Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions Representation of a general composition of Dirac structures Carles Batlle, Imma Massana, Ester Sim´ o Universitat Polit` ecnica de Catalunya IEEE CDC-ECC 2011, Orlando, FL, December 11-15 2011 1 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions Introduction. 1 Dirac structures and port-Hamiltonian systems. 2 General composition of Dirac structures. Kernel and image 3 representations of the composed structure. Examples. 4 Conclusions. 5 2 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions Port-Hamiltonian systems Port-Hamiltonian systems in explicit form (explicit PHS) J ( x ) ∂ x H T + g ( x ) u, x ˙ = g ( x ) T ∂ x H T , = y where x ∈ R n is the state , J T = − J is the interconnection matrix, g is the port matrix and H is the energy function, describe a class of control systems in many areas of interest. Interconnecting several explicit PHS by identifying some of their input/output variables u and y does not, in general, yield again an explicit PHS, but a system of DAE (differential-algebraic equations). 3 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions Dirac structures This resulting DAE is, however a generalized PHS, and it can be described in terms of an underlying mathematical structure, known as a Dirac structure. Circuit theory provides the paradigmatic example of Dirac structure by means of Kirchhoff laws v ∈ R ( D T ) i ∈ N ( D ) , where D is the incidence matrix of the digraph associated with the circuit. From these, Tellegen’s theorem v T i = 0 , stating the power-preserving nature of the interconnection, can be deduced. 4 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions Main contribution of the paper The suitable interconnection of Dirac structures yields again a Dirac structure (ditto for generalized PHS) (Dalsmo & van der Schaft, 1998), but the practical matter of explicitly constructing the resulting Dirac structure has only been addressed recently (Cervera, van der Schaft & Ba˜ nos, 2007) for a simple feedback interconnection of two systems. The main contribution of the present work is the explicit construction of the image/kernel representation of the Dirac structure resulting from an arbitrary interconnection (by means of an interconnecting Dirac structure) of any number of Dirac structures. 5 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions Dirac structures F , a finite-dimensional vector space, the space of flows , and F ∗ , its dual space, the space of efforts . � e | f � , which has dimensions of power, denotes the action of the form e ∈ F ∗ on the vector f ∈ F . An indefinite, non-degenerate symmetric bilinear form can be defined on F × F ∗ by means of ⋖ ( f a , e a ) | ( f b , e b ) ⋗ = � e a | f b � + � e b | f a � . A constant Dirac structure on D ⊂ F × F ∗ is a subspace such that D = D ⊥ , with ⊥ the orthogonal complement with respect to ⋖ | ⋗ . 6 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions If the components of vectors and forms are given with respect to dual basis, then � e | f � = e T f , and then D ⊥ = � e ) ∈ F × F ∗ | ˜ e T f + e T ˜ � ( ˜ f = 0 ∀ ( f, e ) ∈ D f, ˜ . If follows from the definition of D that, for any ( f, e ) ∈ D � e | f � = 0 , which encodes the energy conservation property of Dirac structures, also known as power continuity , and which generalizes Tellegen’s theorem from circuit theory. 7 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions Every Dirac structure D admits a kernel representation D = { ( f, e ) ∈ F × F ∗ | Ff + Ee = 0 } , for linear maps F : F → V and E : F ∗ → V satisfying EF ∗ + FE ∗ = 0 with rank ( F + E ) = dim F , where V is a vector space with dim V ≥ dim F , and F ∗ : V ∗ → F ∗ and E ∗ : V ∗ → F denote the adjoint maps, which are represented, for given basis, by the transposed matrix of the corresponding map. Dirac structures can also be given by means of an image representation D = { ( f, e ) ∈ F ×F ∗ | ∃ u ∈ V ∗ s.t. f = E ∗ u, e = F ∗ u } (1) with the same maps and spaces of the kernel representation. 8 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions Generalized PHS Consider a lumped-parameter physical system defined on a manifold M , with local coordinates x ∈ R n . The total energy of the system is given by the Hamiltonian function H ( x ) , and the system is assumed to have m open ports. For each x we consider F x = T x M × R m and F ∗ x = T ∗ x M × R m , and define a Dirac structure at each x ∈ M , D ( x ) ⊂ F x × F ∗ x . We will write the elements of F x × F ∗ x as ( f x , f b , e x , e b ) , where b stands for boundary, and the f x and e x are power variables associated to state ports, i.e. ports connected to energy storing elements. 9 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions A port-Hamiltonian system on M can be defined by x, f b , ∂ T � � − ˙ ∈ D ( x ) ∀ x ∈ M . x H, e b In a kernel representation one gets � − ˙ � � ∂ T � x x H F ( x ) + E ( x ) = 0 . f b e b This is, in general, a set of differential and algebraic equations (DAE), and may include the definition of some boundary variables as inputs or outputs. 10 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions A sufficient (although not necessary) condition to get an explicit PHS is that F has full rank. In this case the f b are the outputs of the system while the e b are the inputs. The minus sign in f x = − ˙ x is introduced in order to get ˙ x = e T H = − ∂ x H ˙ b f b . Source or dissipative terms can be added to the system through some of the boundary ports. 11 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions Composition of Dirac structures Consider N Dirac structures D i given in kernel/image representation by matrices ˆ ˆ E i = ( E i E iI ) , F i = ( F i F iI ) where the i columns correspond to internal (or maybe open port) flow-effort pairs ( f i , e i ) , f i , e i ∈ R n i , while the iI ones are those of the flow-effort pairs ( f iI , e iI ) , f iI , e iI ∈ R m i that are to be interconnected. We now compose these N Dirac structures by means of an interconnecting Dirac structure, with matrices E C = ( E C 1 · · · E CN ) , F C = ( F C 1 · · · F CN ) , where the columns have been split according to the subsystem to which they are to be attached. 12 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions The set ( D C ) N i =1 D i is made up of those ( f 1 , e 1 , . . . , f N , e N ) for which there exist interconnecting variables ( f 1 I , e 1 I , . . . , f NI , e NI ) ∈ D C such that ( f i , f iI , e i , e iI ) ∈ D i , ∀ i = 1 , . . . , N . The following result can then be proved ( D C ) N i =1 D i is a Dirac structure This result was established in (Dalsmo & van der Schaft, 1998), and proved again, for a simple feedback interconnection, in (Cervera, van der Schaft & Ba˜ nos, 2007) using different techniques, which have been generalized in the present work. 13 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions Kernel and image representations From the matrices corresponding to the interconnected ports of each subsystem and the matrices of the interconnecting Dirac structure one can construct M T i = F Ci E T iI + E Ci F T iI , i = 1 , . . . , N, and M T = ( M T · · · M T · · · M T N ) . 1 i Now choose a matrix L such that R ( M ) = N ( L ) . This is equivalent to LM = 0 , or M T L T = 0 . 14 / 19 IEEE CDC-ECC 2011
Introduction Dirac structures and port-Hamiltonian systems General composition of Dirac structures Examples Conclusions From the columns of L corresponding to each subsystem and the columns of the matrices of the individual subsystems corresponding to the ports that are not connected one can construct E = ( L 1 E 1 · · · L N E N ) , ( L 1 F 1 · · · L N F N ) . F = Then the main result of this paper can be proved: E , F provide a kernel/image representation for the Dirac structure of the interconnected system. The proof follows that in (Cervera, van der Schaft & Ba˜ nos, 2007), with the appropriate generalizations due to the fact that more than two subsystems are considered, and with an arbitrary interconnection instead of a simple feedback one. 15 / 19 IEEE CDC-ECC 2011
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