Transfer Function Approach to the Model Matching Problem of Nonlinear Systems

Transcription

1 Tranfer Function Approach to the Model Matching Problem of Nonlinear Stem Mirolav Halá Ülle Kotta Claude H. Moog Intitute of Control and Indutrial Informatic, Fac. of Electrical Engineering and IT, Slovak Univerit of Technolog, Ilkovičova 3, 829 Bratilava, Slovakia ( Intitute of Cbernetic, Tallinn Univerit of Technolog, Akademia tee 2, 268 Tallinn, Etonia ( IRCCN, UMR C.N.R.S. 6597, rue de la Noë, BP 920, 4432 Nante Cedex 3, France ( Abtract: The maintream for the anali and nthei of nonlinear control tem i the o-called tate pace approach. The Laplace tranform of a nonlinear differential equation i non tracktable and an tranfer function approach wa not developed until recentl. Herein, we how that one ma ue uch mathematical tool to recat and olve the model matching problem. Note that the latter wa original tated for linear time invariant tem, in term of equalit of the tranfer function of both the model and the compenated tem. Keword: nonlinear tem, model matching problem, tranfer function, tranfer equivalence, output feedback. INTRODUCTION The model matching problem wa olved within the tate pace approach b variou author, uing problem tatement which are lighlt different from one contribution to the other (Benedetto and Iidori (984); Benedetto (990); Huijbert (992); Conte et al. (2007)). In the model matching problem preented in thi paper we tud the problem of deigning a compenator for a nonlinear control tem under which the input-output map of the compenated tem become tranfer equivalent to a prepecified model. B tranfer equivalence one mean that the tem admit the ame irreducible input-output differential equation (Conte et al. (2007)). In general, the model i alo aumed to be nonlinear. The model matching problem i a tpical deign problem in the ene that it pla a role in variou other problem like the input-output linearization and the (diturbance) decoupling. In the linear cae, one require the equalit of the tranfer function of the model and of the compenated tem. However, the tranfer function formalim wa recentl developed alo for nonlinear tem, ee Zheng and Cao (995); Halá and Huba (2006); Halá (2008). Such a formalim generalize well known reult valid for linear time invariant tem and wa, for intance, alread emploed in Perdon et al. (2007) to invetigate ome tructural propertie of nonlinear tem. Tranfer function approach to the model matching problem, of coure, repreent a ver natural tool. The nonlinear model matching problem ha been tudied earlier (Benedetto and Iidori (984); Benedetto (990); Huijbert (992); Conte et al. (2007)) in the tate-pace Partiall upported b Slovak Reearch and Development Agenc (LPP ), Slovak Grant Agenc (Grant No. VEGA /3089/06) and Etonian Science Foundation (Grant No. 6922). etting. Our problem tatement and olution conider, however, a more general cae, ince neither the control tem itelf, nor the model and the compenator are required to be realizable in the tate-pace form. In particular, thi give u a chance to find realizable compenator for nonlinear tem not having the tate-pace realization. The paper tudie both feedforward and feedback compenator. In cae of a feedforward compenator we how that, in contrat to what happen in linear cae, a cla of nonlinear tem for which the olution exit i quite retricted. In cae of a feedback compenator the approach preented here ha contact point with that of Rudolph (994) and of Glad (990) where the input-output behaviour of the nonlinear tem and a model are decribed b the differential polnomial and differential algebraic tool, particularl Ritt remainder algorithm have been emploed to find the controller equation. 2. TRANSFER FUNCTIONS OF NONLINEAR SYSTEMS We will ue the algebraic formalim of Halá and Huba (2006); Halá (2008) which introduce tranfer function of nonlinear tem. Conider the SISO nonlinear tem defined b an inputoutput equation of the form (n) = ϕ(, ẏ,..., (n ), u, u,..., u (m) ) () where ϕ i aumed to be an element of the field of meromorphic function K. Remark. Even if one tart with a tate-pace repreentation it i alwa poible to eliminate the tate variable to get an input-output equation of the form (), ee Conte et al. (2007).

2 The left kew polnomial ring K[] of polnomial in over K with the uual addition, and the (non-commutative) multiplication given b the commutation rule a = a + ȧ (2) where a K, repreent the ring of linear ordinar differential operator that act over vector pace of oneform E = pan K {dξ; ξ K} in the following wa ( k ) k a i i v = a i v (i) i=0 i=0 for an v E. The commutation rule (2) actuall repreent the rule for differentiating. Lemma 2. (Ore condition). For all non-zero a, b K[], there exit non-zero a, b K[] uch that a b = b a. Thu, the ring K[] can be embedded to the noncommutative quotient field K b defining quotient (Halá and Huba (2006); Halá (2008)) a a b = b a The addition and multiplication in K are defined a a + a 2 = β 2a + β a 2 b b 2 β 2 b where β 2 b = β b 2 b Ore condition and a a2 = α a 2 (3) b b 2 β 2 b where β 2 a = α b 2 again b Ore condition. Due to the non-commutative multiplication (2) the, of coure, differ from the uual rule. In particular, in cae of the multiplication (3) we, in general, cannot impl multipl numerator and denominator, nor cancel them in a uual manner. We neither can commute them a the multiplication in K i non-commutative a well. Example 3. Conider two quotient, Then + 2 2ẏ/ = 2 ( + ẏ/) and = 2 ẏ = 2 Once the fraction of two kew polnomial i defined we can introduce the tranfer function of the nonlinear tem () a an element F () K uch that d = F ()du. After differentiating () we get n d (n) ϕ (i) d(i) = or alternativel i= where a() = n n i= a(), b() K[]. Then m i=0 a()d = b()du ϕ du(i) u (i) ϕ i, b() = m (i) i=0 F () = b() a() ϕ u (i) i and Example 4. Conider the tem ẏ = u After differentiating dẏ = ud + du ( u)d = du and the tranfer function i F () = u Remark 5. The tranfer function i defined b emploing the tandard algebraic formalim of differential form, following the line in Conte et al. (2007) which introduce the notion of a one-form in a formal and abtract wa. Hence, it i not necear to deal here with the linearization of the tem along a trajector uing the Kähler-tpe differential which lead to a time-varing linear tem. In the linear cae to each proper rational function an input-output differential equation of a control tem can be aociated. However, thing are different in the nonlinear cae. Though one can alwa aociate to a b() a() proper rational function F () = a correponding input-output differential form, ω = a()d b()du, thi one-form i not necearil integrable. If the inputoutput differential form i integrable, or can be made integrable, then there exit an input-output differential equation of the form () uch that the tranfer function of thi input-output equation i F (). In other word, not ever quotient of kew polnomial necearil repreent a control tem. Of coure, thi will pla a crucial role in deigning compenator. Tranfer function of nonlinear tem atif man propertie we expect from tranfer function (Halá (2008)): The characterize a nonlinear tem uniquel; that i, each nonlinear tem ha a unique tranfer function, no matter what tate-pace realization one tart with. The characterize onl acceible and obervable ubtem. The provide an input-output decription. The allow u to ue tranfer function algebra when combining tem in erie, parallel or feedback connection. Some additional tructural propertie of tranfer function of nonlinear tem are dicued in Perdon et al. (2007). In term of the model matching problem we remark that Two nonlinear tem are locall tranfer equivalent (admit the ame irreducible input-output differential equation) if and onl if the have the ame tranfer function. It i now ea to conclude that in the nonlinear model matching problem one, a in linear cae, require the equalit of the tranfer function of the model and that of the compenated tem. 3. FEEDFORWARD COMPENSATOR One of the baic tak in olving the model matching problem i to be intereted in finding olution in term of a feedforward compenator.

3 G() {}}{ v R() Fig.. Compenated tem u F () Problem tatement. Conider a nonlinear tem F and a model G decribed b the tranfer function F () = b F () a F () G() = b G() a G () repectivel. Find a (proper) feedforward compenator R (not necearil tate-pace realizable), decribed b the tranfer function R() = b R() a R () uch that the tranfer function of the compenated tem coincide with that of the model G G() = F () R() The ituation i depicted in Fig.. Theorem 6. Given F () 0 and G(), there exit a feedforward compenator R() which olve the model matching problem if a R ()du b R ()dv i integrable, where b R () a R () = F () G(). Proof. B the tranfer function algebra (Halá (2008)) we get G() = F () R() Hence, the compenator R() = F () G() (4) Clearl, the exitence of uch a compenator i determined b the integrabilit of the compenator equation a R ()du = b R ()dv. If one i intereted in finding a olution in a cla of proper compenator, it i necear to retrict the relative degree of the model. Propoition 7. R() i proper (caual) if and onl if rel deg G() rel deg F () Proof. Sufficienc: Aume rel deg G() rel deg F (). Since rel deg G() = deg a G () deg b G () rel deg F () = deg a F () deg b F () we have deg a G () deg b G () deg a F () deg b F () Now, a deg a G () = deg a F ()+deg a R () and deg b G () = deg b F () + deg b R () we get deg a R () deg b R () which mean a proper R(). Neceit: Aume a proper R(), deg a R () deg b R (). Since all previou tep can be done in revere order, rel deg G() rel deg F (). Example 8. Given the tem F ẏ = u + u 2 with the tranfer function F () = + 2u Conider the following three model G() = G () = + G () = + 2 B (4) and (3) we get the following tranfer function of the compenator R() = R () = R () = + 2u + 2u + 2u = + 2u + = + 2 = ( + )( + 2u) ẏ/ ( + 2 ẏ/)( + 2u) While R() and R () reult in the integrable compenator R : R : u + u 2 = v ü + 2u u + u + u 2 = v R () doe not. We can alo eail check, ee Conte et al. (2007), that both R and R are tate-pace realizable. Finall, note that in multipling tranfer function one alwa ha to follow the rule (3) which, in general, ield a different reult from the uual multiplication, a can be een for intance in the cae of R (). Example 9. Conider the tem from Example 4 F () = u Let the deired dnamic be given b the tranfer function G() = The compenator read R() = F () G() = u which after rearrangement ield a non-integrable compenator equation. So, in contrat to what happen in the linear time-invariant or time-varing tem (Marinecu and Bourlè (2003)), a cla of nonlinear tem for which the olution in term of a feedforward compenator exit i, due to the integrabilit condition, quite retricted. Hence, it i natural to be intereted in finding a olution in a (more general) cla of feedback compenator. 4. FEEDBACK COMPENSATOR Problem tatement. Conider a nonlinear tem F and a model G decribed b the tranfer function

4 v G() {}}{ R() Fig. 2. Compenated tem u F () = b F () a F () G() = b G() a G () F () repectivel. Find a (proper) feedback compenator R (not necearil tate-pace realizable), decribed b the tranfer function R v () = b Rv() (7) a R () R () = b R() (8) a R () with du = R v ()dv + R ()d uch that the tranfer function of the compenated tem coincide with that of the model G G() = F ()R v() F ()R () The ituation i depicted in Fig. 2. Theorem 0. Given F () 0 and G(), there exit a feedback compenator R() which olve the model matching problem. Proof. B the tranfer function algebra (Halá (2008)) we get G() = F ()R v() F ()R () When conidering (5), (7) and (8) G() = b F () a F () brv() a R () b F () a F () br() a R a () R () af () b F () b R() After matching the latter to (6) we can et b G () = b Rv () = b Rv () a G () = a R () af () b F () b R() If a R () i choen to be q()b F () then a G () = q()a F () b R () Aume, without lo of generalit, that deg a G () deg a F (). Then one can think of q() a a right kew polnomial quotient and of b R () a a kew polnomial remainder of kew polnomial a G () and a F (). So, from given a G () and a F () one can, b Euclidean diviion algorithm, determine the compenator with du = R v ()dv + R ()d (5) (6) a R ()du = b Rv ()dv + b R ()d (9) a G () = q()a F () b R () a R () = q()b F () b Rv () = b G () In comparion to what happen in cae of a feedforward model matching problem, now uch a compenator i alwa integrable. Equation (9) can be retated a q()b F ()du = (q()a F () a G ())d + b G ()dv q()(b F ()du a F ()d) = b G ()dv a G ()d One-form b F ()du a F ()d, b G ()dv a G ()d are clearl exact and appling q() to an exact one-form reult in an exact one-form a well. Remark. Aumption deg a G () deg a F () in the proof i clearl necear to get a reaonable olution to Euclidean diviion algorithm of a G () and a F (). However, it i not retrictive, for if we have a model G() = b G() a G () with deg a G () < deg a F () then we can, without lo of generalit, ue the model G () = k b G () k a G () uch that deg k a G () deg a F (). Clearl, the model G () i tranfer equivalent to the model G(). Roughl peaking, in the ene of tranfer equivalence there alwa exit a feedback compenator which olve the model matching problem. In cae one i intereted in finding a olution in a cla of proper compenator, the ituation i the ame a in cae of a feedforward compenator. Propoition 2. R() i proper (caual) if and onl if rel deg G() rel deg F () Proof. Sufficienc: Aume rel deg G() rel deg F (). Since rel deg G() = deg a G () deg b G () rel deg F () = deg a F () deg b F () we have deg a G () deg b G () deg a F () deg b F () Now, a a G () = q()a F () b R () we get deg q() + deg a F () deg b G () deg a F () deg b F () deg q() + deg b F () deg b G () Or, b taking into account that a R () = q()b F () and b G () = b Rv () deg a R () deg b Rv () which mean a proper R(). Neceit: Aume a proper R(), deg a R () deg b Rv (). Since all previou tep can be done in revere order, rel deg G() rel deg F (). Example 3. Conider the tem and the model from Example 9 where we were not able to find a olution in term of a feedforward compenator. F () = u

5 G() = Now, we have b F () =, a F () = u, b G () =, a G () = From Euclidean diviion algorithm we get q() = u b R () = u 2u u 2 uch that a G () = q()a F () b R (). The compenator i determined b that i b Rv () = b G () = b R () = u 2u u 2 a R () = q()b F () = + ẏ u a R ()du = b Rv ()dv + b R ()d d u + ẏdu +2du + udu = dv d ud 2ud u 2 d Note that ẏ i not independent, ẏ = u, and after ubtituting, the lat equation repreent the differential of u + 2u + + u 2 = v The compenator ha the following tate-pace realization ξ = 2ξ ξ 2 + v u = ξ Example 4. Conider the tem and the model from Example 8 F () = + 2u G() = G () = + G () = + 2 We get now the following compenator R : u + u 2 = v R : u + u 2 = v R : u + u 2 = v 2 all of them tate-pace realizable. Note alo that while R doe not differ from it feedforward counterpart R doe. 5. MODEL MATCHING PROBLEM FOR NONREALIZABLE SYSTEMS The input-output approach to the model matching problem, a preented here, ha one trong point. It i, in fact, more general in that ene that it i applicable to nonlinear tem not having the tate-pace realization. We do not require thi from the original tem equation neither from compenator equation. So there i a chance to find realizable compenator for nonrealizable tem in both feedforward and feedback cae, a demontrated b the following example. Example 5. Conider the tem ÿ = + u 2 + u which ha, according to Conte et al. (2007), no tate-pace realization. The tranfer function i F () = 2 u Let the deired dnamic be given b the tranfer function G() = 2 To find a feedforward compenator we compute R() = F () G() = u + 2 = (2 u + ) The compenator equation i integrable 2ü uu (3) + ü = v + v and ha the following tate-pace realization ξ = ξ 2 ξ + v ξ 2 = ξ 3 ξ 3 = v u = ξ In cae we are intereted in finding a feedback compenator we get and finall q() = b R () = b Rv () = b G () = a R () = q()b F () = 2 u + (2 u + )du = dv + d u 2 + u = v + Thi time, the compenator ha the tate-pace realization ξ = v + ξ u = ξ 6. CONCLUSIONS In thi paper the tranfer function formalim wa emploed to recat and olve the model matching problem of ingle-input ingle-output nonlinear control tem. Thi reulted in deigning compenator, both feedforward and feedback, under which the input-output map of the compenated tem become tranfer equivalent to a prepecified model. It wa hown that the exitence of a feedforward compenator require a retrictive integrabilit condition. A feedback compenator exit whenever the tem i nontrivial, that i F () 0. Obvioul, the properne of the compenator require the tandard inequalit on the relative degree of the tem and that of the model. It i argued that the tranfer function approach i the mot natural one, in comparion with the tate pace approache. In addition, it ma be applied alo to nonrealizable nonlinear tem where it i poible to find realizable compenator. Reult of thi paper ma be extended from a everal

6 point of view. Obvioul, when following Halá and Kotta (2007) thi approach carrie over quite eail to the nonlinear dicrete-time tem. Another natural extenion conit in appling idea of thi work to the cae of nonlinear time-dela tem. For the correponding tranfer function formalim, ee Halá (2007). One of the future tak i to appl the tranfer function approach to olve the model matching problem for quare multi-input multioutput tem uing invere of tranfer function matrice. Another topic i the model matching problem with tabilit. Clearl, to get a table olution to the model matching problem the aumption that the et of the untable zero of the tem i included in the et of the untable zero of the model, in both feedforward and feedback cae, ha to be met. In addition, in cae of a feedforward compenator, one ha to aume that the tem i table itelf, for the olution i, in fact, baed on the zero-pole cancellation. Y. Zheng and L. Cao. Tranfer function decription for nonlinear tem. Journal of Eat China Normal Univerit (Natural Science), 2:5 26, 995. REFERENCES M.D. Di Benedetto. Nonlinear trong model matching. IEEE Tranaction on Automatic Control, 35:35 355, 990. M.D. Di Benedetto and A. Iidori. The matching of nonlinear model via dnamic tate feedback. In 23rd IEEE Conf. on Deciion and Control, La Vega, Nevada USA, 984. G. Conte, C.H. Moog, and A.M. Perdon. Algebraic Method for Nonlinear Control Stem. Theor and Application. Communication and Control Engineering. Springer-Verlag, London, 2nd edition, S.T. Glad. Nonlinear regulator and Ritt remainder algorithm. In Colloque International Sur L Anale de Steme Dnamique Controle, 990. M. Halá. Ore algebra: a polnomial approach to nonlinear time-dela tem. In 9th IFAC Workhop on Time-Dela Stem, Nante, France, M. Halá. An algebraic framework generalizing the concept of tranfer function to nonlinear tem. Automatica, 44, Scheduled for the Ma. M. Halá and M. Huba. Smbolic computation for nonlinear tem uing quotient over kew polnomial ring. In 4th Mediterranean Conference on Control and Automation, Ancona, Ital, M. Halá and Ü. Kotta. Extenion of the concept of tranfer function to dicrete-time nonlinear control tem. In European Control Conference, Ko, Greece, H.J.C. Huijbert. A nonregular olution of the nonlinear dnamic diturbance decoupling problem with an application to a complete olution of the nonlinear model matching problem. SIAM Journal of Control Optimization, 30: , 992. B. Marinecu and H. Bourlè. The exact model-matching problem for linear time-varing tem: an algebraic approach. IEEE Tranaction on Automatic Control, 48:66 69, A.M. Perdon, C.H. Moog, and G. Conte. The pole-zero tructure of nonlinear control tem. In 7th IFAC Smpoium NOLCOS, Pretoria, South Africa, J. Rudolph. Viewing input-output tem equivalence from differential algebra. J. Math. Stem Etim. Control, 4: , 994.