OBSERVER DESIGN USING A PARTIAL NONLINEAR OBSERVER CANONICAL FORM

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1 Int J Appl Math Comput Sci, 26, Vol 16, No 3, OBSERVER DESIGN USING A PARTIAL NONLINEAR OBSERVER CANONICAL FORM KLAUS RÖBENACK,ALAN F LYNCH Technische Universität Dresden Department o Mathematics, Institute o Scientiic Computing Mommsenstr 13, D 162 Dresden, Germany klaus@roebenackde University o Alberta Department o Electrical and Computer Engineering Edmonton AB T6G 2V4, Canada alanl@ieeeorg This paper proposes two methods or nonlinear observer design which are based on a partial nonlinear observer canonical orm POCF Observability and integrability existence conditions or the new POCF are weaker than the well-established nonlinear observer canonical orm OCF, which achieves exact error linearization The proposed observers provide the global asymptotic stability o error dynamics assuming that a global Lipschitz and detectability-like condition holds Examples illustrate the advantages o the approach relative to the existing nonlinear observer design methods The advantages o the proposed method include a relatively simple design procedure which can be broadly applied Keywords: observer design, canonical orm, detectability 1 Introduction We consider the observer design problem or a SISO system ẋ = x+gx, u, y = hx 1 with smooth vector ields : R n R n, g : R n R R n, and smooth output unctions h : R n R Exact error linearization is a well-established observer design method based on an observer canonical orm OCF which yields linear time-invariant error dynamics in some state coordinates Since the initial work in Bestle and Zeitz, 1983; Krener and Isidori, 1983, many variations on and extensions to this design method have been proposed Kazantzis and Kravaris, 1998; Krener and Respondek, 1985; Krener et al, 1991; Krener and Xiao, 22, Lynch and Borto, 21; Marino and Tomei, 1995; Phelps, 1991, Respondek et al, 24; Rudolph and Zeitz, 1994; Wang and Lynch, 25;26; Xia and Gao, 1988;1989 In the single-output case, the aorementioned work relies on the assumption dim span{dh, dl h,,dl n 1 h}x =n 2 or all x in a suitable set The unction L h = h x in 2 is the Lie derivative o h along Repeated Lie derivatives are deined as L k h = L L k 1 h, k 1 with L h = h The dierential or gradient o a unction λ : R n R is denoted by dλ and has a local coordinate description dλ = λ x =λ x 1,, λ x n The condition 2 ensures a orm o observability or the unorced system Hermann and Krener, 1977, and is necessary to ensure the existence o the OCF Krener and Isidori, 1983 It is well known that OCF-based methods can be diicult to apply due to restrictive existence conditions Also, the condition 2 does not always hold globally or even on a suiciently large set to avoid a singular observer gain in many canonical orm designs In an eort to address these drawbacks, we propose an observer based on a partial nonlinear observer canonical orm POCF which requires a weaker condition dim span{dh, dl h,,dl r 1 h}x =r, 1 r<n 3 to hold or all x in a suitable set Additionally, less restrictive integrability conditions than those or an OCF will be required To ensure the convergence o the estimate error, we impose Lipschitz and detectability-like conditions Jo and Seo 22 also consider observer design with the weaker observability condition 3 They propose an

2 334 observer design based on ż = A z + γ y, u, ż = A z + y, z +γ y, u, 4a 4b y = c T z, 4c where A R r r and c R r 1 are in a dual Brunovsky orm Brunovsky, 197: 1 A =, 1 c T = 1 5 The system 4 is divided into two parts: the irst subsystem 4a is isolated rom the second one and is in an OCF On the other hand, the second subsystem 4b contains the term which allows or a nonlinear dependence on both the second subsystem state z and the output The output depends linearly on the irst subsystem state z Although the existence conditions or 4 are weaker than the OCF, in this paper we propose a POCF which exists under less restrictive conditions and is suitable or observer design Two observer designs based on POCF coordinates are proposed The irst design has an advantage o a simpler gain expression The second design leads to a simpler error convergence proo but involves a more complicated gain calculation This paper is organized as ollows: Section 2 presents the existence conditions or the POCF Section 3 presents two observers and a theorem or the global asymptotic convergence o their error dynamics Section 4 presents examples 2 Partial Nonlinear Observer Canonical Form POCF First, we investigate the existence conditions or a dieomorphism T transorming 1 into a partial nonlinear observer canonical orm POCF o index r {1,,n 1}: ż = Az+ αy, z r+1,,z n,u, y = c T z, 6a 6b with z =z 1,,z n T, and α = α 1 z α n z n is a smooth vector ield The matrix A R n n and the vector c R n 1 have the orm A A = and c T = c T, where c and A are deined in 5 K Röbenack and AF Lynch We recall the ollowing result on simultaneous rectiication: Theorem 1 Nijmeijer and van der Schat, 199, Thm 236 Let X 1,,X r be linearly independent vector ields deined on a neighbourhood o ξ R n Suppose that on a neighbourhood U R n o ξ [X i,x j ]=, 1 i, j r Then there exist coordinates x 1,,x n deined on U such that on U X i =, 1 i r x i We remark that when applying Theorem 1 later we will choose n r linearly independent vector ields X i,r+ 1 i n to X i, 1 i r such that about ξ [X i,x j ]=, 1 i, j n This choice is nonunique and aects the expressions or the system in the new coordinates The observer design method presented in Jo and Seo, 22 imposes additional constraints on the choice o X i, r +1 i n, which are not required here These additional constraints can limit the applicability o that approach In order to deine the POCF, we need to deine the so-called starting vector ield Ir<n, the matrix Q r = dh dl r 1 h 7 is called the reduced observability matrix When n = r, we call 7 the observability matrix A smooth solution v o Q r v = 1 =: e r R r 8 is called the starting vector ield Beore giving suicient conditions or the existence o the POCF 6, we deine some notation The Lie bracket o two vector ields and g is deined as [,g] = g x xg Repeated Lie brackets are deined as ad k g =[,ad k 1 g], k 1 with ad g = Theorem 2 There exists a dieomorphism T : U R n deined on a neighbourhood U o x transorming 1 into POCF 6 o index r i C1 rank Q r = r, C2 [ad i v, adj v]=, i, j r 1,

3 Observer design using a partial nonlinear observer canonical orm 335 C3 [g, ad i v] =, i r 2, in some neighbourhood o x The dieomorphism T is global i the conditions C1 C3 hold on R n and, in addition, C4 ad i v, i, j r 1 are complete vector ields Proo The proo is divided into two parts In Part A we show that there exists a change o coordinates ζ =Ψx which transorms 1 into ζ = Aζ + ηζ r,ζ r+1,,ζ n,u, y = c T ζ + βζ r+1,,ζ n, 9a 9b with a smooth vector ield η = η 1 ζ η n ζ n, a smooth map β, and ζ =ζ 1,,ζ n T In Part B we construct a second coordinate system in which β Part A: Assume that the conditions C1 C3 o Theorem 2 are satisied The condition C1 implies that 8 has a solution v deined on some neighbourhood o x R n Equation 8 can be rewritten as { L v L i or i r 2, h = 1 or i = r 1 From Isidori, 1995, Lem 412, this implies that dh v ad v ad r 1 v dl r 1 h 1 = 1 1 in a neighbourhood o x Thereore, the vector ields v, ad v,,ad r 1 v are linearly independent in some neighbourhood o x Using the condition C2 and Theorem 1, we deduce that there exists a local dieomorphism ζ =Ψx such that Ψ ad i v = ζ i+1, i r 1, 11 where Ψ = Ψ/x For clarity, the representations o,g, and h in the ζ-coordinates are denoted by ζ =Ψ x x=ψ 1 ζ, ḡζ,u=ψ gx, u x=ψ 1 ζ, hζ =hx x=ψ 1 ζ Owing to 1, we have L ad i vh = { h = ζ i+1 or i r 2, 1 or i = r 1 Thereore, the gradient o h has the orm h ζ = 1, 12 where the leading one on the right-hand-side o 12 appears in the r-th column Hence, in the ζ-coordinates the output map h has the orm given in 9b Next, we consider the drit vector ield ζ = 1 ζ + + ζ n ζ 1 ζ n Due to 11, or 1 i r 1 we have ζ i+1 =Ψ ad i v =Ψ [,ad i 1 v] =[ Ψ,Ψ ad i 1 v] =[ Ψ, =[, = n j=1 ζ i ] ζ i ] Comparing both sides o 13 yields j 13 ζ i ζ j j ζ i =or 1 j n, j = i +1, 1 i r 1, i+1 ζ i =1or 1 i r 1 This means that the Jacobian matrix o has the orm 14 1 ζ ζ = 1 15

4 336 Finally, we consider the input-dependent vector ield ḡ Because o the condition C3 and 11, or i r 2 we have This implies =Ψ [g, ad i v] =[Ψ g, Ψ ad i v] [ ] = ḡ, = ζ i+1 n j=1 ḡ j ζ i+1 ζ j ḡ j =, 1 j n, i r 2 16 ζ i+1 Hence, the Jacobian matrix o ḡ looks like ḡ ζ ζ,u= 17 From 14 and 16 or, equivalently, 15 and 17, we can conclude that the right-hand side o the transormed system has the orm 9 Part B: In this part we construct a second change o coordinates transorming 9 into 6 Let z =Φζ be a global dieomorphism deined by From 9b, we have 6b: z i = ζ i, i r, 1 i n, z r = ζ r + βζ r+1,,ζ n y = c T z The dynamics transorm into 6a with α i y, z r+1,,z n,u = η i zr βz r+1,,z n,z r+1,,z n,u, i r, 1 i n, α r y, z r+1,,z n,u = η r zr βz r+1,,z n,z r+1,,z n,u + n j=r+1 β η j ζ,u ζ j ζ=φ 1 z K Röbenack and AF Lynch Thereore, the dieomorphism T which transorms 1 into the POCF 6 is a composition o the transormations given in Part A and B: T =Φ Ψ Part A ixes the dependence o the system on the irst r coordinates without speciying the dependence on the remaining n r coordinates Part B only changes the dependence in the r-th coordinates to ensure that the output equals z r I the conditions C1 C3 hold globally, the condition C4 on the completeness o the vector ields implies the existence o a global dieomorphism Respondek, 1986 We remark that, i r = n, the conditions in Theorem 2 are the same as those o the OCF Krener and Isidori, 1983 Evidently, or r < n the proposed existence conditions are satisied by a larger class o systems than those admitting an OCF When n =2, we can only have a POCF o index r =1 In this case, only the condition C1 ie, dh must be checked since C2 and C3 are always satisied As is mentioned in the proo o Theorem 2, the condition C1 implies that a solution o 8 exists but is not unique This nonuniqueness can be used to simpliy the vector ields ad i v, 1 i r 1 Simpler expressions or these vector ields lead to a less complex observer design A particular solution o 8 is given by v = Q + r e r, where Q + r =QT r Q r 1 Q T r denotes the Moore-Penrose inverse Moore, Observer Design and Error Convergence We consider two observer designs which are based on the POCF 6 The irst design has an advantage o a simpler expression or its gain The second design requires the knowledge o the POCF coordinates to compute its gain When discussing observers and their convergence, it is convenient to introduce an alternative notation or the POCF We split 6 into two subsystems: ż 1 = A z 1 + α 1 y, z 2,u, ż 2 = α 2 y, z 2,u, y = c T z 1, where z 1 denotes the irst r components o z, and z 2 stands or the last n r components o z Similarly, α 1 denotes the irst r components o α, and α 2 signiies the last n r components o α 31 Observer Design No 1 We consider a Luenberger-like observer structure ˆx = ˆx+gˆx, u+kˆx y hˆx, 18

5 Observer design using a partial nonlinear observer canonical orm 337 where the gain vector k depends on the estimated state alone Assuming that the system 1 satisies the conditions o Theorem 2, we can express the observer 18 in the POCF coordinates ẑ1 ẑ 2 = A ẑ 1 + α 1 ŷ, ẑ 2,u α 2 ŷ, ẑ 2,u +S ẑ 1 k Sẑ y hˆx, 19 where S = T 1, S = x/z and ŷ = c T ẑ 1 We consider the choice k Sẑ = S ẑ l 2 with l =p,,p r 1 T, and below, in Section 33, we will appropriately assign the roots o det λi A lc T = p +p 1 λ+ +p r 1 λ r 1 +λ r 21 Substituting 2 into 19, we obtain ẑ 1 = A ẑ 1 + α 1 ŷ, ẑ 2,u+ly c T ẑ1, ẑ 2 = α 2 ŷ, ẑ 2,u 22a 22b The estimation error z = z ẑ o this observer is governed by z 1 =A lc T z 1 + α 1 y, z 2,u α 1 ŷ, ẑ 2,u, z 2 = α 2 y, z 2,u α 2 ŷ, ẑ 2,u 23a 23b An observer is typically implemented in the original x- coordinates and, ideally, to simpliy the design procedure, the gain k can be computed without requiring expressions or the POCF coordinates or related unctions α 1 and α 2 Since S is the inverse o T, we can rewrite 11 in the orm ad i vx =S T x e i+1, i r 1 Hence rom 2 we have a simple expression or the observer gain: kˆx =p vˆx+p 1 ad vˆx+ + p r 1 ad r 1 vˆx Observer Design No 2 I we choose the observer structure ˆx = ˆx+gˆx, u+kˆx, y, u 25 and require a cascade or triangular orm error dynamics then this implies that in the z-coordinates the observer is ẑ 1 = A ẑ 1 + α 1 y, ẑ 2,u+ly c T ẑ 1, ẑ 2 = α 2 y, ẑ 2,u, and the gain in 25 is k Sẑ,y,u 27a 27b = S ẑ αy, ẑ 2,u αŷ,ẑ 2,u+lc T z 1, 28 where the constant gain vector l is chosen below, in Section 33, to assign the roots o 21 Comparing 22 and 27, we remark that the observers dier in that the second one uses y in place o ŷ From this one might expect that the second design uses more exact system inormation and might lead to better convergence 33 Error Dynamics Convergence Next, we demonstrate the convergence o the observers 18, 24 and 25, 28 We treat the convergence o the observers in separate theorems and consider 18, 24 irst 331 Observer Design No 1 We begin with the ollowing assumptions: A1 The input u is bounded, ie, there exists a positive constant γ such that ut γ,t A2 The map α 1 is globally Lipschitz in y and z 2, uniormly in u, ie, there exist positive constants γ 1,γ 2 such that α 1 y, z 2,u α 1 ŷ, ẑ 2,u γ 1 ỹ + γ 2 z 2 or all y, ŷ R, z 2, ẑ 2 R n r, and any bounded u As in Amicucci and Monaco, 1998, we require a steadystate solution property o the system The next assumption is the uniorm robust steady-state solution property with respect to y: A3 There exist a positive deinite matrix P 2 R n r n r and positive constants γ 3,γ 4 such that or V 2 z 2 = z 2 T P 2 z 2 we have V 2 z 2 z 2 α2 y, z 2,u α 2 ŷ, ẑ 2,u =2 z 2 T P 2 α2 y, z 2,u α 2 ŷ, ẑ 2,u z 1 =A lc T z 1 + α 1 y, z 2,u α 1 y, ẑ 2,u, z 2 = α 2 y, z 2,u α 2 y, ẑ 2,u, 26a 26b γ 3 ỹ 2 γ 4 z or all y, ŷ R, z 2, ẑ 2 R n r, and any bounded u

6 338 The unction V 2 is also called an exponential-decay output-to-state stable OSS Lyapunov unction Sontag and Wang, 1997 Beore stating the convergence theorem, we introduce a lemma rom Röbenack and Lynch, 24 which is a slightly dierent orm o a result in Gauthier et al, 1992 Lemma 1 Given A and c deined in 5, consider the Lyapunov equation A T P θ+p θa + θpθ =c c T, 3 where θ is a positive number and P R r r Then there exists θ > such that the Lyapunov equation 3 has a positive deinite solution P θ > with P 2 θ P θ, θ θ 31 Proo It can directly be veriied that the i, j-th entry o P satisying 3 is given by p ij = 1i+j θ 2r i j+1 2r i j!, 1 i, j r r i! r j! 32 Moreover, this solution o 3 is unique and positive deinite Thereore, all eigenvalues o P are real and positive Due to 32, all entries o P converge to as θ Hence, the eigenvalues o P also converge to as θ and there exists θ > such that the eigenvalues o P are less than 1 or all θ θ Theorem 3 Consider the system 1 together with the observer 18 and the observer gain 24 Assume that the conditions C1 C4 hold and, under Assumptions A1 A3, there exists a vector l R r such that lim ˆxt xt = t or all initial values x and ˆx o 1 and 18, respectively Proo Our proo is based on the work Gauthier et al, 1992 Assuming that the conditions C1 C4 hold, convergence can be analysed in the POCF coordinates We have to show that the equilibrium z =o 23 is globally asymptotically stable Let P R r r be a positive deinite matrix which will be speciied later, and take the positive deinite matrix P 2 rom Assumption A3 Then the candidate Lyapunov unction with V z 1, z 2 =V 1 z 1 +V 2 z 2 V 1 z 1 = z T 1 P z 1 and V 2 z 2 = z T 2 P 2 z 2 K Röbenack and AF Lynch is positive deinite and radially unbounded The time derivative o V 1 along 23a is d dt V 1 z 1 23a = z 1 T [ A lc T T P + P A lc T ] z 1 +2 z T 1 P [ α 1 y, z 2,u α 1 ŷ, ẑ 2,u ] 33 We choose the gain vector as Hence we have l = ν 2 P 1 c with ν> 34 A lc T T P + P A lc T =AT P + PA νc c T 35 Using A2, we obtain 2 z T 1 P [ α 1 y, z 2,u α 1 ŷ, ẑ 2,u ] 2 z T 1 P [ α 1 y, z 2,u α 1 ŷ, ẑ 2,u ] 2 P z 1 α 1 y, z 2,u α 1 ŷ, ẑ 2,u 2 P z 1 γ 1 ỹ + γ 2 z 2 2γ 1 P z 1 c T z 1 +2γ 2 P z 1 z 2 36 γ1 2 z 1 T P 2 z 1 + z 1 T c c T z 1 + γ2 2 zt 1 P 2 z 1 + z 2 T z 2 37 γ1 2 + γ2 2 z 1 T P 2 z 1 + z 1 T c c T z 1 + z 2 T z 2 38 or all > Going rom 36 to 37 we have used ab δa 2 +b/δ 2, δ R\{},a,b R Combining 33, 35, and 38 results in d dt V 1 z 1 23a z 1 T A T P + PA z1 + z 2 T z 2 + z 1 T γ1 2 + γ2 2 P 2 ν 1c c T z 1 39 Using Assumption A3, a bound on the time derivative o V 2 along 23b is given by 29: d dt V 2 z 2 γ 3 ỹ 2 γ 4 z b γ 3 z T 1 c c T z 1 γ 4 z T 2 z 2 4 From 39 and 4 we collect the terms with z 2 2 : γ 4 z

7 Observer design using a partial nonlinear observer canonical orm 339 This quadratic orm is negative deinite or any,γ 4 Next, we collect the terms with z 1 occurring in 39 and 4 and obtain [ z 1 T A T P + PA ν 1 γ 3 c c T + γ 1 + γ2 ] z 2 P Take θ rom Lemma 1 and choose { } θ>max θ, γ 1 + γ2 2 and ν>γ 3 Using Lemma 1, the matrix P is the unique solution o A T P θ+p θa + θpθ =c c T Then the quadratic orm 42 can be bounded as z T 1 [ A T P + PA ν 1 γ 3 c c T + γ 1 + γ2 2 P 2] z 1 z T 1 z T 1 z T 1 [ γ 1 + γ2 2 P 2 θp ν γ 3 c c T ] z 1 [ ] γ 1 + γ2 2 P 2 θp z 1 [ ] γ 1 + γ2 2 θp P z 1, 43 where we employed 31 Since 41 and 43 are both negative deinite, we conclude that V z 1, z 2 < or z 1, z 2, 23 Thereore, V is a Lyapunov unction o 23 and the equilibrium z 1, z 2 =, is globally asymptotically stable 332 Observer Design No 2 We require Assumption A1 and the ollowing two modiied versions o Assumptions A2 and A3: A4 The map α 1 is globally Lipschitz in z 2 uniormly in y and u, ie, there exists a positive constant γ 2 > such that α 1 y, z 2,u α 1 y, ẑ 2,u γ 2 z 2 or all y R, z 2, ẑ 2 R n r, and any bounded u A5 There exist a positive deinite matrix P 2 R n r n r and a positive constant γ 4 such that or V 2 z 2 = z T 2 P 2 z 2 we have V 2 z 2 α 2 y, z 2,u α 2 y, ẑ 2,u z 2 =2 z 2 T P 2 α 2 y, z 2,u α 2 y, ẑ 2,u γ 4 z 2 2 or all y R, z 2, ẑ 2 R n r, and any bounded u 44 The convergence result or the error dynamics 26 is given by the ollowing theorem, whose proo is based on Theorem 3 Theorem 4 Consider the system 1 together with the observer 25, where the observer gain is given by 28 Assume that the conditions C1 C4 hold Under Assumptions A1, A4, and A5, there exists a vector l R r such that lim ˆxt xt = t or all initial values x and ˆx o 1 and 25, respectively Proo The proo is identical to that o Theorem 3 with γ 1 = γ 3 = Hence we require { } θ>max θ, γ2 2, and ν>2 and, as beore,,γ 4 With the values o θ and ν satisying these inequalities, we can compute l using 3 and 34 It is important to note that although the stability results in Theorem 3 and 4 are stated globally, ollowing the results in Gauthier et al, 1992 or Shim et al, 21, we can obtain semi-global stability results with weaker conditions, suicient or most practical applications In particular, we do not require a global Lipschitz assumption or a semi-global result 4 Examples 41 Synchronous Machine Neglecting damper windings, armature resistance, time derivatives o stator lux linkages and back-em in stator voltage expressions, a synchronous motor can be expressed in state space orm as ollows Birk and Zeitz, 1988; Keller, 1986; Mukhopadhyay and Malik, 1972: ẋ 1 = x 2, ẋ 2 = B 1 A 1 x 2 A 2 x 3 sin x B 2 sin2x 1, ẋ 3 = u D 1 x 3 + D 2 cos x 1, y = x 1, 45

8 34 kˆx, y = p 1 A 1 y ˆx 1 p A 1 p 1 + A 2 1 y ˆx 1 A 2ˆx 3 sin y sin ˆx 1 B 2 D 2 cos y cos ˆx 1 K Röbenack and AF Lynch sin2y sin2ˆx1 48 The measured output and the irst state component x 1 denote the rotor position, x 2 is the rotor velocity, and x 3 is the ield winding lux linkage The control u is proportional to the voltage applied to ield winding The observability matrix Q 3 x 1 = 1 A 2 x 3 cos x 1 B 2 cos2x 1 A 1 A 2 sin x 1 is not regular or x 1 πz The unique starting vector ield or Q 3 satisying 8 is vx = 1, A 2 sin x 1 which is not deined or x 1 πz Since [ad 1 v, ad 2 v], the integrability condition or the OCF is not ulilled Krener and Isidori, 1983 Further, adding an output transormation does not lead to an OCF We consider the observer design proposed in Section 31 with the index r =2 We remark that, in general, the proposed method allows or a range o choice or r The reduced observability matrix Q 2 has the orm Q 2 = 1 1 A starting vector ield satisying 8 is v = Q + 2 e 2 =, 1, T This v results in ad v =1, A 1, T We supplement this vector with the vector w 1 =,, 1 T so that the Jacobian matrix S z = 1 1 A 1 1 is nonsingular We compute the transormations x = Sz and z = T x that are linear: x 1 = z 2 x 2 = z 1 A 1 z 2 x 3 = z 3 and z 1 = A 1 x 1 + x 2, z 2 = x 1, z 3 = x 3 Applying this transormation to 45 yields ż 1 z 1 ż 2 = 1 z 2 ż 3 + z 3 B 1 A 2 z 3 sin z 2 B 2 2 sin2z 2 A 1 z 2, } u D 1 z 3 + D 2 cos z 2 {{ } αz 2,z 3,u y = z 2 46 The second subsystem has the orm ż 3 = u D 1 z 3 + D 2 cos z 2 47 This system is linear i we consider the signals u and z 2 as time-dependent inputs Its unorced dynamics have an asymptotically stable equilibrium at z 3 =or D 1 = 3222 > The observer gain 28 has the orm 48 For the simulation parameters A 1 =273, A 2 =121, B 1 =3919, B 2 = 484, D 1 =3222, D 2 =19, and u 1933 were used The initial conditions are x = 8, 1, 1 T and ˆx =,, T all variables are per unit The observer eigenvalues were placed at 1, ie, p = 1 and p 1 = 2 The simulation results are shown in Fig 1 The slow convergence o the proposed observer is due to exp D 1 t resulting rom the second subsystem 47 It is important to note that the example does not admit an OCF Krener and Isidori, 1983 or a partial nonlinear observer orm Jo and Seo, 22 Also, extended Luenberger observer design leads to very large expressions Birk and Zeitz, 1988 We remark that the observability condition 2 is only satisied locally and there are advantages to not having the observer depend on the inverse o the observability matrix as this avoids singularities in the observer gain This inverse appears in most high-gain designs and other related methods based on canonical orms Finally, the example illustrates the computationally simple nature o the design 42 Magnetic Levitation System Under standard modelling assumptions, a one degree-o-reedom mag-

9 Observer design using a partial nonlinear observer canonical orm 341 v = ξ 3,wehave [v, g]x = ξ 3 x1 ψ x 2 mx 2 Fig 1 Trajectories o the motor example netic levitation system can be modelled by x 1 x 3 Rx 1x 2 x 2 x 2 2β x= x 3 g βx2, 2β gx, u= u, 49 1 mx 2 2 y = hx =x 2 Here x 1 is the coil current, x 2 the shited rotor position, x 3 the rotor velocity Schweitzer et al, 1994, and g, m, R, β are positive constants As the rotor makes physical contact with the coil at x 2 = c>, we must have x 2 c An OCF does not exist or the system 49 This can be seen by irst transorming the system to observable orm coordinates ξ 1 = ψx 2,ξ 2 = L ξ 1 = ψ x 2 x 3,ξ 3 = L 2 ξ 2, which include an output transormation denoted by ψ Krener and Respondek, 1985 We transorm the input vector ield g into the observable orm coordinates gx = ξ x gx = x 1ψ x 2 mx 2, where g is the representation o g in the ξ =ξ 1,ξ 2,ξ 3 T coordinates Since the Jacobian matrix ξ x has the orm ξ x = ψ x 2 ψ x 2 x 3 ψ x 2, we necessarily have ξ3 x 1 or ξ x to be nonsingular Since the starting vector in observable orm coordinates is Thereore an OCF including an output transormation does not exist Krener and Respondek, 1985 We consider a transormation to the POCF o index r =2 We have v = Q + 2 e 2 =,, 1 T and ad v = x 1 /x 2, 1, T Deining the complete vector ield w 1 = x 2,, T as the last column o the Jacobian matrix S z = x 1 /x 2 x 2 1 1, we ensure S to be nonsingular or x 2 c and [v, w 1 ]=, [ad v, w 1 ]= Letting Ψ t v x denote the low o the vector ield v, wehave Ψ z1 v x = x 1 x 2 z 1, x 1 z 2 + x 2 Ψ z2 ad v x x = 2 z 2 + x 2, x 3 Ψ z3 w 1 x = x 2 z 3 + x 1 x 2 x 3 Taking the composition o these lows and letting x =,c, T, we obtain x = Sz =Ψ z1 v z = T x = x 2 c, x 1 /x 2 Ψ z2 ad v Ψz3 w 1 x = x 3 z 3 z 2 + c z 2 + c z 1, see Nijmeijer and van der Schat, 199, Thm 236 The transormation T is a dieomorphism on {x R 3 :

10 342 x 2 >c} Transorming 49 into a POCF, we obtain ż 1 z 1 ż 2 = 1 z 2 ż 3 + y = z 2 z 3 g βz2 3 m u Rc + z 2 z 3, } 2β {{ } αz 2,z 3,u We consider the second observer design described in Section 32 The second subsystem is ż 3 = u Rc + z 2z 3 2β 5 and, since z 2, 5 has an exponentially stable equilibrium at z 3 = when u =, and hence it satisies Assumption A5 Although Assumption A4 is not satisied globally, we have ensured global error convergence as the irst error dynamics subsystem is LTI driven by a decaying input, β z 1 l 1 z 1 = + 2m z2 3 ẑ3 2, z 2 1 l 2 z 2 and hence or all z 1, z 2 T R 2, z 1, z 2 T as t Simulations were perormed using estimated state eedback to implement state state eedback linearizing control which tracks a square wave-like reerence trajectory shown in Fig 2 The parameter values were identiied rom an actual physical system: g = 981 m/s 2, β = 766 kg m 3 /s 2 A 2, c =4mm, m =68 kg, R =11Ω The observer eigenvalues were taken at 5 which leads to p = and p 1 = 1 The initial conditions were taken at x = 5 A,, T The corresponding estimate error trajectories are shown in Fig 3 5 Conclusion This paper has presented two observer designs or nonlinear systems based on a new partial nonlinear observer canonical orm POCF, a detectability condition, and a Lipschitz assumption The POCF exists under weaker conditions than the well-established OCF Krener and Isidori, 1983 and the existing partial observer canonical orms Jo and Seo, 22 Two observer designs are provided The irst design has an advantage o a simple gain Pos mm Trk Err mm Vel mm/s uv A x 1 Pos error mm Vel error mm/s Current error A 2 1 K Röbenack and AF Lynch Time s Fig 2 Trajectories o the magnetic levitation example Pos Re Time s Fig 3 Estimate errors or the magnetic levitation example expression The second design leads to a simpler error convergence proo but requires a more complicated gain ormula Two examples illustrate the design method The synchronous generator example involves an observability matrix which is only locally nonsingular; it illustrates how the proposed design avoids the problems o inverting this matrix This inversion is required in many canonical orm designs and is not possible at points where the system is not observable Hence, the proposed designs can admit a wide region o operation Neither o the examples admits an OCF, as Lie bracket conditions do not hold As the proposed approach involves weaker Lie bracket conditions, it is also more broadly applicable or this reason Reerences Amicucci G L and Monaco S 1998: On nonlinear detectability J Franklin Inst 335B, Vol 6, pp

11 Observer design using a partial nonlinear observer canonical orm 343 Bestle D and Zeitz M 1983: Canonical orm observer design or non-linear time-variable systems Int J Contr, Vol 38, No 2, pp Birk J and Zeitz M 1988: Extended Luenberger observer or non-linear multivariable systems Int J Contr, Vol 47, No 6, pp Brunovsky P 197: A classiication o linear controllable systems Kybernetica, Vol 6, No 3, pp Gauthier J P, Hammouri H and Othman S 1992: A simple observer or nonlinear systems Application to bioreactors IEEE Trans Automat Contr, Vol 37, No 6, pp Hermann R and Krener A J 1977: Nonlinear controllability and observability IEEE Trans Automat Contr, Vol AC-22, No5, pp Isidori A 1995: Nonlinear Control Systems: An Introduction, 3-rd Edn London: Springer Jo N H and Seo J H 22: Observer design or non-linear systems that are not uniormly observable Int J Contr, Vol 75, No 5, pp Kazantzis N and Kravaris C 1998: Nonlinear observer design using Lyapunov s auxiliary theorem Syst Contr Lett, Vol 34, pp Keller H 1986: Entwur nichtlinearer Beobachter mittels Normalormen Düsseldor: VDI-Verlag Krener A and Xiao M 22: Nonlinear observer design in the siegel domain SIAM J Contr Optim, Vol 41, No 3, pp Krener A, Hubbard M, Karaham S, Phelps A and Maag B 1991: Poincaré s linearization method applied to the design o nonlinear compensators Proc Algebraic Computing in Control, Vol 165 o Lecture Notes in Control and Inormation Science, Berlin: Springer, pp Krener A J and Isidori A 1983: Linearization by output injection and nonlinear observers Syst Contr Lett, Vol 3, pp Krener A J and Respondek W 1985: Nonlinear observers with linearizable error dynamics SIAM J Contr Optim, Vol 23, No 2, pp Lynch A and Borto S 21: Nonlinear observers with approximately linear error dynamics: The multivariable case IEEE Trans Automat Contr, Vol 46, No 7, pp Marino R and Tomei P 1995: Nonlinear Control Design: Geometric, Adaptive and Robust London: Prentice Hall Moore E H 192: On the reciprocal o the general algebraic matrix Bull Amer Math Soc, Vol 26, pp Mukhopadhyay B K and Malik O P 1972: Optimal control o synchronous-machine excitation by quasilinearisation techniques Proc IEE, Vol 119, No 1, pp Nijmeijer H and van der Schat A J 199: Nonlinear Dynamical Control Systems New York: Springer Phelps A R 1991: On constructing nonlinear observers SIAM J Contr Optim, Vol 29, No 3, pp Respondek W 1986: Global aspects o linearization, equivalence to polynomial orms and decomposition o nonlinear control systems Proc Algebraic and Geometric Methods in Nonlinear Control, Dordrecht: Reidel, pp Respondek W, Pogromsky A and Nijmeijer H 24: Time scaling or observer design with linearizable error dynamics Automatica, Vol 4, No 2, pp Röbenack K and Lynch A F 24: High-gain nonlinear observer design using the observer canonical orm J Franklin Institute, submitted Rudolph J and Zeitz M 1994: A block triangular nonlinear observer normal orm Syst Contr Lett, Vol 23, pp 1 8 Schweitzer G, Bleuler H and Traxler A 1994: Active Magnetic Bearings Zürich: VDF Shim H, Son Y I and Seo J H 21: Semi-global observer or multi-output nonlinear systems Syst Contr Lett, Vol 42, No 3, pp Sontag E D and Wang Y 1997: Output-to-state stability and detectability o nonlinear system Syst Contr Lett, Vol 29, No 5, pp Wang Y and Lynch A 25: Block triangular observer orms or nonlinear observer design Automatica, submitted Wang Y and Lynch A 26: A block triangular orm or nonlinear observer design IEEE Trans Automat Contr, to appear Xia X-H and Gao W-B 1988: Non-linear observer design by observer canonical orm Int J Contr, Vol 47, No 4, pp Xia X H and Gao W B 1989: Nonlinear observer design by observer error linearization SIAM J Contr Optim, Vol 27, No 1, pp Received: 19 April 26 Revised: 16 June 26

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