We study the feedback classification of discrete-time control systems whose linear approximation around an equilibrium is controllable. We provide a normal form for systems under investigation

Hamzi, Boumediene

Tall, Issa Amadou

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Southern Illinois University CarbondaleOpenSIUCMiscellaneous (presentations, translations,interviews, etc) Department of Mathematics12-2003Normal Forms for Nonlinear Discrete TimeControl SystemsBoumediene HamziIssa Amadou TallSouthern Illinois University Carbondale, itall@math.siu.eduFollow this and additional works at: http://opensiuc.lib.siu.edu/math_miscPart of the Control Theory Commons, and the Mathematics CommonsPublished in Hamzi, B., & Tall, I. A. (2003). Normal forms for nonlinear discrete time controlsystems. Proceedings of the IEEE Conference on Decision and Control, v 2, 2003, 1357-1361. doi:10.1109/CDC.2003.1272798. ©2003 IEEE. Personal use of this material is permitted. However,permission to reprint/republish this material for advertising or promotional purposes or for creatingnew collective works for resale or redistribution to servers or lists, or to reuse any copyrightedcomponent of this work in other works must be obtained from the IEEE. This material is presentedto ensure timely dissemination of scholarly and technical work. Copyright and all rights therein areretained by authors or by other copyright holders. All persons copying this information are expectedto adhere to the terms and constraints invoked by each author's copyright. In most cases, these worksmay not be reposted without the explicit permission of the copyright holder.This Article is brought to you for free and open access by the Department of Mathematics at OpenSIUC. It has been accepted for inclusion inMiscellaneous (presentations, translations, interviews, etc) by an authorized administrator of OpenSIUC. For more information, please contactopensiuc@lib.siu.edu.Recommended CitationHamzi, Boumediene and Tall, Issa Amadou, "Normal Forms for Nonlinear Discrete Time Control Systems" (2003). Miscellaneous(presentations, translations, interviews, etc). Paper 32.http://opensiuc.lib.siu.edu/math_misc/32Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawaii  USA, December 2003 Tup1 1-3 Normal Forms for Nonlinear Discrete Time Control Systems Boumediene Hamzi and Issa Amadou Tall, Department of Mathematics, University of California, Davis, CA 95616, email:{hamzi, tall} @math.ucdavis.edu Keywords: discrete-time, normal forms, homogeneous denote by transformations. m Abstract- We study the feedback classification of discrete- time control systems whose linear appmximation around an equilibrium is controllable. We provide a normal form for systems under investigation. I. INTRODUCTION The method of normal forms has been a useful approach in studying the dynamical systems. This method, first in- troduced by Poincare in his Ph.D. thesis (see [lSl), has been successfully applied by the author to vector fields (differential dynamical systems) and maps (discrete-time systems), in order to provide a change of coordinates in which the system is in a "simplest" form (see also [I]). For continuous-time control systems with controllable lin- earization, quadratic normal for& were obtained in 1141 using change of coordinates and feedback. This result has been generalized to normal forms of any degree in [13]. Later on, normal forms for control systems with uncontrol- lable linearization has been derived [12], [16], [ZO], [221. Quadratic and cubic normal forms for discrete-time control systems has been treated in [21, [61, 191, [171. Although this method is formal, it has several applications in control theory. It has been used for the stabilization of systems with uncontrollable linearization, in continnous and discrete-time [4], [7], [5], [8], [IO], [16], [17]. It has led to a complete description of symmetries around equilibrium [19], [26], and allowed the characterization of systems equivalent to feedforward forms [231, [241, [251. In this paper, we propose a normal form, at any degree, for discrete-time control systems whose linearization is h(z)  = h~O](s)+h'1~(s) +h'*](z) +" '=  h'"l(2) m=0 its Taylor series expansion at 0 E W", where h["](s) stands for a homogeneous polynomial of degree m. Similarly, throughout the paper, for a map 4 of an open subset of B" into W" (resp. for a vector field f on an open subset of W"), we will denote by $I["] (resp. by f'"]) the term of degree m of its Taylor series expansion at 0 E W", that is, each component 4Am1 of 41") (resp. f;"] of f["]) is a homogeneous polynomial of degree m. We consider the problem of transforming the discrete-time nonlinear control system n : I+ = f(z,.), s(.) E W" U(.) ER, where z+ = z ( k + l ) ,  and f ( s , u )  = f ( s ( k ) , u ( k ) )  for any k E N, by a feedback transformation of the form 2 = 4(z) : U = -&,U) fi : 2+ = j ( Z , U ) ,  &>U, = f ( 4 - % ) > 7 ( % U ) )  ' to a simpler form. The transformation T brings n to the system whose dynamics are given by We suppose that (0,O) E R" x W is an equilibrium point, that is, f (0 ,O)  = 0, and we denote by nrll : Z+ = F~ + GU , controllable. The paper is organized as following: Section ll deals with its linearization at this point, where basic definitions. In Section JlI, we construct a normal form for discrete-time nonlinear control systems whose We will assume that this linearization is controllable, that is h e a r  approximation is controllable. The proofs are given in Section IV. 11. Notations and definitions. span{F'G: O < i < n - l } = B "  Let us consider the Taylor series expansion nm of the system II, given by All objects, that is, functions, maps, vector fields, conml systems, etc., are considered in a neighborhood of 0 E W" and assumed to be Cm-smooth. For a smooth W-valued function h, defined in a neighborhwd of 0 Bn, we 0-7803-7924-1/03/$17.00 02003 IEEE 1357 m nm : z+=Fx+Gu+C~[~~(Z ,U)  (1) E m=2 Authorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on January 22, 2009 at 12:57 from IEEE Xplore.  Restrictions apply.and the Taylor series expansion transformation T, given by Tm of the feedback Throughout the paper, in particular in formulas (1) and (2), the homogeneity of f["I and y["I will betaken with respect to the variables ( x ,  U)' and ( x ,  U)' respectively. We first notice that, because of the controllability assump- tion, there always exists a linear feedback transformation 2 = Tz " : U = K z + L u  bringing the linear part I['l : x + = F z + G u  into the BrunOvs!@ canonical form (see [ 111) ngb : z+ = A ~  + B ~ .  Then we study, successively for m 2 2, the action of the homogeneous feedback transformations (3) 2 = x + & q X )  Tm : U. = u + r[" ' (z,u) on the homogeneous systems IIlml : z+ = Ax + Bu + f'"I(x, U) . (4) Let us consider another homogeneous system film] : 2' = Ar + Bu + j["l(r, U )  . (5 )  Dejinition 2.1: We say that the homogeneous system IIlm], given by (4).  is feedback equivalent to the homogeneous system filml, given by (5), if there exist a homogeneous feedback transformation Tm. of the form (3), which brings rlw system IIlml into the system fI["l modulo higher order remts. The starting point is the following proposition giving the equivalence conditions. Proposition 2. I :  The homogeneous feedback transforma- tion T", defined by (31, brings the homogeneous sys- tem IIlml, given by (4), into the homogeneous system fi[ml, given by (5). if and only if the following relation q$"I(Ax+Bu)-q4~Y\(x) = $ " l ( x , u ) - f ! m l ( x , u )  & ' l ( A x + B u ) + y l m ] ( x )  = f , ! m l ( x , u ) - ~ m l ( i , u )  hold for all 1 5 j 5 n - 1. 111. M A I N  RESULTS. In this section we will establish OUI main results. Let us denote the control by u = r,+l, and for any 1 5 i 5 n+ 1, z; = (21 ,... ,2,). Our main result for discrete-time nonlinear control systems with controllable linearization is as following. Theorem 3.1: The control system IIm, defined by ( l ) ,  is feedback equivalent, by a formal feedback transforma- tion Tm of the form (2), to the normal form m IIgF : Z+ = A r  + Bu + fmI(z, U )  , m=2 where for any m 2 2, we have if j = n. (6) As the homogeneous feedback transformations Tm leave invariant the terms of degree less than m, Theorem 3.1 follows from a successive application of Theorem 3.2 below. Theorem 3.2: The homogeneous control system HIm], de- fined by (4), is feedback equivalent, by a homogeneous feedback transformation T" of the form (3). to the normal form IIti : z+ = A z + B u +  ?" ' ( z ,u ) ,  where for any m 2 2, the vector field flml(r, u) is given by (6). A. Example Consider the Bressan and Rampazzo pendulum (see (31, [21]) described by the equations e, = 5 2  2 2  = - g s i n z 3 + x 1 x j  x3 = x4 x 4  = U, where XI denotes the length of the pendulum, x2 its velocity, x3  the angle of the pendulum with respect to the horizontal, z 4  its angular velocity, and g the gravity constant. We discretize the system by taking x l  = x: -21 ,  i 2  = x: - 2 2 ,  x3 = x i - x s ,  x q  = x: - 2 4 .  The system above rewrites x: = X l + X 2  x: = x 2 - g s i n x 3 + x l x q  x: = x 3 + 5 4  x: = 2 4  +U. Let us consider the change of coordinates 21 = 21 2 2  = 2 2  + X l  23 = -gsinzs  + 2 x 2  + X I  2 4  = U = 2: -gsin(x4 + 1 3 )  + 3x2 - 2gsinx3 + 2 x 1 4  + z1 whose inverse is such that x4 = h ( q ,  ~ ~ ~ 2 3 ,  r4)  is a smooth function. This change of coordinates takes the system into 1358 Authorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on January 22, 2009 at 12:57 from IEEE Xplore.  Restrictions apply.the form 2: = 22 2: = 2 4  2,' = U. 2: = 2 3 + 2 1 h Z ( ~ l r Z 2 r Z 3 r i l q )  Actually the function h2(r1,z2, 2 3 ,  z4) could be decom- posed as h2(z1, 22rZ3r 24) = hl(z1, Z2rz3) + 24hZ(Zlr  z 2 . 2 3 , 2 4 )  where the 1-jet at 0 of hl is zero and h2(0) = 0. Put The objective is to show that we can get rid of the terms H I ( z I , z ~ , z ~ ) .  Let ns suppose that the k-jet at 0 of HI is zero. Consider the change of coordinates 21 = Z I , ? ~  = ZZ.  23 = 2 3  + H l ( s l , z ~ , t 3 ) , i 4  = 2:. This change of coordinates, completed by the feedback i,' = w, takes the system into the form 2: = iz i: = i: = 5 4  2,' = w, HI (21 I 2 2 ;  23) = 2lhl(zl,22; 2 3 ) .  i 3  + Hl  (il, 22, i 3 )  + i 1 5 4 H 2 ( ~ z ( i l ,  1 2 , 2 3 , i 4 )  where f i l ( i l , ? z , & )  and i f 2 ( i l r i 2 , i 3 , i 4 )  are some smooth functions. It is enough to remark that the (k+2)-jet atOof fil(il,i2,i3) iszerobecause the2-jetof z1z4H2(z) is zero. Then by iteration we can cancel terms H l ( z 1 ,  ~ 2 ~ 2 3 )  and put the system into the desired normal form 2: = 2 2  2,' = 24 24' = U. 2: = ~ 3 + ~ 1 Z 4 H ( ~ I r ~ 2 ~ ~ 3 ~ ~ 4 )  IV. PROOFS In this section we will prove our main result. Before let us state the following useful lemma. Lemma4.1: Ifh["'l(r,u) =hlml(rz,... ,zn,u) isahomo- geneous polynomial depending exclusively on the variables 2 2 , .  . . , zn and the control U, then there is a unique homo- geneous polynomial H [ " l ( t )  = H[ml(zl,. . , z,,) such that The proof of this lemma is straightforward, and hence will be omitted. A. Proof of Theorem 3.2 The proof will be constructive and based on a inductive argument. Let ns consider the system I@] given by H [ ~ I ( A ~  + B ~ )  = hlml(z,u). 2: = 2 2  + f p ( 2 , U )  Applying the feedback U = U + fAm'(z,u). we can anni- hilate the terms f,!"'](z, U), and hence we can assume that p " ' 2 ,  a) = 0. Let us suppose that for some 1 I j 5 n - 1, the system (7) has been taken to the form 2: = 2 2  + f p " l 2 , U )  2: = x3+1 + f'"I(2,U) + -  4m1 Zj+1 - .r,+z + f,+l(...) 2 L  = Z n  + f:?-],(2, U )  2; = 21, ... ... where for any j + 1 5 1 5 n - 1, we have "+I r,'-l(.,a) = 212iPg-~l(zi). i=1+2 We first decompose the component f)"'](z, U) uniquely as follows j+i + cz lz iP j~ -~ l ( z i )  + R y ' ( Z 2  )... , % , U ) .  i=l We consider the feedback transformation 2 = Z + ~ ' " ] ( l )  U = v+y[mI(z, tJ)  r m  : whose components +\"I(z), . . . , +k","](z), and $"l(z, U) are defined as following. Using Lemma 4.1, we define &"(z) such that &qAz+Bu) = -  Rjml(2 %... , X n , U )  (9) and we take &%) = &!1(~2 + B ~ )  #"l(s) = +Lm'(Az + Ba). The components q+\ml(z), . . . , +!'!l(z) could be taken to be zero or arbitraly. Moreover, we can notice that the components @](s), . . . , &"'(z) did not depend on the control U. Actually, for any j 5 1 5 n, we have + p ( Z )  = ~~"l(Zl,..' ,q). Applying Proposition 2.1, we easily deduce that the trans- formation Tm whose components are given by (9)-(10) takes the system (8) into the form 2: = 22 + f p ( Z , u )  23-1 + - z3 + fJ?i(z,a) 2; = ZJ+1 + f l " ' ( 2 , U )  = 2" + f;?l1(2, U )  ... ... z,+ = U ,  1359 Authorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on January 22, 2009 at 12:57 from IEEE Xplore.  Restrictions apply.where for any j 5 1 5 n - 1, we have This achieves the proof of Theorem 3.2 V. REFERENCES [I] Arnold, V. I. (1983). Geometrical Methods in the Theory of Ordinary Differential Equations, Springer Verlag. [21 Barbot, J.-P., S .  Monaco and D. Normand-Cyrot (1997). Quadratic forms and approximative feedback linearization in Discrete Tune, International Journal of Control, 67, 567-586. [31 Bressan. A. and Rampazzo, F, On differential systems with quudraric impulses and their applications to Lagrangian mechanics, in S I A M  Joumal on Control and Optimization, 31, (1993) pp. 1205-1230. [41 Hamzi, B., J.-P. Barbot and W. Kang (1998). Bifnrca, tion and Topology of Equilibrium Sets for Nonlinear Discrete-Time Control Systems, Proc. of the Nonlinear Control Systems Design Symposium (NOLCOS’98). pp. 35-38. [51 Hamzi, B., J.-P.’Barbot and W. Kang (1999). ”Sta- bilization of Nonlinear Discrete-Time Control Sys- tems with Uncontrollable Linearization”. In Modem Applied Mathematics Techniques in Cimuits, Systems and Control”, World Scient$c and Engineering So- ciety Press , pp. 278-283. Also in Prw. of the 3rd IEEwlMACS Mulriconference on Circuits, Systems. Conimunications and Computers, pp. 4581-4586. [61 Hamzi, B., J.-P. Barhot and W. Kang (1999). Normal forms for discrete time parameterized systems with uncontrollable linearization, Proceedings of the 38th IEEE Conference on Decision and Control, vol. 2 , pp. 2035-2038. [71 Hamzi, B., J.-P. Barbot and W. Kang (1999). Bifur- cation for discrete-time parameterized systems with uncontrollable linearization, Pmc. of the 38th IEEE Conference on Decision and Control, vol.1, pp. 684 -688. [81 Hamzi, B. (2M)l). Analyse et communde des systdmes non liniaires non commandables en prenii2re approx- imation duns le cadre de la thiorie des bifurcations, Ph.D. Thesis, University of Paris XI-Orsay, France. 191 Hamzi, B., J.-P. Barbot, S .  Monaco, and D. Normand- Cyrot (2001). Normal Forms versus Naimark-Sacker Bifurcation Control, invited paper to the Nonlinear Control Systenrs Design Symposium (NOLCOS’OI). [IO] Hamzi, B., J:P. Barbot, S .  Monaco, and D. Normand- Cyrot (2001). Nonlinear Discrete-Time Control of Systems with a Naimark-Sacker Bifurcation, Systems and Control Letters, 44, 245-258. [ I  11 Kailatb, T. (1980). Linear Systems. Prentice-Hall. E121 Kang, W. (1995). Quadratic normal forms of nonli-  ear control systems with uncontrollable linearization, Proceedings of the 34th IEEE Conference on Decision and Control, vol. 1, pp. 608 -612. [131 Kang, W. (1996a). Extended Controller Form and Invariants of Nonlinear Control Systems with a Single Input.Journa1 of Marhematical Systems, Estimation, and Control, 6, 27-51. [I41 Kang, W. and A.J. Krener (1992). Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems, Siam J. Control and Optimization, 30, 1319-1337. [I51 Krener, A. J. (1984). Approximate linearization hy state feedback and coordinate change, Systems and Control Letters, 5, 181-185. [16] Krener, A.J., W. Kang, and D.E. Chang (2M)1), Con- trol Bifurcations, submitted to IEEE trans. on Auto- matic Control. [171 Krener, A.J. and L. Li (2002). Normal Forms and Bifurcations of Discrete T i e  Nonlinear Control Sys- tems, SIAM J. on Control and Optimization, 40, 1697- 1723. [I81 PoincarC, H. (1929). Sur les propriCtCs des fonctions dkfinies par les Cquations anx diffkrences partielles. Oeuvres, pp. XCIX-CX, Gauthier-Wllars: Paris. [191 W. Respondek and I. A. Tall, How Many Symmetries Does Admit a Nonlinear Single-Input Control System around Equilibrium, in Proc. of the 40th CDC, pp. 1795-1800, Florida, (2001). [201 I. A. Tall, Classification par bouclage des systkmes de contrbles non linkaires mono-enuke: formes n o d e s ,  formes canoniques, invariants et symkhies. Ph. D. Thesis, INSA de Rouen, (2000). [211 I. A. Tall and W. Respondek, Feedback classification of nonlinear single-input control systems with control- lable linearization: normal forms, canonical forms, and invariants, in SIAM Joumal on Control and Optimiza- rion, pp. 1498-1531, 2002. [221 I. A. Tall and W. Respondek, Normal forms and invari- ants of nonlinear single-input systems with noncontrol- lable linearization, NOLCOS’OI, Petershurg, Russia, [231 I. A. Tall and W. Respondek, Feedback Equivalence to a Strict Feedforward Form for Nonlinear Single-Input Systems, to appear in International Journal of Control. [241 I .  A. Tall and W. Respondek, Transforming a Single- Input Nonlinear System to a Smct Feedforward Form via Feedback, Nonlinear Control in the Year 2000, A. Isidori, F. Lamnabhi, and W. Respondek, (eds.), Springer-Verlag. 2, pp. 527-542, London, England, 1251 I. A. Tall and W. Respondek, Feedback Equivalence to Feedforward Form for Nonlinear Single-Input Sys- tems, Dynamics, Bifurcations and Control, E Colo- nius and L. Gmne (eds.), LNCIS, 273, pp. 269-286, Springer-Verlag. Berlin Heidelberg, (2002). (2001). (2001). 1360 Authorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on January 22, 2009 at 12:57 from IEEE Xplore.  Restrictions apply.1261 I. A. Tall and W. Respondek, Nonlinearizable Ana- lytic Single-Input Control Systems with Controllable Linearization DO Not Admit Stationary Symmetries, Systems and Control Leners, 46 ( I ) .  pp. 1-16, (2002). 1361 Authorized licensed use limited to: Southern Illinois University Carbondale. Downloaded on January 22, 2009 at 12:57 from IEEE Xplore.  Restrictions apply.

Normal Forms for Nonlinear Discrete Time Control Systems

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Normal Forms for Nonlinear Discrete Time Control Systems

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