Observer design for a class of wave equation driven by an unknown periodic input

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1 Milano (Ialy) Augus 8 - Sepember, 11 Observer design for a class of wave equaion driven by an unknown periodic inpu Jonaan Cauvin Absrac: In is paper, exponenially convergen observers are designed for a class of wave equaion driven by an unknown periodic inpu wi only boundary sensing available. Te problem is posed as a problem of designing an inverible coordinae ransformaion of e observer error sysem ino an exponenially sable sysem. Observer gains (oupu injecion funcion) are sown o saisfy a well-posed wave equaion. Moreover, e observer gains are obained in closed form. Keywords: Parial differenial equaions, observer, periodic sysems. 1. INTRODUCTION In is paper, exponenially convergen observers are designed for a class of wave equaion driven by an unknown periodic inpu wi only boundary sensing available. Te moivaion o consider unknown periodic perurbaion arises from e auomoive engine applicaions. Indeed, is periodiciy sems from a fundamenal propery of e engines. 1.1 Auomoive engine problemaic In e auomoive indusry, observer design as been garnering increasing aenion in recen years. Increasing performance and environmenal requiremens ave imposed advanced conrol sraegies wile mainaining e effors oncosreducion. Inis conex, severalcommonreads can be found. In paricular, a various levels of modelling, auomoive engine dynamics can be considered as a sysem being mecanically coordinaed and syncronized by e revoluion of e cranksaf. Terefore, observer design arises in order o ave furer informaion on e combusion process were indusrial sensors are cosly and unreliable from reacing commercial producs. Anexample is combusion esimaion usingaknock sensor. Due o increasing inernal combusion engine arciecure and performance requiremens, e need of engine managemen is igly increased. Tis can be performed roug ig frequency recordings of cylinder pressure. Indeed, is pressure measuremen is a relevan variable for combusion managemen. Unforunaely, e cos and reliabiliy of suc a sensor preven em from reacing commercial producs lines. As aconsequence an ineresing problem is e design of a real-ime combusion observer using e reliable and available knock sensor as only measuremen. Teknock sensor is an indirec measuremen of J. Cauvin is wi e Deparmen of Conrol Signal and Sysem a IFP Energies nouvelles, 1 e 4 Avenue de Bois Préau, 985 Rueil Malmaison, France jonaan.cauvin@ifpen.fr. e combusion, i.e. i measures e combusion vibraion roug e engine block. Tis sensor is classically on producion engine Terefore, is problem can be seen as e esimaion of wave equaion driven by a periodic unknown signal. 1. Problem formulaion To solve is problem, one draws inspiraion from recen papers of Smyslyaev and Krsic Smyslyaev and Krsic (4) and Smyslyaev and Krsic (5) in wic an infinie dimensional backsepping observer is proposed for a class of parabolic parial differenial equaions wiou periodic perurbaions. In Cauvin e al. (7), e case of periodic inpu signals, a could be wrien as a sum of a finie number of armonics, driving aperiodic linear sysem is considered. In is paper, e reconsrucion of periodic inpus is formulaed ino e general framework of class of wave equaion driven by T -periodic signals. Consider e following sysem { u = α xx u β u, x [, 1] x u(,) = qu(,) (1) u(1,) = W () were x is e space coordinae in [, 1], e ime coordinae and W () e T -periodic unknown inpu. α >, β, and q are assumed o be arbirary consans. Suppose e only available measuremen for our sysem is a x =, i.e. y() u(,), e opposie end of e inpu. 1.3 Conribuion and paper organizaion Te conribuion of is paper is a ecnique o esimae e sae and e periodic unknown inpu u(1,) = W () of a wave equaion from e measuremen y() u(,) a e opposie end. Exponenial convergence ofe observer is proven. Copyrig by e Inernaional Federaion of Auomaic Conrol (IFAC) 1333

2 Milano (Ialy) Augus 8 - Sepember, 11 Te paper is organized as follows. In Secion, problem saemen and noaions are deailed. Ten, e observer design is presened following a backsepping for PDEs approac. Finally, e major par of e paper is dedicaed o convergence analysis. Asympoic convergence is proven, wen uning parameers are cosen sufficienly small. In deails, convergence is proven in several seps, following argumens of Cauvin e al. (7), in Secions 3, 4, and 5 wi Proposiion 4. Simulaion resuls are provided in Secion 6.. PROBLEM STATEMENT AND OBSERVER DESIGN ũ = α xx ũ β ũ ǫφ (x)ũ(,) ǫ(φ 1 (x) + βφ (x)) ũ(,) x ũ(,) = p ũ(,) + p ǫφ 1 ()ũ(,) ũ(1,) = ũ k e ıkω ũ k = ǫφ k ()ũ(,), k I+ and ũ k = ũ k (4). Overview of main pracical resuls: observer gains design guidelines Consider e wave equaion driven by an unknown periodic inpu signal W () following sysem (1) described in e inroducion. α >, β, and q are assumed o be arbirary consans. Suppose e only available measuremen for our sysem is a x =, i.e. y() u(,), e opposie end of e inpu. Te goal of our sudy is e esimaion of e T -periodic coninuous inpu signal u(1,) = W () R roug is Fourier decomposiion over a finie number of armonics W () u k e ıkω, ω π T were I σ=1 {ρ(σ), ρ(σ)} indexes e modes, and ρ : N \ {} N is sricly increasing. Noe I + {k I,k }, d card(i ). Wi ese noaions, e reference sysem wries u = α xx u β u, x [, 1] x u(,) = qu(,) u(1,) = u k e ıkω () u k = k I + y() = u(,) were eac vecor u k are complex enries. Le denoe e Hermiian ranspose..1 Observer srucure Corresponding o sae-space model (), one define a Luenberger ype observer û = α xx û β û + ǫφ (x)(y() û(,)) +ǫ(φ 1 (x) + βφ (x))(ẏ() û (,)) x û(,) = qy() p (ẏ() û (,)) ǫφ (ẏ() û (,)) û(1,) = û k e ıkω û k = ǫφ k ()(u(,) û(,)), k I+ and û k = û k (3) were ǫ is a small parameer. Here p, Φ (x), Φ 1 (x), and {φ k } k I are oupu injecion gain-funcions o be designed. Noice a oupu injecion is no only inroduced in e PDE, bu also a e boundary were e measuremen is available. Te observer error ũ u û saisfies e following PDE: Soluion wiou periodic perurbaion Observer gain p sould be now cosen o sabilize e error sysem in e case W (). In a case, e error sysem (4) wries (ǫ is uned o ) { ũ = α xx ũ β u x ũ(,) = p ũ(,) ũ(1,) = Te free parameer p is cosen o sabilize e error sysem, i.e. i acs as e damping on e boundary condiions. Terefore, wen β is sricly posiive no addiional damping is needed and en p =. On e oer and, wen β = is addiional damping is needed. In a case, e parameer p is cosen sricly posiive. Convergence of is wave equaion will be proven in Secion 3. Design of Φ, Φ 1, and {φ k } k I Te design of ese funcions sould be now cosen o guaranee e convergence of e Fourier expansion. For a, le Φ(x 1,x ) cos (x 1ψ(x )) + xp ψ(x sin(x ) 1ψ(x )) cos(ψ(x )) + xp ψ(x sin(ψ(x (5) ) )) βx were ψ(x ) +x α. One can noice a Φ(x 1,x ) is, for all x fixed, e soluion of e following equaion x Φ(x 1,x ) = α x x Φ(x 1,x ) βx Φ(x 1,x ) x1 Φ(,x ) = p x Φ(,x ) (6) Φ(1,x ) = 1 Based on is funcion Φ, one can define φ k () λ k e ıkω Φ(,ıkω ) (7) and Λ {λ k } k I (],+ [) d suc a λ k = λ k. Tese free parameers are used o se e desired observer convergence speed. Ten, Φ 1 (x) e ıkω Φ(x,ıkω )φ k () = λ k Φ(x,ıkω )Φ(,ıkω ) (8) and Φ (x) (ikω )e ıkω Φ(x,ıkω )φ k () = (ikω )λ k Φ(x,ıkω )Φ(,ıkω ) (9) Under is form, e roles of e uning parameers (e gain p, and e parameers Λ) appear disincly. On e one and, p conrols e convergence rae of e error 13333

3 Milano (Ialy) Augus 8 - Sepember, 11 sae ũ wile Λ, along wi ǫ, impacs on e convergence rae of e Fourier coefficiens esimaes. On e oer and, Φ and Φ 1 allow o coordinae bo dynamics. 3. CONVERGENCE PROOF / STEP 1 : CHANGE OF COORDINATES Le z ũ, z k ũ k. Te error dynamics (4) rewrie z = α xx z β z ǫφ 1 (x) z(,) ǫ(φ (x) + βφ 1 (x))z(,) x z(,) = p z(,) + p ǫφ 1 ()z(,) z(1,) = e ıkω z k k I ż k = φ k ()z(,) k I+ (1) Following Javid (198) and Javid (198), a series expansion on (1) w.r.. ǫ is performed. Le z z e ıkω Φ(x,ıkω )z k (11) 3.1 Evoluion of e PDE Firs of all, by a firs ime derivaion, one as z = z ( ) Φ(x,ıkω ) e ıkω z k = z + ǫφ 1 (x)z(,) e ıkω Φ(x,ıkω )(ikω )z k by e definiion of Φ 1 in (8). Ten, z = z + ǫφ 1 (x) z(,) + ǫφ (x)z(,) e ıkω Φ(x,ıkω )(ikω ) z k by e definiion of Φ in (9). Finally, using e wo previous compuaions and Equaion (1), z = α xx z β z + e ıkω ( βıkω + (ıkω ) ) f(x,ıkω )z k e ıkω α xx Φ(x,ıkω )z k k I = α xx z β z because, for all k I, using e definiion of Φ in (6), α xx Φ(x,ıkω ) = ( βıkω + (ıkω ) ) Φ(x,ıkω ) Ten x z(,) = p z(,) because for all k I, x Φ(,ıkω ) = p (ıkω )Φ(,ıkω ) Moreover, for e second boundary condiion, one ges z(1,) = z(1,) e ıkω Φ(1,ıkω )z k = because for all k I, Φ(1,ıkω ) = Toward a decoupled error dynamics In e ( z, {z k } k I + ) coordinaes, sysem (1) reads (afer some compuaions) z = α xx z β z x z(,) = p z(,) z(1,) = ż k = ǫλ k e ıkω Φ(,ıkω ) z(,) ǫ (1) e ı(l k)ω R k,l z l k I + l I were R k,l (λ k Φ(,ıkω )) (λ l Φ(,ılω )) for all (k,l) I. Tis cange of coordinaes sresses e firs par of e dynamics as an asympoically sable sysem parial differenial equaion. Tis cange of coordinaes ransforms e sysem (1) ino sysem (1), i.e. a wave equaion and a periodic linear sysem excied by a boundary of e previous PDE. 4. CONVERGENCE PROOF / STEP : CONVERGENCE OF Z. In is secion, convergence oward of z(x,), defined by (13), is proven { z = α xx z β z x z(,) = p z(,) (13) z(1,) = In e case β >, one can coose p =, and en exponenial convergence is well known. In e case β =, e addiional boundary condiion, using p >, will lead o e exponenial convergence of z oward. Here, e main elemens of e proof are recalled. For a, one draw inspiraion from Lagnese (1988); Morgl (199); Komornik (1991); Krsic and Smyslyaev (8) for e consrucion of a Lyapunov funcion. 4.1 Energy of e sysem 3. Evoluion of e boundary condiions Te firs boundary condiion becomes x z(,) = x z(,) e ıkω x Φ(,ıkω )z k = p z(,) + ǫp Φ 1 ()z(,) e ıkω x Φ(,ıkω )z k = p z(,) e ıkω x Φ(,ıkω )z k +p e ıkω Φ(,ıkω )(ikω )z k Te oal energy a ime of e sysem is defined by e funcional E () 1 α( x z(x,)) + ( z(x,)) dx (14) Differeniaing wi respec o ime leads o Ė () = = 1 α x z x z + z zdx α x z x z + zα xx zdx = αp ( z(,)) using by par inegraion. Te negaiviy of e rig and side sows a e energy E is decreasing wi ime due 13334

4 Milano (Ialy) Augus 8 - Sepember, 11 o e incorporaion of e boundary condiion. Te sysem is us non-energy conserving. Naurally, e quesion arises as weer e sysem is exponenially sable or no. An affirmaive answer is conained in e following proposiion, Proposiion 1. Le z be a soluion of sysem (13) wi z(.,) H 1 [,1], were H 1 [,1] {{[,1] x() R} wi a derivaive locally inegrable / x() d < + and x() + dx d () d < + }, en ere exiss some sricly posiive reals ( M, τ) (],+ [) suc a e energy E, defined by (14), saisfies R + τ E () Me Noe a H 1 [,1]. To prove e exponenial decrease of e energy, one can define a Lyapunov-like funcion V () as e sum of e energy and a cross-erm, i.e. V () E ()+ C (). For a, is cross-erm id defined as C () γ (x 1) x z(x,) z(x,)dx (15) wi γ being a posiive weiging consan. In a firs sep, one proves a e funcional V () is posiive. Ten, e negaiveness of e ime derivaive of V () is deailed. Properies of V will en lead o e exponenial convergence of e energy oward. 4. Proof of proposiion 1 Firs of all, for sufficienly small γ, e funcional V () is posiive. To is end, noe a C saisfies e following inequaliies C () γ min{1,α} E () Tus, e funcional V can be bounded as follows λ m E () V () λ M E () were λ m 1 γ min{1,α} and λ M 1 + γ min{1,α} provided a γ < min{1,α}. We now illusrae a, for for sufficienly small γ, e ime derivaive of V () is negaive. Te derivaive of C wi respec o ime yields Ċ () = γ (x 1)( x z z + x z z)dx Tis can be spli ino wo erms. Firsly, using by par inegraion, we ave (x 1) x z zdx = 1 ( z) dx + 1 ( z(,)) Ten, again, by par inegraion leads o Ten, (x 1) x z zdx = α ( x z) dx + αp ( z(,)) V () = γe () (αp γ (1 + αp ))( z(,)) λ V V () were λ V γ λ m provided a γ < αp. 1+αp Terefore, by defining γ = min{ min{1,α} αp 4, }, e 1+αp previous inequaliies are verified and Proposiion 1 olds using M = V () λ m, and τ = γ 4λ m,β. 4.3 Convergence of z As e funcion z is in H 1 [,1] and z(1,) =, en, from Hardy e al. (1959), z(x,) ( x z) dx Terefore, e following proposiion olds Proposiion. Consider e following sysem { z = α xx z β z x z(,) = p z(,) z(1,) = (16) For any funcion z(.,) H 1 [,1], sysem (16) as a unique classical soluion z(x,) C,1 ([,1] [,+ [). Addiionally, for p >, e origin is exponenially sable. Moreover, i follows a ere exiss ( K, τ) (],+ [) suc a R + τ z(,) Me 5. CONVERGENCE PROOF / STEP 3 : CONVERGENCE OF {Z K } K I + H Gaering {z k } k I + in ζ {z k } k I +, one can finally regroup sysem (1) under e form ζ = ǫa()ζ ǫb() (17) were A() = [a k,l ()] M d,d (C). Is coefficiens are of e form a k,l () e ı(l k)ω R k,l and B() = [b k ()] M d,1(c) is a vecor. Is coefficiens are of e form B k () λ k e ıkω Φ(,ıkω ) z(,) 5.1 Exponenial sabiliy of ζ = ǫa()ζ Te marix A is T -periodic. To sow e exponenial sabiliy of ζ = ǫa()ζ, averaging ecnique is applied. Le Ā 1 T A()d = diag(a k,k ) T and A() = (A(s) Ā)ds. Since A() Ā is T -periodic in and as zero mean, e funcion A is also T -periodic in.hence, A()is a boundedlinear marix. Consider e cange of variables [I ǫa()] ζ ζ Differeniaing bo side wi respec o, [I ǫa()] ζ = ǫ Ā ζ ǫ A() ζ Because e T -periodic marix A() is bounded on [,T ], e marix I ǫa() is nonsingular for sufficienly smallǫ, 13335

5 Milano (Ialy) Augus 8 - Sepember, 11 and [I ǫa()] 1 = I + O(ǫ) Terefore, e sae equaion for ζ is given by ζ = ǫā ζ + ǫ N(,ǫ) ζ (18) were N(,ǫ) is a regular, bounded, and T -periodic w.r... Furer, is expression is (up o second order erms in ǫ) ime-invarian. Equaion (18) is asympoically sable for < ǫ << 1 if and only if e sysem ζ = ǫā ζ (19) is yperbolically sable. All e canges of coordinaes are linear, ime-periodic and smoo, and us uniformly coninuous. Terefore, asympoic convergence oward of ζ leads o e asympoic convergence oward of ζ. Ā is a diagonal marix wi is elemens a k,k = λ kφ(,ıkω ) Φ(,ıkω ) = λ k 1 + p an(ψ(ıkω )) ıkω () ψ(ıkω ) > Te marix Ā is definie posiive and sysem (19) is us yperbolically sable. Ten e following proposiion olds Proposiion 3. Consider sysem ζ = ǫa()ζ. Ten, for ǫ small enoug, ζ converges uniformly asympoically oward. 5. Exponenial sabiliy of ζ = ǫa()ζ + B() Te soluion of (17) wries ζ() = Γ(,)ζ() + Γ(,τ)E z (τ)dτ were Γ(,τ) is e ransiion marix of e sysem ζ = ǫa()ζ beween τ and. As e sysem is linear, Proposiion 3 is equivalen o (k 1,k ) > (,τ) Γ(,τ) k 1 e k( τ) (see Kalil (199)). Ten, ζ() k 1 e k ζ() + Γ(,τ)B(τ)d Moreover, B() R d = λk e ıkω Φ (,ıkω ) z(,) Λ z(,) Λ M e τ Tus, B is square inegrable on [,+ [, i.e., Furer, + lim + B() d = Λ M τ B < + Γ(,τ) dτ = k 1 k Γ < + To prove a ζ ends o, e inegral is spli ino wo pars [, ] and [,]. On e firs inerval, e Caucy- Scwarz inequaliy leads o Γ(,τ)B(τ)dτ ( ) ( ) Γ(,τ) dτ B(τ) dτ ( ) ( + ) k1e k( τ) dτ B(τ) dτ ( k1 k e k B ) By e same way, Γ(,τ)B(τ)dτ Finally, one ges Λ M τ e τ Γ ζ() k 1 e k ζ() + k 1 k e k B + Λ M τ e τ Γ Tus, ζ() converges owards wen ends o +. All e canges of coordinaes are linear, ime-periodic and smoo, and us uniformly coninuous. Terefore, e following proposiion olds Proposiion 4. Consider e reference sysem (1) and e proposed observer (3). For ǫ small enoug, e error dynamics (4) converges asympoically oward in H 1 [,1] C d. 6. NUMERICAL RESULTS In is secion,numerical simulaions resuls of e plan () are presened. Te parameers of e sysem were aken o be α = 1 and q =.5. Two examples are presened. Firs of all, e case β =.5 is described. Ten, e case β = will be presened. Te iniial condiion u(x,) = 4+sin(πx) 1.83sin(.5πx). Te goal is o esimae e following periodic inpu W () = 1 + cos(ω ).5sin(ω ) +cos(ω ) +.4cos(3ω ).sin(3ω ) +.1cos(4ω ) +.5sin(4ω ) were ω = π 1. Te observer is designed based on e PDE model wiou any approximaion (discreizaion), wic makes eapproac esseniallyinfinie dimensional, bu for implemenaion purpose or simulaion, e plan and e observer are bo discreized usingfinie difference meod. Te PDE is discreized ino 1 cells and use a simulaion ime sep of. s. Moreover, a Runge-Kua meod was used o solve e simulaion problem. Observer design Te observer design follows.. Firs of all, define p o guaranee e convergence of sysem (13). For a, p = is used if β >, and p = 1 if β =. Ten, le us focus on e design of {φ k } k I. Firs of all, le {λ k } k [ 4,4] = { 1 k +1 } k [ 4,4] and ǫ =.1. Tese parameers are used o se e desired convergence speed of e coefficien-par of e observer. Te definiion of {φ k } k [ 4,4], Φ (x), and Φ 1 (x) follows.. Simulaion resuls for β > Te plan and e observer are discreized using a finie difference meod. Te observer converges oward e plan as can be seen on Figure 1, wi e knowledge of u(,), one is able o esimae e periodic unknown inpu. Moreover, wi e proposed observer ecnique, one is able o esimae e Fourier decomposiion of e periodic unknown inpu (see Fig ). A good esimaion of bo e sae and e unknown inpu is performed. Moreover, e esimaion algorim for e parameer esimaion is simple o une (only one parameer) and as a simple srucure

6 Milano (Ialy) Augus 8 - Sepember, ũ(x) L ([,1]) Figure 1. Numerical resuls (β =.5): ũ(u,) wi respec o e space x (beween and 1) and e ime. Decomposiion error W() Figure. Numerical resuls (β =.5). Top: inpu esimaion error Ũ(). Boom: k [ 4,4] c k(). Simulaion resuls for β = Te same parameers were kep for e observer design, excep p a becomes o 1. Figure 3 sows exponenial convergence ofe observer for e wave equaion. Again, wi e proposed observer ecnique, esimaion e Fourier decomposiion of e periodic unknown inpu is well performed. 7. CONCLUSION AND FUTURE WORK A consrucive inpu esimaion meod for a class of wave equaion is proposed. Convergence is analyzed using averaging ecniques along wi cange of coordinaes. Moreover, e observer gains are obained in closed form. A good esimaion of bo e sae and e unknown inpu is performed. Moreover, e esimaion algorim for e parameer esimaion is simple o une (only one parameer) and as a simple srucure. Furer publicaions will concenrae on infinie dimensional cases (i.e. wen = + ) and on e more general case of wave equaions Figure 3. Numerical resuls. Exponenial convergence of e observer for e ea equaion (β = ). ACKNOWLEDGEMENTS Te auor is indebed o Nicolas Pei for fruiful discussions abou PDEs sabilizaion and periodic sysems. REFERENCES Cauvin, J., Corde, G., Pei, N., and Roucon, P. (7). Periodic inpu esimaion for linear periodic sysems: auomoive engine applicaions. Auomaica, 43, Hardy, G.H., Lilewood, J.E., and Polya, G. (1959). Inequaliies. Cambridge Universiy Press. Javid, H. (198). Observing e slow saes of singularly perurbed sysem. Proc. in e IEEE Transacions on Auomaic Conrol, 5(5), Javid, H. (198). Sabilizaion of ime-varying singulary perurbed sysems by observer-based slow-sae feedback. Proc. in e IEEE Transacions on Auomaic Conrol, 7(3), Kalil, H. (199). Nonlinear Sysems. Prenice-Hall, Inc. Komornik, V. (1991). Rapid boundary sabilizaion of e wave equaion. SIAM Journal of Conrol and Opimizaion, 9, Krsic, M. and Smyslyaev, A. (8). Boundary Conrol of PDEs: A Course on Backsepping Designs. SIAM. Lagnese, J. (1988). Noe onboundarysabilizaion ofwave equaions. SIAM Journal of Conrol and Opimizaion, 6, Morgl, O. (199). Conrol and sabilizaion of a flexible beam aaced oarigid body. Inernaional Journal of Conrol, 51, Smyslyaev, A. and Krsic, M. (4). Closed-form boundary sae feedbacks for a class of 1-d parial inegrodifferenial equaions. IEEE Transacionson Auomaic Conrol, 49(1), 185. Smyslyaev, A. and Krsic, M. (5). Backsepping observers for a class of parabolic pdes. Sysems & Conrol Leers, 54,