A characterization of normal forms for control systems
aa r X i v : . [ m a t h . O C ] M a r A characterization of normal forms for control systems
Boumediene Hamzi ∗ Jeroen S.W. Lamb Debra LewisDepartment of Mathematics Department of Mathematics Mathematics DepartmentImperial College London Imperial College London UC Santa CruzLondon, SW7 2AZ, UK London, SW7 2AZ, UK Santa Cruz, CA 95064, [email protected] [email protected] [email protected]
The study of the behavior of solutions of ODEs often benefits from deciding on a convenient choiceof coordinates. This choice of coordinates may be used to “simplify” the functional expressions thatappear in the vector field in order that the essential features of the flow of the ODE near a criticalpoint become more evident. In the case of the analysis of an ordinary differential equation in theneighborhood of an equilibrium point, this naturally leads to the consideration of the possibilityto remove the maximum number of terms in the Taylor expansion of the vector field up to agiven order. This idea was introduced by H. Poincar´e in [25] and the “simplified” system is callednormal form. There have been several applications of the method of normal forms particularly inthe context of bifurcation theory where one combines between the method of normal forms andthe center manifold theorem in order to classify bifurcations [9]. This approach was extended tocontrol systems in continuous-time by Kang and Krener ([17], see also [18] for a survey) and Talland Respondek ([23], see [24] for a survey), and by Barbot et al. [2] and Hamzi et al. in discrete-time [15, 16]. The center manifold theorem was extended to control systems by Hamzi et al. [10, 11]and combined with the normal forms approach to analyze and stabilize systems with bifurcationsin continuous and discrete-time [12, 13, 14].On another side, even though in many textbook treatments (see eg [9]) the emphasis is on thereduction of the number of monomials in the Taylor expansion, one of the main reasons for thesuccess of normal forms lies in the fact that it allows to analyze a dynamical system based ona simpler form and a simpler form doesn’t necessarily mean to remove the maximum number ofterms in the Taylor series expansion. This observation, led to introduce the so-called “inner-productnormal forms” in [3, 19, 7]. They are based on properly choosing an inner product that allows tosimplify the computations. This inner-product will characterize the space overwhich one performsthe Taylor series expansion. The elements in this space are the ones that characterize the normalform. Our goal in this paper is to generalize such an approach to control systems.In section §
2, we review some results about normal forms. In section §
3, we develop a newmethod for deriving normal forms for control systems. ∗ Parts of this work were done while at Department of Mathematics, Duke University, Durham, NC 27708, USA. Normal forms near equilibria of ODEs
In this section we briefly review some results on normal forms near equilibria of nonlinear ODEs.Consider the nonlinear ODE in IR n ˙ x = Ax + f ( x ) , (2.1)with f ∈ C r +1 (IR n ; IR n ), f (0) = 0 and A = ∂f∂x | x =0 is in real or complex Jordan form. Without lossof generality the latter condition can be met by application of a linear coordinate transformation.The goal is to find a change of coordinates x = ξ ( y ) , (2.2)with ξ ∈ C r (IR n ; IR n ) in a neighborhood of the origin, such that the Taylor expansion of (2.1) issimple, making essential features of the flow of (2.1) near the equilibrium x = 0 more evident. Thedesired simplification of (2.1) will be obtained, up to terms of a specified order, by constructing anear identity coordinate transformation from a sequence of compositions of coordinate transforma-tions of the form (2.2) with ξ ( y ) = exp( ξ [ k ] )( y ) = y + ξ [ k ] ( y ) + O ( | y | k +1 ) , (2.3)where y ∈ IR n is close to zero, ξ [ k ] ∈ H kn ( k ≥ k in n variables with values in IR n , and exp( ξ [ k ] ) denotes the time-one flow of the ODE˙ y = ξ [ k ] ( y ). We consider a formal power series expansion of f in (2.1) and write f ( x ) = f [2] ( x ) + f [3] ( x ) + . . . , (2.4)with f [ k ] ∈ H kn . From (2.3) we obtain ξ − ( y ) = y − ξ [ k ] ( y ) + O ( | y | k ) . (2.5)Substituting (2.2), (2.3) and (2.5) in (2.1), we get˙ y = Ay + · · · + f [ k − ( y ) + f [ k ] ( y ) − ( L A ξ [ k ] )( y ) + O ( | y | k +1 ) , (2.6)with the Lie derivative L A defined on vector fields f as( L A f )( y ) := ∂f ( y ) ∂y Ay − Af ( y ) . (2.7)In the present context L A is also known as the homological operator.The Lie derivative leaves H kn invariant, L A : H kn → H kn . We denote its range in H kn as R k andlet C k denote a complement of R k in H kn H kn = R k ⊕ C k , k ≥ . (2.8)We define a normal form of f of order r as a Taylor expansion of the vector field with linear partand terms f [ k ] ∈ C k for 2 ≤ k ≤ r . We may associate the choice of complement C k to an innerproduct on H kn , for which it is the orthogonal complement of R k in H kn , i.e. C k := ( R k ) ⊥ . etal. [7], enabling the characterization of expression of C k as the kernel of the Lie derivative of A ∗ (theadjoint of linear part A of the vector field at the equilibrium). Denoting monomials in shorthandnotation as x ℓ := x ℓ · · · x ℓ n n with ℓ ! := ℓ ! · · · ℓ n !, we define an inner product on polynomials p ( x ) = X ℓ p ℓ x ℓ , q ( x ) = X m q m x m , as h p, q i = X m m ! p m q m . (2.9)For vector polynomials we define the corresponding inner product as the sum of the inner productsbetween the polynomials of corresponding vector components. The inner product (2.9) with T ∈ gl ( n, IR) and T ∗ denoting its adjoint (with respect to the standard inner product on IR n ) satisfies[3, 7] h p ◦ T, q i = h p, q ◦ T ∗ i . (2.10)Accordingly, one obtains that the adjoint of L A on H kn with the above defined inner product satisfiesthe following relation [3, 7] ( L A ) ∗ = L A ∗ . (2.11)By application of the Fredholm alternative, it follows that ( R k ) ⊥ = ker( L ∗ A | H kn ). In combinationwith (2.11), this leads us to C k = ker( L A ∗ | H kn ) , as a result of which nonlinear elements of the normal form g satisfy the linear PDE L A ∗ g = 0 . (2.12)This PDE can be solved explicitly using the method of characteristics (for more details on thismethod, see for example [6]).We recall that since L A ∗ is a Lie derivative, it follows that the nonlinear elements of the normalform commute with the group G = { exp( A ∗ t ) | t ∈ IR } . (2.13)We finally note that ker( L kA ∗ ) = ker( L kA ∗ s ) ∩ ker( L kA ∗ n ) , (2.14)where A = A s + A n is the Jordan-Chevalley decomposition of A in its (mutually commuting) semi-simple and nilpotent parts. As A ∗ commutes with A s but not with A n (if nonzero), if A is notsemi-simple, only a subgroup of G (as defined above) is a symmetry group of the normal form. Ingeneral, with the above choices made, the normal form is equivariant with respect to the group G s = { exp( A ∗ s t ) | t ∈ IR } . (2.15)The appearance of this symmetry group is an important feature. The object of this section is to extend the normal form theory set out above to nonlinear controlsystems. We consider the nonlinear control system˙ x = f (˜ x ) , (3.16)3ith ˜ x = ( x, u ) T ∈ IR n × IR m and u ∈ IR m representing the control. In lowest linear order Taylorexpansion, the control system takes the form˙ x = A ˜ x + O ( | ˜ x | ) , with A := (cid:0) A B (cid:1) and A := ∂f (˜ x ) ∂x | ˜ x =0 , B := ∂f (˜ x ) ∂u | ˜ x =0 .We consider the effect of coordinate transformations of the form˜ x = p (˜ y ) = exp( p [ k ] )(˜ y ) = ˜ y + p [ k ] (˜ y ) + O ( | ˜ y | k +1 ) , (3.17)where ˜ y = ( y, v ) T and p [ k ] ∈ S kn + m,n with S kn,m := { ( p [ k ] x , p [ k ] u ) T | p [ k ] x ∈ H kn , p [ k ] u ∈ H kn + m,m } , (3.18)with H kn + m,m denoting the vector space of homogeneous polynomials of degree k from IR n + m toIR m . The skew product form of p [ k ] ∈ S kn,m , p [ k ] (˜ y ) = ( p [ k ] x ( y ) , p [ k ] u (˜ y )) T , guarantees that the controlsystem is transformed to another control system of the same type. If p x would depend on u thenthe coordinate transformation would introduce a relationship involving ˙ u . We obtain˙ y = A ˜ y + · · · + f [ k − (˜ y ) + f [ k ] (˜ y ) − ( L A p [ k ] )(˜ y ) + O ( | ˜ y | k +1 ) , (3.19)where the homological operator L A : S kn,m → H kn + m,n has the form( L A p [ k ] )(˜ y ) = Dp [ k ] x ( y ) A ˜ y − A p [ k ] (˜ y ) = ( L A p [ k ] x )( y ) + Dp [ k ] x ( y ) Bu − Bp [ k ] u (˜ y ) . (3.20)We recognize in this expression the Lie derivative L A , that is equal to L A in case B = 0. Indeed, f ( y,
0) (the part of f that does not depend on u ) can be put into a G s -equivariant normal form,using coordinate transformations of the form p (˜ y ) = (exp( p x )( y ) , u ) only.We now proceed to characterize a normal form by a (choice of) complement of the range of L A . In order to do so in analogy to the theory developed for ODEs, we temporarily take theviewpoint as if the coordinate transformation would be for the ODE ( ˙ x, ˙ u ) = ( f ( x, u ) , h ( x, u )), forsome h : IR m + n → IR m with D x h (0 ,
0) = 0 and D u h (0 ,
0) = 0. The homological operator for thelatter ODE, with coordinate transformations of the form (3.17) takes precisely the form of the Liederivative L A , with A = (cid:18) A B (cid:19) , so that L A = ( L A ,
0) and L A := πL A , with π : IR m + n → IR n denoting the canonical projection π ( x, u ) := x .We may thus choose the complement of the range of L A as the projection under π of theorthogonal complement C k to the range of L A taken with respect to the inner product (2.9) with x n + i = u i , i = 1 , . . . , m , i.e. C k := { q ∈ H km + n | h q, L A p i = 0 , ∀ p ∈ S kn,m } . (3.21)By the Fredholm alternative we have h q, L A p i = h L A ∗ q, p i , (3.22)4o that this complement takes the form C k := { q ∈ H km + n | L A ∗ q ∈ ( S kn,m ) ⊥ } . (3.23)By the definition of the inner product (2.9),( S kn,m ) ⊥ = { q ∈ H km + n | q ( x,
0) = 0 } , i.e. the subset of vector polynomials in H km + n for which each constituting monomial containsa factor u i , i = 1 , . . . , m . The complement to the range of L A characterising the correspondingnormal form is π C k . By writing out the relevant operators, the following result follows immediately. Theorem 3.1 (Control normal form)
Consider a finite order in Taylor expansion of the vectorfield defining the control system (3.16), f (˜ x ) = A ˜ x + N X k =2 f [ k ] (˜ x ) + O ( | ˜ x | k +1 ) , with f [ k ] ∈ H km + n,n . By a choice of coordinates, the nonlinear parts f [ k ] can be made to satisfy ˆ L A ∗ f [ k ] ( x,
0) = 0 (3.24) where ˆ L A ∗ f [ k ] (˜ x ) := D ˜ x f [ k ] (˜ x ) A ∗ x − A ∗ f [ k ] (˜ x ) , (3.25) and ˜ x = ( x, u ) .Remark. We note that by restricting first to coordinate transformations that do not involve u ,we can achieve G s -equivariance of the control system to any desired order. Then we can refine thenormalization further using G s -equivariant coordinate transformations that preserve this equivari-ance. ⊳ To illustrate this method, consider the nonlinear control system Σ in (3.16) with one input, i.e. m = 1, and assume that its linearization is controllable. From linear control theory we know thatthere exists a linear change of coordinates and feedback that allows to transform the linear part inthe Brunovsk`y form, i.e. A = · · ·
00 0 1 · · · · · ·
10 0 0 · · · , B = . (4.26)5n this case, the PDE (3.25) becomes x ∂p ∂x + · · · + x n − ∂p ∂x n + x n ∂p ∂u = 0 x ∂p ∂x + · · · + x n − ∂p ∂x n + x n ∂p ∂u − p = 0... x ∂p n ∂x + · · · + x n − ∂p n ∂x n + x n ∂p n ∂u − p n − = 0 (4.27)that we’ll solve using the method of characteristics. Theorem 4.1
Consider the nonlinear control system Σ given by (3.16). There exist a change ofcoordinates and feedback (3.17) such that Σ writes as ˙ x = x + Φ ( ℓ , · · · , ℓ r +1 ) , ˙ x = x + Φ ( ℓ , · · · , ℓ r +1 ) + Z p ( x, u ) dx x , ... ˙ x n = u + Φ n ( ℓ , · · · , ℓ r +1 ) + Z p n − ( x, u ) dx x , (4.28) with Φ i ( ℓ , · · · , ℓ n ) are functions satisfying Φ i ( ℓ , · · · , ℓ n ) x p (cid:12)(cid:12)(cid:12)(cid:12) x =0 = 0 for p = 0 , · · · , n − i, (4.29)(4.30) and ℓ ( x ) = x , ℓ ( x ) = x − x x , · · · , ℓ i ( x ) = 12 x i + n − p X k =1 ( − k x i − k x i + k for i = 2 , · · · , r + 1 , and r is such that r = n/ if n is even, and r = ( n − / if n is odd (here, x = 0 and x n +1 = u )Proof. In the n − dimensional space of the variables x , x , · · · , x n we determine the curves x i = x i ( s )in terms of a parameter s by means of the system of ordinary differential equations that representthe characteristic curves dx ds = 0 dx ds = x ... dx n ds = x n − duds = x n (4.31)6long the characteristic curves and using the chain rule, the systems of PDEs (3.25) writes as dp ds = dx ds ∂p ∂x + dx ds ∂p ∂x + · · · + dx n ds ∂p ∂x n + duds ∂p ∂u = 0 dp ds = dx ds ∂p ∂x + dx ds ∂p ∂x + · · · + dx n ds ∂p ∂x n + duds ∂p ∂u = p ... dp n ds = dx ds ∂p ∂x + dx ds ∂p ∂x + · · · + dx n ds ∂p ∂x n + duds ∂p ∂u = p n − (4.32)Hence, along the characteristic curves defined by (4.31), the systems of PDEs (3.25) transformsinto a set of ODEs dp ds = 0 dp ds = p ... dp n ds = p n − (4.33)This system of ODEs can be solved explicitly p ( s ) = c p ( s ) = c + Z p ( s ) ds ... p n ( s ) = c n + Z p n − ( s ) ds The “constants of integration”, c i , are the constants along the characteristic curves which arethe trivial first integrals of the system (4.31). One can check that they are given by ℓ ( x ) = x , ℓ ( x ) = x − x x , · · · , ℓ i ( x ) = 12 x i + n − p X k =1 ( − k x i − k x i + k for i = 2 , · · · , r + 1, and r is such that r = n/ n is even, and r = ( n − / n is odd (for notation convenience, x = 0 and x n +1 = u ).From (4.31), we have ds = dx x , and the solution of (4.33) is given by p ( x, u ) = Φ ( ℓ , · · · , ℓ r +1 ) p ( x, u ) = Φ ( ℓ , · · · , ℓ r +1 ) + R p ( x, u ) dx x ... = ... p n ( x, u ) = Φ n ( ℓ , · · · , ℓ r +1 ) + R p n − ( x, u ) dx x (4.34)where Φ i ( ℓ , · · · , ℓ n ), i = 1 , · · · , n , are functions of the variables ℓ , · · · , ℓ n and are thus constantsalong the characteristic curves define in (4.31). Since Φ i ( ℓ , · · · , ℓ n ) and ˜ q i ( x, u ) satisfy the condi-tions Φ i ( ℓ , · · · , ℓ n ) x p (cid:12)(cid:12)(cid:12)(cid:12) x =0 = 0 for p = 0 , · · · , n − i, We can also use ds = dx i +1 x i and in this case the normal form will be parametrized by x i +1 . We can alsoparameterize each component with a different parameterization. p , · · · , p n − are divisible by x . Hence p ( x, u ) , · · · p n ( x, u ) in (4.34) are polynomials. Consider a two dimensional system with controllable linearization. In this case, the linear part of(3.16) writes as ˙ x = x (4.35)˙ x = u (4.36)and the PDE (3.25) writes as x ∂p ∂x + x ∂p ∂u = 0 x ∂p ∂x + x ∂p ∂u = 0 (4.37)Hence, we get dp ds = 0 , (4.38a) dp ds − p = 0 (4.38b)with dx ds = 0 (4.39a) dx ds = x (4.39b) duds = x (4.39c) dp ds = 0 (4.39d) dp ds − p = 0 (4.39e)We thus deduce the following parametrization of the solution x = x , (4.40a) x = x , s + x , (4.40b) u = x , s + x , s + x , (4.40c)The first integrals are ℓ ( x, u ) = x and ℓ ( x, u ) = 2 x u − x . From (4.39d)-(4.39e) we deducethat p ( x, u ) = Φ ( x , x u − x ) (4.41)8 ( x, u ) = Z p ( t ) dt + Φ ( x , x u − x ) (4.42)We can use either (4.39b) or (4.39c) to express the normal form as a function of x or u . Forexample, using (4.39b) we deduce that dt = dx x . Moreover, using (4.29), we obtain conditions onΦ i ( ℓ , ℓ ) and ˜ q i , i = 1 ,
2, Φ ( ℓ , ℓ ) | x =0 = 0 , At the quadratic level these conditions imply thatΦ ( ℓ , ℓ ) | x =0 = φ x + O ( x, u ) (4.43a)Φ ( ℓ , ℓ ) | x =0 = ˜ φ x + ˜ φ (2 x u − x ) + O ( x, u ) (4.43b)Hence p ( x, u ) = φ x + O ( x, u ) (4.44a) p ( x, u ) = φ x x + ˜ φ x + ˜ φ (2 x u − x ) + O ( x, u ) (4.44b)Hence the normal form has the form˙ x = x + φ x + O ( x, u ) (4.45a)˙ x = u + φ x x + ˜ φ x + ˜ φ (2 x u − x ) + O ( x, u ) (4.45b) Now, consider the nonlinear control system Σ in (3.16) with one input, i.e. m = 1, and assumethat the system has r uncontrollable modes. From linear control theory we know that there existsa linear change of coordinates and feedback that allows to write the linear part as˙ z = A z + O ( z, x, u ) , (4.46)˙ x = A x + B u + O ( z, x, u ) (4.47)where z ∈ IR r × , x ∈ IR ( n − r ) × , A ∈ IR r × r , and ( A , B ) ∈ IR ( n − r ) × ( n − r ) × IR ( n − r ) × are in theBrunovsk`y form.In this case, A = (cid:18) A A B (cid:19) , ˜ x = ( z, x, u ) T in the PDE (3.25). Let’s note that when r = 0we recover the case in the preceding section and we can find a general explicit solution. However,when r = 0 a general solution is not as easily found and depends on A . We’ll illustrate the methodthrough an example. 9 .2.1 Example Consider the system whose linear part writes as ˙ z = O ( z, x, u ) , ˙ x = x + O ( z, x, u ) ˙ x = u + O ( z, x, u ) (4.48)This system has uncontrollable linearization and the uncontrollable dynamics corresponds to the z − dynamics.The elements of the normal form satisfy the PDE z ∂p ∂x + x ∂p ∂u = 0 z ∂p ∂x + x ∂p ∂u = 0 z ∂p ∂x + x ∂p ∂u − p = 0 (4.49)The equation of the characteristics is dp ds = 0 dp ds = 0 dp ds = p (4.50)The characteristic equations are z = c , x = c , x = c s + c , u = c s + c s + c . We can eitherparametrize by x or u by writing ds = dx z or ds = dux .The solution of the system of PDEs (4.49) is p = Ψ ( z, x , x − zu ) p = Ψ ( z, x , x − zu ) p = Ψ ( z, x , x − zu ) + R p ( z, x, u ) dx z (4.51)The normal form is thus given by˙ z = Ψ ( z, x , x − zu )˙ x = x + Ψ ( z, x , x − zu )˙ x = u + Ψ ( z, x , x − zu ) + R p ( z, x, u ) dx z (4.52) Given the preceding, one could think about hyper normal forms where instead of normalizing withrespect to the linear term, one normalizes the quadratic term with respect to the linear term, thennormalize the cubic term with respect to the sum of the linear and quadratic terms, and so forth.This direction has been fruitful for systems without control [21, 22] and its extension to the controlcase is the object of future research. Several other extensions are possible for this work. One couldthink about characterizing completely the normal form in the case of systems with uncontrollable10inearization, developing the Hamiltonian case, and computing the coefficients in the normal formdirectly from the original system.
Acknowledgement:
The first author is thankful to Prof. Murdock for useful comments andfor pointing out to references [4, 22]. BH also thanks the European Union for financial supportreceived through an International Incoming Marie Curie Fellowship.
References [1] Arnold, V.I. (1983).
Geometrical Methods in the Theory of Ordinary Differential Equations .Springer.[2] Barbot, J.-P., S. Monaco and D. Normand-Cyrot (1997). Quadratic forms and approximatedfeedback linearization in discrete time, in Int. Journal of Control, , 4, pp 567-587.[3] Belitskii, G. R. (1979). Invariant Normal Forms and Formal Series. Functional Analysis andApplications , ,59-60.[4] Belitskii, G. R. (2002). C ∞ -normal forms of local vector fields, Acta Appl. Math. , 23-41.[5] Chow, S.-N., C. Li, D. Wang (1994). Normal Forms and Bifurcation of Planar Vector Fields .Cambridge University Press.[6] Courant, R. and D. Hilbert (1961).
Methods of Mathematical Physics, vol. II . IntersciencePublishers.[7] Elphick, C., E. Tirapegui, M.E. Brachet, P. Coullet and G. Iooss (1987). A Simple GlobalCharacterization for Normal Forms of Singular Vector Fields.
Physica D , , 95-127.[8] Elphick, C. (1988). Global Aspects of Hamiltonian Normal Forms. Physics Letters A , ,418-424.[9] Guckenheimer, J. and P. Holmes. (1983). Nonlinear Oscillations, Dynamical Systems, andBifurcations of Vector Fields . Springer.[10] Hamzi, B., A. J. Krener and W. Kang,
The Controlled Center Dynamics of Discrete-TimeControl Bifurcations , Systems and Control Letters, , 7, 585-596, 2006.[11] Hamzi, B., W. Kang and A. J. Krener, The Controlled Center Dynamics , SIAM J. on MultiscaleModeling and Simulation, , 4, 838-852, 2005.[12] Hamzi, B., W. Kang and J.-P. Barbot, Analysis and Control of Hopf Bifurcations , SIAM J.on Control and Optimization, , 6, 2200-2220, 2004.[13] Hamzi, B., J.-P. Barbot, S. Monaco, and D. Normand-Cyrot, Nonlinear Discrete-Time Controlof Systems with a Naimark-Sacker Bifurcation , Systems and Control Letters, , pp. 245-258,2001.[14] Hamzi, B. (2001), Quadratic Stabilization of Nonlinear Control Systems with a Double-ZeroControl Bifurcation , Proc. of the 5th IFAC symposium on Nonlinear Control Systems (NOL-COS’2001), pp. 161-166, 2001. 1115] Hamzi, B., J.-P. Barbot and W. Kang,
Normal Forms for Discrete-Time Parameterized Sys-tems with Uncontrollable Linearization , Proc. of the 38th IEEE Conference on Decision andControl, pp. 2035–2039, 1999.[16] Hamzi, B. and I. A. Tall, Normal Forms for Discrete-Time Control Systems, Proc. of the 42ndIEEE Conference on Decision and Control, 2, 1357 - 1361, 2003.[17] W. Kang and A. J. Krener, Extended quadratic controller normal form and dynamic statefeedback linearization of nonlinear systems, SIAM J. Control and Optimization, 30 (1992),1319-1337.[18] Kang, W., and A. J. Krener (2006). Normal Forms of Nonlinear Control Systems, in Chaos inAutomatic Control, W. Perruquetti and J-P. Barbot (Eds.), pp. 345-376.[19] Meyer, K. R. (1984). Normal Forms for the General Equilibrium, Funkcialaj Ekvacioj, , pp.261-271.[20] Meyer, K. R., G. R. Hall, and D. Offin (2009). Introduction to Hamiltonian dynamical systemsand the N-body problem . Springer.[21] Murdock, J. (2003).
Normal Forms and Unfoldings for Local Dynamical Systems . Springer.[22] Murdock, J. (2004). Hypernormal form theory: foundations and algorithms,
Journal of Dif-ferential Equations , , 424-465.[23] Tall, I.A. and W. Respondek (2003), ”Feedback Classification of Nonlinear Single-Input Con-trol Systems with Controllable Linearization: Normal Forms, Canonical Forms, and Invari-ants”,in SIAM Journal on Control and Optimization, 41(5), pp. 1498-1531[24] Tall, I.A. and W. Respondek (2006). Feedback Equivalence of Nonlinear Control Systems: ASurvey on Formal Approach, in Chaos in Automatic Control, W. Perruquetti and J-P. Barbot(Eds.), pp. 137-262.[25] Poincar´e, H. (1885). M´emoire sur les courbes d´efinies par une ´equation diff´erentielle, J. MathsPures Appl.,4