H. MOUNIER, J. RUDOLPH, M. FLIESS, AND P. ROUCHON. interior mass is considered. It is viewed as a linear delay system. A

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1 ESAIM: Control, Optimisation an Calculus of Variations URL: September 998, Vol. 3, 35{3 TRACKING CONTROL OF A VIBRATING STRING WITH AN INTERIOR MASS VIEWED AS DELAY SYSTEM H. MOUNIER, J. RUDOLPH, M. FLIESS, AND P. ROUCHON Abstract. A vibrating string, moelle by thewave equation, with an interior mass is consiere. It is viewe as a linear elay system. A trajectory tracking problem is solve using a new type of controllability.. Introuction The \hybri system" moelling a vibrating string with an interior point mass is analyze in a remarkable recent paper by Hansen an Zuazua [6]. Using Hilbert's uniqueness metho [7, 8], on the one han, an nonharmonic Fourier series [3], on the other han, they give a etaile escription of its exact controllability an stabilization by bounary feeback. The present stuy, which is a companion paper of [3], aims to solve another natural control problem, namely tracking a trajectory of the mass position. We exploit the well known relation [] between the unampe wave equation an linear elay systems. We freely use [3], where relations to several classical structural properties of elay systems are establishe. The present case stuy illustrates in particular the importance of -freeness [3] for tracking control of a elay system, in a similar spirit as the \atness base control" of nonlinear (nite imensional) systems []. See also [] for other examples. In the next Section the elay system moel is erive from the hybri one. Its -freeness is establishe in Section 3. As in [6], two cases are istinguishe: position control on both bounaries or on either one, in which case the other en is xe. The tracking control in the single control case is treate an illustrate by simulations in Section 4. Relate examples may H. Mounier: Departement AXIS, Institut 'Electronique Fonamentale, B^atiment, Universite Paris-Su, 945 Orsay Ceex, France. mounier@ief.u-psu.fr. J. Ruolph: Institut fur Regelungs- un Steuerungstheorie, TU Dresen, Mommsenstr. 3, 6 Dresen, Germany. ruolph@erss.et.tu-resen.e. M. Fliess: Centre e Mathematiques et Leurs Applications, Ecole Normale Superieure e Cachan, 6, avenue u Presient Wilson, 9435 Cachan, France. fliess@cmla.ens-cachan.fr & fliess@lss.supelec.fr. P. Rouchon: Centre Automatique et Systemes, Ecole es Mines e Paris, 6, b. Saint- Michel, 757 Paris Ceex 6, France. rouchon@cas.ensmp.fr. This work was partially supporte by the European Commission's Training an Mobility of Researchers (TMR) Contract # ERBFMRX-CT9737, by the G.D.R. Meicis an by the G.D.R.-P.R.C. Automatique. Receive by the journal June 3, 997. Revise March 6, 998. Accepte for publication August 3, 998. c Societe e Mathematiques Appliquees et Inustrielles. Typeset by LATEX.

2 36 H. MOUNIER, J. RUDOLPH, M. FLIESS, AND P. ROUCHON be foun in [4, ]. In [] the control of an Euler-Bernoulli exible beam is sketche, which uses some of the tools evelope in [3].. Hybri system an elay system moel The evice uner stuy can be moelle as a hybri system: a (oneimensional) wave equation for each interval between the bounary an a point mass together with a secon orer linear orinary ierential equation escribing the motion of the t p x p x [;L ] t R + t q x q x [ L ] t R + : (.b) Here p(x t) an q(x t) represent the eformation at abscissa x an time t of the respective parts of the string, suppose homogeneous, occupying [;L ] an [ L ]. The physical parameters are their ensities an an the tensions,. The position z(t) of the mass M, attache to the string at x =, satises z(t) =p( t)=q( t) t R + (.c) M@ t x p( t) x q( t)= t R + : (.) Applying position controls u(t) an v(t) at the ens leas to Dirichlet bounary conitions p(;l t)=u(t) t R + (.e) q(l t)=v(t) t R + : (.f) The (compatible) initial conitions are p(x ) = p t p(x ) = p (x) x [;L ] (.g) q(x ) = q t q(x ) = q (x) x [ L ] (.h) z() = z = p () = q t z() = z = p () = q (): (.i) (.j) u p(z t) M q(z t) v z ;L L Figure. The vibrating string with an interior point mass The general solution of (.) in the convolution ring D (R) of istributions is well known to be p(x t) =( x )(t)+(;x )(t) (.) q(x t) =( x )(t)+(;x )(t) (.3) ESAIM: Cocv, September 998, Vol. 3, 35{3

3 TRACKING CONTROL OF A VIBRATING STRING WITH AN INTERIOR MASS 37 with D (R), h enoting the Dirac istribution concentrate at h, an = p = = p = : The bounary conitions (.c), (.e), an (.f) then rea (t)+ (t) ; (t) ; (t) = M( ( + )(t)) + ( ( ; )(t)) ; ( ( ; (t)) = (;L )(t)+( L )(t) =u(t) (.4a) (.4b) (.4c) ( L )(t)+(;l )(t) =v(t) (.4) where an enote the erivatives of the Dirac istribution concentrate at. Since (t) = (t) +(t) an (t) = (t) ; (t) give risetothe same solution for all (t), we can choose to have M( ( + ))(t)+ ( ; )(t) ; ( ; )(t) =: (.5) Dening 3. Structural properties = =M = =M an the localize elay operators an of respective amplitues L an L, the preceing equations can be resume as + = + (3.a) t ( + )+ ( ; ) ; ( ; )= (3.b) + ; = u (3.c) ; + = v: (3.) Remark 3.. Here we use the notation for the elay operators rather than as in [3] in orer to avoi any confusion with the Dirac istribution. 3.. Case of two controls The elay system moel is ene as the R[ t ]-moule u v generate by f,,,, u, vg with the relations (3.). We stuy the structure of the elay system u v with two controls rst. Theorem 3.. The system u v is torsion free, but not free. It is -free with basis ( ; u + ). Proof. The presentation matrix of u v associate with (3.) is 3 ; ; P u v ( t )= 6t + t ; ; 7 4 ; 5 : ; Calculating the minors which ene the variety V u v associate to u v, one realizes that P u v (s s s 3 ) has a loss of rank for s = s 3 = an ESAIM: Cocv, September 998, Vol. 3, 35{3

4 38 H. MOUNIER, J. RUDOLPH, M. FLIESS, AND P. ROUCHON s = ; only. This proves that u v is torsion free but not free (see [9] an [3, Proposition 3. an Theorem 3.]). Introuce b = ; u an b = +. The inepenence of b an b over the quotient el R( t ) can be checke on (3.) using elementary linear algebra. This reaily implies their inepenence over R[ t ]. Let us check that b an b are generators of u v. One has b = ; u = ;, whence = ; b, = b + b, u = b ; ( ; )b. Then, using t b + (b + b )= ( ; b ) an = b ; Finally, v = + yiels = t b + b + + b = ; t b ; b + ; b : v = (; ; ) t b + ( ; ; ) b + ( + ) ; +( ; ) b : The R[ t ]-linear inepenence of b an b is a irect consequence of rk R[=t ] u v =. 3.. Case of a single control Let the system be controlle on one bounary only. Therefore, consier v = in (3.), as in [6]. Equations (3.) then yiel + =(; ) + = u t ( + )+ ( ; )+ ( + ) =: (3.a) (3.b) (3.c) Denote as u the R[ t ]-moule generate by f ug with the relations (3.). Proposition 3.3. The system u is torsion free but not free. It is -free, with basis. Proof. The presentation matrix of u associate with (3.) is ; 3 P u( 6 t )= 4 ; 7 5 : t + t ; ( + ) The associate variety of zeros V u is the curve inthes s s 3 space given by +s s = s = ; 3 ; s : 3 It follows that u is torsion free, but not free (see [3], Theorems an 3). ESAIM: Cocv, September 998, Vol. 3, 35{3

5 TRACKING CONTROL OF A VIBRATING STRING WITH AN INTERIOR MASS 39 We now show that u is -free, with basis. The relation v =in (3.) implies = ;. Using this together with (3.a) in (3.c) yiels = ; ( ; ) t +( + ) +( ; ) (3.3a) = ( ; ) t +( ; ) +( + ) (3.3b) u = ( ; ; ) ; ( ; ) t + ; + (3.3c) ( ; + )( ; ) an generates R[ t ; ] R[=t ] u it is then a basis for this latter moule. 4. Mass tracking From Proposition 3.3 an equations (.c) an (.), we get z =(; ) : (4.) Theorem 4.. The system u is ( ; )-free, with basis z. Thus, the control law allowing to track a esire trajectory z of z, following irectly from (3.3c), involves an avance of L. Aitionally, we have from (.) p(x t)=( x )(t)+(;x )(t) x [;L ] q(x t)=( x )(t)+(;x )(t) x [ L ] an an are given in terms of z through (3.3a), (3.3b), an (4.). These formulae allow to compute the explicit solution of (.) an the control law u (t) yieling a esire trajectory z (t) of the mass position, as the one shown in Figure. The corresponing control function u (t) is epicte in Figure 3 an the isplacements along the string are shown on Figure 4 they correspon to =, = p, =, = p,anl =,L = Figure. The esire output z (t) ESAIM: Cocv, September 998, Vol. 3, 35{3

6 3 H. MOUNIER, J. RUDOLPH, M. FLIESS, AND P. ROUCHON Figure 3. The control law u (t) 8 6 p(x,t) q(z,t) Time t Abscissa x.5 Figure 4. Graph of p(x t) (x [; ]) an q(x t) (x [ ]) References [] K.L. Cooke, D.W. Krumme: Dierential-ierence equations an nonlinear initialbounary value problems for linear hyperbolic partial ierential equations, J. Math. Anal. Appl., 4, 968, 37{387. [] M. Fliess, J. Levine, P. Martin, P. Rouchon: Flatness an efect of nonlinear systems: introuctory theory an examples, Internat. J. Control, 6, 995, 37{36. [3] M. Fliess, H. Mounier: Controllability an observability of linear elay systems: an algebraic approach, ESAIM: Control Optimisation an Calculus of Variations, 3, 998, 3{34. ESAIM: Cocv, September 998, Vol. 3, 35{3

7 TRACKING CONTROL OF A VIBRATING STRING WITH AN INTERIOR MASS 3 [4] M. Fliess, H. Mounier, P. Rouchon, J. Ruolph: Controllability an motion planning for linear elay systems with an application to a exible ro, In 34th Conf. Decision Contr. Proc., New Orleans, Lousiana, 995, 46{5. [5] M. Fliess, H. Mounier, P. Rouchon, J. Ruolph: Systemes lineaires sur les operateurs e Mikusinski, ESAIM: Proceeings, 997, 83{93. [6] S. Hansen, E. Zuazua: Exact controllability an stabilization of a vibrating string with an interior point mass, SIAM J. Contr. Opt., 33, 995, 357{39. [7] J.L. Lions: Contr^olabilite exacte, perturbations et stabilisation e systemes istribues,, Masson, Paris, 988. [8] J.L. Lions: Exact controlability, stabilization an perturbations for istribute systems, SIAM Rev., 3, 988, {68. [9] H. Mounier: Proprietes structurelles es systemes lineaires a retars : aspects theoriques et pratiques, These, Universite Paris-Su, Orsay, 995. [] H. Mounier: Algebraic interpretations of the spectral controllability of a linear elay system, Forum Math.,, 998, 39{58. [] H. Mounier, P. Rouchon, J. Ruolph: Some examples of linear systems with elays, RAIRO-APII-JESA, 3, 997, 9{95. [] H. Mounier, J. Ruolph, M. Petitot, M. Fliess: A exible ro as a linear elay system, In European Control Conference Proc., Rome, 995, 3676{368. [3] D.L. Russel: Controllability an stabilizability theory for partial ierential equations recent progress an open question, SIAM Rev.,, 978, 639{739. ESAIM: Cocv, September 998, Vol. 3, 35{3