Nonregular feedback linearization of a class of multi-input nonlinear control systems

11th IFAC Symposium on Nonlinear Control Systems 11th IFAC Symposium on Nonlinear Control Systems 11th IFAC IFAC Symposium on Nonlinear Nonlinear Control Systems Systems Vienna, Austria, Sept. 4-6, 2019 11th Symposium on Control Vienna, Austria, Sept. 4-6, 2019 Available online at www.sciencedirect.com Vienna, Austria, Sept. 4-6, 4-6, 2019 11th IFAC Symposium on Nonlinear Control Systems Vienna, Austria, Sept. 2019 Vienna, Austria, Sept. 4-6, 2019

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IFAC PapersOnLine 52-16 (2019) 286–291

Nonregular feedback linearization of a class Nonregular feedback linearization of a class Nonregular feedback linearization of a class of multi-input nonlinear control systems Nonregular feedback linearization of a class of multi-input nonlinear control systems of multi-input nonlinear control systems ofFlorentina multi-input nonlinear control systems ∗∗∗ Nicolau ∗∗ Witold Respondek ∗∗ ∗∗ Shunjie Li ∗∗∗

Florentina Nicolau ∗∗ Witold Respondek ∗∗ Shunjie Li ∗∗∗ Florentina Nicolau Witold Respondek ∗∗ Shunjie Li ∗∗∗ ∗ ∗∗ ∗∗∗ Shunjie Li ∗ Florentina Nicolau Witold Respondek Laboratory, ENSEA, 95014 Cergy-Pontoise, France (e-mail: ∗ Quartz Quartz Laboratory, ENSEA, 95014 Cergy-Pontoise, France (e-mail: ∗ ∗ Quartz Laboratory,[email protected]). ENSEA, 95014 Cergy-Pontoise, France (e-mail: [email protected]). ∗ ∗∗ Quartz Laboratory, ENSEA, 95014 Cergy-Pontoise, France (e-mail: Universi´ ee,, INSA [email protected]). ∗∗ Normandie INSA Rouen, Rouen, LMI, LMI, France. France. (e-mail: (e-mail: ∗∗ Normandie Universi´ ∗∗ [email protected]). [email protected]) Normandie Universi´ e , INSA Rouen, LMI, France. (e-mail: [email protected]) ∗∗ ∗∗∗ Normandie Universi´ e,and INSA Rouen, NUIST, LMI, France. (e-mail: [email protected]) Statistics, 201124 ∗∗∗ School of Mathematics of Mathematics and Statistics, NUIST, 201124 Nanjing, Nanjing, ∗∗∗ ∗∗∗ School [email protected]) School of Mathematics and Statistics, NUIST, 201124 Nanjing, China (e-mail: [email protected]) China (e-mail: [email protected]) ∗∗∗ School of Mathematics and Statistics, NUIST, 201124 Nanjing, China (e-mail: [email protected]) China (e-mail: [email protected]) Abstract: Abstract: In In this this paper paper we we study study feedback feedback linearization linearization of of multi-input multi-input control-affine control-affine systems systems via a particular class of nonregular feedback transformations, namely by reducing the number Abstract: In this paper we study feedback linearization of multi-input control-affine systems via a particular class of nonregular feedback transformations, namely by reducing the number Abstract: In this paper we study feedback linearization of multi-input control-affine systems via a particular class ofcomplete nonregular feedbackcharacterization transformations, namely byof reducing the requires number of controls by one. A geometric of systems that class of controls by one. A complete geometric characterization of systems of that class requires 0 1 ofnamely 2 via a particular class of nonregular feedback transformations, by reducing the number of controls by one. A complete geometric characterization systems of that class requires special properties of the linearizability distributions D ⊂ case 2 ⊂ · · · . Contrary to the special properties of A thecomplete linearizability distributions D000 ⊂ ⊂D D111 of ⊂D D · · · .ofContrary to the case 2 2 ⊂ of controls by one. geometric characterization systems that class requires of regular feedback linearization, they need not be involutive but the first noninvolutive one special properties of the linearizability distributions D ⊂ D ⊂ D ⊂ · · · . Contrary to the case of regular feedback linearization, they need not be involutive but 0 1 2 the first noninvolutive one special properties of the linearizability distributions D ⊂ D ⊂ D ⊂ · · · . Contrary to the case has to contain a sufficiently large involutive subdistribution. Recently, we solved the problem of regular feedback linearization, they need not be involutive but the first noninvolutive one has to a sufficiently large they involutive subdistribution. Recently, we 0solved the problem 0 contain of regular feedback linearization, needpaper, not beweinvolutive but theoffirst noninvolutive one11 has to contain a sufficiently large involutive subdistribution. Recently, we the problem of D noninvolutive. In the present study the case D involutive but 0 being 0solved of Dto being noninvolutive. In the involutive present paper, we study the case of we D00solved involutive but D D11 0 0 has contain a sufficiently large subdistribution. Recently, the problem of D being noninvolutive. In the present paper, we study the case of D involutive but D noninvolutive. This case is specially interesting because it covers aa big class of mechanical noninvolutive. This case is specially interesting because it covers big 0 0class of mechanical of D being noninvolutive. In the present paper, and we study the casea ofbigD class involutive butclass D1 control systems. We provide geometric necessary sufficient conditions describing our noninvolutive. This case is specially interesting because it covers of mechanical control systems.This We case provide geometricinteresting necessary and sufficient conditions describing our class noninvolutive. is specially because it covers a big class of mechanical of systems that verified by and algebraic operations only. control systems. We be provide geometric necessary and our class of systems that can can verified by differentiation differentiation and sufficient algebraic conditions operationsdescribing only. We We illustrate illustrate control systems. We be provide geometric necessary and sufficient conditions our(static class of systems thatseveral can be verified by differentiation andwith algebraic operationsdescribing only. We illustrate our results by examples and discuss relations other linearizability problems our results by several examples and discuss relations with other linearizability problems (static of systems that can be verified by differentiation and algebraic operations only. We illustrate our results by several examples and discuss relations with other linearizability problems (static invertible feedback linearization or linearization). invertible linearization or dynamic dynamic linearization). our resultsfeedback by several examples and discuss relations with other linearizability problems (static invertible feedback linearization or dynamic linearization). © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. invertible feedback linearization or dynamic linearization). Keywords: Keywords: Feedback Feedback linearization, linearization, nonregular nonregular feedback, feedback, one-fold one-fold reduction. reduction. Keywords: Feedback linearization, nonregular feedback, one-fold reduction. Keywords: Feedback linearization, nonregular feedback, one-fold reduction. 1. INTRODUCTION aa feedback feedback linearizable linearizable system system retain retain the the linearizability linearizability 1. 1. INTRODUCTION INTRODUCTION aproperty feedback linearizable system retain the linearizability if new actuators are added to it? property if new actuators are added to it? 1. INTRODUCTION a feedback linearizable retain the linearizability if new actuatorssystem are added to it? Feedback linearization linearization is aa powerful powerful tool tool for for nonlinear nonlinear property Feedback is The goal of this paper is to study the feedback linearization property if new actuators are added to it? The goal of this paper is to study the feedback linearization Feedback linearization is a powerful tool for nonlinear control systems systems and and has has attracted attracted aa lot lot of of research research in in The goal of this paper is tovia study the feedback linearization control of control-affine systems a particular class of Feedback linearization is attracted a the powerful tool nonlinear control-affine systems via a particular class of nonregnonregcontrol systems and has a of lotBrockett offorresearch in of the recent years. Following work (1979) The goal of this paper is to study the feedback linearization of control-affine systems via a particular class of nonregthe recent years. Following the work of Brockett (1979) ular feedback transformations, namely by reducing the control systems and has attracted lotBrockett of single-input research in ular feedback transformations, namely by reducing the the recent years. Following the worka of (1979) who solved the state feedback linearization for of control-affine systems via(see a particular class of nonregwho solved the state feedback linearization for single-input number of controls by one Sun and Xia (1997); Ge ular feedback transformations, namely by reducing the the recent years. Following the work of Brockett (1979) number of controls by one (see Sun and Xia (1997); Ge who solved the state feedbackclass linearization for single-input systems under a restricted of feedback transformaular feedback transformations, namely by reducing the of feedback transformasystems under a restricted class et al. (2001); Zhong et al. (2007) for other approaches). number of controls by one (see Sun and Xia (1997); Ge who solved the state feedback linearization for single-input et al. (2001); Zhong et al. (2007) for other approaches). systems under a restricted class of feedback transformations, Jakubczyk Jakubczyk and and Respondek Respondek (1980) (1980) and, and, indepenindepen- et number of controls byet one (see static Sun Xiaapproaches). (1997); Ge al.considered (2001); Zhong al. (2007) forand other tions, The systems become invertible feedback systems underand a restricted class of (1980) feedback transformaThe considered systems become static invertible feedback tions, Jakubczyk and Respondek and, independently, Hunt Su (1981) gave geometric necessary and et al. (2001); Zhong et al. (2007) for other approaches). The considered systems become static invertible feedback dently, Hunt and Su (1981) gave geometric necessary and after aa one-fold reduction of aa suitably chosen tions, Jakubczyk and (1980) and, indepenlinearizable aftersystems one-fold reduction suitably chosen dently, Hunt and Su (1981) gave geometric necessary and linearizable sufficient conditions forRespondek linearizing multi-input controlThe considered become staticof feedback controlsufficient conditions for linearizing multi-input control and we say they are linearizable via a one-fold linearizable after a that one-fold reduction ofinvertible a suitably chosen dently, Hunt and Su (1981) gave geometric necessary and control and we say that they are linearizable via a one-fold sufficient conditions for linearizing multi-input controlaffine systems under change of coordinates and invertible linearizable after a one-fold reduction of a suitably chosen and invertible affine systems under change of coordinates reduction. Linearizarion via a one-fold reduction can control and we say that they are linearizable via a one-fold sufficient conditions for linearizing multi-input controlreduction. Linearizarion via are a one-fold reduction can be be affine systems under change of coordinates and invertible static feedback transformations (called also regular feedcontrol and we say that they linearizable via a one-fold reduction. Linearizarion via a one-fold reduction can be static feedback transformations (called also regular feedseen as aa feedback linearization via aa nonregular feedback affine transformations) systems change of coordinates invertible seen as feedback linearization via nonregular feedback static feedbackunder transformations (called alsoand regular feedback modifying both the drift and the reduction. Linearizarion via a one-fold reduction can be seen as a feedback linearization via a nonregular feedback back transformations) modifying both the drift and the transformation which is noninvertible” (that static feedback transformations also drift regular which is ”minimally ”minimally noninvertible” (that back transformations) modifying both the andfeedthe control vector fields. fields. Their main(called structural condition re- transformation seen asrank a feedback linearization via afeedback nonregular feedback control vector Their main structural condition reis, the of the matrix defining the transformatransformation which is ”minimally noninvertible” (that back both theofdrift and the the rank of the matrix the feedback transformacontrol vector fields. Their main sequence structural condition re- is, quirestransformations) the involutivity ofmodifying a nested nested distributions transformation which is defining ”minimally noninvertible” (that quires the involutivity of a sequence of distributions tion equals m − 1, where m is the number of controls, and is, the rank of the matrix defining the feedback transforma0 1vector 2 fields. Their control main structural condition retion equals m − 1, where m is the number of controls, and quires the involutivity of a nested sequence of distributions D 0 ⊂ D 1 ⊂ D 2 ⊂ · · · (called linearizability distributions). is, the rank of the matrix defining the feedback transformation equals m − 1, where m is the number of controls, and ⊂ D ⊂ D ⊂ · · · (called linearizability distributions). D is the maximal possible among all noninvertible matrices). 0 1 2 0 1 involutivity 2 quires the of a nested sequence of distributions is the maximal possible among all noninvertible matrices). D ⊂ D ⊂ D ⊂ · · · (called linearizability distributions). tion equals m − 1, where m is the number of controls, and is the maximal possible among all noninvertible matrices). Thus we deal with a problem of feedback linearization via 0 1 2 In the question: is D this ⊂ Dpaper, ⊂ D we ⊂ ·study · · (called linearizability distributions). we deal with a problem of all feedback linearization via a a In this paper, we study the following following question: is aa given given Thus is the maximal possible among noninvertible matrices). In this paper, we study the following question: is a given nonregular feedback but which remains as close as possible Thus we deal with a problem of feedback linearization via a nonlinear control system an extension (or a perturbation) nonregular feedback but which remains aslinearization close as possible nonlinear control system an extension (or a perturbation) Thus we deal with a problem of feedback via a In this paper, we system studyfeedback the following question: is a given to a regular one. A complete geometric characterization nonregular feedback but which remains as close as possible of a static invertible linearizable one? More nonlinear control an extension (or a perturbation) a regularfeedback one. A but complete geometric characterization of a static invertible feedback linearizable one? More to nonregular which remains as close as possible to a regular one. A complete geometric characterization nonlinear control system an extension (or a perturbation) of systems of that requires special properties of precisely, deal nonlinear control that of a staticwe invertible one? More systems that Aclass class requires special properties of the the precisely, deal with withfeedback nonlinearlinearizable control systems systems that of 0geometric 1 2 a regularof complete characterization of systems of one. that class requires the of a from staticwe invertible feedback linearizable one? More precisely, we deal with nonlinear control systems that to linearizability distributions D D D 0 ⊂special 1 ⊂ properties 2 ⊂ · · · .ofConstem systems with less inputs which are, contrary linearizability distributions D ⊂ D ⊂ D ⊂ · · · . Constem from systems with less inputs which are, contrary 0 special 1 2 0 1 2 of systems of that class requires properties of the precisely, we deal with nonlinear control systems that stem from systems with less inputs which are, contrary trary to the case of regular feedback linearization, they linearizability distributions D ⊂ D ⊂ D ⊂ · · · . Conto the original ones, static invertible feedback linearizable. trary to the case of regularDfeedback they 0 1 linearization, 2 to the original ones, static invertible feedback linearizable. linearizability distributions ⊂ D ⊂ D ⊂ · · · . Constem from systems with inputs which are, contrary trary need not be but first one has to the case of regular linearization, This is of interest since identifying to theproblem original ones, staticless invertible not be involutive involutive but the thefeedback first noninvolutive noninvolutive onethey has This problem is of practical practical interestfeedback since by bylinearizable. identifying need trary to the case of regular linearization, need not beainvolutive but thefeedback first noninvolutive onethey has to the original ones, static invertible feedback linearizable. to contain sufficiently large involutive subdistribution. the inputs that make the system non static invertible This problem is of practical interest since by identifying to contain a sufficiently large involutive subdistribution. the inputs that make the system non static invertible 0 need not be involutive but the first noninvolutive one has This problem is of practical interest since by identifying the inputs that make the system non static invertible Recently, we solved the problem of D being noninvolutive, to contain a sufficiently large involutive subdistribution. feedback linearizable and removing them, one can plan wea solved the problem of D000 being noninvolutive, feedback linearizable andthe removing them, one caninvertible plan and and Recently, to contain sufficiently large involutive subdistribution. the inputs that make system noninvertible static feedback linearizable andreduced removing them, one can feedback plan and see Nicolau et al. (2018). In the present paper, we study Recently, we solved the problem of D being noninvolutive, track trajectories for the static see Nicolau (2018). In the1ofpresent paper, we study track trajectories for the static invertible feedback 0 al. Recently, weDet solved the problem D0 being noninvolutive, Nicolau et (2018). In the paper, we study feedback linearizable andreduced removing them, one can plan and see involutive but D noninvolutive. This case the case of 0 al. 1 present linearizable system. question can be seen as the dual track trajectories forThis the reduced static invertible feedback involutive but D noninvolutive. This case the case of D linearizable system. This question can be seen as the dual 0 1 0 1 see Nicolau et al. (2018). In the present paper, we study involutive but D noninvolutive. This case the case of D track trajectories for the reduced static invertible feedback is specially interesting because it covers a big class of to that considered by Franch and Agrawal (2010): does linearizable system. This question can be seen as the dual is specially interesting because 0 1 it covers a big class of to that considered by Franch and Agrawal (2010): does involutive butsee D Remark noninvolutive. This case thespecially case of D linearizable system. by This question be seen(2010): as the does dual is to that considered Franch andcan Agrawal mechanical control systems, 1 and Section 4. interesting because it covers a big class of mechanical control systems, see Remark 1 and Section 4.  Research specially control interesting because it covers1 and a big class 4. of to that considered by Franch AgrawalNatural (2010):Science does is partially supported by and the National mechanical systems, see Remark Section  Research partially supported by the National Natural Science  Research of  mechanical control systems, see Remark 1 and Section 4. partially supported by the National Natural Science Foundation China (61573192). Research partially supported by the National Natural Science

Foundation of China (61573192).  Research of partially supported by the National Natural Science Foundation of China (61573192). (61573192). Foundation China Foundation of2019, China (61573192). 2405-8963 © IFAC Copyright © 2019 IFAC (International Federation of Automatic Control) 426 Hosting by Elsevier Ltd. All rights reserved. Copyright 2019 IFAC 426 Control. Peer review© responsibility of International Federation of Automatic Copyright © under 2019 IFAC IFAC 426 Copyright © 2019 426 10.1016/j.ifacol.2019.11.793 Copyright © 2019 IFAC 426

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The paper is organized as follows. In Section 2, we formalize the problem. In Section 3, we give our main results. We illustrate our results by several examples in Section 4 and provide the proof of the main theorem in Section 5. 2. PROBLEM STATEMENT Consider the following nonlinear control-affine system: m  Σm : x˙ = f (x) + ui gi (x) = f (x) + g(x)u, (1) i=1

where x is the state defined on an open subset X of Rn (more generally, an n-dimensional manifold X) and u is the control taking values in Rm . The vector fields f , g1 , . . . , gm are smooth (the word smooth will always mean C ∞ -smooth) and g1 , . . . , gm are supposed everywhere independent. The system Σm is linearizable by regular static feedback if it is equivalent, via a diffeomorphism z = φ(x) and an invertible static feedback transformation u = α(x) + β(x)v, to a linear controllable system Λ : z˙ = Az + Bv. The problem of static feedback linearization was solved by Jakubczyk and Respondek (1980) and, independently, by Hunt and Su (1981), who gave geometric necessary and sufficient conditions (recalled in Theorem 1). Set D0 = span {g1 , . . . , gm } and define inductively the sequence of distributions Dj+1 = Dj + [f, Dj ], where [f, Dj ] = {[f, ξ] : ξ ∈ Dj }. In other words, Dj = span {adqf gi , 1 ≤ i ≤ m, 0 ≤ q ≤ j}. Theorem 1. Σm is locally static invertible feedback linearizable, around x0 ∈ X, if and only if for any 0 ≤ j ≤ n − 1, the distributions Dj are of constant rank, around x0 ∈ X, involutive, and Dn−1 = T X.

The geometry of static invertible feedback linearizable systems is given by the following nested sequence of involutive distributions: D0 ⊂ D1 ⊂ · · · ⊂ Dn−1 = T X. Static feedback linearization is a powerful tool in dealing with nonlinear systems and has been applied to many engineering systems, in particular, to the problems of constructive controllability and motion planning. Although, in general, a nonlinear control system is not static feedback linearizable, it may stem, however, from a system with less inputs which is static feedback linearizable. More precisely, the problem that we are addressing in this paper is the existence of a local invertible static feedback transformation of the form u, rank β reg (·) = m, u = β reg (x)˜ bringing the system Σm into m   m : x˙ = f (x) + u ˜i g˜i (x), Σ i=1

with g˜ = gβ reg , where g = (g1 , . . . , gm ) and g˜ = (˜ g1 , . . . , g˜m ), such that the reduced system m−1   m−1 : x˙ = f (x) + u ˜i g˜i (x) Σ i=1

is locally static invertible feedback linearizable. A system Σm satisfying the above property will be called linearizable via a one-fold reduction. Indeed, the sys m−1 is, as indicated by the notation, obtained by tem Σ removing the control u ˜m (for which we put u ˜m ≡ 0) and keeping u ˜i , for 1 ≤ i ≤ m − 1, unchanged. The feedback transformation u = β(x)˜ u that defines the passage 427

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˜ m into Σ  m−1 is given by β consisting of first m − 1 from Σ reg  m−1 with columns of β . Notice that when linearizing Σ ˜ the help of u ˜ = α(x) ˜ + β(x)v, it is actually enough to apply the pure feedback u ˜ = α(x) ˜ + v only (that simply m−1 transforms f into f + i=1 α ˜ i g˜i ) because changing g˜i ’s, for 1 ≤ i ≤ m − 1, can be performed via the initial nonregular feedback β (the one that consists of first m − 1 columns of β reg and allows to eliminate the control u ˜m from the system). In other words, instead of applying a nonregular β (of rank m − 1) followed by an invertible β˜ of rank m − 1, we can just apply one nonregular β (of rank m − 1) that plays a double role. To summarize, our problem of linearization via a one-fold reduction can be equivalently formulated as follows: Problem. When is the nonlinear control system x˙ = f (x)+g(x)u equivalent via a diffeomorphism z = φ(x) and a feedback transformation of the form u = β(x)(v + α ˜ (x)), where rank β(·) = m − 1, to a linear controllable system Λ : z˙ = Az + Bv? The subject of our paper is closely related to the slightly more general problem of linearization via noninvertible feedback transformations, see Sun and Xia (1997); Ge et al. (2001); Zhong et al. (2007). To compare both problems, notice that the class of feedback transformations (for linearization via a one-fold reduction) considered in this paper is not as general as possible. Indeed, we use feedback transformations of the form u = β(v + α), ˜ where rank β(·) = m−1 and α ˜ is an Rm−1 -valued function, that is, to the original system Σm we first apply β and then α ˜ and get x˙ = f + gβ α ˜ + gβv. If the matrix β(·) were invertible, the order in which we apply α and β would play no role but if the matrix β(·) is not invertible, then the order does matter. Indeed, if we apply first α and then β, that is, we put u = α + βv, where rank β(·) = m − 1 but α is an Rm -valued function, then the modified system is x˙ = f + gα + gβv. For both classes of transformations, we choose m − 1 new control vector fields m g˜i , j for 1 ≤ i ≤ m − 1, in the same way as g˜i = j=1 βi gj but, clearly, the second class (which defines all noninvertible feedback transformations) is more general because it allows to modify the drift f by any smooth combination of gi for 1 ≤ i ≤ m, while the first class allows to modify f by adding to it smooth combinations of g˜i for 1 ≤ i ≤ m − 1 only. The first class is, however, more natural in all cases when we have to decrease the number of controls from m to m − 1 and, as a consequence, we are not allowed to use in control strategies (in feedback transformations, for instance) all inputs but only those of the reduced system. We use the first class in the present paper. From now on, we deal only with systems that are nowhere static invertible feedback linearizable. So there exists a smallest integer k ≥ 0 such that Dk is not involutive. We also need to introduce the notion of corank that will be used in the paper. Notation 1. Let A and B be two distributions of constant rank and f a vector field. Denote [A, B] = {[a, b] : a ∈ A, b ∈ B} and [f, B] = {[f, b] : b ∈ B}. If A ⊂ B, the corank of

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the inclusion A ⊂ B, denoted by cork (A ⊂ B), equals the rank of the quotient B/A, i.e., cork (A ⊂ B) = rk (B/A).

Theorem 2 stated in Nicolau et al. (2018) provides necessary and sufficient geometric conditions for linearization via a one-fold reduction. Theorem 2. Σm is locally linearizable, around x0 , via a one-fold reduction if and only if it satisfies around x0 : (R1) There exists an involutive subdistribution N 0 ⊂ D0 , of corank one; (R2) The distributions N j , j ≥ 1, are involutive and of constant rank, where N j = N j−1 + [f, N j−1 ], j ≥ 1; (R3) There exists ρ such that N ρ = T X.

In Nicolau et al. (2018), we discussed in detail the relations between linearization via a one-fold reduction and dynamic linearization via an invertible one-fold prolongation Nicolau and Respondek (2017), we explained how m i we can construct the control u ˜m = i=1 βm (x)ui to be canceled by putting u ˜m ≡ 0 in order to obtain a static feedback linearizable reduced system (the control u ˜m is called the to-be-removed control in Nicolau et al. (2018)) and presented a normal form for systems linearizable via a one-fold reduction. The most important structural condition characterizing systems linearizable via a one-fold reduction is the existence of an involutive subdistribution N 0 of corank one in D0 , see Nicolau et al. (2018). To verify conditions (R1)-(R3) of Theorem 2, we have to check whether the distribution D0 contains an involutive subdistribution N 0 of corank one and identify it in order to calculate N j , for j ≥ 1, and check their involutivity. In Nicolau et al. (2018), we solved the problem of D0 being noninvolutive. In the present paper, we study the case of D0 involutive but D1 noninvolutive (the case when the first noninvolutive distribution is Dk , where k ≥ 2, is similar). In this case, the geometry of systems described by the previous theorem can be summarized by the following sequence of inclusions: D0 ⊂ D1 ∪ ∪ N 0 ⊂ N 1 ⊂ · · · ⊂ N ρ = T X, where all distributions, except D1 , are involutive, as stated by condition (R1), cork (N 0 ⊂ D0 ) = 1 and the corank of the inclusion N 1 ⊂D1 is either one or two. Remark 1. (Mechanical control systems). The considered case of D0 involutive but D1 noninvolutive is specially interesting because it covers a big class of mechanical control systems. Indeed, in local coordinates, the motion of many mechanical control systems can be described by second order differential equations that can be written as the following system of first order differential equations:   x˙ i =yi m  (MS) jk j u bi (x),  y˙ i =−Γi (x)yj yk + di (x)yj + ei (x) + =1

with 1 ≤ i ≤ n and where (x, y) = (x1 , . . . , xn , y1 , . . . , yn ) are local coordinates on the tangent bundle T Q of the configuration manifold Q, and u = (u1 , . . . , um ) are inputs of the system. The summation convention is used, except for terms involving controls. The expression Γjk i (x)yj yk corresponds to Coriolis and centrifugal terms. The terms dji (x)yj correspond to forces linear with respect to velocities, like dissipative forces, e(x) corresponds to an

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uncontrolled force (potential or not) and b1 (x), . . . , bm (x) correspond to controlled forces (assumed independent everywhere) acting on the system. Mechanical control systems form an important class of control systems that has attracted a lot of attention because of their various applications in real life. They form a natural bridge between mechanics and control theory and are studied, for instance, in Bloch (2003); Bullo and Lewis (2004); Ortega et al. (1998); Ricardo and Respondek (2010). Denote by n jk j ∂ ∂ f = i=1 (yi ∂xi + (−Γi (x)yj yk + dni (x)yj + e∂i (x)) ∂yi ) the drift of (MS) and by g = i=1 bi (x) ∂yi , where 1 ≤  ≤ m, its control vector fields. Due to the mechanical structure of the system (MS), we can deduce some properties of the sequence of distributions Dj . Firstly, notice that D0 is always involutive (since [gi , gj ] = 0, for 1 ≤ i, j ≤ m) and D1 is of rank 2m (since the vector fields n ∂ mod V, adf g are of the form adf g = − i=1 gi (x) ∂x i where V denotes the vertical distribution of (MS), that is tangent to the fibers Tx Q, and in (x, y)-coordinates is ∂ , 1 ≤ i ≤ n}). Therefore, for a given by V = span { ∂y i mechanical system that is not static invertible feedback linearizable, the first distribution that fails to satisfy the involutivity condition of Theorem 1 is necessarily Dk with k ≥ 1. Our main result (Theorem 3 in the next section), gives necessary and sufficient verifiable conditions to check whether a nonlinear system Σm (with D0 involutive, but D1 noninvolutive) is locally linearizable via a one-fold reduction, and, in particular, whether a mechanical system (MS) for which D1 is noninvolutive satisfies that property. 3. LINEARIZATION VIA A ONE-FOLD REDUCTION: CASE D1 NONINVOLUTIVE Throughout, we consider the system Σm and assume that the distribution D0 is involutive, but D1 is not involutive. As we have already observed, in order to verify conditions (R1)-(R3) of Theorem 2, we have to check whether D0 contains an involutive subdistribution N 0 of corank one and identify it. Since D0 is involutive, it always contains many involutive subdistributions N 0 of corank one and we would have to verify for each of them whether it satisfies conditions (R1)-(R3) of Theorem 2. There is no an algorithmic way to do it because the family of all involutive subdistributions N 0 of corank one in D0 is parameterized by functional parameters. We will explain next how we may overcome this difficulty and propose verifiable conditions for linearization via a onefold reduction for the case when D1 is noninvolutive. To this end, let r denote the corank of the inclusion D1 ⊂ D1 + [D0 , D1 ], that is r = rk ((D1 + [D0 , D1 ])/D1 ), see Notation 1. The integer r plays an important role in verifying our conditions. The case r = 0 (that is, [D0 , D1 ] ⊂ D1 and the involutivity of D1 is thus lost with a bracket of the form [adf gi , adf gj ]) can be treated in a similar way to the case D0 noninvolutive, and thus from now on we suppose r ≥ 1. To sum up, we assume that: (A1) The distribution D0 is involutive, but D1 is noninvolutive, of constant rank, and, moreover, the corank r is constant and satisfies r = cork (D1 ⊂ D1 + [D0 , D1 ]) ≥ 1.

We define the annihilator of a distribution E by (E)⊥ = {ω ∈ Λ1 (X) : ω, ξ = 0, ∀ξ ∈ E}, where Λ1 (X) is

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the space of smooth differential 1-forms on X. Now, consider the distribution D1 , denote its rank by d, and let ω 1 , . . . , ω r , η 1 , . . . , η s , where s = n − d − r, be differential 1-forms such that locally: (D1 )⊥ = span {ω 1 , . . . , ω r , η 1 , . . . , η s } (D1 + [D0 , D1 ])⊥ = span {η 1 , . . . , η s }. For each ω  , 1 ≤  ≤ r, we define, around a given x0 ∈ X, the map T  : D0 × D0 → R (gi , gj ) → Tij = ω  , [gi , adf gj ].  Notice that T is defined on D0 × D0 (and not on D0 × D1 ) which is important for the construction of the desired subdistribution of D0 . Observe also that with our construction of ω  , for 1 ≤  ≤ r, (they complete the forms η 1 , . . . , η s annihilating D1 + [D0 , D1 ], to the annihilator of D1 ), we are not interested in the brackets [gi , adf gj ] that stay in D1 , but in those that stick out of D1 . It is immediate that T  is a bilinear map (the matrix defined by it will also be denoted by T  ). By applying the Jacobi’s identity, it can be easily proven that [gi , adf gj ] = [gj , adf gi ] mod D1 , 1 ≤ i, j ≤ m, implying that T  is also symmetric. For 1 ≤  ≤ r, define W  = ker T  = {g ∈ D0 : T  (g, ·) = 0} and let B be the distribution given by r  B= W . (2) =1

Although the maps T  , and thus their kernels W  , depend on the choice of ω  ’s, the distribution B does not. In fact we have the following result: Proposition 1. Suppose that the (involutive) distribution D0 = span {g1 , . . . , gm } contains an involutive subdistribution N 0 = span {h1 , . . . , hm−1 } of corank one such that N 1 = N 0 + [f, N 0 ] is also involutive. Then the following holds true: (i) The distribution B does not depend on the choice of ω  ’s and is invariant with respect to feedback transformations of the form f˜ = f + α1 h1 + . . . + αm−1 hm−1 and g˜ = gβ reg , where β reg is an invertible (m × m)-matrix, g = (g1 , . . . , gm ) and g˜ = (˜ g1 , . . . , g˜m ). (ii) Assume r ≥ 2. Then the distribution B is of corank one in D0 , and N 0 is unique and given by N 0 = B. Moreover, in the sum (2), it is actually enough to take only two terms corresponding to any 1 ≤ 1 < 2 ≤ r. (iii) Assume r = 1. Then the distribution B is - either of corank one in D0 and, in that case, N 0 is unique and given by N 0 = B; - or of corank two in D0 and, in that case the map T 1 possesses a zero eigenvalue of multiplicity m − 2 and two nonzero eigenvalues of opposite sign, and N 0 is given either by g1 } N 0 = B + span {˜ or by N 0 = B + span {˜ gm }, where g˜1 , g˜m do not belong to B and satisfy [˜ g1 , adf g˜1 ]∈D1 1 and [˜ gm , adf g˜m ]∈D , their existence being guaranteed by the above condition on the eigenvalues. Moreover, the pair of distributions G˜1 = span {˜ g1 } and G˜m = span {˜ gm } of

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rank 1 does not depend on the choice of ω 1 and is invariant under feedback transformations. Construction of the vector fields g˜1 and g˜m . Suppose that r = 1 (implying that there is only one form ω  = ω 1 and one map T  = T 1 ) and that B = ker T 1 is of corank two in D0 (as in condition (iii) of Proposition 1), and thus rk T 1 = 2. Choose g˜2 , . . . , g˜m−1 such that B = span {˜ g2 , . . . , g˜m−1 } and let g1 and gm be two vector fields such that D0 = B + span {g1 , gm }. Denote by K, the 2 × 2 matrix given by Kij =< ω 1 , [gi , adf gj ] >, for i, j ∈ {1, m}. Notice that, for the above choice of generators, K actually contains all nonzero elements of the matrix T 1 (since, by its construction, B is such that [B, D1 ] ⊂ D1 ) and that the condition on the eigenvalues of T 1 (see item (iii) of Proposition 1) translates into det K < 0. Clearly, K depends on applied feedback transformations as well as on the choice of ω 1 , but (similarly to T  ) exhibits some invariant properties that allow us to define two distinguished vector fields g˜1 and g˜m (actually, rank 1 distributions spanned by them). Lemma 1 assures not only the existence of those vector fields, but gives (through its constructive proof) an explicit way to obtain them. Lemma 1. The matrix K has the following properties: (a) K is symmetric; (b) K does not depend on transformations of the form f˜ = f + α1 g1 + α2 gm mod B; (c) If (˜ g1 , g˜m ) = (g1 , gm )β mod B, with β an invertible ˜ = β T Kβ; (2 × 2)-matrix, then K 1 1 (d) If ω ˜ = hω , with h a nonvanishing smooth function, ˜ then K = hK; (e) det K(x0 ) < 0 (resp. det K(x0 ) > 0, det K(x0 ) = 0) is invariant under the feedback transformations described in conditions (b) and (c), and is independent of the choice of the function h in condition (d); (f ) If det K(x0 ) < 0, then there exists an invertible (2 × 2)-matrix β transforming K into   0 k(x) ˜ = β T Kβ = K k(x) 0 with k(x0 ) = 0. The vector fields (˜ g1 , g˜m ) = (g1 , gm )β mod B satisfy [˜ g1 , adf g˜1 ] ∈ D1 and [˜ gm , adf g˜m ] ∈ D1 and g1 } generate feedback invariant distributions G˜1 = span {˜ gm } of rank 1. and G˜m = span {˜

According to Proposition 1, the distribution B is uniquely attached to the system. Item (i) assures that although the kernels W  depend on the choice of the 1-forms ω 1 , . . . , ω r , the distribution B depends on D0 only but neither on the choice of its generators gi nor on the 1forms ω  . The distribution B is unique by its (explicit) geometric construction and if cork (B ⊂ D 0 ) = 1, then it is the only candidate for the corank one involutive subdistribution N 0 of D0 that linearizes the system Σm via one-fold reduction. If cork (B ⊂ D0 ) = 2, only two candidates for N 0 are possible (N 0 = B + span {˜ g1 } and N 0 = B + span {˜ gm }). Once the unique candidate N 0 = B has been identified (two candidates N 0 = B + span {˜ g1 } and N 0 = B + span {˜ gm }, if cork (B ⊂ D0 ) = 2), we have to check that N 0 is involutive, of corank one in D0 , and that N j , for j ≥ 1, are also involutive. We would like to emphasize that the distribution B, as well as g˜1 and g˜m , can be explicitly computed and, as a consequence, for

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any given control-affine system satisfying cork (D1 ⊂ D1 + [D0 , D1 ]) ≥ 1, the conditions of Theorem 2 are verifiable and we can thus check whether the system is linearizable via a one-fold reduction, as summarized by Theorem 3 below. Moreover, the verification involves differentiation and algebraic operations only, without solving PDE’s or bringing the system into a normal form. Theorem 3. Consider the control system Σm , given by (1), and suppose that the distribution D1 is as in assumption (A1). Then Σm is linearizable via a one-fold reduction, locally around x0 , if and only if Σm satisfies conditions (R1)-(R3) of Theorem 2 with the subdistribution N 0 given by either N 0 = B, or N 0 = B + span {˜ g1 } or N 0 = B + span {˜ gm } (the two latter if cork (B ⊂ D0 ) = 2), where [˜ g1 , adf g˜1 ] ∈ D1 and [˜ gm , adf g˜m ] ∈ D1 . Now we will explain the role of the value of corank r. If r ≥ 2, then there are r ≥ 2 matrices T  and their ranks are either one or two. Therefore, each corresponding kernel W  allows us to identify a subdistribution of D0 of rank either m − 1 or m − 2 and their sum gives always the desired corank one subdistribution in D0 . If r = 1, then there is only one matrix T  = T 1 and again two cases can be distinguished: rk T 1 = 1 and rk T 1 = 2. In the first case, the involutivity of D1 is necessarily lost with a bracket of the form [gi , adf gi ], the kernel of T 1 is of corank one in D0 (it does not contain the vector field gi ) thus defines the distribution B of rank m − 1, which is the only candidate for N 0 . If rk T 1 = 2 and T 1 has two eigenvalues of different sign and a zero eigenvalue of multiplicity m − 2, then there exist two vector fields g˜1 and g˜m such that [˜ g1 , adf g˜m ] ∈ D1 , but [˜ g1 , adf g˜1 ] ∈ D1 1 1 and [˜ gm , adf g˜m ] ∈ D . The kernel of T , which is the distribution B, is of corank two in D0 and it does not contain the vector fields g˜1 and g˜m . Proposition 1 states that, in fact, only two candidates are possible for an involutive corank one subdistribution in D0 whose bracket with f is again involutive: N 0 is given either by N 0 = B + span {˜ g1 } or by N 0 = B + span {˜ gm }, where the vector fields g˜1 and g˜m are uniquely defined by Lemma 1. 4. APPLICATIONS: TWO-INPUT MECHANICAL SYSTEMS WITH 3 DEGREES OF FREEDOM We consider two-input mechanical systems with 3 degrees of freedom of the form (MS), with n = 3 and m = 2 (which, see Remark 1, are the simplest mechanical systems that may satisfy the assumptions under which we work, that is, D0 is involutive, but D1 is non involutive and satisfies [D0 , D1 ] ⊂ D1 ). Now, observe that (MS) can be always transformed via invertible static transformations compatible with the mechanical structure (that is, an extended invertible point transformation Φ = (φ1 , φ2 ) of the form x ˜ = φ1 (x) and y˜ = φ2 (x, y) = Dφ1 (x)y, and an ˜ invertible static feedback transformation u ˜ = β(x)u, with β˜ an invertible matrix) into the following form, for which, for simplicity, we drop the tildes: x˙ 2 = y2 x˙ 3 = y3 x˙ 1 = y1 y˙ 1 = a1 (x, y) + u1 y˙ 2 = a2 (x, y) + u2 y˙ 3 = a3 (x, y) + b(x)u1 j with ai (x, y) = −Γjk i (x)yj yk + di (x)yj + ei (x) and b(x) smooth functions. In these coordinates, a simple calculation shows that all brackets [gi , adf gj ] are of the form

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[gi , adf gj ] = δij (x) ∂y∂ 1 + γij (x) ∂y∂ 3 mod span { ∂y∂ 2 }, where the functions δij and γij depend on x only and can be expressed in terms of Γjk i (x), b(x) and its partial derivatives of first order. Hence, if the distribution D1 is not involutive and [D0 , D1 ] ⊂ D1 , we necessarily have D1 + ∂ ∂ , ∂ + b(x) ∂x , ∂ , 1 ≤ i ≤ 3} and [D0 , D1 ] = span { ∂x 2 ∂x1 3 ∂yi 1 1 0 1 r = cork (D ⊂ D + [D , D ]) = 1. It is easy to see that the 1-forms η 1 and ω 1 can, respectively, be taken as η 1 = dx3 − b(x)dx1 and ω 1 = dy3 − b(x)dy1 mod span {dx1 , dx2 }. Therefore, for the corresponding matrix T 1 = (Tij1 ), we obtain Tij1 (x) = γij (x) − b(x)δij (x), for 1 ≤ i, j ≤ 2. The determinant of T 1 is of the form  3 ∂b det T 1 = ζ0 (x) + i=1 ζi (x) ∂x , where the functions ζi , i for 0 ≤ i ≤ 3, can be expressed in terms of the Γjk i ’s and successive powers of b(x). The partial differential equation det T 1 = 0 always admits solutions, that is, for any triplet of functions (a1 , a2 , a3 ), there are mechanical systems for which det T 1 = 0 and rk T 1 = 1. In that case, the only candidate for N 0 is B and the conditions of Theorem 2 have to be checked for N 0 = B. Generically, det T 1 is however nonzero and, if, in addition, det T 1 < 0, then the control vector fields g˜1 and g˜2 satisfying item (iii) of Proposition 1 can be constructed and there are now two candidates for the distribution N 0 (either N 0 = span {˜ g1 } or N 0 = span {˜ g2 }). Then the conditions of Theorem 2 have to be verified for each of them. Example 1. (The 3-coupled spring-mass system is linearizable via a one-fold reduction). Consider the following mechanical control system:

m1 x ¨1 = −k1 x1 + k2 (x2 − x1 ) + u1 m2 x ¨2 = −k2 (x2 − x1 ) + k3 (x3 − x2 ) + b(x)u1 + cb(x)u2 m3 x ¨3 = −k3 (x3 − x2 ) − k4 x3 + u2 , (3) describing the motion of a 3-coupled spring-mass system, which consists of 3 bodies, where xi is the position of the ith body and mi is its mass (to simplify, we suppose that all masses are normalized, i.e., mi = 1, 1 ≤ i ≤ 3). The bodies are connected by 4 springs (ki being the spring constant of the ith spring). Two external forces u1 and u2 are applied on the first and third body, respectively. The forces act in the same (up to a multiplicative constant c) through a nonlinear function b(x) also on the second body. ∂b ∂b ∂b − c ∂x )(x0 ) = 0 and ( ∂x + Suppose b(x0 ) = 0, ( ∂x 3 1 3 ∂b cb ∂x2 )(x0 ) = 0. Transform (3) into the form (MS). We ∂b ∂b ∂b + ∂x ) ∂ , [g1 , adf g2 ] = (c ∂x + get [g1 , adf g1 ] = 2(b ∂x 2 1 ∂y2 1 ∂b ∂b ∂ ∂b ∂b ∂ 2c ∂x2 + ∂x3 ) ∂y2 and [g2 , adf g2 ] = 2c( ∂x3 + cb ∂x2 ) ∂y2 . Then D1 is clearly noninvolutive and the 1-forms η 1 and ω 1 1 can be taken as η 1 = 1c dx1 − cb dx2 + dx3 and ω 1 = 1c dy1 − 3 ∂b 1 1 i=1 yi ∂xi )dx2 . They lead to a matrix cb dy2 + dy3 − cb2 ( 1 T of rank two (thus to an empty distribution B) whose ∂b ∂b 2 − c ∂x ) (hence its determinant is det T 1 = − c21b2 ( ∂x 3 1 eigenvalues satisfy condition (iii) of Proposition 1). With the help of Lemma 1 we can construct the vector fields gi , adf g˜i ] ∈ D1 . The feedback g˜i , i = 1, 2, satisfying [˜ transformation (βij ) allowing their construction can be ∂b ∂b ∂b ∂b taken as β11 = β12 = 1, β21 = −(b ∂x + ∂x )/( ∂x +cb ∂x ) 2 1 3 2 g1 , g˜2 ) = (g1 + β21 g2 , g1 + β22 g2 ). and β22 = −1/c. We get (˜ We have two candidates for the distribution N 0 : either N 0 = span {˜ g1 } or N 0 = span {˜ g2 }. In the first case, we

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obtain [˜ g1 , adf g˜1 ] ∈ N 1 = span {˜ g1 , adf g˜1 }, thus N 1 is not involutive and the associated reduced system (obtained by putting u ˜2 ≡ 0) is not static feedback linearizable. In the g2 } = span { ∂y∂ 1 − 1c ∂y∂ 3 } second case, we have N 0 = span {˜ and the reduced system (obtained by putting u ˜ 1 = u1 + cu2 ≡ 0) is indeed linear. Example 2. (The PVTOL aircraft is not linearizable via a one-fold reduction). The following model of a planar vertical take off and landing (PVTOL) aircraft was introduced in Hauser et al. (1992) and has attracted a lot of attention in the last years (see, e.g., Lozano et al. (2004); Martin et al. (1996)). The configuration of the system is (θ1 , x1 , y1 ), with θ1 the angle the aircraft makes with the horizontal axis and (x1 , y1 ) the position of its center of mass. After normalisation of m and J, the dynamics of the PVTOL aircraft is given by: θ˙1 = θ2 x˙ 1 = x2 y˙ 1 = y2 θ˙2 = u2 x˙ 2 = −u1 sin θ1 + u2 cos θ1 y˙ 2 = −ag + u1 cos θ1 +u2 sin θ1 , where u1 and u2 correspond, respectively, to the body vertical force (minus the gravity) and to forces on the tips of the wings, ag is the gravity acceleration and  = 0 is a fixed constant related to the geometry of the aircraft. By a direct calculation, we find that both [g1 , adf g1 ] and [g2 , adf g2 ] belong to D1 , but [g1 , adf g2 ] ∈ D1 (i.e., the two distinguished vector fields g˜1 and g˜2 , see condition (iii) of Proposition 1, are simply g˜1 = g1 and g˜2 = g2 ). It follows that there are two candidates for the distribution N 0 : either N 0 = span {g1 } or N 0 = span {g2 }. Let us first ∂ +cos θ1 ∂y∂ 2 }. consider N 0 = span {g1 } = span {− sin θ1 ∂x 2 By computing its successive brackets with the drift, we obtain that all N i are involutive, but rk N 4 = rk N 3 = 3. Hence the associated reduced system (obtained by removing the control u2 ) is not accessible and therefore, not static feedback linearizable. We now turn to the case ∂ N 0 = span {g2 } = span {cos θ1 ∂x + sin θ1 ∂y∂ 2 + ∂θ∂ 2 }. In 2 this case, we obtain [g2 , adf g2 ] ∈ N 1 = span {g2 , adf g2 }, thus N 1 is not involutive and the associated reduced system (now obtained by putting u1 ≡ 0) is not static feedback linearizable. It follows that the PVTOL aircraft is not linearizable via a one-fold reduction. Recall that the PVTOL aircraft is dynamically linearizable via a twofold prolongation of a well chosen control Martin et al. (1996) showing that there is no a simple relation between linearization via reduction and via prolongation. 5. PROOF OF THEOREM 3 Necessity. Consider the control system Σm : x˙ = f (x) + m u g (x) and suppose that Σm is locally linearizable i i i=1 via a one-fold reduction. Thus there exists an invertible feedback transformation u = β reg (x)˜ u (recall that we work with a restricted class of feedback transformations: so we do not apply an initial feedback transformation  m : x˙ = f (x) + on f ), bringing Σm into the form Σ m  m−1 : x˙ = ˜i g˜i (x), such that the reduced system Σ i=1 u m−1 ˜i g˜i (x) is locally static invertible feedback f (x) + i=1 u linearizable. Define the distribution N 0 = span {˜ gi , 1 ≤ i ≤ m − 1}. It is immediate that N 0 ⊂ D0 , where the inclusion is of corank one. Set N j = N j−1 + [f, N j−1 ], for j ≥ 1 and remark that we actually have N j = 431

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span {˜ gi , . . . , adjf g˜i , 1 ≤ i ≤ m − 1}, for j ≥ 1, so N j ˜ m−1 that is are linearizability distributions of the system Σ feedback linearizable. Therefore they are involutive and of constant rank and there exists ρ such that N ρ = T X thus proving that the original system Σm satisfies conditions (R1)-(R3) of Theorem 1. We have just proven that D0 contains an involutive subdistribution N 0 of corank one such that N 1 = N 0 + [f, N 0 ] is also involutive. If in addition, the distribution D1 is as in assumption (A1), then the hypothesis of Proposition 1 are verified, and it follows by item (ii) or item (iii) of that proposition that we have either N 0 = B or N 0 = B+span {˜ g1 } or N 0 = B+ span {˜ gm }, with g˜1 and g˜m with desired properties. Sufficiency. Consider a control system Σm : x˙ = f (x) + m 1 i=1 ui gi (x) whose distribution D is as in assumption (A1). Compute the distribution B associated to D1 . The distribution N 0 (independently of whether defined as N 0 = B or as N 0 = B + span {˜ g1 } or as N 0 = B + span {˜ gm }) is supposed, together with N j = N j−1 + j−1 [f, N ], to satisfy conditions (R1)-(R3) of Theorem 1 and therefore by that theorem the system Σm is thus locally linearizable via a one-fold reduction. REFERENCES Bloch, A. (2003). Nonholonomic Mechanics and Control. SpringerVerlag, New York. Brockett, R. (1979). Feedback invariants for nonlinear systems. IFAC Congress 6, Helsinki, 1115–1120. Bullo, F. and Lewis, A. (2004). Geometric Control of Mechanical Systems. Springer-Verlag, Texts in Applied Mathematics 49, New York. Franch, J. and Agrawal, S.K. (2010). On sufficient conditions to keep differential flatness under the addition of new inputs. Internat. J. Control, 83(4), 829–836. Ge, S., Sun, Z., and Lee, T. (2001). Nonregular feedback linearization for a class of second-order nonlinear systems. Automatica, 37(11), 1819 – 1824. Hauser, J., Sastry, S., and Meyer, G. (1992). Nonlinear control design for slightly non-minimum phase systems: Application to v/stol aircraft. Automatica, 28(4), 665–679. Hunt, L. and Su, R. (1981). Linear equivalents of nonlinear time varying systems. In Proc. MTNS, Santa Monica, CA, 119–123. Jakubczyk, B. and Respondek, W. (1980). On linearization of control systems. Bull. Acad. Polonaise Sci. Ser. Sci. Math., 517–522. Lozano, R., Castillo, P., and Dzul, A. (2004). Global stabilization of the pvtol: real-time application to a mini-aircraft. Int. J. Control, 77(8), 735–740. Martin, P., Devasia, S., and Paden, B. (1996). A different look at output tracking: control of a vtol aircraft. Automatica, 32(1), 101–107. Nicolau, F., Li, S., and Respondek, W. (2018). Multi-input nonlinear control systems linearizable via one-fold reduction. In 37th Chinese Control Conference (CCC), 944–949. IEEE. Nicolau, F. and Respondek, W. (2017). Flatness of multiinput control-affine systems linearizable via one-fold prolongation. SIAM J. Control and Optim., 55(5), 3171–3203. Ortega, R., Loria, A., Nicklasson, P., and Sira-Ramirez, H. (1998). Passivity-based control of Euler-Lagrange systems. Communications and Control Engineering Series, Springer Verlag, Berlin. Ricardo, S. and Respondek, W. (2010). When is a control system mechanical? Journal of Geometric Mechanics, 2(3), 265–302. Sun, Z. and Xia, X. (1997). On nonregular feedback linearization. Automatica, 33(7), 1339 – 1344. Zhong, J., Karasalo, M., Cheng, D., and Hu, X. (2007). New results on non-regular linearization of non-linear systems. Internat. J. Control, 80(10), 1651–1664.