ON FLATNESS OF NONLINEAR IMPLICIT SYSTEMS

ON FLATNESS OF NONLINEAR IMPLICIT SYSTEMS Paulo Sergio Pereira da Silva, Simone Batista Escola Politécnica da USP Av. Luciano Gualerto trav. 03, 158 05508-900 Cidade Universitária São Paulo SP BRAZIL Escola de Ciência e Tecnologia da UFRN Avenida Hermes da Fonseca, n o 1111 59015-001 Tirol Natal-RN BRAZIL Emails: paulo@lac.usp.r, ssimoneatista@ig.com.r Astract This work considers a semi-implicit system, that is, a pair (S, y), where S is a explicit system descried y a state representation ẋ(t) = f(t, x(t), u(t)), where x(t) R n and u(t) R m, which is suject to a set of algeraic constraints y(t) = h(t, x(t), u(t)) = 0, where y(t) R l. An input candidate is a set of functions v = (v 1,..., v s), which may depend on time t, on x, and on u and its derivatives up to a finite order. Forgetting some technical assumptions, one says that this input is proper, if there exists some state z such that the implicit system admits a proper state representation ż = g(t, z, v). Given an implicit system and an input candidate v, the prolem studied in this paper is the question of verifying if v is proper input and if the corresponding state representation is linearizale y regular static-state feedack (considering the state z and the input v). Under some mild assumptions, the main result of this work shows necessary and sufficient conditions for the solution of this prolem. These solvaility conditions may e computed without the knowledge of z, and they rely on an integraility test that regards the explicit system S. The approach of this paper is the infinite-dimensional differential geometric setting of (Fliess et al., 1999). Keywords Nonlinear systems, flatness, exact linearization, state representations, implicit systems, realization theory, DAE s, differential geometric approach, diffieties 1 Introduction The prolem that is considered in this paper is defined in this section. The statement of this prolem is a little it imprecise for the moment, since this section concerns only the main ideas. Let an implicit system e given y : ẋ(t) = f(t, x(t), u(t)) y(t) = h(t, x(t), u(t)) = 0 (1) where 1 x(t) R n, u(t) R m. Let s stands for the set 1, 2,..., s. Given a set of functions v = (v 1,..., v s ), called input candidate, where v i = φ i (t, x(t), u(t),..., u (αi) ), α i N, i s, one says that v is a proper input, if there exists a set z = (z 1,..., z q ) of functions z = ϑ(t, x(t), u(t),..., u (β) ) such that the system admits a state representation ż(t) = g(t, z(t), v(t)) (2) The prolem studied in this paper regards the following aspects (a) To check if v is a proper input. () If (a) is true, then one may try to check if (2) is locally linearizale y regular static-state feedack. 1 Note that x(t) and u(t) are not necessarily the state and the input of the implicit system since the algeraic constraints and their derivatives may induce (differential) relations linking their components. Note that static-state feedack of this prolem considers the state z and is applied to the input v, that is, is of the form v(t) = V(t, z(t), η(t)), where η(t) is the new input. The closed loop system is given y the linear controllale time-invariant system ζ(t) = Aζ(t) + Bη(t) where ζ is a new state. Note that v is not necessarily a physical input of the implicit system, ut it may e a virtual input, as in the context of ackstepping (Krstic et al., 1995). Under some mild assumptions, the main result of this work shows necessary and sufficient conditions for the solution of this prolem. These conditions rely on an integraility test that is computed from the explicit system S that is otained y disregarding the constraints y 0. Furthermore, these conditions may e computed without the knowledge of the state z of (2). The prolem of verifying if v is a proper input of the implicit system has een solved in (Pereira da Silva and Batista, 2010). The results of (Pereira da Silva, 2008) are comined with the ones of (Pereira da Silva and Batista, 2010) in order to solve this prolem. The approach of this paper is the infinitedimensional differential geometric setting of (Fliess et al., 1999). The survey (Pereira da Silva et al., 2008) presents the results aout this approach that are considered here. For completeness, a very rief summary of this approach is presented in appendix A, which introduces some standard vocaulary and notation of this approach. 1426

The field of real numers will e denoted y R. For simplicity, we ause notation, letting (z 1, z 2 ) stand for the column vector (z1 T, z2 T ) T, where z 1 and z 2 are also column vectors. Let x = (x 1,..., x n ) e a vector of functions (or a collection of functions). Then dx stands for the set dx 1,..., dx n. Let S e a system with Cartan field d (see appendix A). The Lie derivative L d η of a function (or a form) η will e denoted y η (or η (1) ) and the k-fold Lie deriva- tive L k d η of η will e denoted y η (k). If η = (η 1,..., η m ) is a set of functions (or forms), then η (k) stands for η (k) = (η (k) 1,..., η(k) m ). A codistriution Ω = span ω i, i A is said to e integrale if the exterior derivatives of each ω i can e expressed as dω i = j F η j ω j for convenient one forms η j, j F, and F is a finite set 2. Cartan s version of the Froenius theorem can e used in the context of diffieties for finite dimensional integrale distriutions Γ, when Γ is nonsingular (see (Pomet, 1995; Pereira da Silva et al., 2008)). 2 Some facts aout implicit systems Consider an implicit system of the form (1), and suppose that also that all the functions defining (1) are smooth. One will call x(t) R n the pseudo-state and u(t) R m will e called the pseudo-input 3. Recall from (3), that S is otained from y disregarding the constraints y 0. Furthermore, the functions y = h(t, x, u) are considered to e outputs of S. Throughout this paper, the system is the implicit system defined y (1), and S is the explicit system given y ẋ(t) = f(t, x(t), u(t)) S : (3) y(t) = h(t, x(t), u(t)) System S can e viewed as a diffiety with Cartan field d and output4 y = h(t, x, u), in the framework of (Fliess et al., 1999) (see appendix A). Then y (k) stands for the function L k d y = dk k y defined on S, which may depend on x, u (0), u (1),.... The following codistriution, defined on the system (diffiety) S given y (3), will e used in the sequel Y = span, (dy (k) : k N). (4) Definition 1 A local output susystem 5 Y for the explicit system S with output y defined y (3), is a diffiety Y and a Lie-Bäcklund sumersion 2 This means that the differential ideal generated y Ω is differentially closed). 3 This terminology is justified in (Pereira da Silva and Batista, 2010; Pereira da Silva et al., 2008) 4 It must e pointed out again that y = h(t, x, u) is regarded an output rather than a constraint. 5 See (Pereira da Silva and Corrêa Filho, 2001; Pereira da Silva et al., 2008). π : U S Y, where U S is an open suset, such that π (Tξ Y ) = Y ξ for all ξ U. A local state representation ((x a, x ), (u a, u )) of S is said to e strongly adapted 6 to the output susystem Y if (A) The Lie-Bäcklund sumersion π is locally given y π(t, x a, x, (u (j) a, u (j) : j N)) = (t, x a, (u (j) a, j N)). (B) The local state equations of S are of the form ẋ a = f a (t, x a, u a ) (5a) ẋ = f (t, x a, x, u a, u ) (5) where (5a) are the local state equations 7 for Y. (C) span dx a, (du (k) a : k N) = span dy (k) : k N. (D) The set of functions x a, u a (k) : k N) is contained in the set y (k) : k N. Remark 1 It is important to point out that the components of the input (u a, u ) are redefined, that is, they are not necessarily a reordering of the original input u of S. The same remark applies to the components of (x a, x ), with respect to the original state x. Let e the suset of the explicit system (diffiety) S defined y the points of S for which all the Lie derivatives y (k) = d k y vanish: = ξ S y (k) (ξ) = 0, k N (6) Definition 2 An implicit system (defined y (1)) is regular if. There exists a local output susystem Y for system S with output y around all ξ. Around all ξ, system S admits local state equations that are strongly adapted to Y. Sufficient conditions for showing that a given implicit system is regular are given in (Pereira da Silva and Corrêa Filho, 2001; Pereira da Silva et al., 2008). They are essentially linked to the notion of zero dynamics that appears in the decoupling theory. For completeness, these results are re-stated in Appendix B, which shows how 6 The weaker definition of adapted state equations considered in Theorem 4.3 of (Pereira da Silva and Corrêa Filho, 2001) is otained if one replaces the assumptions (C) and (D) y the only assumption that Y = span, dx a, (du (k) a : k N). This last theorem also shows that the output susystem is locally unique up to local Lie-Bäcklund isomorphisms. 7 Using (C) and (D), one may show that f a does not depend on t. 1427

to compute the strongly adapted state representation. The following definition regards an implicit system as an immersed sumanifold. Definition 3 Consider an implicit system defined y (1) and the explicit system S defined y (3). A diffiety Γ is said to e equivalent to the implicit system if: There exists a Lie-Bäcklund immersion ι : Γ S. For every solution σ(t) of, there exists a solution ν(t) of Γ such that σ(t) = ι ν(t). The equivalent system Γ is said to e canonical if Γ S, the topology of Γ is the suset topology, ι is the insertion map, and Γ is a control system 8. The concept defined aove is compatile with the notion of equivalence y endogenous feedack. In fact, it is show in (Pereira da Silva et al., 2008) that, if an implicit system is equivalent to Γ 1 and is also equivalent to Γ 2, then Γ 1 and Γ 2 are equivalent y endogenous feedack. It can e shown that a regular implicit system defined y (1) is equivalent to an immersed system in the explicit system S defined y (3). This result is the Proposition 1 elow. Proposition 1 (Pereira da Silva and Corrêa Filho, 2001; Pereira da Silva et al., 2008) Let e a regular implicit system defined y (1) and let S e the explicit system associated to (3). Then the suset S defined y (6) has a canonical structure of immersed (emedded) sumanifold of S such that the canonical insertion ι : S is a Lie-Bäcklund immersion 9. Furthermore admits a local classical state representation around every point ξ. In particular, is canonically equivalent to. The idea of the proof of Proposition 1 is to consider the local state representation (5) that is strongly adapted to the output susystem. It is shown that t, x a, x, U a, U and t, x, U are respectively local coordinates 10 for S and, where U a = u (j) a : j N and U = u (j) : j N. In these coordinates ι(t, x, U ) = (t, 0, x, 0, U ). It must e pointed out that the proof of proposition 1 shows also that the local state equations for are given y ẋ = f (t, 0, x, 0, u ). 8 By definition, Γ is a control system if it locally admits a state representation around every point γ Γ. 9 Since ι injective, it can e shown that Cartan field of may e canonically defined y ι = d ι, where d is the Cartan field of S. 10 Using the same name of x as a set of local coordinate functions of and S is an ause of notation. One could write for instance x and consider that x = x ι. In particular, the implicit system is equivalent to a kind of zero dynamics, as pointed out in (Byrnes and Isidori, 1991; Krishnan and McClamroch, 1994). Note that these notions of equivalence allow one to define a state representation of an implicit system as eing a state representation of an equivalent system. 3 Adapted projections Given any state representation ( x, ũ) defined on U S, with x = (x a, x ) and ũ = (u a, u ) of S that is adapted to the output susystem Y, its clear that the C (U)-module U = span, dx a, dx, (du (k) a, du (k) : k N) is locally decomposed as U = B Y, where Y = span, dx a, (du (j) a : j N) and B = span dx, (du (j) : j N). Here one may regard U = B Y, B, and Y as modules over C (U). Another possile point of view is to work pointwise at some ξ U. Then, U ξ, B ξ, Y ξ ecome vector spaces over R. Recall that a one-form defined on U may e written as n a n ω = α 0 + α i dx ai + β i dx i (7) + m a j=0 k=0 γ jk du (k) m a j + j=0 k=0 ɛ jk du (k) j for adequate smooth functions α i, β i, γ jk, ɛ jk. Around any ν U, a state representation defines a local coordinate system, so there exists some open neighorhood V ν of ν, such that only a finite suset of those functions could e nonzero on V ν. However, for the sake of defining our projection θ, one may consider the previous infinite sum without any prolem. One may define locally the projection θ : B Y B, called adapted projection, which associates a one-form ω to its projection n m θ(ω) = β i dx i + ɛ jk du (k) j. (8) k=0 j=1 This projection is clearly a module morphism (or a linear map etween vector spaces, when one works pointwise). 4 Proper Inputs The notion of a proper input v is now stated in a precise manner. Definition 4 (Proper input) Assume that a implicit system (1) is regular. Consider the explicit system S with output y defined y (3). Let v = (v 1,..., v s ) e a set of functions defined around a point ν S, where is defined 1428

y (6) 11. The set v is the input candidate of the implicit system. The input candidate v is said to e a proper input around some point ν if there exists a local proper state representation ((z a, z), (v a, v)) for the explicit system S, defined around ν, that is strongly adapted to susystem Y. Remark 2 Roughly speaking, in the prolem of testing if v is proper, note that the input candidate v is the data, and the question is to verify the existence of such z. The following theorem gives necessary and sufficient conditions for the solution a input candidate v to e a proper input. Theorem 1 (Pereira da Silva and Batista, 2010) Let ((x a, x ), (u a, u )) e a local proper state representation of system S defined y (3) that is adapted to Y. Let θ e the associated adapted projection (see section 3). Let Y e the codistriution, defined on S y (4). Let γ e the least nonnegative integer 12 such that one may locally write that span dv span dx, du (0),..., du (γ) Y. Then v is a proper input around some ν S if and only if there exists a non-negative integer δ such that, for the codistriution Γ 0 defined y Γ 0 = span dx, du (0),..., du (γ) θ ( dv (0)),..., θ and for the codistriutions Γ k defined y, ( dv (δ)) (9) Γ k = span ω Γ k 1 ω Γ k 1 + Y (10) one has: (i) Γ k is finite dimensional and nonsingular and dim Γ k 1 dim Γ k = dim v for k = 1,..., δ + 1. (ii) Γ δ+1 Y is integrale. (iii) Γ 0 = Γ 1 span θ ( dv (δ)). (iv) The set V (k) = θ(dv (k) ) is locally linearly independent for k = 0,..., δ. An intrinsic version of the last theorem 13 can e found in (Pereira da Silva and Batista, 2010). However, the presented version is much more suitale for computations. 5 A sufficient condition of feedack-linearizaility This section presents the main result of this paper. One will comine the results of (Pereira da 11 Recall that the components of v may depend on t, x, u, u (1),... 12 The existence of the integer γ is assured y the fact that a state representation is a local coordinate system. 13 Intrinsic in the sense that it does not depend on the choice of a particular adapted state representation. Silva, 2008) with the ones of (Pereira da Silva and Batista, 2010) in order to otain a sufficient condition of flatness of the implicit system (1). Theorem 2 Let v e an input candidate that is a local proper input of the regular implicit system 1 around some ξ (see (6)). Assume that δ is the corresponding integer, whose existence is assured y Theorem 1, and let Γ δ+1 e the corresponding codistriution that is defined in that Theorem. Let D 0 = Γ δ+1 and let D k = span ω D k 1 ω D k 1. Assume that (a) The codistriutions D k are nonsingular at ξ for k N. () There exists k N ig enough, such that D k = 0. (c) The codistriutions D k + Y are locally integrale around ξ, for k N. Let (2) e local state equations for the implicit system (1). Then this state representation is linearizale y a regular static state feedack of the form v = V(t, z, η). Proof (Sketch): According to (Pereira da Silva and Batista, 2010), the explicit system S defined y (3) admits a strongly adapted local state representation ((x a, z), (u a, v)) with local state equations given y ẋ a = f a (t, x a, ũ a ) (11a) ż = f z (t, z, v, x a, ũ a ) (11) Recall the context of the proof of proposition 1 (see (Pereira da Silva et al., 2008)). Let ι : S e the Lie-Bäcklund immersion that is locally defined y (t, z, (v (k) : k N)) (t, z, (v (k) : k N), x a, (u (k) a : k N)), where (x a, (u (k) a : k N)) = 0. Since the state representation is strongly adapted, then Y = span dy (k) = span dx a, (du (k) a : k N)). Then it is easy to show that ι Y = 0. From this, it is easy to show from the fact that Γ δ+1 + Y = span, dz + Y (see (Pereira da Silva and Batista, 2010)) that ι (Γ δ+1 + Y) = span, dz. (12) From the fact that ι is Lie-Bäcklund, it follows that, locally, L ι ω = ι L d ω for every oneform ω defined on S, where is the Cartan field of and d is the Cartan-field of S. From this, and from (12) it follows that, if one defines E k = ι D k, k N, then it is easy to show that E k = span ω E k 1 ω E k 1, k N. Then, from the fact that ι is an isomorphism (pointwise) when restricted to D 0 + span, it is easy to see that conditions (a), (), (c) implies that (A) The codistriutions E k are locally nonsingular at for k N. (B) There exists k N ig enough, such that 1429

E k = span. (C) The codistriutions E k are locally integrale around ξ, for k N. The desired result follows easily, since (A), (B), and (C) are the static linearizaility conditions of (Pereira da Silva, 2008, Theo. 2). Remark 3 It is shown 14 in (Pereira da Silva and Batista, 2010) that, under the assumptions of Theorem 3 (see Appendix B), the codistriution Γ 0 = Γ 0 + Y is given y Γ 0 = span dx, du (0),..., du (γ), dv (0),..., dv (δ) + Y and it is also shown that the codistriutions Γ k + Y, k N do not depend on the particular (strongly) adapted state representation that is chosen. In particular, it is easy to show that the conditions of theorem 2 are also independent of this choice. 6 Example Consider the following academic example (see (Pereira da Silva and Batista, 2010)) ẋ 1 (t) = u 1 + 2x 3 u 2 ẋ 2 (t) = tx 3 + 2(x 1 + 1)u 1 + 4x 1 x 3 u 2 ẋ 3 (t) = u 2 (t), y(t) = x 1 x 2 3 = 0 Let x = (x 1, x 2, x 3 ) and u = (u 1, u 2 ) The input( candidate for this example is v = tx 3 + u 1 x1 x3) 2. In (Pereira da Silva and Batista, 2010) it is shown fr this example that is possile to choose x a = y = x 1 x 2 3, u a = ẏ = u 1, x = (x 2, x 3 ) and u = u 2. It is also shown that Γ 0 = span dx, du, d u. Γ 1 = span dx, du, and that Γ 2 = span dx. Γ 3 = span, dx 2 4x 3 x 1 dx 3. To see that Γ 3 + span dy is integrale, it suffices to notice that dy = dx 1 2x 3 dx 3, hence Γ 3 + span dy = span, dx 2 2x 1 dx 1, dy = span, d(x 2 x 2 1), dy. It follows that all the assumptions of Theorem 1 holds. Further computations 15 shows that D 1 = 0. In particular the conditions of Theorem 2 holds. In this relatively simple example, it was possile to compute the state representation 2, otaining (see (Pereira da Silva and Batista, 2010)) ż = v (13) where z is given y x 2 x 2 1. It is ovious that (13) is feedack linearizale and so z is a flat output of the implicit system. It is important to say that, in some cases, it may e very hard to integrate the codistriutions in order to otain z and (2), that is, an explicit state representation of the implicit system. However, the existence conditions of Theorems 1 and 2 are always checkale. 14 See the proof of Theorem 4.9 of (Pereira da Silva and Batista, 2010) 15 These computations have een performed using Matla R / Maple R. A Diffieties and Systems This appendix is a very rief summary of some facts aout the infinite dimensional approach of (Fliess et al., 1999). A survey aout this suject can e found in (Pereira da Silva et al., 2008). R A -Manifolds, Diffieties and Systems. An ordinary diffiety is an R A manifold for which there exists a field d, called Cartan field. A system S is a pair (S, t), where S is an ordinary diffiety, and t : S R is a function, called time, such that d (t) = 1 and such that around any point ξ S there exists local coordinates of S of the form (t, η) 16. State Space Representation and Outputs. A local state representation of a system (S, t) is a local coordinate system ψ = t, x, U, where x = x i, i n, U = u (k) j j m, k N. The set of functions x = (x 1,..., x n ) is called state and the set u = (u 1,..., u m ) is called input. In these coordinates the Cartan field is locally written y d = n t + f i + x i k N, j m u (k+1) j u (k) j (14) It follows from (14) that L d u (k) = d (u(k) ) = u (k+1). So the notation u (k) is consistent with the fact that, along a solution 17, it represents the differentiation of u (k 1) with respect to time. A state representation of a system S is completely determined y the choice of the state x and the input u and will e denoted y (x, u). An output y of a system S is a set of functions defined on S. A state representation is said to e classical (or proper) if f i does not depend on u (α) for α > 1. A control system S is a system such that there exists a local state representation around every ξ S. System associated to differential equations. Now assume that a control system is given y a set of equations ṫ = 1 ẋ i = f i (t, x, u,..., u (αi) ), i n y j = η j (x, u,..., u (αj) ), j p (15) One can always associate to these equations a diffiety S of gloal coordinates ψ = t, x, U and Cartan field given y (14). Solutions. A solution of a system S with Cartan field d is a smooth map σ : (a, ) S, where (a, ) R, such that σ(t) = d (σ(t)). Susystems. A (local) susystem S a of a system S with time notion t is a pair (S a, π), where S a is a system with a time notion τ a and Cartan 16 This is equivalent to saying that the function t is a sumersion, and the fact that d (t) = 1 is equivalent to saying that that the function t is Lie-Bäcklund, when R is regarded as a diffiety with trivial Cartan field. 17 See the definition of solution given in this section. 1430

field a, and π is a Lie-Bäcklund sumersion π : U S S a etween the system U S and S a such that τ a π = t. A local state representation x = (x a, x ), u = (u a, u ) is said to e adapted to a susystem S a if we locally have ẋ a = f a (t, x a, u ) (16a) ẋ = f (t, x a, x, u a, u ) (16) and (x a, u a ) is a local state representation of S a with state equations (16a). B Existence of strongly adapted state equations Theorem 3 (Pereira da Silva et al., 2008) Let S e the system with classical state representation (x, u) and classical output y, defined 18 y (3). Let ν, where S is defined y (6). Assume that there exists a partition y = (ȳ, ŷ), where ȳ is called the independent part, and ŷ is called dependent part of the output. Assume also that there exists some α N such that, locally around ν, one has 1. span dx span span dx span 2. span nonsingular around ξ. 3. span, dȳ (0),..., dȳ (α 1) =, dȳ (0),..., dȳ (α)., dx, dȳ (0),..., dȳ (α 1) is locally, dx, du, dȳ (0),..., dȳ (α) is locally nonsingular around ξ. 4. The set, dȳ (0),..., dȳ (α) is pointwise independent in an open neighorhood of ξ. 5. span dy (0),..., dy (α 1) span, dx, dȳ (0),..., dȳ (α 1). 6. span, dy (0),..., dy (k) is nonsingular for k = α and k = α 1. 7. span dy (α) span, dy (0), dy (1),..., dy (α 1), dȳ (α). 8. span dy (0), dy (1),..., dy (k) is nonsingular around ν for k = α 1 and k = α. Then there exists a local output susystem Y defined around ν that admits a strongly adapted state representation ( x, ũ), where x = (x a, x ) and ũ = (u a, u ). Moreover: (A) One may choose u a = ȳ (α). (B) One may choose x a y (0),..., y (α 1) such that, dx a is a local asis of span dy (0),..., dy (α 1). (C) One may chose x in a way that dx completes, dx a to a local asis of span, dx, dy,..., dy (α 1). (D) One may chose u in order to complete, dx a, dx, du a 18 The state representation (x, u) is classical if span dẋ span, dx, du and the output y is classical if span dy span, dx, du. to a asis, dx a, dx, du a, du of span, dx, du, dy,..., dy (α). In particular, if is nonempty and these assumptions hold around all ξ, then the corresponding implicit system (1) is regular. Furthermore, span dx + Y = span dx Y and span dx, du + Y = span dx, du Y. Proof: The proof of the theorem is an easy consequence of the proof of Theorem 5 of (Pereira da Silva et al., 2008). References Byrnes, C. I. and Isidori, A. (1991). Asymptotic stailization of minimum phase nonlinear systems, IEEE Trans. Automat. Control 36(10): 1122 1137. Fliess, M., Lévine, J., Martin, P. and Rouchon, P. (1999). A Lie-Bäcklund approach to equivalence and flatness of nonlinear systems, IEEE Trans. Automat. Control 44(5): 922 937. Krishnan, H. and McClamroch, N. H. (1994). Tracking in nonlinear differential algeraic control systems with applications to constrained root systems, Automatica J. IFAC 30: 1885 1897. Krstic, M., Kanellakopoulos, I. and Kokotovic, P. (1995). Nonlinear and Adaptive Control Design, John Wiley and Sons. Pereira da Silva, P. S. (2008). Some remarks on static-feedack linearization for time-varying systems, Automatica J. IFAC 44(12): 3219 3221. Pereira da Silva, P. S. and Batista, S. (2010). On state representations of nonlinear implicit systems, International Journal of Control 83(3): 441 456. http://dx.doi.org/10.1080/00207170903193441. Pereira da Silva, P. S. and Corrêa Filho, C. (2001). Relative flatness and flatness of implicit systems, SIAM J. Contr. Optimiz. 39: 1929 1951. Pereira da Silva, P. S., Silveira, H. B., Correa Filho, C. and Batista, S. (2008). An infinite dimensional differential-geometric approach for nonlinear systems : Part II - system theory. sumitted, availale in www.lac.usp.r/ paulo/. Pomet, J.-B. (1995). A differential geometric setting for dynamic equivalence and dynamic linearization, in e. a. B. Jackuczyk (ed.), Geometry in Nonlinear Control and Differential Inclusions, Banach Center Pulications, Warsaw, pp. 319 339. 1431