1. Introduction. The problem of transforming the nonlinear control singleinput. ξ = f(ξ)+g(ξ)u. x = φ(ξ), u = α(ξ)+β(ξ)v

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1 SIAM J CONTROL OPTIM Vol 0, No 0, pp c 2003 Society for Industrial and Applied Mathematics FEEDBACK CLASSIFICATION OF NONLINEAR SINGLE-INPUT CONTROL SYSTEMS WITH CONTROLLABLE LINEARIZATION: NORMAL FORMS, CANONICAL FORMS, AND INVARIANTS ISSA AMADOU TALL AND WITOLD RESPONDEK Abstract We study the feedback group action on single-input nonlinear control systems We follow an approach of Kang and Krener based on analyzing, step by step, the action of homogeneous transformations on the homogeneous part of the same degree of the system We construct a dual normal form and dual invariants with respect to those obtained by Kang We also propose a canonical form and a dual canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms resp, their dual canonical forms coincide We give an explicit construction of transformations bringing the system to its normal, dual normal, canonical, and dual canonical forms We illustrate our results by simple examples on R 3 and R 4 Key words feedback equivalence, normal forms, canonical forms, invariants AMS subect classifications 93B11, 93B17, 93B27 PII S Introduction The problem of transforming the nonlinear control singleinput system Σ : ξ = fξ+gξu by a feedback transformation of the form Γ : x = φξ, u = αξ+βξv to a simpler form has been extensively studied during the last twenty years transformation Γ brings Σ to the system The Σ : ẋ = fx+ gxv, whose dynamics are given by f = φ f + gα, g = φ gβ, where for any vector field f and any diffeomorphism φ we denote φ fx =dφφ 1 x fφ 1 x A natural question to ask is whether we can find a transformation Γ such that the Received by the editors November 28, 2000; accepted for publication in revised form April 21, 2002; published electronically DATE Institut National des Sciences Appliquées de Rouen, Laboratoire de Mathématiques de l INSA, Pl Emile Blondel, Mont Saint Aignan Cedex, France tall@insa-rouenfr, wresp@insarouenfr The first author worked on this paper while on leave from the Department of Mathematics, Université Cheikh Anta Diop, Dakar, Senegal 1

2 2 ISSA AMADOU TALL AND WITOLD RESPONDEK transformed system Σ is linear, that is, whether we can linearize the system Σ via feedback Necessary and sufficient geometric conditions for this to be the case have been given in [13] and [18] Those conditions, except for the planar case, turn out to be restrictive, and a natural problem that arises is to find normal forms for nonlinearizable systems Although natural, this problem is very involved and has been extensively studied during the last twenty years Four basic methods have been proposed for studying feedback equivalence problems The first method is based on the theory of singularities of vector fields and distributions, and their invariants, and using this method a large variety of feedback classification problems have been solved; see, eg, [4], [7], [14], [15], [18], [19], [27], [29], [32], [38] The second approach, proposed by Gardner [9], uses Cartan s method of equivalence [6] and describes the geometry of feedback equivalence [10], [11], [12], [28] The third method, inspired by the Hamiltonian formalism for optimal control problems, was developed by Bonnard [3], [4] and Jakubczyk [16], [17] and has led to a very nice description of feedback invariants in terms of singular extremals Finally, a very fruitful approach was proposed by Kang and Krener [26] and then followed by Kang [21], [22] Their idea, which is closely related with Poincaré s classical technique for linearization of dynamical systems see, eg, [1], is to analyze the system Σ and the feedback transformation Γ step by step and, as a consequence, to produce a simpler equivalent system Σ also step by step Our paper is deeply inspired by those of Kang and Krener [26], [21] and can be considered as a completion of their results In [21], Kang constructed a normal form for single-input nonlinear control systems with controllable linearization using successively homogeneous feedback transformations, and he proved that the homogeneous terms of a given degree of his normal form are unique under homogeneous feedback transformations of the same degree He also showed that a nonlinear system can admit different normal forms under feedback resulting from the action of lower order terms of the feedback transformation on higher order terms of the system The main goal of our paper is to propose a canonical form for the class of single-input systems with controllable linearization and to prove that two systems are equivalent, via a formal feedback, if and only if their canonical forms coincide In [26] Kang and Krener constructed two normal forms for the quadratic part of a single-input system In the first normal form, all components of the linear part of the control vector field are annihilated and all nonremovable quadratic nonlinearities are grouped in the drift; in the second normal form, all quadratic terms of the drift are annihilated and all nonremovable nonlinearities are present in the control vector field Kang normal form is a generalization, for higher order terms, of the first normal form In this paper, we generalize the second one and produce a dual normal form for higher order terms We also construct dual invariants of homogeneous feedback transformations They contain the same information, as Kang invariants, encoded in a different way We also give a dual canonical form and prove that two systems are equivalent, via formal feedback, if and only if their dual canonical forms coincide The third aim of the paper is to construct explicit homogeneous feedback transformations which bring the homogeneous part of the system of the same degree into its normal, or dual normal, form For any fixed degree, our transformations are easily computable via differentiation and integration of polynomials A successive application of those transformations gives formal feedbacks that bring any system to its normal form, dual normal form, canonical form, and dual canonical form The theory of normal forms initialized and developed by Kang and Krener [26] and Kang [21], [22] and continued in the present paper and in [33], [34] has proved to be very useful in analyzing structural properties of nonlinear control systems It

3 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 3 has been used to study bifurcations of nonlinear systems [23], [24], [25], has led to a complete description of symmetries around equilibrium [30], [31], and has allowed us to characterize systems equivalent to feedforward forms [35], [36], [37] The paper is organized as follows In section 2 we will introduce, following [21] and [26], homogeneous feedback transformations We give a normal form obtained by Kang and discuss invariants of homogeneous transformations, also obtained by him We provide an explicit construction of transformations bringing the system to Kang normal form In section 3 we construct a canonical form and give one of our main results stating that two control systems are feedback equivalent if and only if their canonical forms coincide Proofs of results presented in sections 2 and 3 are given in section 4 Section 5 dualizes the main results of section 2: we give a dual normal form, explicitly construct transformations bringing the system to that form, and define dual invariants of homogeneous transformations Similarly to normal forms, a given system can admit different dual normal forms In section 6 we thus dualize the results of section 3 by constructing a dual canonical form and proving that two control systems are feedback equivalent if and only if their dual canonical forms coincide Section 7 contains proofs of results presented in sections 5 and 6 Throughout the paper, we illustrate our results by simple examples on R 3 and R 4 2 Normal form and m-invariants All obects, that is, functions, maps, vector fields, control systems, etc, are considered in a neighborhood of 0 R n and assumed to be C -smooth Let h be a smooth R-valued function By hx =h [0] x+h [1] x+h [2] x+ = h x we denote its Taylor series expansion at 0 R n, where h x stands for a homogeneous polynomial of degree m Similarly, for a map φ of an open subset of R n to R n resp, for a vector field f on an open subset of R n we will denote by φ resp, by f the homogeneous term of degree m of its Taylor series expansion at 0 R n, that is, each component φ of φ resp, f of f is a homogeneous polynomial of degree m in x We will denote by H x the space of homogeneous polynomials of degree m of the variables x 1,,x n and by H m x the space of formal power series of the variables x 1,,x n starting from terms of degree m Analogously, we will denote by R x the space of homogeneous vector fields whose components are in H x and by R m x the space of vector fields formal power series whose components are in H m x Consider the Taylor series expansion of the system Σ given by m=0 21 Σ : ξ = Fξ + Gu + m=2 f ξ+g [m 1] ξu, where F = f ξ 0 and G = g0 We will assume throughout the paper that f0=0 and g0 0 Consider also the Taylor series expansion Γ of the feedback transformation Γ

4 4 ISSA AMADOU TALL AND WITOLD RESPONDEK given by 22 Γ : x = Tξ + φ ξ, m=2 u = Kξ + Lv + m=2 α ξ+β [m 1] ξv, where T is an invertible matrix and L 0 Let us analyze the action of Γ on the system Σ step by step To start with, consider the linear system ξ = Fξ + Gu Throughout the paper we will assume that it is controllable It can be thus transformed by a linear feedback transformation of the form Γ 1 : x = Tξ, u = Kξ + Lv to the Brunovský canonical form A, B; see, eg, [20] Assuming that the linear part F, G, of the system Σ given by 21, has been transformed to the Brunovský canonical form A, B, we follow an idea of Kang and Krener [26], [21] and apply successively a series of transformations 23 Γ m : x = ξ + φ ξ, u = v + α ξ+β [m 1] ξv for m =2, 3, A feedback transformation defined as a series of successive compositions of Γ m, m =1, 2,, will also be denoted by Γ because, as a formal power series, it is of the form 22 We will not address the problem of convergence and will call such a series of successive compositions a formal feedback transformation Observe that each transformation Γ m for m 2 leaves invariant all homogeneous terms of degree smaller than m of the system Σ, and we will call Γ m a homogeneous feedback transformation of degree m We will study the action of Γ m on the following homogeneous system: 24 Σ : ξ = Aξ + Bu + f ξ+g [m 1] ξu Consider another homogeneous system, Σ, given by 25 Σ : ẋ = Ax + Bv + f x+ g [m 1] xv We will say that the homogeneous system Σ is feedback equivalent to the homogeneous system Σ if there exists a homogeneous feedback transformation of the form 23, which brings Σ into Σ modulo terms in R m+1 x, v

5 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 5 Notation Because of various normal forms and various transformations that are used throughout the paper, we will keep the following notation We will denote, respectively, by Σ and Σ the following systems: Σ : ξ = Aξ + Bu + f ξ+g [m 1] ξu, Σ : ξ = Aξ + Bu + f [k] ξ+g [k 1] ξu k=2 The systems Σ and Σ will stand for the systems under consideration Their state vector will be denoted by ξ and their control by u The system Σ resp, the system Σ transformed via feedback will be denoted by Σ resp, by Σ Its state vector will be denoted by x, its control by v, and the vector fields, defining its dynamics, by f [k] and g [k 1] Feedback equivalence of homogeneous systems Σ and Σ will be established via a smooth feedback, that is, precisely, via a homogeneous feedback Γ m On the other hand, feedback equivalence of systems Σ and Σ will be established via a formal feedback Γ We will introduce two kinds of normal forms, Kang normal forms and dual normal forms, as well as canonical forms and dual canonical forms The bar symbol will correspond to the vector field f defining the Kang normal forms Σ NF and Σ NF and the canonical form Σ CF as well as to the vector field ḡ[m 1] defining the dual normal forms Σ DNF and Σ DNF and the dual canonical form Σ DCF Analogously, the m-invariants resp, dual m-invariants of the system Σ will be denoted by a,i+2 resp, by b [m 1] and the m-invariants resp, dual m-invariants of the normal form Σ NF resp, dual normal form Σ DNF byā,i+2 [m 1] resp, by b The starting point is the following result, proved by Kang [21] Proposition 1 The homogeneous feedback transformation Γ m, defined by 23, brings the system Σ, given by 24, into Σ, given by 25, if and only if the relations L Aξ φ φ +1 ξ = f ξ f ξ, 26 L B φ n L Aξ φ n L B φ + α ξ = ξ = g [m 1] f n ξ+β [m 1] ξ = g [m 1] n ξ g [m 1] ξ, ξ f n ξ, ξ g n [m 1] ξ hold for any 1 n 1, where φ are the components of φ This proposition represents the essence of the method developed by Kang and Krener and used in our paper The problem of studying the feedback equivalence of two systems Σ and Σ requires, in general, solving a system of first order partial differential equations On the other hand, if we perform the analysis step by step, then the problem of establishing the feedback equivalence of two systems Σ and Σ reduces to solving the algebraic system 26 Using the above proposition, Kang [21] proved the following result Theorem 1 The homogeneous system Σ can be transformed, via a homogeneous feedback transformation Γ m, into the normal form

6 6 ISSA AMADOU TALL AND WITOLD RESPONDEK 27 Σ NF : ẋ 1 = x 2 + n x 2 i P [m 2] 1,i x 1,,x i, i=3 n ẋ = x +1 + x 2 i P [m 2],i x 1,,x i, i=+2 ẋ n 2 = x n 1 + x 2 np [m 2] n 2,n x 1,,x n, ẋ n 1 = x n, ẋ n = v, where P [m 2],i x 1,,x i are homogeneous polynomials of degree m 2 depending on the indicated variables In order to construct invariants of homogeneous feedback transformations, let us define X m 1 i ξ = 1 i ad i Aξ+f ξ B + g[m 1] ξ and let X [m 1] i be its homogeneous part of degree m 1 By π i we will denote the proection on the subspace 28 W i = {ξ R n : ξ i+1 = = ξ n =0}, that is, π i ξ =ξ 1,,ξ i, 0,,0 Following Kang [21], we denote by a,i+2 ξ the homogeneous part of degree m 2 of the polynomials CA 1 [ X m 1 i,x m 1 ] i+1 πn i ξ, where C =1, 0,,0 T R n and, i N N, defined by ={, i N N : 1 n 2, 0 i n 2} The homogeneous polynomials a,i+2 for, i will be called m-invariants of Σ The following result of Kang [21] asserts that m-invariants a,i+2 for, i are complete invariants of homogeneous feedback and, moreover, illustrates their meaning for the homogeneous normal form Σ NF Consider two homogeneous systems Σ and Σ and let and { a,i+2 :, i } { ã,i+2 :, i } denote, respectively, their m-invariants The following theorem was proved by Kang [21] Theorem 2 The m-invariants have the following properties:

7 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 7 i Two homogeneous systems Σ and Σ are equivalent via a homogeneous feedback transformation Γ m if and only if a,i+2 =ã,i+2 for any, i ii The m-invariants ā,i+2 of the homogeneous normal form Σ NF, defined by 27, are given by 29 ā,i+2 x = 2 x 2 n i x 2 n ip [m 2],n i x 1,,x n i for any, i Our first aim is to find explicitly feedback transformations bringing the homogeneous system Σ to its normal form Σ NF Define the homogeneous polynomials ψ [m 1],i ξ by setting ψ [m 1],0 ξ =ψ [m 1] 1,1 ξ =0, n i 210 ψ [m 1],i ξ = CA 1 ad n i Aξ g[m 1] + 1 t ad t 1 Aξ ad A n i t Bf if 1 <i n and t=1 211 ψ [m 1],i ξ =L A n i Bf 1 π iξ + L Aξ ψ [m 1] 1,i π iξ + ψ [m 1] 1,i 1 π i 1ξ + ξi 0 L A n i+1 Bψ [m 1] 1,i π iξdξ i if 1 i, where ψ [m 1],i π i ξ is the restriction of ψ [m 1],i ξ to the submanifold W i Define the components φ of φ for 1 n and the feedback α,β [m 1] by 212 φ ξ = φ n i=1 ξi 0 ξ =f n 1 ξ+l Aξφ α ξ = β [m 1] ξ = f n g [m 1] n ψ [m 1],i π i ξdξ i, 1 n 1, n 1 ξ, ξ+l Aξ φ n ξ ξ+l B φ n ξ, We have the following result Theorem 3 The homogeneous feedback transformation Γ m : x = ξ + φ ξ, u = v + α ξ+β [m 1] ξv, where α, β [m 1], and the components φ of φ are defined by 212, brings the homogeneous system Σ into its normal form Σ NF given by 27 Proof of Theorem 3 Denote by Σ : ẋ = Ax + Bv + f x+ g [m 1] xv

8 8 ISSA AMADOU TALL AND WITOLD RESPONDEK the system Σ transformed via the feedback transformation Γ m defined by 212 From the expressions of α ξ and β [m 1] ξ given by 212 and the last two equations of 26, we get f n x = 0 and g [m 1] n x =0 Plugging φ, defined by 212, into the second equation of 26 gives ψ [m 1],n x = g [m 1] x g [m 1] x, which, by 210, implies g [m 1] x = 0 for 1 n 1 Now we consider the first equation of 26 From the expression of φ n we get f n 1 x = 0, and for any 1 i n, we obtain by differentiating 213 f x i = f x i φ + L Ax x i + φ +1 φ x i 1 x i In the above formula, the term φ x i 1 is not present in the case i =1 If i +1,weget f x i π i 1 x = f x i = f x i φ + L Ax x i Hence, by an induction argument, we obtain f x i + φ +1 φ x i 1 x i π i 1 x + L Ax ψ [m 1],i π i 1 x + ψ [m 1],i 1 π i 1x ψ [m 1] +1,i π i 1x π i 1 x π i 1 x + L Ax ψ [m 1],i π i 1 x + ψ [m 1],i 1 π i 1x ψ [m 1] +1,i π i 1x=0 and, finally, we get 214 f π i 1 x=0 x i If 1 i, then, using 212 and 213, we obtain f x i π i x = f x i = f + x i xi 0 φ + L Ax x i + φ +1 φ x i 1 x i π i x π i x + L Ax ψ [m 1],i π i x + ψ [m 1],i 1 π i 1x ψ [m 1],i x i 1 π i x dx i ψ [m 1] +1,i π ix

9 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 9 Using the expression 211, it follows that 215 f π i x=0 x i From the relations 214 and 215, we conclude that f x = x 2 i P [m 2],i x 1,,x i, i=+2 which proves that Σ is a normal form satisfying 27 Thus the system Σ given by 24 is feedback equivalent to the normal form Σ NF given by 27 Example 1 To illustrate results of this section, we consider the system Σ, given by 24 on R 3 Theorem 1 implies that the system Σ is equivalent, via a homogeneous feedback transformation Γ m defined by 212, to its normal form Σ NF see 27 ẋ 1 = x 2 + x 2 3P [m 2] x 1,x 2,x 3, ẋ 2 = x 3, ẋ 3 = v, where P [m 2] x 1,x 2,x 3 is a homogeneous polynomial of degree m 2 of the variables x 1,x 2,x 3 We would like now to discuss the interest of Theorem 3 As we have already mentioned, Poincaré s method allows us to replace a partial differential equation by solving successively linear algebraic equations defined by the homological equation 26; see [26] and [21], and Proposition 1 The solvability of this equation was proved in [26] and [21], while Theorem 3 provides an explicit solution in the form of the transformations 212, which are easily computable via differentiation and integration of homogeneous polynomials to the homological equation As a consequence, for any given control system, Theorem 3 gives transformations bringing the homogeneous part of the system to its normal form For example, if the system is feedback linearizable, up to order m 0 1 see [27], then a diffeomorphism and a feedback compensating all nonlinearities of degree lower than m 0 can be calculated explicitly without solving partial differential equations More generally, by a successive application of transformations given by 212 we can bring the system, without solving partial differential equations, to its normal form given in Theorem 4 below Consider the system Σ of the form 21 and recall that we assume the linear part F, G to be controllable Apply successively to Σ a series of transformations Γ m, m =1, 2,, such that each Γ m brings Σ to its normal form Σ NF ; for instance we can take a series of transformations defined by 212 Successive repeating of Theorem 1 gives the following result of Kang [21] Theorem 4 There exists a formal feedback transformation Γ which brings the

10 10 ISSA AMADOU TALL AND WITOLD RESPONDEK system Σ to a normal form Σ NF given by ẋ 1 = x 2 + x 2 i P 1,i x 1,,x i, 216 Σ NF : i=3 ẋ = x +1 + x 2 i P,i x 1,,x i, i=+2 ẋ n 2 = x n 1 + x 2 np n 2,n x 1,,x n, ẋ n 1 = x n, ẋ n = v, where P,i x 1,,x i are formal power series depending on the indicated variables Example 2 Consider a system Σ defined on R 3 whose linear part is controllable Theorem 4 implies that the system Σ is equivalent, via a formal feedback transformation Γ, to its normal form Σ NF ẋ 1 = x 2 + x 2 3P x 1,x 2,x 3, ẋ 2 = x 3, ẋ 3 = v, where P x 1,x 2,x 3 is a formal power series of the variables x 1,x 2,x 3 3 Canonical form As proved by Kang and recalled in Theorem 2, the normal form Σ NF is unique under homogeneous feedback transformation Γm The normal form Σ NF is constructed by a successive application of homogeneous transformations Γ m for m 1 which bring the corresponding homogeneous systems Σ into their normal forms Σ NF Therefore a natural and fundamental question which arises is whether the system Σ can admit two different normal forms, that is, whether the normal forms given by Theorem 4 are in fact canonical forms under a general formal feedback transformations of the form Γ It turns out that a given system can admit different normal forms, as the following example of Kang [21] shows The main reason for the nonuniqueness of the normal form Σ NF is that, although the normal form Σ NF is unique, homogeneous feedback transformation Γ m bringing Σ into Σ NF is not It is this small group of homogeneous feedback transformations of order m that preserve Σ NF described by Proposition 2 below, which causes the nonuniqueness of Σ NF The aim of this section is thus to construct a canonical form for Σ under feedback transformation Γ Example 3 Consider the system 31 ξ 1 = ξ 2 + ξ3 2 2ξ 1 ξ3, 2 ξ 2 = ξ 3, ξ 3 = u on R 3 Clearly, this system is in Kang normal form compare with Theorem 4 The

11 feedback transformation Γ 3 : FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 11 x 1 = ξ 1 ξ ξ3 2, x 2 = ξ 2 2ξ 1 ξ 2, x 3 = ξ 3 2ξ2 2 + ξ 1 ξ 3 2ξ 2 ξ3, 2 u = v +6ξ 2 ξ 3 +12ξ 1 ξ 2 ξ 3 4ξ3 3 +2ξ 1 +2ξ1 2 +2ξ 2 ξ 3 v brings the system 31 into the form ẋ 1 = x 2 + x 2 3, ẋ 2 = x 3, ẋ 3 = v modulo terms in R 4 x, v Applying successively homogeneous feedback transformations Γ m given, for any m 4, by 212, we transform the above system into the normal form 32 ẋ 1 = x 2 + x x 2 3P x, ẋ 2 = x 3, ẋ 3 = v, where P is a formal power series whose 1-et at 0 R 3 vanishes The systems 31 and 32 are in their normal forms and, moreover, feedback equivalent, but the latter system does not contain any term of degree 3 As a consequence, the normal form Σ NF is not unique under formal feedback transformations Consider the system Σ of the form 21 Since its linear part F, G is assumed to be controllable, we bring it, via a linear transformation and linear feedback, to the Brunovský canonical form A, B Let the first homogeneous term of Σ which cannot be annihilated by a feedback transformation be of degree m 0 As proved by Krener [27], the degree m 0 is given by the largest integer p such that all distributions D k = span {g,,ad k 1 f g} for 1 k n 1 are involutive modulo terms of order p 2 We can thus, due to Theorems 1 and 2, assume that, after applying a suitable feedback, Σ takes the form ξ = Aξ + Bu + f [m0] ξ+ m=m 0+1 f ξ+g [m 1] ξu, where A, B is in Brunovský canonical form and the first nonvanishing homogeneous vector field f [m0] is of the form ξ f [m0] i 2 P [m0 2],i ξ 1,,ξ i, 1 n 2, ξ = i=+2 0, n 1 n Let i 1,,i n s, where i i n s = m 0 and i n s 2, be the largest, in the lexicographic ordering, n s-tuple of nonnegative integers such that for some 1 n 2, we have m0 [m0] f 0 ξ i1 1 ξin s

12 12 ISSA AMADOU TALL AND WITOLD RESPONDEK Define { = sup =1,,n 2 : m0 [m0] f ξ i1 1 ξin s } 0 We have the following result Theorem 5 The system Σ given by 21 is equivalent by a formal feedback Γ to a system of the form 33 Σ CF : ẋ = Ax + Bv + m=m 0 f x, where, for any m m 0, 34 f x = additionally, we have 35 i=+2 and, moreover, for any m m 0 +1, x 2 i P [m 2],i x 1,,x i, 1 n 2, 0, n 1 n; m0 [m0] f x i1 1 xin s = ±1 36 m0 f x 1, 0,,0=0 x i1 1 xin s The form Σ CF satisfying 34, 35, and 36 will be called the canonical form of Σ The name is ustified by the following Theorem 6 Two systems Σ 1 and Σ 2 are formally feedback equivalent if and only if their canonical forms Σ 1,CF and Σ 2,CF coincide Proofs of Theorems 5 and 6 are given in section 4 Kang [21], generalizing [26], proved that any system Σ can be brought by a formal feedback into the normal form 33 for which 34 is satisfied He also observed that his normal forms are not unique; see Example 3 Our results, Theorems 5 and 6, complete his study We show that for each degree m of homogeneity we can use a onedimensional subgroup of feedback transformations which preserves the triangular structure of 34 and at the same time allows us to normalize one higher order term The form of 35 and 36 is a result of this normalization These one-dimensional subgroups of feedback transformations are given by the following proposition Proposition 2 The transformation Γ m given by 23 leaves invariant the system Σ defined by 24 if and only if 37 φ = a m L 1 Aξ ξm 1, 1 n, α = a m L n Aξξ m 1, β [m 1] = a m L B L n 1 Aξ ξm 1,

13 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 13 where a m is an arbitrary real parameter Proof of Proposition 2 Observe that, following Proposition 1, the transformation Γ m leaves invariant the system Σ if and only if it satisfies the following system of equations: L Aξ φ φ +1 ξ =0, 1 n 1, L B φ =0, 1 n 1, L Aξ φ n + α ξ =0, L B φ n + β [m 1] ξ =0 In order to solve the above system, let us remark, using the second equation of the system, that for any such that 1 n 1, the component φ does not depend to the variable ξ n Putting = n 2 into the first equation, we get φ n 2 ξ 1 ξ φ n 2 ξ n 1 ξ n = φ n 1 Since φ n 1 and φ n 2 do not depend on the variable ξ n, we conclude that φ n 2 does not depend on the variable ξ n 1 An inductive argument shows that φ 1 depends only on the variable ξ 1, that is, φ 1 ξ =a m ξ1 m Now, all equations of 37 follow easily Theorem 5 establishes an equivalence of the system Σ with its canonical form Σ CF via a formal feedback Its direct corollary yields the following result for equivalence under a smooth feedback of the form Γ : x = φξ, u = αξ+βξv, up to an arbitrary order Corollary 1 Consider a smooth control system Σ : ξ = fξ+gξu For any positive integer k we have the following: i There exists a smooth feedback Γ transforming Σ, locally around 0 R n, into its canonical form Σ k CF given by Σ k CF : ẋ = Ax + Bv + k m=m 0 f x, modulo Ox, v k+1, where f x, for any m 0 m k, satisfies 34, 35, 36 ii Feedback equivalence of Σ and Σ k CF, modulo Ox, vk+1, can be established via a polynomial feedback transformation Γ k of degree k iii Two smooth systems Σ 1 and Σ 2 are feedback equivalent modulo terms of order Ox, v k+1 if and only if their canonical forms Σ k 1,CF and Σ k 2,CF coincide This corollary follows directly from Theorem 5 and its proof, given in section 4, which provides explicit polynomial transformations bringing, step by step, the system into its canonical form

14 14 ISSA AMADOU TALL AND WITOLD RESPONDEK We will illustrate results of this section by two examples Example 4 Let us reconsider the system Σ given by Example 2 It is feedback equivalent to the normal form ẋ 1 = x 2 + x 2 3P x 1,x 2,x 3, ẋ 2 = x 3, ẋ 3 = v, where P x 1,x 2,x 3 is a formal power series Assume, for simplicity, that m 0 =2, which is equivalent to the following generic condition: g, ad f g, and [g, ad f g] are linearly independent at 0 R 3 This implies that we can express P = P x 1,x 2,x 3 as P = c + P 1 x 1 +x 2 P 2 x 1,x 2 +x 3 P 3 x 1,x 2,x 3, where c 0 and P 1 0 = 0 Observe that any P x 1,x 2,x 3, of the above form, gives a normal form Σ NF In order to get the canonical form Σ CF, we use Theorem 5, which ensures the existence of a feedback transformation Γ of the form x = φx, v = αx+βxṽ, which normalizes the constant c and annihilates the formal power series P 1 x 1 More precisely, Γ transforms Σ into its canonical form Σ CF, x 1 = x 2 + x 2 P 3 x 1, x 2, x 3, x 2 = x 3, x 3 =ṽ, where the formal power series P x 1, x 2, x 3 is of the form P x 1, x 2, x 3 =1+ x 2 P2 x 1, x 2 + x 3 P3 x 1, x 2, x 3 Now, we give an example of constructing the canonical form for a physical model of variable length pendulum Example 5 Consider the variable length pendulum of Bressan and Rampazzo [5] see also [2] and [8] We denote by ξ 1 the length of the pendulum, by ξ 2 its velocity, by ξ 3 the angle with respect to the horizontal, and by ξ 4 the angular velocity The control u = ξ 4 = ξ 3 is the angular acceleration The mass is normalized to 1 The equations are compare [5] and [8] ξ 1 = ξ 2, ξ 2 = g sin ξ 3 + ξ 1 ξ4, 2 ξ 3 = ξ 4, ξ 4 = u, where g denotes the gravity Notice that if we suppose to control the angular velocity ξ 4 = ξ 3, which is the case of [5] and [8], then the system is three-dimensional but the control enters nonlinearly At any equilibrium point ξ 0 =ξ 10,ξ 20,ξ 30,ξ 40 T =ξ 10, 0, 0, 0 T, the linear part of the system is controllable Our goal is to produce, for the variable length pendulum,

15 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 15 a normal form and the canonical form as well as to answer the question whether the systems corresponding to various values of the gravity constant g are feedback equivalent In order to get a normal form, put The system becomes x 1 = ξ 1, x 2 = ξ 2, x 3 = g sin ξ 3, x 4 = gξ 4 cos ξ 3, v = gξ4 2 sin ξ 3 ug cos ξ 3 ẋ 1 = x 2, x 1 ẋ 2 = x 3 + x 2 4 g 2 x 2, 3 ẋ 3 = x 4, ẋ 4 = v, which gives a normal form Indeed, we rediscover Σ NF, given by 216, with P 1,3 =0, P 1,4 = 0, and P 2,4 = x 1 g 2 x 2 3 In order to bring the system to its canonical form Σ CF, first observe that m 0 =3 Indeed, the function x 2 x 1 4 starts with third order terms, which corresponds to the g 2 x 2 3 fact that the invariants a [2],i+2 vanish for any 1 2 and any 0 i 2 The only nonzero component of f [3] is f [3] 2 = x 2 4P [1] 2,4 Hence = 2 and the only, and thus largest, quadruplet i 1,i 2,i 3,i 4 of nonnegative integers, satisfying i 1 +i 2 +i 3 +i 4 =3 and such that 3 f [3] 2 x i1 1 xi4 4 0, is i 1,i 2,i 3,i 4 =1, 0, 0, 2 In order to normalize f [3] 2, put x i = a 1 x i, 1 i 4, ṽ = a 1 v, where a 1 =1/g We get the following canonical form for the variable length pendulum: x 1 = x 2, x 2 = x 3 + x 2 x x 2, 3 x 3 = x 4, x 4 =ṽ Independently of the value of the gravity constant g, all systems are feedback equivalent to each other

16 16 ISSA AMADOU TALL AND WITOLD RESPONDEK 4 Proofs of Theorems 5 and 6 Proof of Theorem 5 The proof of this theorem will be done in two steps In the first step we will deal with terms of degree m 0 Then we will prove the general step by an induction argument First step Let us consider the system Σ given by 21 and let m 0 be the degree of the first nonlinearizable homogeneous part We can assume that see Theorems 1 and 2, after applying a suitable feedback transformation, the system Σ given by 21 takes the form 41 ξ = Aξ + Bu + f [m0] ξ+ m=m 0+1 f ξ+g [m 1] ξu, where A, B is in Brunovský canonical form and the first nonvanishing vector field f [m0] is of the form f [m0] ξ = i=+2 Notice that the linear feedback transformation ξ 2 i P [m0 2],i ξ 1,,ξ i, 1 n 2, 0, n 1 n Γ 1 : x = a 1 ξ, u = 1 a 1 v, where a 1 R and a 1 0, brings the system 41 into the following one: ẋ = Ax + Bv + 1 a m0 1 1 f [m0] x+ m=m 0+1 By the definitions of i 1,,i n s and,wehave m0 [m0] f 0, x i1 1 xin s and thus we can suitably choose the parameter a 1 such that f x+ g [m 1] xv m0 [m0] f = ±1 x i1 1 xin s General step Now, we assume that, for some l 1, the system Σ given by 21, takes the form 42 m 0+l 1 Σ : ξ = Aξ + Bu + m=m 0 f ξ+f [m0+l] ξ+g [m0+l 1] ξu + rξ,u, where rξ,u R m0+l+1 ξ,u and the vector fields f ξ for any m such that m 0 m m 0 + l 1 satisfy the conditions 34, 35, and 36 We will construct a transformation Γ which preserves all terms of degree smaller than m 0 + l while taking those of degree m 0 + l into the canonical form defined by 34 and 36

17 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 17 Consider the following feedback transformation 43 Γ : x = ξ + u = v + m=l+1 m=l+1 φ ξ, α ξ+β [m 1] ξv, where, for any m such that m 0 m m 0 + l 1, the triplet φ,α,β [m 1] is given by 37 and φ =0,α = 0, and β [m 1] = 0 for m m 0 + l +1 The transformation Γ is actually a polynomial transformation Γ m0+l and can be viewed as a composition of a transformation Γ m0+l 1 and a homogeneous transformation Γ m0+l defined, respectively, by 44 and 45 Γ m0+l 1 : Γ m0+l : m 0+l 1 y = ξ + φ ξ, u = w + m=l+1 m 0+l 1 m=l+1 x = y + φ [m0+l] y, α ξ+β [m 1] ξw w = v + α [m0+l] y+β [m0+l 1] yv Let us denote by Σ the system Σ, given by 42, transformed via Γ m0+l 1 Since f [m0] ξ = f [m0] y φ [l+1] y = f [m0] [m0] f y φ [l+1] y+r 1 y, y where r 1 y R m0+l+1 y and for any m m 0 +1, f ξ = f y φ [l+1] y = f y+r 2 y, where r 2 y R m0+l+1 y, we get 46 Σ : m 0+l 1 ẏ = Ay + Bw + f y+ f [m0+l] y+ g [m0+l 1] yw + r 3 y, w, m=m 0 where r 3 y, w R m0+l+1 y, w and f [m0+l] = f [m0+l] +[ f [m0],φ [l+1] ], g [m0+l 1] = g [m0+l 1] Let { a [m0+l],i+2 } :, i

18 18 ISSA AMADOU TALL AND WITOLD RESPONDEK and { ã [m0+l],i+2 } :, i denote, respectively, the sets of m 0 + l-invariants associated with the homogeneous parts of degree m 0 + l of the systems 42 and 46 We have 47 where ã [m0+l],i+2 = a [m0+l],i+2 +â [m0+l],i+2, [ i [ â [m0+l],i+2 = CA 1 1 i+k ad A i Bad i k Aξ ad A k B f [m 0],φ [l+1]] π n i ξ k=0 i 1 ] [ + 1 i+k ad A i+1 Bad i k 1 ad A k B f [m 0],φ [l+1]] π n i ξ k=0 Since the identity ad k A n 1 B adi Aξh = ad i Aξad k A n 1 B h holds for any vector field h and any k, i 0, we get by differentiating 48 L i1+l An 1Bâ[m0+l],i+2 [ i [ = CA 1 1 i+k ad A i Bad i k Aξ ad A Bad i1+l k A n 1 B f [m 0],φ [l+1]] π n i ξ k=0 i 1 ] [ + 1 i+k ad A i+1 Bad i k 1 Aξ ad A Bad i1+l k A n 1 B f [m 0],φ [l+1]] π n i ξ k=0 Due to the definition of the n s-tuple i 1,,i n s, we obtain [ ad i1+l A n 1 B f [m 0],φ [l+1]] = c 1 [ad i1a f n 1B [m0],ad l A n 1 B φ[l+1]] 49 [ + c 2 ad i1 1 f [m0] A n 1 B,ad l+1 A n 1 B φ[l+1]], where c 1 and c 2 are strictly positive integers From the relations ad l A n 1 B φ[l+1] = a l+1 l + 1!ξ 1,ξ 2,,ξ n T, ad l+1 A n 1 B φ[l+1] = a l+1 l + 1!1, 0,,0 T, one can easily deduce that identity 410 can be rewritten as [ ad i1+l A n 1 B f [m 0],φ [l+1]] = γ l ad i1 f [m0] A n 1 B, where we set γ l = a l+1 l + 1! c 1 m 0 i 1 +1+c 2 Plugging the above identity into the formula 48, we obtain L i1+l An 1Bâ[m0+l],i+2 [ i = γ l CA 1 1 i+k ad i1 A n 1 B ad A i Bad i k Aξ ad f [m0] A k B π n i ξ k=0 i 1 ] + 1 i+k ad i1 A n 1 B ad A i+1 Bad i k 1 ad f [m0] A k B π n i ξ k=0 Aξ Aξ

19 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 19 Since f [m0] is of the form 34, we get for any k such that 0 k i 1, ad Ak B f [m0] π n i ξ=0 and for any t 0, ad t Aξad A k B f [m0] π n i ξ=0 Thus, we can deduce the relation i1+l â [m0+l],i+2 = γ l CA ξ i1+l 1 1 i1+2 f [m0] ξ i1 1 ξ2 n i π n i ξ, which leads, after differentiating and setting = and i = s, to the following one: m0+l 2 â [m0+l],s+2 ξ i1+l 1 ξ i2 2 ξin s 2 m0 [m0] f = γ l ξ i1 1 ξi2 2 ξin s Differentiating 47 and using the above identity, we get m0+l 2 ã [m0+l],s+2 ξ i1+l 1 ξ i2 2 ξin s 2 m0 [m0] f + γ l 2 ξin s 2 ξ i1 1 ξi2 2 ξin s = m0+l 2 a [m0+l],s+2 ξ i1+l 1 ξ i2 n s We can choose suitably the parameter a l+1 recall the definition of γ l such that we obtain m0+l 2 ã [m0+l],s+2 =0 ξ i1+l 1 ξ i2 2 ξin i 2 Now, transforming the homogeneous part of degree m 0 + l of the system 46 to its normal form via a homogeneous transformation Γ m0+l and taking into account Theorem 2, we bring the system 46 into the form Σ : ẋ = Ax + Bv + m 0+l m=m 0 f x+rx, v, where rx, v R m0+l+1 x, v and the vector fields f, for any m such that m 0 m m 0 + l, satisfy the conditions 34, 35, and 36 This completes the proof of Theorem 5 In our proof of Theorem 6, we will use the following result Lemma 1 A transformation Γ leaves invariant all terms of degree smaller than m 0 + l of the system 42 if and only if Γ is of the form 410 Γ : x = ξ + u = v + m=l+1 m=l+1 φ ξ, α ξ+β [m 1] ξv, where, for any m such that m 0 m m 0 + l 1, the triplet φ,α,β [m 1] is given by 37

20 20 ISSA AMADOU TALL AND WITOLD RESPONDEK Proof of Lemma 1 We have shown, when proving Theorem 5, that the transformation Γ, defined by 410 and 37, leaves invariant all terms of degree smaller than m 0 + l of the system 42 Conversely, assume that the transformation Γ leaves invariant all terms of degree smaller than m 0 + l of the system 42 Without loss of generality, we can take Γ : x = ξ + φ ξ, u = v + m=k+1 m=k+1 α ξ+β [m 1] ξv, where k+1 denotes the smallest degree among degrees of all nonvanishing components φ of the transformation Γ There is nothing to prove if k +1 m 0 + l Wethus focus our attention on the case k +2 m 0 + l Since Γ leaves invariant all terms of degree smaller than m 0 + l of the system 42, in particular it leaves invariant terms of degree k +1, which implies that φ [k+1],α [k+1],β [k] satisfies the condition 37 By induction, we show that φ,α,β [m 1] also satisfies the condition 37 for any m such that k +1 m m 0 + k 1 Thus it remains only to prove that k l Assume this is false; that is, suppose k l 1 We can see that the transformation Γ brings the system 42 into the following one: 411 m 0+k 1 ẋ = Ax + Bv + f x+ f [m0+k] x+rx, v, m=m 0 where rx, v R m0+k+1 x, v and the vector field f x, for any m such that m 0 m m 0 + k 1, is of the form 34 and 36 and f [m0+k] = f [m0+k] +[ f [m0],φ [k+1] ] Since the transformation Γ leaves invariant all terms of degree smaller than m 0 + l of the system 42, in particular it leaves invariant all terms of degree m 0 + k, which is equivalent to [ f [m0],φ [k+1] ]=0 Repeating the calculations done in the proof of Theorem 5 we deduce, by differentiating, the identity m0+k CA 1 [ f [m0],φ [k+1] ] x i1+k 1 x i2 2 xin s Thus, due to the fact that m0 [m0] f x i1 1 xi2 2 xin s m0 [m0] f = γ k x i1 1 xi2 2 xin s we obtain γ k = 0 and hence φ [k+1],α [k+1],β [k] = 0, which contradicts the definition of k + 1 As a conclusion, it follows that the transformation Γ is of the form 410 and 37 0, =0

21 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 21 Proof of Theorem 6 Let us consider two systems Σ 1 and Σ 2 and let Σ 1,CF : ẋ = Ax + Bv + m=m 0 f x and Σ 2,CF : ż = Az + Bw + m=m 1 f z denote, respectively, their canonical forms, where m 0 and m 1 denote the degrees of the first nonlinearizable homogeneous parts It is obvious that Σ 1 and Σ 2 are feedback equivalent if their canonical forms Σ 1,CF and Σ 2,CF coincide To prove the converse, we assume that the systems Σ 1 and Σ 2 are feedback equivalent while their canonical forms fail to coincide Since Σ 1 and Σ 2 are feedback equivalent, so are their canonical forms Σ 1,CF and Σ 2,CF It means that there exists a transformation Γ which brings Σ 1,CF into Σ 2,CF First remark that, from the definition of the integers m 0 and m 1, we necessarily have m 0 = m 1 Then, Theorem 2 and the fact that the components f [m0] [m0] and f are normalized see 35 ensure that f [m0] = f [m1] Let l be the largest integer such that for any i l, wehave f [m0+i 1] = f [m0+i 1] This means that the transformation Γ leaves invariant all terms of degree smaller than m 0 + l of the system Σ 1,CF Then Lemma 1 shows that the transformation Γ is of the form 410 Since the transformation Γ brings Σ 1,CF into Σ 2,CF, we deduce that 412 f [m0+l] = f [m0+l] +[ f [m0],φ [l+1] ] Following arguments in the proof of Theorem 5, we obtain where m0+l 2 ã [m0+l],s+2 x i1+l 1 x i2 2 xin s 2 m0 [m0] f + γ l, 2 xin s 2 x i1 1 xi2 2 xin s = m0+l 2 ā [m0+l],s+2 x i1+l 1 x i2 n s { ā [m0+l],i+2 } :, i and { ã [m0+l],i+2 } :, i denote, respectively, the set of m 0 + l-invariants associated with the homogeneous parts of degree m 0 + l of the systems Σ 1,CF and Σ 2,CF Using Theorem 2, the last identity can be rewritten as 413 Since m0+l [m0+l] f x i1+l 1 x i2 2 xin s = m0 [m0+l] f x 1, 0,,0 = x i1 1 xin s m0+l [m0+l] f x i1+l 1 x i2 2 xin s m0 f [m0+l] x i1 1 xin s m0 [m0] f + γ l x i1 1 xi2 2 xin s x 1, 0,,0=0,

22 22 ISSA AMADOU TALL AND WITOLD RESPONDEK the identity 413 gives γ l m0 [m0] f x i1 1 xi2 2 xin s which implies γ l =0, that is recall the definition of γ l, we have a l+1 = 0, and consequently φ [l+1],α [l+1],β [l] =0 Then the identity 412 reduces to f [m0+l] = f [m0+l], =0, which contradicts the definition of l We conclude that the canonical forms Σ 1,CF and Σ 2,CF coincide 5 Dual normal form and dual m-invariants In the normal form Σ NF given by 27, all the components of the control vector field g [m 1] are annihilated and all nonremovable nonlinearities are grouped in f Kang and Krener in their pioneering paper [26] showed that it is possible to transform, via a transformation Γ 2 of degree 2, the homogeneous system Σ [2] : ξ = Aξ + Bu + f [2] ξ+g [1] ξu into a dual normal form In that form the components of the drift f [2] are annihilated, while this time all nonremovable nonlinearities are present in g [1] The aim of this section is to propose, for an arbitrary m, a dual normal form for the system Σ and a dual normal form for the system Σ Our dual normal form on the one hand generalizes, for higher order terms, that given in [26] for second order terms, and on the other hand dualizes the normal form Σ NF The structure of this section will follow that of section 2: we will give the dual normal form, then define and study dual m-invariants; finally, we give an explicit construction of transformations bringing the system into its dual normal form Our first result asserts that we can always bring Σ to a dual normal form Theorem 7 feedback transformation Γ m, to the dual normal form Σ DNF 51 Σ DNF : The homogeneous system Σ is equivalent, via a homogeneous given by ẋ 1 = x 2, ẋ 2 = x 3 + vx n Q [m 2] 2,n x 1,,x n, ẋ = x +1 + v i=n +2 x i Q [m 2],i x 1,,x i, ẋ n 1 = x n + v n x i Q [m 2],i x 1,,x i, ẋ n = v, i=3 where Q [m 2],i x 1,,x i are homogeneous polynomials of degree m 2 depending on the indicated variables Theorem 7 follows from Theorem 9, which gives explicit transformation bringing Σ to its dual normal form Σ DNF, and thus we omit its proof

23 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 23 Now we will define dual m-invariants To start with, recall that the homogeneous vector field X [m 1] i is defined by taking the homogeneous part of degree m 1ofthe vector field X m 1 i = 1 i ad i Aξ+f B + g [m 1] By X [m 1] i π i ξ we will denote the vector field X [m 1] i evaluated at the point π i ξ = ξ 1,,ξ i, 0,,0 of the submanifold W i = {ξ R n : ξ i+1 = = ξ n =0} Consider the system Σ and, for any such that 2 n 1, define the polynomial b [m 1] by setting b [m 1] 1 + = g [m 1] k=1 L B L k 1 Aξ f k i=1 ξi L B L 1 Aξ CX [m 1] n i π i ξdξ i 0 The homogeneous polynomials b [m 1] for 2 n 1 will be called the dual m- invariants of the homogeneous system Σ Consider two systems Σ and Σ of the form 24 and 25 Let and { b [m 1] : 2 n 1 } [m 1] { b : 2 n 1 } denote, respectively, their dual m-invariants The following result gives a dualization of Theorem 2 Theorem 8 The dual m-invariants have the following properties: i Two systems Σ and Σ are equivalent via a homogeneous feedback transformation Γ m if and only if b [m 1] [m 1] = b for any 2 n 1 ii The dual m-invariants 51, are given by b [m 1] of the dual normal form Σ DNF, defined by b[m 1] x = i=n +2 x i Q [m 2],i x 1,,x i for any 2 n 1 The above result asserts that the dual m-invariants, as do the m-invariants, form a set of complete invariants of the homogeneous feedback transformation Notice, however, that the same information is encoded in both sets of invariants in different ways We will give a proof of Theorem 8 in section 7

24 24 ISSA AMADOU TALL AND WITOLD RESPONDEK 52 Now, we define the following homogeneous polynomials: ξi φ 1 = CX [m 1] n i π i ξdξ i, i=1 φ +1 = f α = β [m 1] = f 0 + L Aξ φ, 1 n 1, n + L Aξ φ n, + L B φ n g [m 1] n The next result gives an explicit construction of feedback transformations bringing the system Σ to its dual normal form Σ DNF Theorem 9 The feedback transformation Γ m : x = ξ + φ ξ, u = v + α ξ+β [m 1] ξv, where α, β [m 1], and the components φ of φ are defined by 52, brings the system Σ into its dual normal form Σ DNF given by 51 6 Dual canonical form Consider the system Σ of the form 21 and assume that its linear part F, G is controllable Apply successively to it a series of transformations Γ m, m =1, 2,, such that each Γ m brings Σ to its dual normal form Σ DNF ; for instance we can take a series of transformations defined by 52 Successive repeating of Theorem 9 gives the following dual normal form Theorem 10 The system Σ can be transformed via a formal feedback transformation Γ into the dual normal form Σ DNF given by 61 Σ DNF : ẋ 1 = x 2, ẋ 2 = x 3 + vx n Q 2,n x 1,,x n, ẋ = x +1 + v x i Q,i x 1,,x i, ẋ n 1 = x n + v ẋ n = v, i=n +2 x i Q,i x 1,,x i, i=3 where Q,i x 1,,x i are formal power series depending on the indicated variables Naturally, as with normal forms, a given system can admit different dual normal forms We are thus interested in constructing a dual canonical form Assuming that the linear part F, G of the system Σ, of the form 21, is controllable, we denote by m 0 the degree of the first homogeneous term of the system Σ which cannot be annihilated by a feedback transformation Thus, using Theorems 8 and 9, we can assume, after applying a suitable feedback, that Σ takes the form Σ : ξ = Aξ + Bu +ḡ [m 0 1] ξu + f ξ+g [m 1] ξu, m=m 0+1

25 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 25 where A, B is in Brunovský canonical form and the first nonvanishing homogeneous vector field ḡ [m0 1] is of the form ξ i Q [m0 2] ḡ [m0 1],i ξ 1,,ξ i, 2 n 1, ξ = i=n +2 0, = 1 and = n Define { = inf =2,,n 1 : ḡ [m0 1] } ξ 0 and let i 1,,i n such that i 1 + +i n = m 0 1 be the largest, in the lexicographic ordering, n-tuple of nonnegative integers such that m0 1 ḡ [m0 1] 0 ξ i1 1 ξin n We get the following result Theorem 11 There exists a formal feedback transformation Γ which brings the system Σ into the following one: where for any m m 0, 62 Moreover, ḡ [m 1] = Σ DCF :ẋ = Ax + Bv + i=n +2 m=m 0 ḡ [m 1] xv, x i Q [m 2],i x 1,,x i, 2 n 1, 0, =1and = n 63 and for any m m 0 +1 m0 1 ḡ [m0 1] x i1 1 xin n = ±1, 64 m0 1 ḡ [m 1] x 1, 0,,0=0 x i1 1 xin n The form Σ DCF, which satisfies 62, 63, and 64, will be called the dual canonical form of Σ The name is ustified by the following Theorem 12 The two systems Σ 1 and Σ 2 are formally feedback equivalent if and only if their dual canonical forms Σ 1,DCF and Σ 2,DCF coincide Example 6 Let us consider the system Σ : ξ = fξ+gξu, ξ R 3,u R,

26 26 ISSA AMADOU TALL AND WITOLD RESPONDEK whose linear part is assumed to be controllable Theorem 10 ensures that the system Σ is formally feedback equivalent to the dual normal form Σ DNF given by ẋ 1 = x 2, ẋ 2 = x 3 + vx 3 Qx 1,x 2,x 3, ẋ 3 = v, where Qx 1,x 2,x 3 is a formal power series of variables x 1,x 2,x 3 Assume for simplicity that m 0 = 2, which is equivalent to the condition that g, ad f g, and [g, ad f g] are linearly independent at 0 R 3 This implies that we can represent Q = Qx 1,x 2,x 3 by Q = c + x 1 Q 1 x 1 +x 2 Q 2 x 1,x 2 +x 3 Q 3 x 1,x 2,x 3, where c R, c 0 Observe that any Q of the above form gives a dual normal form Σ DNF In order to get the dual canonical form we use Theorem 11, which ensures that the system Σ is formally feedback equivalent to its dual canonical form Σ DCF defined by x 1 = x 2, x 2 = x 3 +ṽ x 3 Q x1, x 2, x 3, x 3 =ṽ, where Q x 1, x 2, x 3 is a formal power series such that Q x 1, x 2, x 3 =1+ x 2 Q2 x 1, x 2 + x 3 Q3 x 1, x 2, x 3 7 Proofs of dual results In this section, we prove our dual results The proof of Theorem 7 will be omitted because in the proof of Theorem 9 we give an explicit homogeneous feedback transformation bringing a given homogeneous system into its dual normal form Theorem 10 follows from a successive application of Theorem 7 We will thus prove Theorems 8, 9, 11, and 12 Proof of Theorem 8 i We will prove that if the system Σ is equivalent to b [m 1] Σ via a transformation Γ m, then their dual m-invariants b [m 1] and The action of Γ m can be decomposed into that of a pure feedback of the form followed by that of a diffeomorphism u = v + α ξ+β [m 1] ξv x = ξ + φ ξ coincide of the state space Since the first n 1 components of the vector fields f and g [m 1], as well as those of X [m 1] n i, are invariant under pure feedback, we can conclude that the functions b [m 1] for 2 n 1 are invariant under pure feedback It remains to prove that they are also invariant under any diffeomorphism x =Φξ of the form Φξ =ξ + φ ξ The diffeomorphism Φ brings the system Σ into the form Σ : ẋ = Ax + Bu + f x+ g [m 1] xu,

27 FEEDBACK CLASSIFICATION OF NONLINEAR SYSTEMS 27 where b [m 1] f = f +[Ax, φ ], g [m 1] = g [m 1] + L B φ Denoting by b [m 1] and respectively, with the homogeneous systems Σ and Σ,weget where and ˆb[m 1] for 2 n 1 the dual m-invariants associated, b[m 1] x =L B φ i=1 = b [m 1] 2 x+ L B L 1 Ax k=0 xi 0 [m 1] + ˆb, L B L k 2 Ax CA k ad Ax φ C [m 1] ˆX n i π i xdx i ˆX [m 1] n i x = 1 n i ad n i Ax+[Ax,φ ] B + L Bφ = 1 n i Φ ad n i Aξ Bx = A n i B + L A n i Bφ x We can deduce that ˆb[m 1] which gives x =L B φ i=1 = L B φ 1 x+ L B L 1 Ax k=1 xi 0 L B L k Ax φ k 1 x+ L B L k k=1 k=2 L A n i Bφ 1 π i xdx i Ax φ k b[m 1] L B L k Ax φ k L B L k k=2 = b [m 1] Ax φ k L B L 1 Ax φ 1 x =0, Thus the functions b [m 1] are invariant under any diffeomorphism of the form x = Φξ =ξ + φ ξ Therefore they remain invariant under the transformation Γ m The fact that two homogeneous systems, whose dual m-invariants coincide, are feedback equivalent follows clearly from item ii of the theorem, which will be proved below Indeed, by item ii, both systems coincide when transformed to their canonical forms [m 1] ii Denote by b for 2 n 1 the dual m-invariants associated with the dual normal form Σ DNF They are given by b[m 1] =ḡ [m 1] i=1 xi L B L 1 [m 1] Ax C X n i π i xdx i, 0

28 28 ISSA AMADOU TALL AND WITOLD RESPONDEK where the components ḡ [m 1] C are given by 51 and X [m 1] n i = 1 n i Cad n i Ax ḡ[m 1] X [m 1] n i It suffices to observe see Lemma 2 below that, on the one hand, C is a linear combination of functions L s Axḡ[m 1] for 0 s n i and 1 n i + 1 and, on the other hand, ḡ [m 1] π i x = 0 for all such that 1 n i +1 We thus conclude that C π i x=0, which implies X [m 1] n i b[m 1] =ḡ [m 1] for any such that 2 n 1 Proof of Theorem 9 Denote by Σ : ẋ = Ax + Bv + f x+ g [m 1] xv the system Σ transformed via a homogeneous feedback transformation Γ m defined by 52 From Proposition 1, it follows that for Σ we have 71 f = 0 for 1 n, g [m 1] = 0 for = 1 and = n, g [m 1] = g [m 1] + L B φ for 2 n 1 It thus suffices to show that the components g [m 1] for 2 n 1 are in the dual normal form 51 We prove easily by an induction argument that which allows us to show that g [m 1] +1 = g [m 1] +1 + Now, from the identity φ +1 = L B L Aξ φ 1 = k=1 L k Aξ f k k=1 k L k k=0 L B L k 1 Aξ f L A Bφ k 1 = CX [m 1] π n k ξ we can deduce that k g [m 1] +1 = g [m 1] +1 + k=1 Aξ + L Aξ φ 1, 1, Aξ L A k Bφ k+1 + i=n k+1 L B L k 1 f k+1 k L k Aξ k=1 i=n k+1 ξi 0 k=0 ξi 0 k L k 1 Aξ L A k Bφ CX [m 1] n i π i ξ dξ i, ξ n k k L k Aξ CX k k=0 CX [m 1] n i ξ n k π n k ξ π i ξ dξ i