On nonlinear equivalence and backstepping observer

Kybernetika J. de Leon; I. Souleiman; Alain Glumineau; G. Schreier On nonlinear equivalence and backstepping observer Kybernetika, Vol. 37 (2001), No. 5, [521]--546 Persistent URL: http://dml.cz/dmlcz/135425 Terms of use: © Institute of Information Theory and Automation AS CR, 2001 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz K Y B E R N E T I K A — V O L U M E 3 7 ( 2 0 0 1 ) , N U M B E R 5, P A G E S 521-546 ON NONLINEAR EQUIVALENCE AND BACKSTEPPING OBSERVER* J . DELEON^ I. SOULEIMAN, A. G L U M I N E A U AND G. SCHREIER An observer design based on backstepping approach for a class of state affine systems is proposed. This class of nonlinear systems is determined via a constructive algorithm applied to a general nonlinear Multi Input-Multi Output systems. Some examples are given in order to illustrate the proposed methodology. 1. INTRODUCTION It is well-known that when a state control law is designed its application is limited if the components of the state vector are not all measurable. This problem can be overcome by using observers. For linear systems, it is traditionally solved by using either a Luenberger observer or Kalman-filter. Moreover, the observability property for linear systems does not depend on the input. However, the observability property of nonlinear systems does depend on the input. There are some inputs for which the system could become unobservable (for more details see [1, 8, 10]). Hence, the inputs which render the system unobservable should be considered when observer is constructed. For these reasons, the observer problem for nonlinear systems remains an interesting field of research. Although the problem of observer synthesis for linear systems is solved, no general methodology exists for the observer design for nonlinear systems. However, some results have been obtained in this direction ([8, 10, 12, 13, 16, 18, 20]), where the observer design has been investigated for a class of nonlinear system which can be transformed into another observable form. Several authors (see for instances [13, 14]) have considered the case when a nonlinear system can be transformed into a linear system up to input-output injection. On the other hand, a straightforward approach verifying and computing the linearization condition for those systems have been given in ([15, 17]). The design of an observer for a class of nonlinear systems can be solved via a change of coordinates which transforms the system into another nonlinear system for which an observer can be constructed (see [10, 14, 20]). Some results related to "This work was supported by CONACYT-MEXICO 26498-A. t Corresponding author. 522 J. DE LEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER the coordinate transformation of a nonlinear system into a state affine systems have been obtained (see for instances [1, 8, 10, 14, 18]). The design of an observer for these state affine systems has been studied in [3]. Furthermore, necessary and sufficient conditions transforming a nonlinear system into a state affine system has been proposed in [2, 10]. However, no construction procedure characterizing such systems exits so far for multi-input-multi-output case. On the other hand, a constructive methodology for the single output case, comput­ ing the change of coordinates, is presented in [14]. This paper deals with the observer synthesis of nonlinear systems via their equiv­ alence to state affine systems. Necessary and sufficient conditions are given to char­ acterize a class of nonlinear systems, which can be transformed into a class of multivariable state affine systems up to input-output injection. Furthermore, for the class of state affine systems an observer is designed using a backstepping observer approach. The paper is organized as follows. In Section 3, a computation algorithm is described which allows the transformation of a nonlinear system into a multi-output affine system. In Section 4, the unmeasurable components of the vector state are estimated using a backstepping observer. For this observer, conditions are given to characterize the inputs which render the system observable. In Section 5, some examples illustrating the proposed methodology are given. Finally, some conclusions are given. 2. PRELIMINARIES Now, consider the following nonlinear system x = f(x,u) E: (1) У = Ңx) where x G Mn is the state, u G Mm is the input, y G Mp is the controlled output, / and h are meromorphic functions of their arguments. Assume that there exists a change of coordinates transforming £ into the state affine system of the form ZІ --Jafrine • where Zi = col (z^\,... AІ y% = = 0 0 Ai(u,y)zi CiZU Ai G MkiXki , z^k{), /0 0 = OІ,I(«) 0 0 Oi,2(«,«) 0 0 0 + фi(u,y) (2) i = 1,... ,p, are matrices of the form \ (3) aiM-Лu,y) 0 On Nonlinear Equivalence and Backstepping 523 Observer ; andOi = ( 1 0 ... O ) l x k f ; ť = l,. where the ki denote observability index related with the output yi,which are ordered as k\ > k2 > .. • > kp and Y7i=i k% = n. Remark 1. In order to simplify the notation and without loss of generality, the outputs are reordered in function of the observability indices; i.e. the output yi is associated to the index observability ki, for i = 1,... ,p. All definitions and results given in the paper can be written locally around a n regular point xo of M, an open subset of M . If this property is generically satisfied, it means that this property is satisfied locally around a regular point Xo of M. Let O denote the generic observability space defined by (see [16]). (4) o = xn(y + u) 1 where X = S p a n ^ d x } , y = S p a n ^ d * / ^ , w > 0}, U = Span^{duW,uv > 0}, (Span^ is a space spanned over the field X of meromorphic functions of x and a finite number of time derivatives of u). The system S is generically observable if Definition 1. dim O = n. The first goal of this paper is to find a state coordinate transformation z = $(x), such that system £ is locally equivalent to system Eaffine in order to design an observer. The approach consists in checking that the Input-Output (I/O) differential equation associated to the observable system S, which is given by ylki) =Pt(y1,y1,...y[kl-1),... ,yP,.-. Jpk"~l),u,u,u,... ,u^~% (5) has the same I/O differential equation as Saffine, which verifies ylki)=Pio = Ftki(aitl,...,ai,n.1) (6) ki-1 + E 4 - r - l f r f e - n - . ,fli,*.-l>V>i,*.-r) + Kik._1F^(ipifki) r=l = Fi.(aiyl,... ,aiyU-i) + T^fai,. •. , ^ - 1 , ^ , 1 , . • - ,<^,*J where Kj. = aiy0 . . . aiyT = l\rj=0 ai}j, and a^o = 1- T h e functions Fr*, r = 0 , . . . , kf, are given as a sum of monomials depending on (ylПi)Ţ and ( í 4 m i ) ) S ' , for ť = 1,... ,P; 524 J. DE LEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER where n*, rrii = 0 , . . . ., ki\ represent the order of derivation of the outputs and the inputs respectively; and <ft, s; = 0 , 1 , . . . ; are the exponents of the outputs and the inputs and their derivatives, respectively. These parameters satisfy the following relation ^2niqi i R e m a r k 2. and (u\mt} J + X/ m i 5 i = r; f° r ^ — 1J- • • >P- i The functions F* involves monomials depending on functions ly\n ) of degree £ • n ^ + £ ; ™>%8i = (&i ~ r ) - On the other hand, the proposed results are obtained from the analysis of I/O differential equations. The observable nonlinear system £ in the state space representation will be transformed into a set of higher-order differential equations depending on the inputs and outputs. These equations are obtained by using state elimination techniques (see [5]). Moreover, considering the assumption of generic observability of the system, the elimination problem has a solution (see [15, 19]). Hence, the state affine transformation problem is solved as a realization problem. The classification problem of nonlinear systems which can be steered by a change of coordinates to some observable form has received significant attention during the last years. In [7] and [8], locally uniformly observable systems are studied. Necessary and sufficient conditions have been stated to guarantee the transformation of nonlinear systems into state affine systems (see [1, 10, 11]). These conditions guarantee the existence of a vector field transforming the system into another observable one. However, this vector field cannot be computed directly and hence, the application of this methodology is limited (see [1]). On the other hand, a constructive methodology for the single output case, computing the change of coordinates, is presented in [14]. In this paper, using the results given in [14], an extension for the class of multivariable systems will be considered. 3. STATE AFFINE TRANSFORMATION ALGORITHM The problem of verifying the equivalence between a nonlinear system and state affine system is considered in this section. Necessary and sufficient conditions allowing to characterize a class of nonlinear systems, which are diffeomorphic to state affine systems, are given. These conditions are obtained using the exterior differential system theory ( for more details see [4, 9, 14, 16]). Now, the algorithm allowing us to know if a diffeomorphism exists between (1) and (2) is given. Let 5j = {&i,A;2,... ,kj} be the set of observability indices such that kj satisfies the following inequality K/j „-** ACi /v for a given k. Denote d* the number of outputs whose observability index is greater than ki — ky as dJ = Card{fci,*2,... , * ; } . (7) On Nonlinear Equivalence and Backstepping 525 Observer AlgorithmStep 1. Computation of the functions ciij. Let Po = Vi , i = 1 , . . . ,p; be the I/O differential equation obtained from the nonlinear system S. Let ulk be the one-form defined by jfc a i= i Q2pi " ^§<^ a dw .fc i m Q2pi ^lri?#^ d "' (8) for A: = 1 , . . . , ki — 1; with c\ = . . . = ck._2 = 1 and clk._1 = 0. Now, in order to verify if it is possible to find an equivalence between £ and Saffine, it is necessary to check the following conditions: — Case d\ < p. If duJk A du ^ 0 or dulk A dydfc+1 A • • • A dyp ^ 0; then, there is no solution. — Case d\ = p: If du>k 7-= 0, then the problem has no solution. Otherwise, let the a^* functions be any solution of jfc d c i .fc fflpi a i m 4 = * £ «w.S.-.)^+£ £ fl2pi (9) OWOIM^ fly) ; dyj ' ' i = 1 / = 1 0u) 'fly) where the right-hand side of this equation is deduced from the I/O differential equation P^ 0 , which is computed from system Saffine. i=1 This ends the Step 1. On the other hand, the previous one-forms do not allow to know the functions (Pi,k- Then, in order to identify the functions ipij, all aij obtained from Step 1 will be used to determine the ipij, as it is presented in the next step. Step 2. Determination ofVi,k». Consider PQ as in Step 1, and let (10) p} = p}-i-K-r+i> for r := 1 , . . . , ki — 1; where the Fk._r+1 are functions as in (6). Let aJ* the one-form given by + *-*{S#** f.s!M (n) . 526 J. DE LEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER where r a a Kr = i,l • • - i,r = J_J_ aiji i=o and ai )0 = 1- Now, in order to compute the functions c^,-, we check the following conditions: — Case d\ < p. If duJr Adu ^ 0 or dulr Ady^+i A • • • Ady p ^ 0, then, the problem has no solution. — Case d\ = p. If dul ^ 0, then the problem has no solution. Otherwise, if duJj. = 0, for Vr = 1 , . . . , ki - 1; then <piiT is a solution of ______ í f _£__áy,ЉЏLdu __!_ fèŞb-dyj+V- Oӣi^ dr ^ m / <-T І-V -. m n дu э 3 1 | (12) And for r = ki, P ki = aiA • • • aiM-WiM l End of the = Klkiipiyki. (13) Algorithm. This Algorithm allows to establish the following theorem. T h e o r e m 1. The system S is locally equivalent by state coordinates transformation to the system £affine if and only if the following conditions are verified: 1. For d\ < p, duk A du = 0, and duk A dydk+1 A • • • A dyp = 0, (14) duk A du = 0, and duJk A dydk+1 A • • • A dyp = 0. 2. Fordf=p, dc4 = 0, and du)k = 0; where uk and uJ^are one-forms defined in (8) and (11). If the conditions of Theorem 1 are satisfied, system £ is locally equivalent to system £affinej and the state coordinates transformation z = _>(x) is given by *i,i *i,2 ZІJ = Vi __ __i = — jfí = ^ { V i ( a O - V м (",!/)} , tor j = o , . . . , ki (15) On Nonlinear Equivalence and Backstepping 527 Observer where Zi = col(z^i . . . zi.fc,) and jDi _ jsi dP , In Pk = Kk^cp^k + k-l dt dK , „ nfix k-l + z^k (loj dt for k = 1 , . . . ,fcj,aiiki = 0 and P[ = (piti. Proof of Theorem 1 (see Appendix B). This result gives the conditions to transform system _ into system —affine (2). The next section introduces a procedure to design a backstepping observer for this class of systems. 4. BACKSTEPPING OBSERVER The propose of this section is to design an observer for the class of state affine systems (2) based on the backstepping approach. From the structure of the state affine system, which is represented by state affine subsystems, an observer will be designed for each subsystem independently. For this reason, consider the following class of single output state affine systems which are in the observable form ±i = ax(u,y)x2 +g1(u,x1) ±i = a{(u,y)xi+1 +gi(u,x1,... in = fn(x) i = 2,... ,n- 1; ,x{), (17) +gn(u,x), y = Cx = x1. It is clear that system (17) is uniformly observable if the applied inputs are persistently exciting. For instance, there are some inputs which render the unmeasured states unobservable. Then, in order to design an observer for the unmeasured states the inputs must be satisfy some observability conditions (see [11]). The observer for the class of systems considered is described by i i =a1(u,y)z2 z{ = ai(u,y)zi+1 +g1(u,z1) - zi) +ip1(z)(x1 + gi(u,zx,z2,... ,z{) +ipi(z)(x1 - zr), for i = 2,... ,n-l Zn = fn(z) + gn(u,z) + ^n(z)(x1 (18) - ZX ) where z = col(z1,z2,... , zn) is the estimated state and ipi(z), i = 2,... , n — 1; are the observer gains which must be determined in order to guarantee the convergence of the observer. Defining the estimation error e* = Xi — zi, for i = 1 , . . . ,n; whose dynamics is given by ei =ax(u,y)e2 -ip1(z)e1 e{ = a{(u,y)ei+1 +gi(u,x1,... ,x{) - gi(u,z1,z2,... ,zi) -ipi(z)e1, for i = 2 , . . . , n - 1 en = fn(x) - fn(z)+gn(u,x) -gn(u,z) -ipn(z)e1. (19) 528 J. DE LEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER Using similar arguments given in [12], we will find the observer gains ^i(z)yi = 1 , . . . ,n, such that the estimation error tends to zero as t -> oo. Now, in order to design the observer the following assumptions are introduced. Al) There exist positive constants c\ and C2, where 0 < ci < C2 < oo, such that for all x G Mn; 0 < ci < \di(u,y)\ < c2 < oo, i = 1 , . . . ,n - 1 A2) The functions gi(u} y,... , Xi), i = 2 , . . . , n, are globally Lipschitz with respect to ( x i , . . . ,Xi), and uniformly with respect to u and y. R e m a r k 3. The condition (20) corresponds to a characterization of "good" inputs, which are required to recover state observability. Let be 0(e)k a function of z and e for k > 0 such that for z G S C iR n , there exist constants N > 0, e > 0 such that |0(e) f c | < AM|e||\ V||e||<e, Vz G 3 . Now, consider the following variables Si for i = 1 , . . . , n + 1; si = e i s2 =c1s1 -Fsi-FO(e) 2 Si = Si-2 +Ci-iSi-i (20) 2 + Si-i + 0 ( e ) , for t = 3 , . . . ,n + 1, where the parameters Ci are positive constants s,nd the error terms are chosen so that s is a linear function of the error e. Next, writing the above equations in terms of the error e, we obtain i si+i = ^2(bt+lyi - Ki-iKi-i^i-i+i) e{ + Kiei+i, for 1 = 1,... , n - l (21) i=l and for I = n, SnFl = z2 ( bn+l>i " Kn-iKi-ilpn-i+l + Kn-\ Í -JT^ ) ) ei (22) where bz+i,i and J ^ - i for i = 1 , . . . ,/; and / = 1 , . . . ,n; aire given in Appendix C. Furthermore, let Up be the /^-neighborhood of C an open subset of iR n , there exists constants Ai > 0 and A2 > 0 such that for all z G Up, a compact subset,with e and 5 G C , the following inequality is satisfied Ai||e||<||S||<A2||e||, (23) where s = col (si, $2, • • • > 5 n) and e = col (ei, e 2 , . . . , e n ). Then we can establish the following result. 529 On Nonlinear Equivalence and Backstepping Observer T h e o r e m 2. Consider the system are satisfied. For any subset C C constants Ai, A2 > 0; e > 0; 7 > 0 system (18) is a locally exponential error (17), and assume that assumptions Al and A2 Mn of the dynamical system (17) there exist such that if x(0) G C and ||e(0)|| < e then the observer for system (17). Thus, the estimation 2 l|e(i)||<^||e(0)||exp- ^ converges exponentially to zero as t tends to 00. P r o o f . Defining the following Lyapunov function n n 1 z=l i=l Taking the time derivative of V along (20), we obtain n V = - ] T Cis] + snsn+i + 0(e)3. i=l Next, the observer gains ^ , i = 1,... , n; are chosen as follows , bn+in-i+i fa = -77 77 Kn-i&i-l Kn-i + -77 77 Ki-iKn-i ( o dfn \ r • t J , for 2 = 1 , . . . , n, \OZn-i+l J where bn+\^ and Kn-i are given in Appendix C. Then, from (38) the term sn+i is equal to 0 (see Appendix C). Hence, we obtain n 2 3 V = -Y/cis i+0(e) . (24) i=l Now, let Up be the p-neighborhood of C an open subset of iR n , then its closure Up is a compact subset. Hence there exist constants TV > 0, e > 0 such that the error term (24) satisfies |O(e)3|<iV||e||3 for all z E Up, and ||e|| < e. Next, let be e = min (p, e). From s = M(bij,tpi)e where s is a linear function of e (see equation (20) and Appendix C), we know that there exists constants Ai > 0,A2 > 0 such that for all z G Up, and e, 8 G C , the following inequality is satisfied Ai||e|| < | | S | | < A 2 | | e | | . Since Ci > 0, there exists a constant 7 > 0 such that 47lN|2<Ec^І=l (25) 530 J. DELEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER Hence, there exist an e > 0 sufficiently small such that the error term in (24) satisfies \0{e?\<\J2ciS* 1 .=1 for all z € Up, and ||e|| < e. For these z and e, we have 2 2 is i<-21V. V = -\Y,c (26) i=l And using Gronwall's inequality V(t) < y ( 0 ) e x p - 2 7 f . Using the inequality (25), we have IKl)ll<^IK0)||exp- 2 ^. Then, the estimation error converges exponentially to zero as t -> oo. This ends the proof. • 5. EXAMPLES Example 1. Single Output Case. Consider the dynamics of a rigid body ii \ / 71^2^3 x2 I = I 72^1^3 ±3 / y \ 73^1^2 = xi in which £1, X2 and £3 are the components of the angular velocity with respect to the principal axes of inertia, J\, J2 and J3 the moments of inertia with respect to the principal axes of inertia 71 = j 3 j ~ j 2 , 72 = JljJ* a n d 73 — J ? J J l • Assume that the angular velocity x\ is measured. The observation problem is the estimation of the angular velocities £2 and £3. Now, we apply the Algorithm presented in Section 3, to check if there exists a transformation for the above system. Step 1. Determination ofai. Applying the proposed algorithm, the I/O differential equation (5), for i = 1 and fci = 3 is given by (2) • 3 (2) y( ) = P,j(v>»>l/ ) = — + 472732/22/ y = F$ + F2 + K\Fi + K2F0 On Nonlinear Equivalence and Backstepping 531 Observer where F2 = F0 = 0. On the other hand, the I/O differential equation of the affine system is given by ya3) = yal>* (lnai - l n a i a 2 In a n +ya2^ (lnai + In a\a2 J - ( l n a i - l n a i a 2 — \na\ipi + (p1 — (lnaia 2 J (pi — ai(lnaia 2 +\na\)tp2 = F3a + F2a + K\Fla \na\)(p\ + a\(p2 + a\a2ips + K2F0a where F0a = (p3, Fia = - ( l n a i a 2 + \nai)ip2 + <p2 +\na\(p2, F2a = - ( l n a i - l n a i a 2 \na\)(pi - l n a i ^ i + (px - (ina x a 2 J <p\, Fsa = ya (lnai - In a x a 2 lnai J +2/i (lnai + l n a i a 2 J . Prom equation (8), the one-form U\ is given by ui = -dy. y Now, for k = 2, the one-form u2 is given by u2 = -dy. y It is easy to see that the one-form ui verify the conditions (14). Now, computing one-form uia, we have ^22/«3) fo51oSa1 , 91ogaia2\ A In the same way, u2a = uia. Then, in order to determine the a^'s, it is necessary to solve the following equation r 2 «91ogai dlogaia2 dy dy H Notice that the function ai depends on y, then the proposed algorithm can be extended to a large class of nonlinear systems where a^i depends on u and y. However, for this class of systems the algorithm gives several solutions for a given system. For example, setting the arbitrary choice a 1 1 = — a • 2 It follows that a solution is of the form a i =y, 1 Q>2 = -o- y2 532 J. DELEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER Step 2. Determination oftpi. Consider I/O differential equation Fb and F3, then V —JL Pi = P0 - F3 = P 0 y = 472732/V Computing the one-form cJi from equation (12), we obtain ZJ\ = 0. ai \ dy ai\dy) ] -'(£)-' Since, ai ^ 0, then, this implies that ipi = 0. Next, to determine cO2, using equation for r = 2, we have F2 — P\ — P2 — P\ since F2 = 0, then 1 дP u2 = a ľa 2 then, we have *A A —dy дy -£1 -U2 -1^[^E1A 2A = 47273У dy (—i\A a2 \ dy J ~ a2\ dy V \ ] = d f — ) = 4727 3 2/ 2 dy. Solving the above equation, we obtain <£2 = 72732Z2Now, for r = 3, and from (13) F3 = aia2(p3. Since F3 = 0, it follows that </?3 = 0. After computation, the change of coordinates obtained is z2 =-= zi=xi, 71Z2Z3 Xi 2 2 zs = 7i72^ ^ + 7173^?^2 + l2xjxl + 7273X1. Then, the transformed system Eaffine 1n the new coordinates is given by M = í° i 2 iз / -\ ° \o У 0 0 0 1 ғ0 z2 J 4- I 7273S/2 I • (27) On Nonlinear Equivalence and Backstepping Observer 533 An observer backstepping for t h e above system can be design as follows. /i, \ Фi(ž) N \ Ыѓ) ( Фз(ž) )' *i ~ ѓ i ) (28) where t h e observer gains are given by 7/>l(z) =2/b4,3 / /-\ ^4>2 fo(z) = - y yl fait) =- 2/64,1 where Kx = y, K2 = ±, gi = 0, g2 = 0, g3 = 0, and 62,1 = ci 63.1 = 1 + c 2 (ci - tpi) - (d - ipi)tpi - -j-(^i) dr 1 / \ dy 63.2 = 2 / ( c 2 + c i ) + — 64,1 = ci - ^ 1 + 03(63,1 - 2/^2) - (63,1 - 2/^2)^1 + 37(63,1 ~ 2/^2) dt -(63,2 ~ 2/^1)^2 + 3^(^3,2 ~ 2/^i) Ч 2 = ž/ + cз(č>з,2 -yфi) + yb3,i 1 1 , d 04,3 = CЗ- + T 62 3 , 2 + У У E x a m p l e 2. M u l t i - I n p u t M u l t i - O u t p u t . Consider t h e following multivariable system: ( xi \ ( = — u2e~-x2 UXi 2 U X$ + UXi X4 { xъ ) \ XiXзЄ~X2 X2 xз ueX2 \ 2/1 = xi, 2 X XĄ ) 2/2 = #4. It is easy t o verify t h a t t h e system is observable with indices of observability given by ki = 3 a n d k2 = 2. Moreover, t h e I / O differential equations (5) of this system 534 J. DE LEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER are 53> = - y { 2 ) + ln(uyi)y[2) Уì y(2) -2-(y2- + Inuýi - Iniuy^lnuýi - lnluy^u 3 + u3 + u2y\ uyi) + u2y\y2 + uyx + uyi. Next, the I/O differential equations associated to the equivalent state affine system are Vi*l = Vi]l (lnai,i - l n a i , i a i , 2 l n a M ) + y[2)a (lnai,i + l n a u a i , 2 ) - (lnai,i - l n a n a i 2 l n a n ) ^ i ? i - lnanv?i,i + <pltl - ^ l n a ^ i a i ^ J (fi,i - a i , i ( l n a i , i a i j 2 + l n a i , i ) ^ i ) 2 +ai,i</?i,2 + ai,iai>2<£i,3 and yK2a = lnd2,i(2/2 - </>2A) + a2,i^2,2 + V?2A. Now, we apply the algorithm Step 1. Computation ofaij. For i = 1, the I/O differential equation P0 is given by P1 - 7/(3) M) — 2/l = ~y[2) + ln(uyi)y[2) u + Inuyi - ln(wyi) In uyi - ln(uyi)u3 + u3 + u2y\. For k = 1, it follows that the number of output that verify condition (7) is given d\ = 1. Now, computing the one-form u\, which is derived from (8), we obtain 2 1 1, _ ui = —dyi H— du. 2/1 u It is clear that do;} = 0. Then, this implies that du\ A du = 0 and du\ A dy2 = 0. Next, for k = 2, and following the same procedure as above, we compute the one-form u\, which is given by 1 1J (jj2 = — d y i 2/1 1, H—du. u Then, checking the condition of the theorem, it follows that duj\ A du = 0, da; A dy2 = 0 and dcul = 0. Given that the conditions of the theorem are verified, now we identify the unknown functions a t j from the I/O differential equation Pa0 := y^a. 535 On Nonlinear Equivalence and Backstepping Observer Now, computing the one-form from the I/O differential equation P^ 0 , we obtain LJl = — (hl^U'V^\ rj 1 dyi\ait2(u,y)J x + — (^hl dii \ a u + ^hl] ai,2/ du The above equation allows to compute the functions ai t i and a\j2. Finally, after straightforward computation, we obtain ai f i = u and 01,2 = j/iNow, for i = 2, the corresponding one-form obtained from P0 = y^ W 2 2 l = ^2-1 is given by -* 1 ~dW' = u Similarly, the one-form obtained from the I/O differential equation P20 := y^2^ i s given by u21 = — (^±) du du \a2ti) Comparing both one-forms, we can deduce that a solution is 02,1 = ^ 2 - Step 2. Computation ofcpij. Now, the components of the vector <\>i = col( (piti are determined. ... ip%,ki ) for each subsystem For i = 1 and r = 1, we have that p i _ p i _ i-d = - (ln(uyi)J (inuyi) - \]n(uyi)j u3 + u3 + u2y\. Computing the one-form oj{, it is easy to verify that u\ = 0 , and this implies the function cpiyi = 0. Now, for i = 1 and r = 2,it follows that P}=Pi-F$ = Pl since F\ = 0 . Hence, the one-form Jj\ is given by -1 6^2 = 1 01,101,2 2, (u*\i — ) dyi + u du. \2/i/ Comparing with following the I/O differential equation -•? o o /-? °1,2 [pí % 9U Oi,2 l ^ % du 536 J. DELEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER This implies t h a t (fi^ = u2T h e last iteration for this o u t p u t leads to uyi<Pi,s = -P31 = p2 - Fl = uy\. Repeating t h e same procedure for z = 2, it follows t h a t P2 = P2 - F2 = 2 - ( - m / i ) + u2y2y2 u + uy\ + uyx a n d t h e one-form u\ is given by -2 1_ 1 _, cji = — ayi H du. u 2/1 By comparison with t h e I / O differential equation, we obtain t h a t ¥>2,i =uy\- Second iteration yields u2y\y2. a2,i<f°2.2 = Pi = Finally, we obtain </?2,2 = 2/i2/2T h e n t h e transformed system is of t h e form ii.i \ ii,2 ii,3 / = / 0 0 \ 0 u 0 0 r 0 2/1 0 21,1' \ *1,2 21,3 // /( 0 2 u \^, Щ\ M ( (£)=(» o X s M s J 2/l=zlA, > 2/2=z2A. T h e s t a t e coordinate transformation is 21.1 = xi, zi,2 = e X2 , Z2,\ z2,2 = X$. = X4, z1>3 = x3 T h e observer for t h e system (29) is given by ( *.i \ = / 0 0 V 0 _ / 0 u 0 0 0 \ _i 0 / / £ltl \ / 0 \ / Vi,i(^i) + «2 + ^,2^) zli2 \ z1)3 / V "J/i / V ^1.3(^1) u 2 \ / z 2>1 \ + / «yi 2 \ + / ^2jl(z2) \ , . ( *M - i / Z2 1 Z2 1 . " I 0 0 J {z2,2 ) [y y2 ) { V 2(2 (i 2 ) ) < - " ' ) u ) On Nonlinear Equivalence and Backstepping 537 Observer where the observer gains are given by / ^4,3 /- \ , /,. x &4,2 , /. v &4A V>i,u^i) = — , ^1,2(^1) = - £ - , V>i,3^i) = — uyi u2 uyi 6 ^3,2(^2) ,*x 3,l(*2) Ф2AҺ) = —^-õ , ^2,2(z2) and for the first subsystem, we obtain uyi, 0 M = 0, pi,2 = u2, 51,3 = uyi; K{ =u,K\= & 2,1 = c iA 63.1 = X + C 1,2(C1,1 ~ </>l,l) - (Ci,i - ^ 1 . 1 ) ^ 1 , 1 ~ , ?1 du x 63.2 = ^ ( C l , 2 + C i , i ) + &4,1 = ft 4,2 = 6 C 1 A - ^1,1 + — l,3(&3,1 ~ ^ 1 , 2 ) ~ (63,1 ~ 1 ^ 1 , 2 ) ^ 1 . 1 + ^ ( f e 3 A - ^1,2) - 1x^1,1)^1,2 + m / 1 7 ^ + ^ ( & 3 , 2 - ™/>l,l) (b\f2 u C ^(^l,l) + Cl,3(&3,2 ~ W ^l,l) + ^3,1 4.3 = C1.3W/1 + yi&3, 2 + ^ (txyi) • And for the second subsystem, we have Kl = ^2> 52,1 = txyi, 52,2 = y?y2; 2A = C2,l & b 3A = 1 + C2,2(C2,1 - </>2,l) ~ (C2,l ~ ^2 f l)^2,l ~ "77 W>2,l) dí d ţ ^ Ь 3,2 = W 2 ( c 2 , 2 + C 2 , l ) + dí ' 6. CONCLUSIONS The observer synthesis for nonlinear systems has been considered in this paper. Based on their equivalence to state affine systems, necessary and sufficient conditions have been given to characterize a class of nonlinear systems which can be transformed into a multivariable state affine form up to input-output injection. For this class of systems a backstepping observer approach has been presented in order to design an observer. Several examples have been given in order to illustrate the proposed methodology. 538 J. DELEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER APPENDIX A Let /C the field of meromorphic functions of a G Mx and b G Mp. u e S p a n x : ( a 6 ) { d a i , . . . , d a A , d b i , . . . ,db p }. Definition A l . A one-form u is closed if da; = 0. Definition A 2 . u = dip. A one-form u is exact if there exists a function ip(a,b) such that Proposition A 3 . Any exact one-form is closed. Lemma de Poincare A 4 . Let a; be a closed one-form of the form u e S p a n ^ ^ ^ ^ d a i , . . . , d a A , d b i , . . . ,dbp} . Then u is locally exact if and only if du = 0. T h e o r e m A 5 . Given u one-form, there exist a function ip such that SpanA:{o;} = Span K: {d'0} if and only if du A u = 0. T h e o r e m A6 (Frobenius Theorem). Let V V = Span^{cJi,... ,un} be a subspace of £. V is closed if and only if du A U\ A . . . A un, for any i = 1 , . . . , n. APPENDIX B P r o o f of T h e o r e m 1. Necessity. Assume that there exists a state transformation z = T(x) transforming system S into system Eaffine- Thus, the I/O differential equation of the system S, P$ = y\ *' is equal to P*0 ~yia% \ P a0 =Fki(aiAi-- ^ . n - O + r ^ " ^ ^ , ! , . . . , 0 ^ . - 1 - ^ , 1 - . . . ,¥>»,*<). On Nonlinear Equivalence and Backstepping Observer 539 Notice that the first term of the right hand does not depends on <£iA, • • • , <pik. ? can be written as Ejk.Ki,--- = ,OÍ,„-I) v (i)/___iL + yl(^).__ + ancj íí 1 ' ki-2 + l^Уi І=2 ) d t i +Ò (30) ІЛ where the <J*-Xj (•) are functions which depend only on functions y"' and u^l\ with / < j . The functions Fl._,, j = 1,... , k{ — 1, have the following form | л Ќ-І = *i* - + (^*'-'- 1) M (*.-J--) / • __í . dť ¥>j ) \ d2/ž (31) for j = 1,... , ki - 1; and the function F02 = </?&.t Then, the I/O differential equation can be written as d 1 pi _ (*.-i) /i,i , (i) f-*'" /?,!^ , A n ^a0-2/i ^ + V< ^ d^t._i J + ^ j where A(-) = r ^ ' " ^ , ! , . . . , 0 ^ . - 1 , ^ , 1 , . . . ,¥>»,*<) + ^1)<*j,i, and A represents to all monomials with a degree less than k{ — 2. 1 Notice that d H,i 0fitl ^dfiA. -dr = -ey-y+i:i-9UTui dkt x ~ fji _ _i°__i_,,(fa-i) , v^ ____ fl (*.-i) Ćty 1=1 Now, let us apply the first step of the algorithm. For k = 1, the one-form is given by d *' m d2Pi д*PІ0 dll "1 _C ~ (1)0 (J..-i) W _C - (f)«"S-i) i d = = 1 f^дfU, Уj iu\Ђ^ = -n-&ň,\(u,y)Ii,1 + ҳrдfІгl Ћ^àщ м 540 J. DELEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER Thus, the one-form u\ is given by Then, the conditions of Theorem 1, for d\ < p, du\ A du = 0 and du\ A dydi+1 A • • • A dyp = 0 are verified directly. The proof for 2 < k < ki — 1 follows the same lines as for k = 1. Substituting the aij functions in F£. in (30), and from equation (31), F£.. verifies ki j) pi - __LA*i-i) , +2^ V dWJ u *ki-jdyyj du i - « {^-^-V--* + E ^"1"""}+»--<(•) where the functions 0^_j(-) involves monomials depending on functions yW and u(l\ with £ < A;^ — j . Applying Step 2 for r = 1, P22 is computed as follows pi — pi _ pi _ «.(*«) en =^l--' + t^}--» - * { g ^ ^ - ' + 1 ^"i"-}+«-.<•) and set if{ = a^i. Computing the one-form u\ as follows 1 I * dPi m dPi 1 Jv^^i . f 9^1, (pi \^d\ogaiyl d ^dlogaiyi = — < __<--- dyj+__.--- d u i — \ i^—-— yj+__^—z—~dui mi \p[ dyj frf 9ut oi.1 yp[ dyj f^ dm Thus, UJ\ = d I J, and it is easy to see that the conditions \ai,lJ duJ\ Adu = 0 and du\ A dydk+1 A • • • A dyp = 0 541 On Nonlinear Equivalence and Backstepping Observer are satisfied. The necessary condition of Theorem 1 is proved for the first iteration. For proving the iterations r = 2 , . . . , ki, a similar procedure can be followed. Sufficiency: Step 1. Determination ofciij. Consider the nonlinear system S and suppose that the conditions dtJk A du = 0, and dulk A dydk+1 A • • • A dyp = 0 are satisfied. The one-form uk given by .fc ui=4 jfc d2P l m d2P1' £ %fi#^dtt+£ £ i^p 1 *" satisfies the above conditions. Then, wj. G Span{dyi,... ,dydfc}. On the other hand, the one-form obtained from the I/O differential equation P*0, satisfies the following relation .fc .fc d 2 P* <4a = 4 I ] (fc) i = 1 5y) ^ m d 2 P* (!,_*)<% + ^ £ o (fc)~ (£-fc)dli*' i=11=1 an} ;ay} ; Solving the set of (d* — 1) partial differential equations, it is possible to obtain the aij functions. This ends the proof of Step 1. Step 2. Determination ofipij. In order to obtain the functions (fiijy we assume the dij are known from Step 1, and for r = 1, replacing the function a^i, the one-form UJ\ is given by -» = 1 J v ^ ^ i . " ^ \£ * ,V^9(r°i^ m ¥>i J v ^ 5 1 o g a i f i £**•*--- \ £ ^ r a ^dlogai,i £~^rd"' On the other hand, the one-form cJjj. obtained from the I/O differential equation of the nonlinear system S and the conditions dujlk A du = 0 and duJj. A dydfc+1 A • • • A dy p = 0 allows to conclude that u\ e Span{dyi,... , dydk}. Then, the (fij can be determined as follows. Let Zi = c o l ^ i . . . Z{9kt) £ Mk\ for i = 1 , . . . .,p; and zi,i = j/i = /ii(x), where /i* is the ith component of the output equation y = h(x). 542 J. DELEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER Now, for k = 2 , . . . , ki, let be zi,k = Zi,k-1 - <Piyk-l • «i,k-l which represent the ki — 1 first dynamics of S. To compute the last dynamic equation i^*.., we note that ylh) = zitk+lKi + Pi where and diyki = 0 by construction and Pf = ( ^ i . Thus the last dynamic equation obtained as follows (ki — 1) - ^ i , f c . - l __ Vj _ zi,k-l *>^i jyi ~1A;.-1 _, r_-_ Oi,*.-l #£,__ Taking the time derivative of the above equation, it follows that _ (»!''' - f . , - i ) «.,_• - (»!'''" - q , - i ) g;,-i After substitution of the function PJ:._-, one finally gets zi,fc; = ^ i , k . - This ends the proof. APPENDIX C Let be 8/+i = Y1 ( Ь '+М " # _ - i # i - i ^ - i + i ) ^ + Äie/+i (32) i=l where s = col(8i,8 2 ,. •• ,8/,s_+i), e = c o l ( e i , e 2 , . . . ,e/+i). Now, writing in terms of the estimation error, we obtain s = M(biJ,xl>i)e (33) On Nonlinear Equivalence and Backstepping Observer 543 á(b ІJ>1>i) = 0 1 b2,i - гþi ( Ьз,i ~ Kxгþ2 ЪĄУI - K2гþз bПyi - Ьз,2 - 0 0 Kiгþi Ki-iгþn bly2 - (Kг) гþ2 Ki-зKľгþi-2 Ь/+i,2 - Kt-2Kiгþi-i ^ 0 2 ЬĄ,2 - Ki-2гþn-i \ Ьn+1,1 ~ 0 Ki . •• Ki-i bt+ij -Ki-iгþ^ (34) ) where Kr = Д ÜІ i=0 and ao = 1; the bij = bij(z) are given by, for i = 2 д 7. . . 9i 02A = Ci + - — azi (35) for i = 3 03,1 = 1 + C2(°2,l - Фl) + (02,1 - tfl) ( | f ^ - tfl) + ^(62,1 - V>l) + R l | ^ (36) ř, IV . U . d Ä "- . IV 03,2 = IÍ1C2 + Oi6 2 ,l + " — - + Ä i ð #2 — for i = 4 04.1 = 62,i -ipi + c 3 (o 3 ,i - # 1 ^ 2 ) + (03,1 - Kiip2) l -^- - Ví j + ^ ( ° 3 , i - -^1^2) + (63,2 - KM ( ^ - V2) + K2|g (37) 04.2 = O! + C 3 (03,2-Kl^l) + ^ 6 3 , 1 + (°3,2 - J f l ^ l ) ^ + ^ ( h , 2 -KM + R 2 b4,3 = C3K2 + G2&3,2 + -77 (-^2) + - Í 2 ň ^ for 4 < z < n + 1 hi = i-2A - Ki-^i-3 b + Ci-i (6Í_I,I - KÍ^Í-2) + — (bi-1,1 - + E (•»-«- - «•«*—> (gf - *) + *« (%r) KÍ^Í-2) | g 544 J. DE LEON, I. SOULEIMAN, A. GLUMINEAU AND G. SCHREIER f dgi-i \ + I^i-2 ( ß I+ bij = bi-2,i - Ki-jsKj-iipi-j-2 + Ci-i (bi-ij - Ki-j-2Ki-2il)i-j-i) üj-ibi-ij-i + -£ (&i-i,j - Ki-j-2Kj-i\l)i-j-i) ( T^J + ^2(bi-iik - Ki-k-2Kk-i^i-k-i) b^i-2 = Ki-3 + Ci-i (bi_i,i_2 ~ Kisfa) + — (bi_l,i_2 - + ai_ 3 bi_i,i_3 + (&i-i,i-2 - Kisipx) Ki-31pi) l ^ - - = - J + Ki-2 f ^ - - J &i,i-l = ifi-2Ci-l + ai_ 2 bi-l,i-2 + Ki-2 f 7 ^ j + -r-Ki-2- When / = n , where n is the dimension of the system, it is easy to see that 8n+l = Yl f &n+l,i ~ Kn-iKi-i^n-i+l + Kn-1 f "^- 2 j j e{. (38) In order to determine the gains of the observer we make the last above equation equal to zero, i. e. bn+l,i - Kn-iKi-i^n-i+l + Kn-i ( - ^ J = 0, for i = 1 , . . . , n. Then, it follows that , &n+l,i tpn-i+1 = -7Z Kn-iKi-X , K -1 + — n Kn-iKi-r fdfn\ . r -r— , for I = 11, . . . , n\ \dzij or equivalently , _ ^n+l,n-j+l Kn-jKj-i Kn-1 Kn-jKj-i ( dfn \dzn-j+i \ J ' ._ . . 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