On state representations of time-varying nonlinear systems

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1 On state representations of time-varying nonlinear systems Paulo Sérgio Pereira a Silva a, Simone Batista a, a University of São Paulo, Escola Politécnicca PTC Av. Luciano Gualberto trav. 03, 158, São Paulo SP BRAZIL Abstract This work consiers a nonlinear time-varying system escribe by a state representation, with input u an state x. A given set of functions v, which is not necessarily the original input u of the system, is the (new) input caniate. The main result presents necessary an sufficient conitions for the existence of a local classical state space representation with input v. These conitions rely on integrability tests that are base on a erive flag. As a byprouct, one obtains a sufficient conition of ifferential flatness of nonlinear systems. Key wors: Nonlinear systems; state representations; ifferential flatness; ifferential geometric approach. 1 Introuction A classical state representation is essentially a set of ifferential equations of the form ż(t) = g(t, z(t), v(t)) (1) where z(t) IR s is the state an v(t) IR q is the input. When the state representation epen on the erivatives of the input, that is, when the state equations are of the form ẇ(t) = η(t, w(t), v (0) (t), v (1) (t),..., v (α) (t)) (2) with η epening on v (α) for α > 0, then the state representation (with state w an input v) is sai to be generalize (see Fliess et al. (1993b)). An output y of (2) given by y = h(t, w(t), v (0) (t),..., v (β) (t)) is sai to be classical with respect to the state representation (2), if y oes not epen on v () for > 0, that is, y may be written as y = ĥ(t, w(t), v(t)). The problem of realization of input/output nonlinear ifferential equations is the problem of giving classical Corresponing author P. S. Pereira a Silva. Tel. +55(11) Fax +55(11) The first author is supporte by Conselho Nacional e Desenvolvimento Científico e Tecnologico CNPq, Brazil aresses: paulo@lac.usp.br (Paulo Sérgio Pereira a Silva), ssimonebatista@ig.com.br (Simone Batista). state representations (1) for input-output equations. This problem was extensively stuie in the literature (see for instance van er Schaft (1987); Gla (1988); Crouch an Lamnabhi-Lagarrigue (1988); van er Schaft (1990); Liu an Moog (1994); Moog et al. (2002); Jakubczyk (1986)). A comparison between some of these works can been foun in Kotta an Mullari (2005). Close relate to this problem is the one of seeking generalize state transformations z = φ(t, w, v (0),..., v (α) ) in a way that (2) is converte to (1), that is, the input erivatives are eliminate. The time-invariant version of this problem is solve in Delaleau an Responek (1995). Recall that, in the behavioral approach of Willems (1992), the input an the output are not chosen a priori. The same point of view is share by the approach of Fliess et al. (1999), an this fact is in accorance of what is foun in physical systems. For instance, the DC motor can be represente by the following moel 1 Li (1) + Ri + [K(i)]θ (1) = E, Jθ (2) + Bθ (1) = [K(i)]i τ (3a) (3b) where θ is the shaft angle, i is the armature current, L is is the armature inuctance, E is the external voltage, R is the armature resistance of the motor, τ is the external 1 The moel (3) may be foun in Atay (2000). One may isregar the effect of magnetic saturation, consiering that K(i) is a constant. This may be a ba approximation for high current values (see Atay (2000)). The linear version of this example is ue to Emmanuel Delaleau. Preprint submitte to Automatica 19 December 2008

2 torque (or the loa torque), J is the oint inertia of the armature an shaft, an K(i) is the function that is relate to the epenence of the flux on the current. When the input is the voltage E, the isturbance input is the loa torque τ, an the output is the shaft angle θ, the evice is working as a motor an one may give the following state equations t i = (R/L)i (K(i)/L)θ(1) + (1/L)E, t θ(0) = θ (1), t θ(1) = (K(i)/J)i (B/J)θ (1) (1/J)τ, y = θ (0) (4a) (4b) (4c) (4) with state x = (i, θ (0), θ (1) ), input u = (E, τ) an output y = θ (0). If one consiers that the evice is working as a voltage generator, one may take the external torque τ as an input, the current i on the loa as a isturbance input, an the voltage E as an output, an one obtains the following state equations t θ(0) = θ (1), t θ(1) = (K(i)/J)i (B/J)θ (1) (1/J)τ, y 1 = Kθ (1) + Ri + Li (1) (5a) (5b) (5c) with state z = (θ (0), θ (1) ), input v = (τ, i) an output y 1 = E. Note that the imension of the states of (4) an (5) are ifferent. The equations (3) are the same for the motor an the generator, since they represent the moel of the same physical system, but one may choose a ifferent set of inputs an outputs, giving rise to ifferent state equations. Motivate by this example, one may state the following problem 2. Given a system that is efine by its state representation ẋ(t) = f(t, x(t), u(t)) (6) where x(t) IR n is the state, an u(t) IR m is the input, when oes this system amit another classical state representation (1), with a given input caniate v? Now, v is a new input, an so the original state x an the new state z may have ifferent imensions. For instance, given the system efine by (4) with input u = (E, τ) an state x = (i, θ, θ (1) ), one may ask if v = (τ, i) is a possible input of (4), corresponing to classical state equations. When the output y 1 is isregare, the expecte answer is yes, as (5a-b) is an alternate state representation of the same physical system (3), an this may 2 In this section one will only motivate the problem. See the precise statements an efinitions in section 2. be confirme by the application of Theo. 1 of section 2. In section 2, necessary an sufficient conitions for the solution of this problem are given. In section 3 a version of this problem for systems with outputs is consiere. As a byprouct, a sufficient conition of ifferential flatness is shown in section 4. Our approach will follow the infinite imensional geometric setting introuce in control theory by Fliess et al. (1993a); Pomet (1995); Fliess et al. (1999). 2 Notations, Preliminaries an Problem Statement The stanar notations of ifferential geometry will be consiere in the finite an infinite imensional case. The fiel of real numbers will be enote by IR. The set of natural numbers {1,..., k} will be enote by k. For simplicity, we abuse notation, letting (z 1, z 2 ) stan for the column vector (z T 1, z T 2 ) T, where z 1 an z 2 are also column vectors, an the transpose of z i is z T i, i=1,2. Let x = (x 1,..., x n ) be a vector of functions (or a collection of functions). Then {x} stans for the set {x 1,..., x n }. Some notations an efinitions of Fliess et al. (1999); Pereira a Silva an Corrêa Filho (2001) are use along the paper. (e. g. the efinition of a system as a iffiety, an the efinition of state representation as a local coorinate system). The survey Pereira a Silva (2008a) presents an elementary introuction to this approach. If S is a iffiety with Cartan fiel t, an u is a function efine on S, then the Lie erivative t u of a function u will be enote by u (or u (1) ) an the k- fol Lie erivative k t (u) of u will be enote by u (k). If u = (u 1,..., u m ) is a set of functions efine on S then u (k) = (u (k) 1,..., u(k) m ). If ω = p k=1 α kφ k is a one-form efine on S, then ω stans for L ω = p t k=1 α kφ k + α k φ k. A nonsingular coistribution Γ that is generate by a linearly inepenent set {ω 1,..., ω r } is integrable if ω i ω 1... ω r = 0 for i r. In the rest of this paper, one will refer to system S, as state in the next efinition. Definition 1 Consier the state equations (6), where t IR is the time, x(t) IR n is the state, an u(t) IR m is the input, an f is smooth with respect to its arguments. The system S associate to (6) is the iffiety with (global) coorinates {t, x, (u (k) : k IN)} an the Cartan fiel t is given by t = t + n k IN m u(k+1) u (k) i=1 f i x i +. A local state representation (w, v) of S is a set of functions {w 1,..., w s, v 1,..., v q }, efine on some open set of S, such that {t, w, (v (k) : k IN)} is a local coorinate system 3 of S. In this 3 The approach of Fliess et al. (1999) consiers an intrinsic efinition of state representation. See Pereira a Silva (2008a) for a comparison with the present efinition. 2

3 v (k) case, the Cartan fiel will be locally given by t = t + s i=1 η i z i + k IN q v(k+1) an the corresponing local state equations 4 will be given by (2), where η = (η 1,..., η s ). A local state representation is sai to be classical if α = 0 in (2), that is, if η i oes not epen on v () for > 0, an for i q. Now, the main problem consiere in this paper may be precisely state. Problem 1 Let v = (v 1,..., v q ) be a set of functions efine on 5 the iffiety S. The Problem of Classical state Representation with input v is solvable aroun some ξ S, if S amits a local classical state representation (z, v) efine in an open neighborhoo U of ξ. Remark 1 Note that there is no loss of generality in assuming that the original system S is efine by (6). For instance, if the original equations 6 were in the form (2), then one may take the ynamic extension x = (w, v,..., v (α 1) ) an u = v (α). In this case, one will obtain a classical state representation (6). The following lemma will be useful in this paper. Lemma 1 (Pereira a Silva, 2008a, Lemma 1) Let (x, u) an (z, v) be two local classical state representations of the system S efine on an open neighborhoo U of ξ S. (1) If b IN is such that span {v} span {t, x, u (0),..., u (b) }, then span {z} span {t, x, u (0),..., u (b 1) } 7. (2) Let β IN be the smallest non-negative integer such that there exists an open neighborhoo V of ξ such that 8 span {u} span { t, z, v (0),..., v (β)} on V. If span {z, v} span { t, x, u (0),..., u (γ)}, then β n + mγ, where n = im x an m = im u. Now, the main result of the section will be presente. Theorem 1 Consier the system S efine by (6). Let v = (v 1,... v m ) be a set of functions locally efine aroun ξ S. Let γ be the least non-negative integer 4 The Def. 1 mimics the concept of enogeneous feeback. Note that the imension of an input v is the ifferential imension of S. In particular, q = m (see Fliess et al. (1999)). 5 Note that each v i may be a function of (t, x, u (0), u (1),..., u (α) ) for some α. If Problem 1 is solvable, one must have q = m (see footnote 4). 6 If the choice of v of Problem 1 coincies with the v of (2), then one recovers a time-varying version of the Problem state in Delaleau an Responek (1995). 7 When b = 0, then span {t, x, u (0),..., u (b 1) } stans for span {x}. 8 The integer β always exists (locally), since {t, z, (v (k), k IN)} is a local coorinate system. such that span {v} span { t, x, u (0),..., u (γ)} (locally). For a given δ IN, one may efine erive flag Γ k, k IN on the iffiety S by { Γ 0 = span t, x, u (0),..., u (γ), v (0),..., v (δ)} (7a) Γ k = span {ω Γ k 1 ω Γ k 1 } (7b) Then Problem 1 is solvable if an only if there exists an integer δ, with 0 δ n + γ(1 + m) such that (i) Γ k is nonsingular for k = 0,..., δ+1, an (im Γ k 1 im Γ k ) = m, for k = 1,..., δ + 1. (ii) Γ δ+1 is integrable. (iii) Γ 0 = Γ 1 span { v (δ)}. (iv) The set {v (k) } is locally linearly inepenent for k = 0,..., δ. The conitions of Theorem 1 will be briefly iscusse before proving this result. Note first that, as one seeks z such that {t, z, (v (i), i IN)} is a local coorinate system, it is clear that the set B = {v (0),..., v (k) } is linearly inepenent for all k IN. Uner certain assumptions 9 the inepenence of B for all k woul imply that there exits local coorinates {t, ζ, (v (i) : i IN)}. This means that the system woul amit a generalize state representation ζ = F (t, ζ, v,..., v (α) ). Then one woul apply, for instance, the conitions of Delaleau an Responek (1995) to the last state representation, obtaining solvability conitions of Problem 1. This coul be a way of proving Theorem 1, but this may lea to some technical ifficulties. Note that conition (iv) is weaker than the inepenence of B for all k IN. The conitions of Theorem 1 may appear to be technical, an so Examples 4 an 5 will present some situations where such conitions may fail. The proof of necessity of Theorem 1 relies on the next Lemma, whose proof is straightforwar by irect computation (see Liu an Moog (1994); Batista (2006)): Lemma 2 Let S be a system with local classical state representation (z, v) an state equations (1). Fix some δ IN. Let Γ 0 = span {t, z, v,..., v (δ) } an let Γ k = span {ω Γ k 1 ω Γ k 1 }. Then one locally has Γ k = span { t, z, v,..., v (δ k)} an Γ δ+1 = span {t, z}. The proof of sufficiency is base on the following result. Lemma 3 (Pereira a Silva (2008a)) Let (x, u) be a local classical state representation of a system S aroun some ξ S, an let z = (z 1,..., z s ) an v = (v 1,..., v q ) be sets of functions efine on the iffiety S such that the set S = {t, z, v,..., v (α) } is (locally) linearly inepenent. Assume that span {x} 9 See for instance Pereira a Silva (2008a) for several special versions of the inverse function theorem for iffieties. 3

4 span { t, z, v,..., v (α 1)}, an span {ż, u} span { t, z, v,..., v (α)} aroun ξ. Suppose also that span {z, v} span {t, x, u}. Then (z, v) is also a local state representation of S aroun ξ. Proof of Theorem 1. Necessity. Let (z, v) be a local classical state representation of system S efine by (6) with state equations (1). Since {t, z, (v (k) : k IN)} is a local coorinate system of the iffiety S, then x = x(t, z, v (0),..., v (β 1) ) an u = u(t, z, v (0),..., v (β) ) for β 0 big enough (see Lemma 1). Hence, u (k) span {t, z, v (0),..., v (β+k) } for all k IN. In the same way, one may (locally) write z = z(t, x, u (0),..., u (r 1) ) an v = v(t, x, u (0),..., u (r) ) for a convenient r IN. Now take δ = β + r an γ = r. By construction, it follows that span {z} span { t, x,..., u (γ 1)}, (or span {z} span {t, x}, if γ = 0). Hence, span { t, x, u,..., u (γ)} span {t, z, v,..., v (δ) }. In particular, one shows that, for γ = r an δ = β + r, one has Γ 0 = span {t, z, v,..., v (δ) }. The proof of necessity then follows from Lemma 2. The fact that δ n + γ(1 + m) follows easily from part 2 of Lemma 1. Sufficiency. It will be shown first that { span v,..., v (δ k)} Γ k, k = 0,..., δ { Γ k = Γ k+1 span v (δ k)}, k = 0,..., δ (8a) (8b) The property (8a) is a straightforwar consequence of the efinition (7b). The equation (8b) will be shown by inuction. By the assumption (iii) of Theorem 1, (8b) hols for k = 0. Assume that it hols for some k, with 0 k < δ in some V S. By reuction to the absur, one assumes that Γ k+2 span { v (δ k 1)} ν {0} for some ν V. By (iv) an (8a), one can construct a local basis of Γ k+1 of the form {ω 1,..., ω s, v (δ k 1) } aroun ν. Let ω = s i=1 α iω i + m =1 β v (δ k 1) be a smooth one form in Γ k+2 such that ω ν Γ k+2 span { v (δ k 1)} ν. This is equivalent to have α i ν = 0, i s an some β ν 0. It follows from (7b) that ω = s i=1 α i ω i + m =1 β v (δ k) + γ Γ k+1, where γ = s i=1 α iω i + m β =1 v (δ k 1) that m i=1 β v (δ k) Γ k+1. One conclues ν Γ k+1 span { v (δ k)} ν. This contraicts the inuction hypothesis. Now, by (i) an (iv), one shows (8b) from simple imensional arguments. Now, let ω = t. Since ω = 0, it follows that t Γ k, k = 0,... δ +1. In particular, span {t} Γ δ+1. Since Γ δ+1 is nonsingular an integrable, from Frobenius theorem, there exists (locally) a set of functions {z 1,..., z p } such that Γ δ+1 = span {t, z}. By (8b), it is clear that the set S = {t, z, v,..., v (δ) } is (locally) linearly inepenent an Γ 0 = span {S}. By (8b) for k = δ, an from (7b), one conclues that span {ż} span {t, z, v}. By construction, from the fact that span {ẋ} span {t, x, u} Γ 0, an span { u (i)} Γ 0 for i = 0,..., γ, it follows that span { ẋ, u (0)..., u (γ)} Γ 0. So, by Lemma 3 applie to the classical state representation ( x, ũ), where x = (x, u,..., u (γ 1) ) an ũ = u (γ), it follows that (z, v) is a local state representation for the system. Since span {ż} span {t, z, v}, this local state representation is classical. From part 2 of Lemma 1, it follows that δ n + γ(1 + m). 3 Systems with outputs Consier system S efine by (6) with output y given by 10 y(t) = h(t, x(t), u(t)) (9) Given a set of functions v efine on S, one may state the following problem. Problem 2 When there exists a local classical state representation (z, v) with state equations (1), an the output is also classical (with respect to (z, v))? Recall that the output is classical (with respect to (z, v)) if one may locally write y = h(t, z(t), v(t)) The following theorem answers this question. Theorem 2 Consier system S efine by (6) with output y given by (9). Problem 2 is solvable if an only if the assumptions of Theorem 1 hol for some δ IN, δ n + γ(1 + m), an furthermore, span {y} Γ δ+1. Proof. Straightforwar from the proof of Theo A sufficient conition of flatness In this section one will show that, uner some conitions, one may choose a (virtual) input v in a way that there exists a local classical state representation (z, v) such that (1) is linearizable by regular static-state feeback v = ψ(t, z, µ), where µ is the new input. Since v is not the actual input of the system, this state feeback is enogenous, but is not really a static-state feeback. 10 There is no loss of generality in consiering that (x, u) is a classic state representation an y is a classic output, that is, y oes not epen on u () for > 0 (see Remark 1). 4

5 In particular, the system is linearizable by enogenous feeback, an so it is flat (see Fliess et al. (1999)). Although v is not the actual input of the system, one may evelop a flatness base control. Now, consier some results an efinitions of Pereira a Silva (2008c). Let S be the system efine by (6). A time-varying regular staticstate feeback (TVRSSF) is a local iffeomorphism φ such that (t, x, u) (t, z, v), where (t, x) (t, z) is also a local iffeomorphism, calle state transformation. The close loop equations are given by ż(t) = f(t, z(t), v(t)) (10) where f(t, [ ] z, v) = φ t + φ x f(t, x, u) (t,x,u)=φ 1 (t,z,v). The linearization problem by TVRSSF seeks a TVRSSF such that the close loop system locally reas ż(t) = Az(t) + Bv(t) where this last system is time-invariant, linear an controllable (an hence it is flat, as shown in Fliess et al. (1999)). This problem is solve for instance in Pereira a Silva (2008c), where it is shown that Prop. 1 generalizes the results of (Arana-Bricaire et al. (1995); Jakubczyk an Responek (1980)). Proposition 1 Let 0 = span {t, x} an let k = span {ω k 1 ω k 1 }. Then the linearization problem by TVRSSF is locally solvable aroun ξ S if an only if: (1) The coistributions k are nonsingular at ξ for k IN. (2) There exists k IN big enough, such that k = span {t}. (3) The coistributions k are locally integrable aroun ξ, for k IN. The following result is a sufficient conition of flatness of a nonlinear system. Theorem 3 Consier the system S efine by (6). Let v = (v 1,..., v m ) be a collection of functions efine on S. Let δ an γ, be integers, an let Γ 0 be the coistribution efine in Theorem 1. Assume that the conitions of Theorem 1 hol aroun some ξ S. Let 0 = Γ δ+1 an let k = span {ω k 1 ω k 1 }, k IN. Then if the conitions 1, 2 an 3 of Proposition 1 are satisfie for k, k IN, then the system is locally flat aroun ξ. Proof. By the proof of Theorem 1, system S amits a local classical state representation (z, v) aroun ξ an 0 = span {t, z}. In particular, the conitions of Proposition 1 are satisfie for the state representation (z, v). In particular, the system amits a flat output y that is a function of z an t. Note that a given system S is not necessarily static-feeback linearizable with respect to the original state representation (x, u) (see example 2). 5 Examples In orer to illustrate the results of this work, five very simple examples are given. Example 1. Consier the system ẋ 1 = x 2, ẋ 2 = t x 2 x 3 x 1 u an ẋ 3 = u. Let v = x 3. Take Γ 0 = span {t, x 1, x 2, x 3, u, v, v} = span {t, x 1, x 2, v, v}. Simple calculations give Γ 1 = span {t, x 1, x 2, v} an Γ 2 = span {t, x 1, x 2 + x 1 v}. Hence, the conitions of Theorem 1 hol. Taking z 1 = x 1 an z 2 = x 2 + x 1 v, it follows that Γ 2 = span {t, z 1, z 2 } an so (z, v) is a local state representation for the system, where z = (z 1, z 2 ). The corresponing state equations are given by ż 1 = z 2 z 1 v an ż 2 = t. Example 2. Consier the system S given by ẋ 1 = e x3 u 1, ẋ 2 = u 1 an ẋ 3 = u 2. For this system one may take v = (v 1, v 2 ), where v 1 = u 1 an v 2 = e x3 u 1. Consiering u 1 0, it is easy to show that, by taking γ = 0 an δ = 1, one fins Γ 0 = span { t, x, u, v, v (1)}, Γ 1 = span {t, x, u 1 }, Γ 2 = span {t, x 1, x 2 }, an Γ 3 = span {t}. It is easy to see that the conitions of theorems 1 an 3 hol. In this case it is easy to integrate Γ 2, furnishing the state representation ż 1 = v 2, ż 2 = v 1, where z 1 = x 1 an z 2 = x 2. It is then clear that y = (x 1, x 2 ) is a flat output of S. Note that S is a classical example of a system that is not staticfeeback linearizable (with respect to the original state representation (x, u)), but it becomes static-feeback linearizable when one puts one integrator in series with the first input u 1. This last situation may be regare in the present approach, by choosing v = ( u 1, u 2 ) an applying Theorems 1 an 3. Example 3. Consier now equation (5). Following the iea of Remark 1, one consiers a ynamic extension in orer to get a classical state representation (x, u), since y 1 epens on the erivative of the input i. Define x = (x 1, x 2, x 3 ) = (θ (0), θ (1), i) an u = (τ, i (1) ) = (u 1, u 2 ), obtaining the following example of equation (6): ẋ 1 = x 2, ẋ 2 = (K(x 3 )/J)x 3 (B/J)x 2 (1/J)u 1, ẋ 3 = u 2, y 1 = K(x 3 )x 2 + Rx 3 + Lu 2. Let v = (τ, i) = (v 1, v 2 ). One obtains Γ 0 = span {t, x, u, v,..., v (δ) }, an for k δ, Γ k = span{t, x, u 1,..., u (δ k+1) 1, u 2,..., u (δ k) 2 }, an Γ δ+1 = span {t, x, u (0) 1 }, for a given δ IN. Since span {y 1 } Γ δ+1, by Theorem 2, Problem 2 is not solvable with input v an output y 1. Example 4. Let ẋ 1 = x 2, ẋ 2 = x 1, ẋ 3 = u. Choose v = x 1. As this example is linear, the nonsingularity an integrability conitions of Theorem 1 hol. Note that γ = 0. Given δ IN, let Γ 0 = span { t, x, u, v,..., v (δ)}. Simple calculations show that, Γ 0 = span {t, x 1, x 2, x 3, u}, Γ 1 = span {t, x 1, x 2, x 3 }, Γ 2 = span {t, x 1, x 2 } = Γ k, k IN. In particular, conition (iii) oes not hol, since v (δ) span {x} Γ 1. In fact, as x 1 is a solution of the ifferential equation ẍ 1 + x 1 = 0 = ẋ 2 + x 1, the 5

6 function x 1 cannot be an input for this system. In this example, the conition that im Γ k im Γ k+1 = m oes not hol for k > 2. Example 5. Let ẋ 1 = x 2, ẋ 2 = u, ẋ 3 = x 2 2. Choose v = x 1. It is easy to show that, for any δ IN, then Γ δ+1 = span {t, x 3 2x 2 x 1 } which is not integrable. By Theorem 1, Problem 1 is not solvable. However, in this case the system amits a state representation ẋ 3 = v 2, which is not classical. 6 Conclusions The main result of this paper checks if there exists another classical state representation (z, v) of (6), consiering v as the new input. A negative answer will be obtaine if v is not the input of any state representation, classical or not. This will happen for instance, if there exists a ifferential equation linking the components of v (see Example 4). A simple trick, namely, to inclue the ifferential t in Γ 0, allow to consier time-varying systems in a very natural way. Our results also contain, with a proper choice of v, the linearization by extening the state with integrators (see Examples 1 an 2). Note that im z may be smaller than im x (see Example 1), an im z may be greater than im x (see Example 2, in the case where v = ( u 1, u 2 )). The results of this paper may be generalize for input-output equations an implicit systems Pereira a Silva (2008b). Acknowlegements The authors are inebte to Ülle Kotta an to three anonymous reviewers for many interesting suggestions. References Arana-Bricaire, E., Moog, C. H., Pomet, J. B., A linear algebraic framework for ynamic feeback linearization. IEEE Trans. Automat. Control 40, Atay, F. M., Magnetic saturation an steay-state analysis of electrical motors. Applie Mathematical Moelling 24 (11), Batista, S., Realizações e sistemas não-lineares implícitos. Ph.D. thesis, Universiae e São Paulo, São Paulo. Crouch, P. E., Lamnabhi-Lagarrigue, F., State space realizations of nonlinear systems efine by input-output equations. Lect. Notes in Cont. an Inf. Sci 111, Delaleau, E., Responek, W., Lowering the orers of erivatives of controls in generalize state space systems. J. Math. Systems Estim. Control 5 (3), Fliess, M., Lévine, J., Martin, P., Rouchon, P., 1993a. Linéarisation par bouclage ynamique et transformations e Lie-Bäcklun. C. R. Aca. Sci. Paris Sér. I Math. 317, Fliess, M., Lévine, J., Martin, P., Rouchon, P., A Lie-Bäcklun approach to equivalence an flatness of nonlinear systems. IEEE Trans. Automat. Control 44 (5), Fliess, M., Lévine, J., Rouchon, P., 1993b. Generalize state variable representation for a simplifie crane escription. Int. J. Control 58, Gla, S. T., Nonlinear state space an input output escriptions using ifferential polynomials. In: Descusse, J., e. a. (E.), New Trens in Nonlinear Control Theory. Springer, pp Jakubczyk, B., Local realizations of nonlinear causal operators. SIAM J. Control 24 (2), Jakubczyk, B., Responek, W., On linearization of control systems. Bull. Aca. Pol. Sc., Ser. Sci. Math. 28, Kotta, U., Mullari, T., Equivalence of ifferent realization methos for high orer nonlinear inputoutput ifferential equations. European Journal of Control 11 (3), Liu, P., Moog, C. H., A local minimal realization algorithm for SISO nonlinear systems. In: Proc. American Contr. Conf., Baltimore. pp Moog, C. H., Zheng, Y., Liu, P., Input-output equivalence of nonlinear systems an their realizations. In: Proc. CD of 15th IFAC Worl Congress. Pereira a Silva, P. S., e. a., 2008a. An infinite imensional ifferential-geometric approach for nonlinear systems : Part II - system theory. submitte, available in paulo/. Pereira a Silva, P. S., e. a., 2008b. On state representations of nonlinear implicit systems. Submitte. Available in paulo/. Pereira a Silva, P. S., 2008c. Some remarks on staticfeeback linearization for time-varying systems. Automatica J. IFAC 44 (12), Pereira a Silva, P. S., Corrêa Filho, C., Relative flatness an flatness of implicit systems. SIAM J. Contr. Optimiz. 39, Pomet, J.-B., A ifferential geometric setting for ynamic equivalence an ynamic linearization. In: B. Jackubczyk, e. a. (E.), Geometry in Nonlinear Control an Differential Inclusions. Banach Center Publications, Warsaw, pp van er Schaft, A. J., On realization of nonlinear systems escribe by higher-orer ifferential equations. Mathematical Systems Theory 19, van er Schaft, A. J., Transformations an representations of nonlinear systems. In: B. Jacubczyk, e. a. (E.), Perspectives in Control Theory. Birkäuser, Boston, pp Willems, J. C., Paraigms an puzzles in the theory of ynamical systems. IEEE Trans. Automat. Control 36,