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Modelling of piezoelectric structures–a Hamiltonian approach

Modelling of piezoelectric structures–a Hamiltonian approach Mathematical and Computer Modelling of Dynamical Systems Vol. 14, No. 3, June 2008, 179–193 a b a M. Scho¨ berl *, H. Ennsbrunner and K. Schlacher Institute of Automatic Control and Control Systems Technology, J.K. University, Linz, Austria; Fronius International GmbH, Wels, Austria (Received 8 February 2007; final version received 17 August 2007) This contribution is dedicated to the geometric description of infinite-dimensional port Hamiltonian systems with in- and output operators. Several approaches exist, which deal with the extension of the well-known lumped parameter case to the distributed one. In this article a description has been chosen, which preserves useful properties known from the class of port controlled Hamiltonian systems with dissipation in the lumped scenario. Furthermore, the introduced in- and output maps are defined by linear differential operators. The derived theory is applied to the piezoelectric field equations to obtain their port Hamiltonian representation. In this example, the electrical field strength is assumed to act as distributed input. Finally it is shown, that distributed inputs, that are in the kernel of the input map act similarly on the system as certain boundary inputs. Keywords: infinite dimensional systems; Hamiltonian formulation; differential geometry; differential operators 1. Introduction Port controlled Hamiltonian systems with dissipation (PCHD-systems) are well known in the context of modelling and control, especially in the lumped parameter case, see for example [1] and references therein. The main advantage of the PCHD systems is that the mathematical description separates structural properties, storage elements, dissipative parts and ports. Furthermore, in the time invariant lumped parameter case the stability analysis can often be reduced to the investigation of the Hamiltonian. Distributed parameter systems are described by partial differential equations and this leads to many difficulties not known from the lumped parameter case. Especially the theorem concerning the existence and uniqueness of the initial value problem for explicit differential equations helps to link the properties of the differential equations with the properties of their solution. As no analogous theorem exists for partial differential equations in this contribution, only the formal properties of the equations will be discussed, which means that we will not focus on the properties of the solutions. The extension of lumped parameter PCHD systems to the distributed parameter case is neither straight-forward nor unique. The determination of a geometrical description of infinite-dimensional port Hamiltonian systems with dissipation, also called I-pHd systems, is an actual field of research. Several publications on that topic *Corresponding author. Email: markus.schoeberl@jku.at ISSN 1387-3954 print/ISSN 1744-5051 online 2008 Taylor & Francis DOI: 10.1080/13873950701844824 http://www.informaworld.com 180 M. Schoberl et al. as for example [2–6] visualize that different Hamiltonian representations are available. The description used in this contribution is based on the demand that useful properties known from the class of PCHD Systems in the lumped scenario should be preserved, see [4]. Furthermore, we will analyse the case where the in- and output maps are given by linear differential operators and give an interpretation of ports on the domain and on the boundary in this case, which is an extension to the theory shown in [4]. As an example, the piezoelectric field equations, a problem with two physical domains, is presented to show how the derived theory can be used for modelling. The contribution is organized as follows. In Section 2 the subsequently applied mathematical framework is introduced. The geometrical description of infinite- dimensional port Hamiltonian systems using first order differential operators as input maps is the content of the third part of this contribution, where also the impact of differential input operators on the corresponding boundary ports of the infinite- dimensional system is investigated. Finally, the derived theory is applied to the geometric representation of the piezoelectric field equations. Here, we will consider nonlinear constitutive relations. Some remarks on further extensions of the introduced framework and possible applications close this contribution. 2. Mathematical framework This contribution uses the language of differential geometry. An introduction and much more detail concerning differential geometry can be found in many textbooks for example in [5,7,8]. In the sequel we will summarize only some important constructions, which will be of frequent use in the following. The notation is similar to the one presented in [7]. 2.1. Manifolds and bundles A fibred manifold is a triple ðE; p; BÞ with the total manifold E, the base manifold B and the surjective submersion p : E! B. For each point ðp 2BÞ, the subset p ðpÞ¼ E is called the fibre over p. If the fibres are diffeomorphic to a so-called typical fibre, then ðE; p; BÞ is a bundle. In the following, a triple ðE; p; BÞ will always denote a bundle. We can i a introduce the adapted coordinates (X , x )to E at least locally with the independent i a coordinates X , i ¼ 1,. . ., r and the dependent ones x , a ¼ 1,. . ., s. Often, we will write E instead of ðE; p; BÞ, whenever the projection p and the base manifold B follow from the context. Bundles, whose fibres are vector spaces, are referred to as vector bundles. A section s of E is a map s : B! E such that p  s ¼ id is met, where id denotes the B B identity map on B. We do not require that a section s exists globally and write for the set of all sections  ðEÞ. From now on we use Latin indices for the independent and Greek indices for the dependent variables. Additionally a domain of integration is defined as an orientable, bounded manifold D with global volume form together with a coherently oriented boundary manifold @D. Let M be a smooth m-dimensional manifold, then its tangent and cotangent bundles are denoted by TðMÞ and T ðMÞ. These vector bundles possess the a a a coordinates ðx ; x_ Þ and ðx ; x_ Þ, respectively. Using local coordinates, we write a a v @ 2 ðÞ TðMÞ ; o dx 2 ðÞ T ðMÞ ; a ¼ 1; ... ; m for sections of TðMÞ; T ðMÞ; a a a a b b where x_ ¼ v ðx Þ and x_ ¼ o ðx Þ are met. Furthermore, we already applied the Einstein a a convention for sums to keep the formulas short and readable. From these vector bundles Mathematical and Computer Modelling of Dynamical Systems 181 one derives further bundles, like the exterior k-form bundle ^ðÞ T ðMÞ or other tensor bundles. We denote the exterior algebra over M by ^TðÞ ðMÞ , d : ^ðÞ T ðMÞ!^ ðÞ T ðMÞ k kþ1 is the exterior derivative and c : TðMÞ ^ ðT ðMÞÞ ! ^ðÞ T ðMÞ kþ1 k is the interior product written as vco with v 2 ðÞ TðMÞ and o 2^ ðÞ T ðMÞ . The kþ1 symbol ^ denotes the exterior product of the exterior algebra ^TðÞ ðMÞ . The Lie derivative of o 2^ðÞ T ðMÞ along the field f 2T ðMÞ is written as f (o). Additionally, we will use Stokes’s theorem [8] Z Z do ¼  o; o 2^ ðÞ T ðMÞ ð1Þ m1 M @M whereby the manifold and its boundary are related using the inclusion mapping : @M!M. A vector field v 2 ðÞ TðEÞ is said to be p-projectable, if there exists a field o 2 ðÞ TðBÞ such that p  v ¼ o  p is met, where p denotes the push forward along the map p. We say v is p-vertical in the case of p  v ¼ 0. It is easy to show that the set of all p-vertical vector fields – the vertical tangent bundle VðEÞ – is a subbundle of TðEÞ. The vertical bundle VðEÞ is i a a equipped with the induced coordinates ðX ; x ; x_ Þ with respect to the holonomic fibre base {@ }. 2.2. Jet manifolds i a Let g be a smooth section of a bundle ðE; p; BÞ with adapted coordinates (X , x ), i ¼ 1,. . ., r, a ¼ 1,. . ., s. The kth order partial derivatives of g will be denoted by a a a g ¼ @ g ¼ g ; j j J 1 r 1 r ð@X Þ ð@X Þ with J ¼ j  j , and k ¼ #J ¼ j . J is nothing else than an ordered multi-index [9]. 1 r i i¼1 The special index J ¼ j  j , j ¼ d , l 2 {1,. . ., r} will be denoted by 1 and J þ 1 is a 1 r i il l l shorthand notation for j þ d with the Kronecker symbol d . Using adapted coordinates i il il we can extend g to a map 1 i a a j ðgÞ : X !ðÞ X ; g ðXÞ;@ g ðXÞ ; the first jet of g. One can provide the set of all first jets of sections ðEÞ with the structure of a differentiable manifold, which is denoted by J ðEÞ. An adapted coordinate system of E 182 M. Schoberl et al. 1 i a a induces an adapted system on J ðEÞ, which is denoted by ðX ; x ; x Þ with the r  s new a 1 coordinates x . The manifold J ðEÞ has two natural projections 1 1 1 1 p : J ðEÞ ! B; p : J ðEÞ ! E; 1 1 1 1 which correspond to the bundles J ðEÞ; p ; B and ðJ ðEÞ; p ;EÞ. Analogously to the first jet of a section g, we define the nth jet j (g)of g by n i a a j ðgÞ¼ X ; g ðXÞ;@ g ðXÞ ; #J ¼ 1; ... ; n: The nth jet manifold J ðEÞ of E may be considered as a container for nth jets of sections of E. Furthermore, an adapted coordinate system of E induces an adapted system n i a on J ðEÞ with ðX ; x Þ; a ¼ 1; ... ; s; #J ¼ 0; ... ; n. These jet manifolds are connected by the following sequence n 1 n1 0 n n1 1 0 J ðEÞ ! J ðEÞ !  ! J ðEÞ ¼ JðEÞ ! J ðEÞ ¼ E ! B: The unique operator d , which meets nþ1 n ðd fÞ j ðsÞ¼ @ fjðÞ ðsÞ i i 1 n 1 for all functions f 2 C ðÞ J ðEÞ and sections s 2 ðEÞ, is the vector field d 2TðÞ J ðEÞ . It is called the total derivative with respect to the independent coordinate X and is defined by @ @ a J J d ¼ @ þ x @ ;@ ¼ ;@ ¼ ð2Þ i i i Jþ1 a a a i i @X @x i a in adapted coordinates (X ,x ). The introduction of the total derivative d enables us to introduce the horizontal derivative d through nþ1 n n ðj sÞðÞ d ðoÞ¼ dðÞ ðj sÞ ðoÞ ; o 2^ðÞ J ðEÞ ð3Þ or in local coordinates d ¼ dX ^ d (see e.g. [6]). Furthermore, we have h i Z Z Z nþ1 n n j ðsÞ ðd oÞ¼ djðÞ ðsÞ ðoÞ¼ ðÞ j ðsÞ ðoÞ ; D D @D n; i 0 for o ¼ h @ cdX 2 p ^ T ðEÞ , which is nothing else than Stokes’ theorem adapted to 0 r1 bundles. In the sequel we will suppress the pull backs and write ^ T ðEÞ instead of r1 n; p ^ T ðEÞ for instance if the pull back is clear from the context. 0 r1 3. Geometrical structure of I-pHd systems The state of a distributed parameter system is given by a certain set of functions defined on the bounded base manifold D. Therefore, it is obvious that we have to use a bundle to Mathematical and Computer Modelling of Dynamical Systems 183 describe the state in the infinite dimensional case. We use the local coordinates (X ), i ¼ 1,. . ., r for D, where these coordinates will represent the independent spatial coordinates according to the analysed plant. Let ðE; p; DÞ denote the state bundle with i a a local coordinates (X ,x ), a ¼ 1,. . ., s, where x represents the dependent coordinates. a a Consequently, a section s 2 ðEÞ defines a state of the system by x ¼ s (X). From the state bundle E we derive four important structures. The nth jet manifold J ðEÞ with i a a adapted coordinates ðX ; x ; x Þ, the vertical tangent bundle VðEÞ with coordinates i a a (X ,x ,x_ ), and the exterior bundles 0 1 a ^ðÞ T ðEÞ¼ span fdXg; ^ðÞ T ðEÞ¼ span fdx ^ dXg r r i a i a with coordinates (X ,x ,w), ðX ; x ; w_ Þ and the volume form 1 r dX ¼ dX ^  ^ dX : The interior product a b a v @ co dx ^ dX ¼ v o dX a b a induces the canonical product 1  0 VðEÞ  ^ðÞ T ðEÞ!^ðÞ T ðEÞ : r r The Hamiltonian functional H is a map H :ðEÞ ! R which is given as m 1 m HðsÞ¼ ðj sÞ ðh dXÞ; h 2 CðÞ J ðEÞ ; ð4Þ 0 0 where in the general case the Hamiltonian also depends on the jet coordinates x #J  m with m 4 0. Let us consider an evolutionary vectorfield v ¼ v @ 2 ðÞ VðEÞ ; a 1 n v 2 C ðÞ J ðEÞ , which corresponds to the set of partial differential equations i a a X ¼ 0; x_ ¼ v : This field v does not generate a flow on E but it may generate a semi group f that maps sections to sections of the bundle ðE; p; DÞ, i.e. f : R  ðEÞ ! ðEÞ. In general the semi flow and the evolutionary vectorfield are linked by a n a v  j ðsÞ¼ @ f ðsÞ : ð5Þ t¼0 In the sequel, we restrict ourselves to the case of first-order Hamiltonians. 3.1. First-order Hamiltonian 1 1 We confine ourselves to the case h 2 C J ðEÞ in equation (4) such that n ¼ 1holds in the relation (5). The change of the functional H (s) along the semi flow f canbecomputedas Z Z 2 1 2 1 ðj sÞ j ðvÞðh dXÞ ¼ ðj sÞ j ðvÞcdðh dXÞ ; ð6Þ 0 0 D D 184 M. Schoberl et al. for first order Hamiltonians, where the first prolongation j (v) 1 a a 1 j ðvÞ¼ v @ þ d ðv Þ@ a i is used. Let us inspect the expression 1 a a 1 j ðvÞcdðh dXÞ¼ðv @ h þ d ðv Þ@ h ÞdX 0 a 0 i 0 and integration by parts leads to 1 a a 1 a 1 i i j ðvÞcdðh dXÞ¼ðv @ h  v d @ h ÞdX þ d ðv @ h ÞdX: ð7Þ 0 a 0 i 0 i 0 a a Using the variational derivative d and the horizontal derivative d the equation (7) can be rewritten as 1 a 1 a j ðvÞcdðh dXÞ¼ vcd h dx ^ dX þ d ðvc@ h dx ^ @ cdXÞ; 0 a 0 h 0 i where the variational derivative d is a map 0  1 d : ^ðÞ T ðEÞ!^ðÞ T ðEÞ r r which has the coordinate expression a 1 h dX !ðd Þh dx ^ dX; d ¼ @  d ð@ Þ: 0 a 0 a a i It is easily seen that the total derivative d splits into the variational derivative d and an exact form. Furthermore, the additional map d can be introduced with @ 0  1 d : ^ðÞ T ðEÞ!^ ðÞ T ðEÞ ; r r1 which in coordinates is given as @ @ 1 a h dX ! d ðh dXÞ; d ðh dXÞ¼ @ h dx ^ @ cdX: 0 0 0 0 i 3.2. Evolutionary equations We propose the following set of equations x_ ¼ðJ <ÞðÞ dðh dXÞþ BðuÞ; ð8Þ y ¼ BðÞ dðh dXÞ ; ð9Þ i a together with X ¼ 0 and dðh dXÞ¼ d ðh Þdx ^ dX. The maps J, < are of the form 0 a 0 J; < : ^ðÞ T ðEÞ!VðEÞ; which are differential operators (see [9]) in general. As the input space we choose the i & vector bundle ðU; p ; DÞ with local coordinates (X ,u ), & ¼ 1,..., m and basis {e }. Of U & course, the output space Y¼ U is given by the dual vector bundle, where we use the Mathematical and Computer Modelling of Dynamical Systems 185 i & coordinates (X ,y ) and the basis {e  dX}. Furthermore, we conclude that there exists a bilinear map U Y! ^ðÞ TðEÞ given by the interior product & g & u e cy e  dX ¼ u y dX: & g & The input map reads as B : U! VðEÞ and B denotes the adjoint output map B : ^ ðT ðEÞÞ ! Y. Here we confine ourselves to the case, where J, < are linear maps and thus no differential operators. The map J is assumed to be skew symmetric i.e. Jðw Þcw ¼Jðw Þcw a b b a and < to be a symmetric positive semidefinite map defined by <ðw Þcw ¼<ðw Þcw ; <ðw Þcw 0: a b b a a a The in- and output maps B () and B*() are given by linear differential operators of first order. We make use of linear differential operators of the form g ai g ai 1 Bðu e Þ¼ d ðB u Þ@ ; B 2 C ðEÞ ð10Þ g i a g g as introduced for example in [5]. These operators meet Bðau þ bu Þ¼ a Bðu Þþ b Bðu Þ; u ; u 2U; a; b 2 R 1 2 1 2 1 2 as well as B ða o þ b o Þ¼ a B ðo Þþ b B ðo Þ; o ; o 2^ðÞ T ðEÞ ; a; b 2 R 1 2 1 2 1 2 due to their linearity. Additionally, their adjoint map is defined by i ai g BðuÞco ¼ ucB ðoÞþ dX ^ d ðB u @ co Þ i a B 1 1 with u 2U; o 2^ðÞ T ðEÞ and o 2^ ðÞ T ðEÞ . Using the horizontal derivative d we B h r r1 obtain ai g BðuÞco ¼ ucB ðoÞþ d ðB u @ co Þ: ð11Þ h a B Already this definition of the in- and output maps visualizes, that the use of differential in- and output operators introduces additional boundary conditions to the system. It is worth mentioning that the application of the total derivative in equation (10) is essential, as this guaranties a clear geometrical interpretation of the used differential operator. Let us consider an extended Hamiltonian density of the form h dX; h ¼ h  h u e e 0 x 186 M. Schoberl et al. then it is obvious that for h 2J ðEÞ the variational derivative dh reads as x e x x 1 x d ðh Þ d ðh u Þ¼ d ðh Þ @ ðh Þu þ d @ ðh Þu b 0 b x b 0 b x i x which shows that the input map contains a differential operator and this justifies the choice made in equation (10). The constructions presented so far can be visualized in the following commutative diagram where the pull backs and the projections have been omitted. 3.3. Infinite-dimensional Hamilton operator and collocation Let the p-vertical operator (this operator is not a vector field, but a submanifold 2 a a on ðp Þ VðEÞ parametrized in u) v ¼ x_ @ with x_ from equation (8) denote the h a Hamilton operator. The Lie derivative of H along the Hamilton operator of the corresponding I-pHd system which is the total time derivative of the Hamiltonian functional along the solution 2 1 d H ¼ ðj sÞ j ðv Þcdðh dXÞ t h 0 as in equation (6) consequently leads to 2 a d H ¼ ðj sÞðÞ <ðÞ dðh dXÞcdðh dXÞþ BðuÞcd h dx ^ dX t 0 0 a 0 2  ½1 þ  ðj sÞ ðv c@ ðh Þdx ^ @ cdXÞ : h 0 i @D Mathematical and Computer Modelling of Dynamical Systems 187 Let us apply the relations (11). Then the domain expression reads as 2 g a ðj sÞ <ðÞ dðh dXÞcdðh dXÞþ u e cB ðd h dx ^ dXÞ ð12Þ 0 0 g a 0 with a n B ðd h dx ^ dXÞ¼ y e dX a 0 n and the boundary term follows to 2  1 a ai g a ðj sÞ v c@ ðh Þdx ^ @ cdX þ B u @ cd h dx ^ @ cdX : ð13Þ h 0 i a a 0 i a g @D The equations (12) and (13) state, that the dissipative operator <, the pairing u y , which is a port distributed over D, and the boundary term 2 b 1 a ar g a ðj s Þ ðx_  Þ@ c @ ðh Þ  dx ^ dX þ B u @ cd h dx ^ dX ð14Þ 0 a a 0 a g @D with b 1 a l ¼ðx_  Þ@ c @ ðh Þ  dx ^ dX @ b 0 determine the Lie derivative of the Hamiltonian functional H. In equation (14) the boundary bundle ðE; p;@DÞ with E¼  E, the boundary section s  : @D! E and the r –1 boundary volume form r1 1 r1 dX ¼ @ cdX ¼ð1Þ dX ^  ^ dX are used as geometric objects, where the inclusion map i is assumed to be given by j j j r : ðX Þ! ðX ¼ X ; X ¼ const:Þ; j ¼ 1; ... ; r  1: 3.4. Boundary ports The form l stated in equation (14) is now assumed to equal the natural pairing of the boundary in- and outputs. In contrary to the determination procedure of the collocated output y on the domain, as stated in equation (9), it is no more possible to give a unique separation of the in- and output variables at the boundary visualized by the use of ðu; yÞ and ðy~; u~Þ in the following expression g g l ¼ u y dX ¼ ~y u~ dX: @ g g To overcome this problem we investigate two cases of boundary pairings on vector bundles. 188 M. Schoberl et al. Let us consider the boundary input vector bundle ðU; Z ;@DÞ with local coordinates j g ðX ; u Þ; j ¼ 1; ... ; ðr  1Þ; g ¼ 1; ... ; m  and the basis fe g and its dual – the boundary j g output vector bundle ðY; Z ;@DÞ with local coordinates ðX ; y Þ and basis fe  dXg.We a & make use of the tensor B e  @ and formulate a boundary input map B whichisdefinedby g a & BðuÞ¼ u e cB e  @ g a & a a ¼ u B @ ¼ðx  Þ@ : a a The adjoint map is then clearly given by 1 b & 1 a r  r B ð@ h  Þ¼ B e  @ cð@ h  Þdx ^ dX 0 b 0 a & a 1 a & & ¼ð@ h  ÞB e dX ¼ y e  dX 0 & a & and we obtain l ¼ u y dX as desired. The second pair is given by the boundary input @ g ~ ~ ~ vector bundle ðU; Z ;@DÞ with local coordinates ðX ; u Þ; j ¼ 1; ... ; ðr  1Þ; g ¼ 1; ... ; m ~ g and the basis fe~ g and its dual – the boundary output vector bundle ðY; ~Z ;@DÞ with local j g g a coordinates ðX ; y~ Þ and basis fdX  e~ g. The tensor B dx ^ dX  e~ is used to define the g g boundary input B map to construct g a & ~ ~ Bðu~Þ¼ B dx ^ dX  e~ cu~ e~ g & g a 1 a ¼ u~ B dx ^ dX ¼ð@ h  Þdx ^ dX g 0 a a and consequently the adjoint map is given by a b g a ~ ~ _ _ ~ B ðx  Þ¼ ðx  Þ@ cB dx ^ dX  e b g a g g ¼ðx_  ÞB dX  e~ ¼ y~ dX  e~ : g g a 1 If one vector or form part of l vanishes, that is x_   ¼ 0or @ h   ¼ 0 for a certain @ 0 a, then the corresponding pairing does not represent a port anymore. Now we are able to conclude, that the evolution of the Hamiltonian functional along the solution (here we assume the existence and uniqueness of the solution of the I-pHd systems) of a first order I-pHd system with in- and output operators is determined by the internal damping, the collocation of the in- and output on the domain and boundary and an additional term ar g B u d h dX a 0 g ai g on the boundary due to the application of an input operator Bðu e Þ¼ d ðB u Þ@ . g i a It is worth mentioning, that the adjoint map of the considered input map also becomes a differential operator with a non-trivial kernel. If one applies an input to the systems that leads to a collocated output lying in the kernel of the output map, then this input influences the evolution of the system through the corresponding boundary conditions, that is this input acts similarly to a boundary input. To provide this mathematical construction with a physical example, we investigate the piezo-electric field equations in the derived framework. Mathematical and Computer Modelling of Dynamical Systems 189 4. Application – the piezoelectric field In this contribution, we consider models of linearized elasticity, linearized quasi static electrodynamics combined with nonlinear constitutive relations. Let D denote the domain of the three-dimensional mechanical structure equipped with the Euclidean coordinates (X ), i ¼ 1,2,3, which are used to mark the positions of the mass points. The actual a a i a position of a mass point X is given by u þ d X , where u , a ¼ 1,2,3 are the displacements. The state of the elastic structure, is given by the positions, or equivalently by the a 1 displacements u , and linear momenta p ¼ rd u_ with the mass density 0 < r 2 C ðDÞ. b ba The total manifold E of the state bundle ðE; p; DÞ is equipped with the local coordinates i a ðX ; u ; p Þ; a; g ¼ 1; 2; 3: We assume, that there exists a stored energy density e dX, which meets ab c de ^ dX ¼ðs de  D dE Þ^ dX; b ¼ 1; 2; 3 ð15Þ S ab c with the stress ab s ¼ s @  @ ; a; b ¼ 1; 2; 3;@ ¼ a b b @u the strain a b i k j e ¼ e du  du ; e ¼ u d d þ u d d ; ab ab ar bk 1 b 1 a i j c c the electrical field strength E ¼ E dX and the electric displacement D ¼ D @ cdX. This c c assumption guaranties due to the exactness of de ^ dX that the stored energy is purely defined by the actual state of the system (see e.g. [10]). Here we introduce the nonlinear constitutive equations of the form ab ab s ¼ s ðe; EÞ c c D ¼ D ðe; EÞ Remark 1: A subclass of these equations are the well-known linear constitutive equations of piezoelectric materials given by ab abtd abc s ¼ C e  G E td c c tdc cn D ¼ G e þ F E ; td n abtd abc cn 1 with t, d, n ¼ 1, 2, 3, and C ; G ; F 2 C ðDÞ. These relations supply ab a de ^ dX ¼ s de  D dE ^ dX S ab a abtd abc tdc cn ¼ðC e  G E Þde ðG e þ F E ÞdE ^ dX td c ab td n c and finally 1 1 abtd abc cn de ^ dX ¼ d C e e  G E e  F E E ^ dX S td ab c ab n c 2 2 190 M. Schoberl et al. abtd batd abdt tdab abc bac cn nc if the integrability conditions C ¼ C ¼ C ¼ C , G ¼ G , F ¼ F are met. The kinetic energy density e dX is defined by gZ e dX ¼ p d p dX; Z ¼ 1; 2; 3 K g Z 2r with r 2 C ðDÞ. Finally, we are able to determine the exterior derivative of the Hamiltonian h as the sum of the exterior derivative of the stored and kinetic energy i.e. 1 1 gZ ab i k j c dh ¼ p d dp þ s d u d d þ u d d  D dE : g Z ar bk c 1 b 1 a i j r 2 The electrical field strength is considered as input and the variational derivative of the Hamiltonian density hdX can be rewritten in the form 1 m d ðhdXÞ¼ @ cdh  d @ cdh dx ^ dX; m ¼ 1 .. . 6 m m i m a with x ¼ (u , p ), whereby it is visualized that the exterior derivative of h is sufficient in the determination of the variational derivative. Consequently, there is no need to know the stored energy function – its existence and its exterior derivative already enables the determination of the equations of motion. i a Remark 2: The choice of the coordinates (X ,u ,p ) obviously leads to a Hamiltonian, which contains first-order jet variables. This should be compared with the approach presented in [11] where the authors consider infinite dimensional systems and avoid the use of jet variables in the Hamiltonian, by considering the map J as a differential operator. In the piezoelectric case, this approach leads to the choice of different state variables, namely the strain instead of the displacement. In this case additional partial differential equations appear as restrictions. Therefore, one has to deal with restricted I-pHd systems. The equations of motion are given by "# 1 ab i ba i a ag d s ðe; EÞd þ s ðe; EÞd u_ 0 d 2 b b p_ d 0 ga using the mechanical coordinates (u ,p ). The achieved Hamiltonian representation does not currently qualify as a port Hamiltonian representation, as the input – the electric field strength E – acts on the system in a nonlinear fashion. This problem can be solved by the introduction of new and less general constitutive relations ab ab abc s ðe; EÞ¼ C ðeÞ G E : This restriction leads us to ! "#"# ag ab a i abc u_ d C ðeÞd 0 d i b d G E d i c ¼ þ : p_ 1 d 0 g ag r Mathematical and Computer Modelling of Dynamical Systems 191 Consequently, we obtained the I-pHd structure x ¼ JðÞ dðh dXÞþ BðEÞ where the free Hamiltonian h meets 1 1 gZ ab r i k j dh ¼ p d dp þ C ðeÞd u d d þ u d d 0 g Z ar bk 1 b 1 a i j r 2 and the input map of interest appears in the form of the operator BðEÞ¼ i : abc d d G E d ga i c From these investigations we see that the input map meets the specifications of equation (10). It is worth mentioning that in the case where the piezoelectric material is an insulator d D ¼ 0 has to be met, because the volume charge density has to vanish. This relation has been omitted in the calculations above. Finally, the Lie derivative of the Hamiltonian functional stated in equation (6) leads to Z Z 2 w  2 1 w ðj sÞ BðEÞcd h dx ^ dX þ  ðj sÞ ðv c@ ðh Þdx ^ @ cdXÞ ð16Þ w 0 h 0 i D @D because no dissipative effects have been taken into account. The integral over the domain in equation (16) can be written as Z Z 2  w 2  gbc i ðj sÞ BðEÞcd h dx ^ dX ¼ ðj sÞ d ðG E d Þ p dX : w 0 i c g D D Let us apply the the definition of the adjoint operator from equation (11), which enables us to obtain Z Z 2 w 2 abc i ðj sÞ BðEÞcd h dx ^ dX ¼ ðj sÞ E G d d p dX ; w 0 c i a D D where the adjoint map reads as 1 1 abc i B p dX ¼G d d p ; g i a r r because we use the electric field strength as the distributed input. The boundary expression from the relation (16) is given as 2 vi v ðj sÞ p @ c C ðeÞ dx ^ @ cdX ; v v i @D which represents the boundary ports, where 1 1 vi i i @ ðh Þ¼ @ cðdh Þ¼ cðdh Þ¼ C ðeÞ; w ¼ 1; .. . ; 6; v ¼ 1; .. . ; 3 0 0 0 w w v @u i 192 M. Schoberl et al. is used and an additional boundary term arises due to the application of the adjoint operator, which reads as 2  gbc i v ðj sÞ G E d @ c p dx ^ @ cdX : c g v i @D If p dX is in the kernel of the output map B*(), then the domain port generated by the input map vanishes completely. In the case of piezoelectric systems this is for example given by p ¼ const: In contrary to the domain port, the boundary port generated by the input operator does not vanish and consequently a domain input could act on the system like a boundary input does. These investigations show, that the spatial shape of the distributed input and collocated output is mainly responsible for its appearance within the field equations, boundary conditions and evolution of the free Hamiltonian h . 5. Conclusions Piezoelectric materials enable fascinating new ways of interaction (actuation and sensing) between control equipment and flexible structures. To derive passivity based control strategies a geometric description of the system in a port Hamiltonian setting is of main interest. This contribution introduces a geometrical representation of infinite-dimensional port Hamiltonian systems with in- and output maps using differential operators. It is shown, that the extension of the description shown in [4] results in the appearance of additional boundary conditions in the Lie derivative of the Hamiltonian functional. As the presented approach is a formal one, based on differential geometric considerations, several aspects from functional analysis are missing. For example, Sobolev norms on linear spaces and manifolds have not been introduced, see for example [12,13], also the existence of solutions has not been discussed. The analysis of the piezoelectric field equations on the introduced I-pHd framework yields a very interesting explanation of the frequently used method of ‘‘electrode shaping’’ for piezoelectric devices. The existence of a linear differential input operator enables the use of spatial output distributions such that the output is in the kernel of the domain output operator. Consequently, the distributed domain input acts in a similar fashion on the system as a boundary input. It is obvious that linear input operators of higher order provide more complex output kernels, and consequently, additional degrees of freedom in the application of control action are given. Finally, it is worth mentioning, that the mappings J, < could also be replaced by appropriate differential operators. Such an extension will enable the treatment of coupled field problems like piezo-thermo-elasticity on the basis of port Hamiltonian systems. This represents the content of future investigations. Acknowledgements Partly, this work has been done in the context of the European sponsored project GeoPlex with reference code IST-2001-34166. Further information is available at http://www.geoplex.cc. Mathematical and Computer Modelling of Dynamical Systems 193 References [1] A.J. van der Schaft, L -Gain and Passivity Techniques in Nonlinear Control, Springer, London, [2] A. Macchelli, Port Hamiltonian Systems – A unified approach for modeling and control, Ph.D. Thesis, University of Bologna, Italy, 2002. [3] A.J. van der Schaft and B.M. Maschke, Hamiltonian formulation of distributed-parameter systems with boundary energy flow, J. Geom. Phys. 42 (2002), pp. 166–194. [4] H. Ennsbrunner and K. Schlacher, On the geometrical representation and interconnection of infinite dimensional port controlled Hamiltonian systems. 44th IEEE, Conference on Decision and Control and European Control Conference, Sevilla, Spain, 2005. [5] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986. [6] G. Giachetta, G. Sardanashvily, and L. Mangiarotti, New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific, New York, 1994. [7] D.J. Saunders, The Geometry of Jet Bundles, Cambridge University Press, Cambridge, 1989. [8] W.M. Boothby, An Introduction to Differentiable Manifolds and Riemanian Geometry, Academic Press, Orlando, 1986. [9] J.F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, New York, 1978. [10] A. Kugi, Non-linear Control Based on Physical Models, Springer, London, 2001. [11] A. Macchelli and C. Melchiorri, Control by interconnection and energy shaping of the Timoshenko beam, J. Math. Comput. Model. Dynamical Syst. (MCMDS) 10 (2004), pp. 231– [12] E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Institute of Mathematical Sciences, New York, 2000. [13] E. Zeidler, Applied Functional Analysis, Springer, New York, 1995. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical and Computer Modelling of Dynamical Systems Taylor & Francis

Modelling of piezoelectric structures–a Hamiltonian approach

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Mathematical and Computer Modelling of Dynamical Systems Vol. 14, No. 3, June 2008, 179–193 a b a M. Scho¨ berl *, H. Ennsbrunner and K. Schlacher Institute of Automatic Control and Control Systems Technology, J.K. University, Linz, Austria; Fronius International GmbH, Wels, Austria (Received 8 February 2007; final version received 17 August 2007) This contribution is dedicated to the geometric description of infinite-dimensional port Hamiltonian systems with in- and output operators. Several approaches exist, which deal with the extension of the well-known lumped parameter case to the distributed one. In this article a description has been chosen, which preserves useful properties known from the class of port controlled Hamiltonian systems with dissipation in the lumped scenario. Furthermore, the introduced in- and output maps are defined by linear differential operators. The derived theory is applied to the piezoelectric field equations to obtain their port Hamiltonian representation. In this example, the electrical field strength is assumed to act as distributed input. Finally it is shown, that distributed inputs, that are in the kernel of the input map act similarly on the system as certain boundary inputs. Keywords: infinite dimensional systems; Hamiltonian formulation; differential geometry; differential operators 1. Introduction Port controlled Hamiltonian systems with dissipation (PCHD-systems) are well known in the context of modelling and control, especially in the lumped parameter case, see for example [1] and references therein. The main advantage of the PCHD systems is that the mathematical description separates structural properties, storage elements, dissipative parts and ports. Furthermore, in the time invariant lumped parameter case the stability analysis can often be reduced to the investigation of the Hamiltonian. Distributed parameter systems are described by partial differential equations and this leads to many difficulties not known from the lumped parameter case. Especially the theorem concerning the existence and uniqueness of the initial value problem for explicit differential equations helps to link the properties of the differential equations with the properties of their solution. As no analogous theorem exists for partial differential equations in this contribution, only the formal properties of the equations will be discussed, which means that we will not focus on the properties of the solutions. The extension of lumped parameter PCHD systems to the distributed parameter case is neither straight-forward nor unique. The determination of a geometrical description of infinite-dimensional port Hamiltonian systems with dissipation, also called I-pHd systems, is an actual field of research. Several publications on that topic *Corresponding author. Email: markus.schoeberl@jku.at ISSN 1387-3954 print/ISSN 1744-5051 online 2008 Taylor & Francis DOI: 10.1080/13873950701844824 http://www.informaworld.com 180 M. Schoberl et al. as for example [2–6] visualize that different Hamiltonian representations are available. The description used in this contribution is based on the demand that useful properties known from the class of PCHD Systems in the lumped scenario should be preserved, see [4]. Furthermore, we will analyse the case where the in- and output maps are given by linear differential operators and give an interpretation of ports on the domain and on the boundary in this case, which is an extension to the theory shown in [4]. As an example, the piezoelectric field equations, a problem with two physical domains, is presented to show how the derived theory can be used for modelling. The contribution is organized as follows. In Section 2 the subsequently applied mathematical framework is introduced. The geometrical description of infinite- dimensional port Hamiltonian systems using first order differential operators as input maps is the content of the third part of this contribution, where also the impact of differential input operators on the corresponding boundary ports of the infinite- dimensional system is investigated. Finally, the derived theory is applied to the geometric representation of the piezoelectric field equations. Here, we will consider nonlinear constitutive relations. Some remarks on further extensions of the introduced framework and possible applications close this contribution. 2. Mathematical framework This contribution uses the language of differential geometry. An introduction and much more detail concerning differential geometry can be found in many textbooks for example in [5,7,8]. In the sequel we will summarize only some important constructions, which will be of frequent use in the following. The notation is similar to the one presented in [7]. 2.1. Manifolds and bundles A fibred manifold is a triple ðE; p; BÞ with the total manifold E, the base manifold B and the surjective submersion p : E! B. For each point ðp 2BÞ, the subset p ðpÞ¼ E is called the fibre over p. If the fibres are diffeomorphic to a so-called typical fibre, then ðE; p; BÞ is a bundle. In the following, a triple ðE; p; BÞ will always denote a bundle. We can i a introduce the adapted coordinates (X , x )to E at least locally with the independent i a coordinates X , i ¼ 1,. . ., r and the dependent ones x , a ¼ 1,. . ., s. Often, we will write E instead of ðE; p; BÞ, whenever the projection p and the base manifold B follow from the context. Bundles, whose fibres are vector spaces, are referred to as vector bundles. A section s of E is a map s : B! E such that p  s ¼ id is met, where id denotes the B B identity map on B. We do not require that a section s exists globally and write for the set of all sections  ðEÞ. From now on we use Latin indices for the independent and Greek indices for the dependent variables. Additionally a domain of integration is defined as an orientable, bounded manifold D with global volume form together with a coherently oriented boundary manifold @D. Let M be a smooth m-dimensional manifold, then its tangent and cotangent bundles are denoted by TðMÞ and T ðMÞ. These vector bundles possess the a a a coordinates ðx ; x_ Þ and ðx ; x_ Þ, respectively. Using local coordinates, we write a a v @ 2 ðÞ TðMÞ ; o dx 2 ðÞ T ðMÞ ; a ¼ 1; ... ; m for sections of TðMÞ; T ðMÞ; a a a a b b where x_ ¼ v ðx Þ and x_ ¼ o ðx Þ are met. Furthermore, we already applied the Einstein a a convention for sums to keep the formulas short and readable. From these vector bundles Mathematical and Computer Modelling of Dynamical Systems 181 one derives further bundles, like the exterior k-form bundle ^ðÞ T ðMÞ or other tensor bundles. We denote the exterior algebra over M by ^TðÞ ðMÞ , d : ^ðÞ T ðMÞ!^ ðÞ T ðMÞ k kþ1 is the exterior derivative and c : TðMÞ ^ ðT ðMÞÞ ! ^ðÞ T ðMÞ kþ1 k is the interior product written as vco with v 2 ðÞ TðMÞ and o 2^ ðÞ T ðMÞ . The kþ1 symbol ^ denotes the exterior product of the exterior algebra ^TðÞ ðMÞ . The Lie derivative of o 2^ðÞ T ðMÞ along the field f 2T ðMÞ is written as f (o). Additionally, we will use Stokes’s theorem [8] Z Z do ¼  o; o 2^ ðÞ T ðMÞ ð1Þ m1 M @M whereby the manifold and its boundary are related using the inclusion mapping : @M!M. A vector field v 2 ðÞ TðEÞ is said to be p-projectable, if there exists a field o 2 ðÞ TðBÞ such that p  v ¼ o  p is met, where p denotes the push forward along the map p. We say v is p-vertical in the case of p  v ¼ 0. It is easy to show that the set of all p-vertical vector fields – the vertical tangent bundle VðEÞ – is a subbundle of TðEÞ. The vertical bundle VðEÞ is i a a equipped with the induced coordinates ðX ; x ; x_ Þ with respect to the holonomic fibre base {@ }. 2.2. Jet manifolds i a Let g be a smooth section of a bundle ðE; p; BÞ with adapted coordinates (X , x ), i ¼ 1,. . ., r, a ¼ 1,. . ., s. The kth order partial derivatives of g will be denoted by a a a g ¼ @ g ¼ g ; j j J 1 r 1 r ð@X Þ ð@X Þ with J ¼ j  j , and k ¼ #J ¼ j . J is nothing else than an ordered multi-index [9]. 1 r i i¼1 The special index J ¼ j  j , j ¼ d , l 2 {1,. . ., r} will be denoted by 1 and J þ 1 is a 1 r i il l l shorthand notation for j þ d with the Kronecker symbol d . Using adapted coordinates i il il we can extend g to a map 1 i a a j ðgÞ : X !ðÞ X ; g ðXÞ;@ g ðXÞ ; the first jet of g. One can provide the set of all first jets of sections ðEÞ with the structure of a differentiable manifold, which is denoted by J ðEÞ. An adapted coordinate system of E 182 M. Schoberl et al. 1 i a a induces an adapted system on J ðEÞ, which is denoted by ðX ; x ; x Þ with the r  s new a 1 coordinates x . The manifold J ðEÞ has two natural projections 1 1 1 1 p : J ðEÞ ! B; p : J ðEÞ ! E; 1 1 1 1 which correspond to the bundles J ðEÞ; p ; B and ðJ ðEÞ; p ;EÞ. Analogously to the first jet of a section g, we define the nth jet j (g)of g by n i a a j ðgÞ¼ X ; g ðXÞ;@ g ðXÞ ; #J ¼ 1; ... ; n: The nth jet manifold J ðEÞ of E may be considered as a container for nth jets of sections of E. Furthermore, an adapted coordinate system of E induces an adapted system n i a on J ðEÞ with ðX ; x Þ; a ¼ 1; ... ; s; #J ¼ 0; ... ; n. These jet manifolds are connected by the following sequence n 1 n1 0 n n1 1 0 J ðEÞ ! J ðEÞ !  ! J ðEÞ ¼ JðEÞ ! J ðEÞ ¼ E ! B: The unique operator d , which meets nþ1 n ðd fÞ j ðsÞ¼ @ fjðÞ ðsÞ i i 1 n 1 for all functions f 2 C ðÞ J ðEÞ and sections s 2 ðEÞ, is the vector field d 2TðÞ J ðEÞ . It is called the total derivative with respect to the independent coordinate X and is defined by @ @ a J J d ¼ @ þ x @ ;@ ¼ ;@ ¼ ð2Þ i i i Jþ1 a a a i i @X @x i a in adapted coordinates (X ,x ). The introduction of the total derivative d enables us to introduce the horizontal derivative d through nþ1 n n ðj sÞðÞ d ðoÞ¼ dðÞ ðj sÞ ðoÞ ; o 2^ðÞ J ðEÞ ð3Þ or in local coordinates d ¼ dX ^ d (see e.g. [6]). Furthermore, we have h i Z Z Z nþ1 n n j ðsÞ ðd oÞ¼ djðÞ ðsÞ ðoÞ¼ ðÞ j ðsÞ ðoÞ ; D D @D n; i 0 for o ¼ h @ cdX 2 p ^ T ðEÞ , which is nothing else than Stokes’ theorem adapted to 0 r1 bundles. In the sequel we will suppress the pull backs and write ^ T ðEÞ instead of r1 n; p ^ T ðEÞ for instance if the pull back is clear from the context. 0 r1 3. Geometrical structure of I-pHd systems The state of a distributed parameter system is given by a certain set of functions defined on the bounded base manifold D. Therefore, it is obvious that we have to use a bundle to Mathematical and Computer Modelling of Dynamical Systems 183 describe the state in the infinite dimensional case. We use the local coordinates (X ), i ¼ 1,. . ., r for D, where these coordinates will represent the independent spatial coordinates according to the analysed plant. Let ðE; p; DÞ denote the state bundle with i a a local coordinates (X ,x ), a ¼ 1,. . ., s, where x represents the dependent coordinates. a a Consequently, a section s 2 ðEÞ defines a state of the system by x ¼ s (X). From the state bundle E we derive four important structures. The nth jet manifold J ðEÞ with i a a adapted coordinates ðX ; x ; x Þ, the vertical tangent bundle VðEÞ with coordinates i a a (X ,x ,x_ ), and the exterior bundles 0 1 a ^ðÞ T ðEÞ¼ span fdXg; ^ðÞ T ðEÞ¼ span fdx ^ dXg r r i a i a with coordinates (X ,x ,w), ðX ; x ; w_ Þ and the volume form 1 r dX ¼ dX ^  ^ dX : The interior product a b a v @ co dx ^ dX ¼ v o dX a b a induces the canonical product 1  0 VðEÞ  ^ðÞ T ðEÞ!^ðÞ T ðEÞ : r r The Hamiltonian functional H is a map H :ðEÞ ! R which is given as m 1 m HðsÞ¼ ðj sÞ ðh dXÞ; h 2 CðÞ J ðEÞ ; ð4Þ 0 0 where in the general case the Hamiltonian also depends on the jet coordinates x #J  m with m 4 0. Let us consider an evolutionary vectorfield v ¼ v @ 2 ðÞ VðEÞ ; a 1 n v 2 C ðÞ J ðEÞ , which corresponds to the set of partial differential equations i a a X ¼ 0; x_ ¼ v : This field v does not generate a flow on E but it may generate a semi group f that maps sections to sections of the bundle ðE; p; DÞ, i.e. f : R  ðEÞ ! ðEÞ. In general the semi flow and the evolutionary vectorfield are linked by a n a v  j ðsÞ¼ @ f ðsÞ : ð5Þ t¼0 In the sequel, we restrict ourselves to the case of first-order Hamiltonians. 3.1. First-order Hamiltonian 1 1 We confine ourselves to the case h 2 C J ðEÞ in equation (4) such that n ¼ 1holds in the relation (5). The change of the functional H (s) along the semi flow f canbecomputedas Z Z 2 1 2 1 ðj sÞ j ðvÞðh dXÞ ¼ ðj sÞ j ðvÞcdðh dXÞ ; ð6Þ 0 0 D D 184 M. Schoberl et al. for first order Hamiltonians, where the first prolongation j (v) 1 a a 1 j ðvÞ¼ v @ þ d ðv Þ@ a i is used. Let us inspect the expression 1 a a 1 j ðvÞcdðh dXÞ¼ðv @ h þ d ðv Þ@ h ÞdX 0 a 0 i 0 and integration by parts leads to 1 a a 1 a 1 i i j ðvÞcdðh dXÞ¼ðv @ h  v d @ h ÞdX þ d ðv @ h ÞdX: ð7Þ 0 a 0 i 0 i 0 a a Using the variational derivative d and the horizontal derivative d the equation (7) can be rewritten as 1 a 1 a j ðvÞcdðh dXÞ¼ vcd h dx ^ dX þ d ðvc@ h dx ^ @ cdXÞ; 0 a 0 h 0 i where the variational derivative d is a map 0  1 d : ^ðÞ T ðEÞ!^ðÞ T ðEÞ r r which has the coordinate expression a 1 h dX !ðd Þh dx ^ dX; d ¼ @  d ð@ Þ: 0 a 0 a a i It is easily seen that the total derivative d splits into the variational derivative d and an exact form. Furthermore, the additional map d can be introduced with @ 0  1 d : ^ðÞ T ðEÞ!^ ðÞ T ðEÞ ; r r1 which in coordinates is given as @ @ 1 a h dX ! d ðh dXÞ; d ðh dXÞ¼ @ h dx ^ @ cdX: 0 0 0 0 i 3.2. Evolutionary equations We propose the following set of equations x_ ¼ðJ <ÞðÞ dðh dXÞþ BðuÞ; ð8Þ y ¼ BðÞ dðh dXÞ ; ð9Þ i a together with X ¼ 0 and dðh dXÞ¼ d ðh Þdx ^ dX. The maps J, < are of the form 0 a 0 J; < : ^ðÞ T ðEÞ!VðEÞ; which are differential operators (see [9]) in general. As the input space we choose the i & vector bundle ðU; p ; DÞ with local coordinates (X ,u ), & ¼ 1,..., m and basis {e }. Of U & course, the output space Y¼ U is given by the dual vector bundle, where we use the Mathematical and Computer Modelling of Dynamical Systems 185 i & coordinates (X ,y ) and the basis {e  dX}. Furthermore, we conclude that there exists a bilinear map U Y! ^ðÞ TðEÞ given by the interior product & g & u e cy e  dX ¼ u y dX: & g & The input map reads as B : U! VðEÞ and B denotes the adjoint output map B : ^ ðT ðEÞÞ ! Y. Here we confine ourselves to the case, where J, < are linear maps and thus no differential operators. The map J is assumed to be skew symmetric i.e. Jðw Þcw ¼Jðw Þcw a b b a and < to be a symmetric positive semidefinite map defined by <ðw Þcw ¼<ðw Þcw ; <ðw Þcw 0: a b b a a a The in- and output maps B () and B*() are given by linear differential operators of first order. We make use of linear differential operators of the form g ai g ai 1 Bðu e Þ¼ d ðB u Þ@ ; B 2 C ðEÞ ð10Þ g i a g g as introduced for example in [5]. These operators meet Bðau þ bu Þ¼ a Bðu Þþ b Bðu Þ; u ; u 2U; a; b 2 R 1 2 1 2 1 2 as well as B ða o þ b o Þ¼ a B ðo Þþ b B ðo Þ; o ; o 2^ðÞ T ðEÞ ; a; b 2 R 1 2 1 2 1 2 due to their linearity. Additionally, their adjoint map is defined by i ai g BðuÞco ¼ ucB ðoÞþ dX ^ d ðB u @ co Þ i a B 1 1 with u 2U; o 2^ðÞ T ðEÞ and o 2^ ðÞ T ðEÞ . Using the horizontal derivative d we B h r r1 obtain ai g BðuÞco ¼ ucB ðoÞþ d ðB u @ co Þ: ð11Þ h a B Already this definition of the in- and output maps visualizes, that the use of differential in- and output operators introduces additional boundary conditions to the system. It is worth mentioning that the application of the total derivative in equation (10) is essential, as this guaranties a clear geometrical interpretation of the used differential operator. Let us consider an extended Hamiltonian density of the form h dX; h ¼ h  h u e e 0 x 186 M. Schoberl et al. then it is obvious that for h 2J ðEÞ the variational derivative dh reads as x e x x 1 x d ðh Þ d ðh u Þ¼ d ðh Þ @ ðh Þu þ d @ ðh Þu b 0 b x b 0 b x i x which shows that the input map contains a differential operator and this justifies the choice made in equation (10). The constructions presented so far can be visualized in the following commutative diagram where the pull backs and the projections have been omitted. 3.3. Infinite-dimensional Hamilton operator and collocation Let the p-vertical operator (this operator is not a vector field, but a submanifold 2 a a on ðp Þ VðEÞ parametrized in u) v ¼ x_ @ with x_ from equation (8) denote the h a Hamilton operator. The Lie derivative of H along the Hamilton operator of the corresponding I-pHd system which is the total time derivative of the Hamiltonian functional along the solution 2 1 d H ¼ ðj sÞ j ðv Þcdðh dXÞ t h 0 as in equation (6) consequently leads to 2 a d H ¼ ðj sÞðÞ <ðÞ dðh dXÞcdðh dXÞþ BðuÞcd h dx ^ dX t 0 0 a 0 2  ½1 þ  ðj sÞ ðv c@ ðh Þdx ^ @ cdXÞ : h 0 i @D Mathematical and Computer Modelling of Dynamical Systems 187 Let us apply the relations (11). Then the domain expression reads as 2 g a ðj sÞ <ðÞ dðh dXÞcdðh dXÞþ u e cB ðd h dx ^ dXÞ ð12Þ 0 0 g a 0 with a n B ðd h dx ^ dXÞ¼ y e dX a 0 n and the boundary term follows to 2  1 a ai g a ðj sÞ v c@ ðh Þdx ^ @ cdX þ B u @ cd h dx ^ @ cdX : ð13Þ h 0 i a a 0 i a g @D The equations (12) and (13) state, that the dissipative operator <, the pairing u y , which is a port distributed over D, and the boundary term 2 b 1 a ar g a ðj s Þ ðx_  Þ@ c @ ðh Þ  dx ^ dX þ B u @ cd h dx ^ dX ð14Þ 0 a a 0 a g @D with b 1 a l ¼ðx_  Þ@ c @ ðh Þ  dx ^ dX @ b 0 determine the Lie derivative of the Hamiltonian functional H. In equation (14) the boundary bundle ðE; p;@DÞ with E¼  E, the boundary section s  : @D! E and the r –1 boundary volume form r1 1 r1 dX ¼ @ cdX ¼ð1Þ dX ^  ^ dX are used as geometric objects, where the inclusion map i is assumed to be given by j j j r : ðX Þ! ðX ¼ X ; X ¼ const:Þ; j ¼ 1; ... ; r  1: 3.4. Boundary ports The form l stated in equation (14) is now assumed to equal the natural pairing of the boundary in- and outputs. In contrary to the determination procedure of the collocated output y on the domain, as stated in equation (9), it is no more possible to give a unique separation of the in- and output variables at the boundary visualized by the use of ðu; yÞ and ðy~; u~Þ in the following expression g g l ¼ u y dX ¼ ~y u~ dX: @ g g To overcome this problem we investigate two cases of boundary pairings on vector bundles. 188 M. Schoberl et al. Let us consider the boundary input vector bundle ðU; Z ;@DÞ with local coordinates j g ðX ; u Þ; j ¼ 1; ... ; ðr  1Þ; g ¼ 1; ... ; m  and the basis fe g and its dual – the boundary j g output vector bundle ðY; Z ;@DÞ with local coordinates ðX ; y Þ and basis fe  dXg.We a & make use of the tensor B e  @ and formulate a boundary input map B whichisdefinedby g a & BðuÞ¼ u e cB e  @ g a & a a ¼ u B @ ¼ðx  Þ@ : a a The adjoint map is then clearly given by 1 b & 1 a r  r B ð@ h  Þ¼ B e  @ cð@ h  Þdx ^ dX 0 b 0 a & a 1 a & & ¼ð@ h  ÞB e dX ¼ y e  dX 0 & a & and we obtain l ¼ u y dX as desired. The second pair is given by the boundary input @ g ~ ~ ~ vector bundle ðU; Z ;@DÞ with local coordinates ðX ; u Þ; j ¼ 1; ... ; ðr  1Þ; g ¼ 1; ... ; m ~ g and the basis fe~ g and its dual – the boundary output vector bundle ðY; ~Z ;@DÞ with local j g g a coordinates ðX ; y~ Þ and basis fdX  e~ g. The tensor B dx ^ dX  e~ is used to define the g g boundary input B map to construct g a & ~ ~ Bðu~Þ¼ B dx ^ dX  e~ cu~ e~ g & g a 1 a ¼ u~ B dx ^ dX ¼ð@ h  Þdx ^ dX g 0 a a and consequently the adjoint map is given by a b g a ~ ~ _ _ ~ B ðx  Þ¼ ðx  Þ@ cB dx ^ dX  e b g a g g ¼ðx_  ÞB dX  e~ ¼ y~ dX  e~ : g g a 1 If one vector or form part of l vanishes, that is x_   ¼ 0or @ h   ¼ 0 for a certain @ 0 a, then the corresponding pairing does not represent a port anymore. Now we are able to conclude, that the evolution of the Hamiltonian functional along the solution (here we assume the existence and uniqueness of the solution of the I-pHd systems) of a first order I-pHd system with in- and output operators is determined by the internal damping, the collocation of the in- and output on the domain and boundary and an additional term ar g B u d h dX a 0 g ai g on the boundary due to the application of an input operator Bðu e Þ¼ d ðB u Þ@ . g i a It is worth mentioning, that the adjoint map of the considered input map also becomes a differential operator with a non-trivial kernel. If one applies an input to the systems that leads to a collocated output lying in the kernel of the output map, then this input influences the evolution of the system through the corresponding boundary conditions, that is this input acts similarly to a boundary input. To provide this mathematical construction with a physical example, we investigate the piezo-electric field equations in the derived framework. Mathematical and Computer Modelling of Dynamical Systems 189 4. Application – the piezoelectric field In this contribution, we consider models of linearized elasticity, linearized quasi static electrodynamics combined with nonlinear constitutive relations. Let D denote the domain of the three-dimensional mechanical structure equipped with the Euclidean coordinates (X ), i ¼ 1,2,3, which are used to mark the positions of the mass points. The actual a a i a position of a mass point X is given by u þ d X , where u , a ¼ 1,2,3 are the displacements. The state of the elastic structure, is given by the positions, or equivalently by the a 1 displacements u , and linear momenta p ¼ rd u_ with the mass density 0 < r 2 C ðDÞ. b ba The total manifold E of the state bundle ðE; p; DÞ is equipped with the local coordinates i a ðX ; u ; p Þ; a; g ¼ 1; 2; 3: We assume, that there exists a stored energy density e dX, which meets ab c de ^ dX ¼ðs de  D dE Þ^ dX; b ¼ 1; 2; 3 ð15Þ S ab c with the stress ab s ¼ s @  @ ; a; b ¼ 1; 2; 3;@ ¼ a b b @u the strain a b i k j e ¼ e du  du ; e ¼ u d d þ u d d ; ab ab ar bk 1 b 1 a i j c c the electrical field strength E ¼ E dX and the electric displacement D ¼ D @ cdX. This c c assumption guaranties due to the exactness of de ^ dX that the stored energy is purely defined by the actual state of the system (see e.g. [10]). Here we introduce the nonlinear constitutive equations of the form ab ab s ¼ s ðe; EÞ c c D ¼ D ðe; EÞ Remark 1: A subclass of these equations are the well-known linear constitutive equations of piezoelectric materials given by ab abtd abc s ¼ C e  G E td c c tdc cn D ¼ G e þ F E ; td n abtd abc cn 1 with t, d, n ¼ 1, 2, 3, and C ; G ; F 2 C ðDÞ. These relations supply ab a de ^ dX ¼ s de  D dE ^ dX S ab a abtd abc tdc cn ¼ðC e  G E Þde ðG e þ F E ÞdE ^ dX td c ab td n c and finally 1 1 abtd abc cn de ^ dX ¼ d C e e  G E e  F E E ^ dX S td ab c ab n c 2 2 190 M. Schoberl et al. abtd batd abdt tdab abc bac cn nc if the integrability conditions C ¼ C ¼ C ¼ C , G ¼ G , F ¼ F are met. The kinetic energy density e dX is defined by gZ e dX ¼ p d p dX; Z ¼ 1; 2; 3 K g Z 2r with r 2 C ðDÞ. Finally, we are able to determine the exterior derivative of the Hamiltonian h as the sum of the exterior derivative of the stored and kinetic energy i.e. 1 1 gZ ab i k j c dh ¼ p d dp þ s d u d d þ u d d  D dE : g Z ar bk c 1 b 1 a i j r 2 The electrical field strength is considered as input and the variational derivative of the Hamiltonian density hdX can be rewritten in the form 1 m d ðhdXÞ¼ @ cdh  d @ cdh dx ^ dX; m ¼ 1 .. . 6 m m i m a with x ¼ (u , p ), whereby it is visualized that the exterior derivative of h is sufficient in the determination of the variational derivative. Consequently, there is no need to know the stored energy function – its existence and its exterior derivative already enables the determination of the equations of motion. i a Remark 2: The choice of the coordinates (X ,u ,p ) obviously leads to a Hamiltonian, which contains first-order jet variables. This should be compared with the approach presented in [11] where the authors consider infinite dimensional systems and avoid the use of jet variables in the Hamiltonian, by considering the map J as a differential operator. In the piezoelectric case, this approach leads to the choice of different state variables, namely the strain instead of the displacement. In this case additional partial differential equations appear as restrictions. Therefore, one has to deal with restricted I-pHd systems. The equations of motion are given by "# 1 ab i ba i a ag d s ðe; EÞd þ s ðe; EÞd u_ 0 d 2 b b p_ d 0 ga using the mechanical coordinates (u ,p ). The achieved Hamiltonian representation does not currently qualify as a port Hamiltonian representation, as the input – the electric field strength E – acts on the system in a nonlinear fashion. This problem can be solved by the introduction of new and less general constitutive relations ab ab abc s ðe; EÞ¼ C ðeÞ G E : This restriction leads us to ! "#"# ag ab a i abc u_ d C ðeÞd 0 d i b d G E d i c ¼ þ : p_ 1 d 0 g ag r Mathematical and Computer Modelling of Dynamical Systems 191 Consequently, we obtained the I-pHd structure x ¼ JðÞ dðh dXÞþ BðEÞ where the free Hamiltonian h meets 1 1 gZ ab r i k j dh ¼ p d dp þ C ðeÞd u d d þ u d d 0 g Z ar bk 1 b 1 a i j r 2 and the input map of interest appears in the form of the operator BðEÞ¼ i : abc d d G E d ga i c From these investigations we see that the input map meets the specifications of equation (10). It is worth mentioning that in the case where the piezoelectric material is an insulator d D ¼ 0 has to be met, because the volume charge density has to vanish. This relation has been omitted in the calculations above. Finally, the Lie derivative of the Hamiltonian functional stated in equation (6) leads to Z Z 2 w  2 1 w ðj sÞ BðEÞcd h dx ^ dX þ  ðj sÞ ðv c@ ðh Þdx ^ @ cdXÞ ð16Þ w 0 h 0 i D @D because no dissipative effects have been taken into account. The integral over the domain in equation (16) can be written as Z Z 2  w 2  gbc i ðj sÞ BðEÞcd h dx ^ dX ¼ ðj sÞ d ðG E d Þ p dX : w 0 i c g D D Let us apply the the definition of the adjoint operator from equation (11), which enables us to obtain Z Z 2 w 2 abc i ðj sÞ BðEÞcd h dx ^ dX ¼ ðj sÞ E G d d p dX ; w 0 c i a D D where the adjoint map reads as 1 1 abc i B p dX ¼G d d p ; g i a r r because we use the electric field strength as the distributed input. The boundary expression from the relation (16) is given as 2 vi v ðj sÞ p @ c C ðeÞ dx ^ @ cdX ; v v i @D which represents the boundary ports, where 1 1 vi i i @ ðh Þ¼ @ cðdh Þ¼ cðdh Þ¼ C ðeÞ; w ¼ 1; .. . ; 6; v ¼ 1; .. . ; 3 0 0 0 w w v @u i 192 M. Schoberl et al. is used and an additional boundary term arises due to the application of the adjoint operator, which reads as 2  gbc i v ðj sÞ G E d @ c p dx ^ @ cdX : c g v i @D If p dX is in the kernel of the output map B*(), then the domain port generated by the input map vanishes completely. In the case of piezoelectric systems this is for example given by p ¼ const: In contrary to the domain port, the boundary port generated by the input operator does not vanish and consequently a domain input could act on the system like a boundary input does. These investigations show, that the spatial shape of the distributed input and collocated output is mainly responsible for its appearance within the field equations, boundary conditions and evolution of the free Hamiltonian h . 5. Conclusions Piezoelectric materials enable fascinating new ways of interaction (actuation and sensing) between control equipment and flexible structures. To derive passivity based control strategies a geometric description of the system in a port Hamiltonian setting is of main interest. This contribution introduces a geometrical representation of infinite-dimensional port Hamiltonian systems with in- and output maps using differential operators. It is shown, that the extension of the description shown in [4] results in the appearance of additional boundary conditions in the Lie derivative of the Hamiltonian functional. As the presented approach is a formal one, based on differential geometric considerations, several aspects from functional analysis are missing. For example, Sobolev norms on linear spaces and manifolds have not been introduced, see for example [12,13], also the existence of solutions has not been discussed. The analysis of the piezoelectric field equations on the introduced I-pHd framework yields a very interesting explanation of the frequently used method of ‘‘electrode shaping’’ for piezoelectric devices. The existence of a linear differential input operator enables the use of spatial output distributions such that the output is in the kernel of the domain output operator. Consequently, the distributed domain input acts in a similar fashion on the system as a boundary input. It is obvious that linear input operators of higher order provide more complex output kernels, and consequently, additional degrees of freedom in the application of control action are given. Finally, it is worth mentioning, that the mappings J, < could also be replaced by appropriate differential operators. Such an extension will enable the treatment of coupled field problems like piezo-thermo-elasticity on the basis of port Hamiltonian systems. This represents the content of future investigations. Acknowledgements Partly, this work has been done in the context of the European sponsored project GeoPlex with reference code IST-2001-34166. Further information is available at http://www.geoplex.cc. Mathematical and Computer Modelling of Dynamical Systems 193 References [1] A.J. van der Schaft, L -Gain and Passivity Techniques in Nonlinear Control, Springer, London, [2] A. Macchelli, Port Hamiltonian Systems – A unified approach for modeling and control, Ph.D. Thesis, University of Bologna, Italy, 2002. [3] A.J. van der Schaft and B.M. Maschke, Hamiltonian formulation of distributed-parameter systems with boundary energy flow, J. Geom. Phys. 42 (2002), pp. 166–194. [4] H. Ennsbrunner and K. Schlacher, On the geometrical representation and interconnection of infinite dimensional port controlled Hamiltonian systems. 44th IEEE, Conference on Decision and Control and European Control Conference, Sevilla, Spain, 2005. [5] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986. [6] G. Giachetta, G. Sardanashvily, and L. Mangiarotti, New Lagrangian and Hamiltonian Methods in Field Theory, World Scientific, New York, 1994. [7] D.J. Saunders, The Geometry of Jet Bundles, Cambridge University Press, Cambridge, 1989. [8] W.M. Boothby, An Introduction to Differentiable Manifolds and Riemanian Geometry, Academic Press, Orlando, 1986. [9] J.F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, New York, 1978. [10] A. Kugi, Non-linear Control Based on Physical Models, Springer, London, 2001. [11] A. Macchelli and C. Melchiorri, Control by interconnection and energy shaping of the Timoshenko beam, J. Math. Comput. Model. Dynamical Syst. (MCMDS) 10 (2004), pp. 231– [12] E. Hebey, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Institute of Mathematical Sciences, New York, 2000. [13] E. Zeidler, Applied Functional Analysis, Springer, New York, 1995.

Journal

Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis

Published: Jun 1, 2008

Keywords: infinite dimensional systems; Hamiltonian formulation; differential geometry; differential operators

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