0 its observability normal orm (8) satises: 2 = ax2 2 + bx 2 + c i n=2, where a; b; c are any smooth unctions o x 1 dened in a neighborhood o x 0 3 = ax2 3 + bx 2 x 3 + cx2 2 + dx 3 + ex 2 + i n =3,where a; b; c; d; e; are any smooth unctions o x 1 dened in a neighborhood o x 0 satisying a = 1 4 b 1 8 b2 c = d 1 2 bd: (9) ProooTheorem 1 Suciency From the denition o g, it ollows that, which is a solution o { 0 i 06 6 n 2; ( 1) n 1 i = n 1; satises d span{dh} Thereore, ad g = sn 1 ad g(mod span{g;:::;ad 1 g}); or any n 1 We thus conclude that { 0 i 06 6 n 2; gh = (10) ( 1) n 1 i = n 1: It is well known (see eg Nimeier & Van der Schat, 1990; Isidori, 1989) that (ii) implies that we can nd a local dieomorphism z = (x) such that g n (z)=az + Kz 1 ; where A = ; K = k 1 k n ; which simply means that z = (x) linearizes the control system ẋ = (x) + g(x)u Moreover, by (10) we have that in z-coordinates, h = z 1 Since = s(z 1 ), it ollows that in z-coordinates reads as = s(z 1)(Az + Kz 1 ); y= z 1 : Necessity Assume that there exists z = (x) bringing into ż 1 = s(z 1 )(z 2 + k 1 z 1 ); y= z 1 ; ż n 1 = s(z 1 )(z n + k n 1 z 1 ); ż n = s(z 1 )k n z 1 : We have h = z 1, L h = s(z 2 + k 1 z 1 ), L 2 h = s2 z 3 + s sz z 2a 2 (z 1 )+b 2 (z 1 ), or some smooth unctions a 2, b 2, where sts or the derivative with respect to z 1 Itis straightorward to prove by an induction argument that L h = s z +1 + l s s 1 z 2 z +z a (z 1 )+b (z 1 ;:::;z 1 );

5 W Respondek et al / Automatica 40 (2004) or some smooth unctions a, b It thus ollows that g = (1=s n 1 )(@=@z n ) that L n h = s n k n z 1 + l n s s n 1 z 2 z n + z n a n (z 1 )+b n (z 1 ;:::;z n 1 ); or some smooth unctions a n, b n Hence L g L n h = sa n + l n s z 2 + a n (z 1 ) dl g L n h = l n s 2 mod span{dh}; which gives = s =s The system { 0 i 06 6 n 2; ( 1) n 1 i = n 1; is thus the ordinary dierential equation (z 1 )= s (z 1 ) s(z 1 ) = (log s(z 1)) : (11) We have (z 1 ) = log s(z 1 )+d, where d is a constant, which gives a 1-parameter amily o solutions s c (z 1 )=cexp (z 1 ), where c R, c 0, is a multiplicative constant It is clear that =(1=s c ) =(1=c)Az g =(s c ) n 1 g = c n n satisy (ii), or 0 6 i 6n, which, actually, is another way o expressing the act that ż = (z)+ g(z)u=(1=c)az + c n 1 (@=@z n )u is a linear system or any c R, c 0 ProooTheorem 2 The proo ollows the same line, the only dierence being that the commutation relation (ii), satised or 0 6 i 6 n 1, is a necessary sucient condition or the observed dynamics ẋ=(x) y=h(x) to be equivalent to the nonlinear observer orm ż =Az +(Cz), y = Cz (see Krener & Isidori, 1983) Notice that in both cases is calculated via (11) uptoan additive constant, which means that s = exp is calculated up to a multiplicative constant This is in agreement with the obvious observation that i a time rescaling d= = s(y) works then any rescaling d= =cs(y), where c 0, works as well 5 Time scaling versus change ooutput coordinates We can notice that Theorems 1 2 recall analogous results or, respectively, linearization linearization up to output inection under a state space output space dieomorphisms (see Krener & Respondek, 1985) We will compare these two classes o transormations in two three dimensions or the problem o linearization up to output inection For the system :ẋ = dx = (x); y= h(x); we look or a (local) dieomorphism z = (x) in the state space a (local) dieomorphism y = (ỹ) in the output space transorming into = Az + (Cz); ỹ = Cz; (12) which, o course, is equivalent to transorming via a state-space dieomorphism z = (x) only into = Az + (Cz); y= (Cz): (13) Notice that the unction s(y) determining g in Theorems 1 2, dened as s = exp in (ii) o Theorem 1, can be equivalently expressed as a solution o the ollowing system o rst-order partial dierential equations: { 0 i 06 6 n 2; gs = (14) ( 1) n 1 s i = n 1; where is dened by (i) o Theorem 1 In Krener Respondek (1985) (see also Plestan & Glumineau, 1997; Respondek, 1985) it is proved that is, locally around x 0, equivalent under a state-space dieomorphism z= (x) an output-space dieomorphism y= (ỹ) to system (12) i only i in a neighborhood o x 0 it satises [ad i g; ad g]=0, or 0 6 i 6n 1, where g= sg, with s being a solution o the system (14), in which is replaced by dened by dl g L n h = n dl h mod span{dh} We will show that or the case n = 2 the conditions o the above result o Theorem 2 coincide while or n = 3 the classes o systems satisying the conditions o these two theorems are, in general, dierent Observe that the conditions o both theorems claim the existence o a commuting rame ad i g, or 0 6 i 6 n 1, in the case o time-rescaling, ad i g, or 0 6 i 6 n 1, in the case o output transormations There are, however, two dierences in constructing these commuting rames Firstly, the rame is constructed with =(1=s) g = s n 1 g, in Theorem 2, with g = sg, in the other case Secondly, although the unctions s s are determined by systems o linear dierential equations o the same orm, the unctions, dening those equations, dier by a multiplicative constant (actually, =(l n =n)) hence s s are dierent A particular situation takes place or n = 2 Firstly, l 2 =2 implying = thus s= s Secondly, in this case, a simple calculation shows that the commutativity o the vector elds g ad g is equivalent to the commutativity o g ad g Starting rom n = 3, the classes o systems satisying, respectively, the conditions o Theorem 2 o the above result o Krener Respondek (1985) are dierent To see this, we discuss the two ollowing examples Example 1 Consider the system ẋ 1 =e x1 x 2 ; ẋ 2 =e x1 x 3 ; ẋ 3 =0; y = x 1 : (15)

6 282 W Respondek et al / Automatica 40 (2004) Clearly, this system is transormable to the observer orm (12), with ỹ = y, by the time scaling d = s(x 1 ), where s(x 1 )=e x1 To conrm this observation, apply Theorem 2 By a direct calculation, condition (i) o Theorem 2 implies that = 1 Thereore, s = ce x1 ; where c R, c 0, thus have = 1 ( ) s x 2 + x 2 g = s 2 g = c : The vector elds g are linear constant, respectively, so no dieomorphism is needed Obviously, they satisy [ad i g; ad g]=0; or 0 6 i 6 2 Now we will show that system (15) cannot be transormed into observer orm (12) via a dieomorphism a change o output coordinates To this end, we apply the result o Krener Respondek (1985) recalled in the beginning o this section We have = 4 3, thereore s = ce (4=3)x1 ; where c R, c 0 we thus obtain g = s g = ce (2=3)x1 : A straightorward calculation shows that [ad g; ad 2 g]= 4 3 x 2c e 5=3x1 0: This implies that system (15) cannot be transormed to the observer orm (12) via a change o state outputs coordinates, although as we have shown earlier, it can be transormed to that orm via a time rescaling The goal o the next example is to show that the converse inclusion between the two classes o systems does not hold either Example 2 Consider the system ẋ 1 =e x1 x 2 ; ẋ 2 = x 3 ; ẋ 3 =0; y = x 1 : (16) By a change o state output coordinates this system is transormable to the observer orm (12) (actually to a linear system) To see this, apply the result o Krener Respondek (1985) We have =1: It ollows that s = ce x1 ; where c R, c 0 Thus we have g = sg = 9 9x 3 hence [ g; ad g]=[ad g; ad 2 g]=0: This implies that system (16) can be transormed to the observer (actually, even linear) orm (12) Indeed, we can use the transormation z 1 = e x1 +1; z 2 = x 2 ; z 3 = x 3 ; ỹ = e y +1: We will now show that system (16) is not transormable to the observer orm via a state-space dieomorphism a time scaling, that is, it is not transormable via a state-space dieomorphism to orm (6) Condition (i) o Theorem 2 implies that = 3 4 : Thereore, s = ce (3=4)x1 ; where c R, c 0, we thus have = 1 ( ) s x 2 e (1=4)x1 + x 3 e 2 g = s 2 g = c e (1=2)x1 : Straightorward calculations shows that [ad g; ad2 g]= 3 4 x 2c e 3=4x1 0 implying that the system cannot be brought to the observer orm via a state-space dieomorphism a time rescaling, that is, it cannot be transormed via a state-space dieomorphism to orm (6) The same conclusion can also be deduced directly rom Proposition 1 Note also that or n=3 3 as in Proposition 1, system (8) can be linearized up to output inection via state output transormations provided a = 1 3 b 1 9 b2 c = d 1 3 bd:

7 W Respondek et al / Automatica 40 (2004) (see Krener & Respondek, 1985; Keller, 1987; Phelps, 1991) This indicates that the conditions or linearization up to output inection via state transormation time rescaling via both state output transormation are indeed dierent 6 Motivating example continuation Now, we will come back to the model o reactor o Section 2 illustrate our main result, that is Theorem 2, with that model To start with, notice that system (3), (4), whose observation is y = T, is not observable decomposes into two subsystems: the observable system evolving on three-dimensional (c A ;c B ;T)-space a completely unobservable system evolving on 2-dimensional (c C ;c D )-space This is clear in view o the act that the evolution o c A, c B, T depends neither on c C nor on c D Due to the mass conservation, c A + c B + c C + c D = const hence, i the sum o initial concentrations is known a priori, the system can be dened on a our-dimensional linear maniold that allows to decrease the dimension o the unobservable part Consider the case when the sum o initial concentrations is unknown We will show that we can bring the maximal observable subsystem, that is the system dc A = (k 1 e E=RT + k 2 e E=RT )c A ; dc B = k 1 e E=RT c A k 3 e E=RT c B ; dt = J 1 k 1 e E=RT c A + J 2 k 2 e E=RT c A + J 3 k 3 e E=RT c B + QH v =%c % V; with the observation y = h = T, to orm (6) The above system can be expressed as dx Obs =e E=RT (F Obs x Obs + Obs (T )) y = C Obs x Obs ; where F Obs C Obs are suitable constant matrices (which are, actually, the restrictions o the matrices F C, respectively, o (3) (4) to the observable subspace R 3, equipped with the coordinates x Obs =(c A ;c B ;T) t ) Denote C 1 x Obs = C Obs x Obs, C 2 x Obs = C Obs F Obs x Obs, C 3 x Obs = C Obs (F Obs ) 2 x Obs Recall that l =(( 1)=2)+1, or any 2 We have L h = l E RT 2 e E=RT (C x Obs )(C 2 x Obs )+e E=RT C +1 x Obs + C x Obs a (C 1 x Obs )+b (C 1 x Obs ;:::;C 1 x Obs ); or , or some suitable unctions a b It ollows that the vector eld g is uniquely dened by L g C 1 x Obs =0; L g C 2 x Obs =0; L g C 3 x Obs =e 2E=RT : Thus we have dl g L 3 h = 4E RT 2 e E=RT d(c 2 x Obs ) mod span{dh} dl h =e E=RT d(c 2 x Obs ) mod span{dh}: Since l 3 = 4, by condition (i) o Theorem 2 we have = E RT 2 ; which yields the ordinary dierential equation on the output space ds dt = E RT 2 s; whose solution is s(t)=ce E=RT ; where c R, c 0 Clearly, the time rescaling d = s(t) = ce E=RT brings the system to the observer orm dx Obs = F Obs x Obs + Obs (T) d y = T: Obviously, ad i g, or 0 6 i 6 2, orm a commuting rame since = 1 s = 1 c ee=rt e E=RT (F Obs x Obs + Obs (T)) g = s 2 g implying that g is dened by L g C 1 x Obs =0; L g C 2 x Obs =0; L g C 3 x Obs = C 2 :

8 284 W Respondek et al / Automatica 40 (2004) Conclusions In this paper we considered the problem o transorming, via coordinate change time scaling, dynamics with output to a orm which admits an observer with a linear error dynamics The class o admissible time scalings used in this paper is given by the equation d = s(y(t)) ; where s is a real-valued positive unction which depends only on the output o the observed dynamical system This condition is necessary or practical implementation o an observer designed by the proposed technique We proposed necessary sucient conditions ensuring the existence o an appropriate (local) coordinate change output-dependent time scaling such that in the new coordinates with respect to the new time the system admits one o the two normal orms (either linear or linear up to a nonlinear additive output inection) or which the observer design problem can be easily solved Our conditions involve solving one ordinary dierential equation on the output space (whose solution actually determines the needed time rescaling) calculating Lie brackets We also compared the class o systems that can be put into observer orm under a state-space dieomorphism time scaling with that or which we use a state-space an output-space dieomorphism Starting rom n = 3 those classes are dierent a natural problem or uture investigations is to consider the group consisting o all those transormations Acknowledgements The authors are grateul to Natalia Shutina Jean-Marie Cosmao or helpul discussions They are also grateul to an anonymous reviewer or pointing out an error in the rst version o the motivating example Reerences Bestle, D, & Zeitz, M (1983) Canonical orm observer design or non-linear time-varying systems International Journal o Control, 38, Gauthier, J P, Hammouri, H, & Othman, S (1992) A simple observer or nonlinear systems: Application to bioreactors IEEE Transactions on Automatic Control, 37, Gauthier, J P, & Kupka, I (2001) Deterministic observation theory applications New York, NY: Cambridge University Press Guay, M (1999) An algorithm or orbital eedback linearization o single-input control ane systems System Control Letters, 38, Guay, M (2001) Observer linearization by output dieomorphism output-dependent time-scale transormations In NOLCOS 01, (pp ) Saint Petersburg, Russia Hammouri, H, & Gauthier, J P (1988) Bilinearization up to output inection System Control Letters, 11, Isidori, A (1989) Nonlinear control systems (2nd ed) Berlin: Springer Keller, H (1987) Nonlinear observer design by transormation into a generalized observer canonical orm International Journal o Control, 46, Krener, A J, & Isidori, A (1983) Linearization by output inection nonlinear observers System Control Letters, 3, Krener, A J, & Respondek, W (1985) Nonlinear observers with linearizable error dynamics SIAM Journal o Control Optimization, 23, Moya, P, Ortega, R, Netto, L S, Praly, L, & Pico, J (2001) Application o nonlinear time-scaling or robust controller design o reaction systems International Journal o Nonlinear Robust Control, to appear Nimeier, H, & Van der Schat, A J (1990) Nonlinear dynamical control systems Berlin: Springer Phelps, A R (1991) On constructing nonlinear observers SIAM Journal o Control Optimization, 29, Plestan, F, & Glumineau, A (1997) Linearization by generalized output inection System Control Letters, 31, Respondek, W (1985) Linearization, eedback Lie brackets In B Jakubczyk, W Respondek, K Tchon (Eds), Geometric theory o nonlinear control systems, (pp ) Wroclaw: Technical University o Wroclaw Respondek, W (1998) Orbital eedback linearization o single-input nonlinear control systems In NOLCOS 98, Enschede, The Netherls (pp ) Respondek, W, Pogromsky, A, & Nimeier, H (2001) Time-scaling or linearization o observable dynamics In NOLCOS 01; (pp ) St Petersburg, Russia Sampei, M, & Furuta, K (1986) On time scaling or nonlinear systems: application to linearization IEEE Transactions on Automatic Control, 31, Xia, X-H, & Gao, W-B (1989) Nonlinear observer design by observer error linearization SIAM Journal o Control Optimization, 27, Witold Respondek received PhD degree rom the Institute o Mathematics, Polish Academy o Sciences in 1981 He has held positions at the Technical University o Warsaw at the Polish Academy o Sciences Since 1994 he has been at INSA de Rouen, France His main research concerns are geometry singularities o nonlinear control systems Pa equations He is a co-editor o 6 books author o several papers devoted to linearization o nonlinear control systems, nonlinear observers, classication o control systems Pa equations, dynamic eedback, high-gain eedback, systems with nonholonomic constraints, recently, symmetries biurcations o nonlinear control systems He has held visiting research positions at the University o Caliornia, Davis, Twente University, Enschede, the Netherls, Rome University Tor Vergata, Rome, Italy, Mittag-Leer Institute, Stockholm, Sweden He was an associate editor o Applicationes Mathematicae SIAM Journal on Control Optimization, currently he is an associate editor o the Central European Journal o Mathematics Alexer Pogromsky was born in Saint Petersburg, Russia, in 1970 He received the MSc degree rom the Baltic State Technical University, Russia in 1991 the PhD (Cidate o Science) degree rom the St Petersburg Electrotechnical University in 1994 From 1995 till 1997, he was with the Laboratory Control o Complex Systems (Institute or Problems o Mechanical Engineering, St Petersburg, Russia) From 1997 to 1998, he was a Research Fellow with the Department o Electrical Engineering, Division on Automatic Control, Linkoping University, Sweden then he was a Research Fellow

9 W Respondek et al / Automatica 40 (2004) in Eindhoven University o Technology Currently he is an Assistant Proessor o Mechanical Engineering Department o Eindhoven University o Technology, The Netherls His research interests include theory o nonlinear control, nonlinear oscillations Dr Pogromsky was awarded the Russian Presidential Fellowship or young scientists in 1997 Henk Nimeier (1955) obtained his MSc degree PhD degree in Mathematics rom the University o Groningen, Groningen, the Netherls, in , respectively From 1983 until 2000 he was aliated with the Department o Applied Mathematics o the University o Twente, Enschede, the Netherls Since, 1997 he was also part-time aliated with the Department o Mechanical Engineering o the Eindhoven University o Technology, Eindhoven, the Netherls Since 2000, he is ull-time working in Eindhoven, chairs the Dynamics Control section He has published a large number o ournal conerence papers, several books, including the classical Nonlinear Dynamical Control Systems (Springer Verlag, 1990, co-author AJ van der Schat) Henk Nimeier is editor in chie o the Journal o Applied Mathematics, corresponding editor o the SIAM Journal on Control Optimization, board member o the International Journal o Control, Automatica, European Journal o Control, Journal o Dynamical Control Systems, SACTA, International Journal o Robust Nonlinear Control, the Journal o Applied Mathematics Computer Science He is a ellow o the IEEE was awarded in 1987 the IEE Heaviside premium