Representation of a general composition of Dirac structures

Transcription

1 Representation of a general composition of Dirac structures Carles Batlle, Imma Massana and Ester Simó Abstract We provide explicit representations for the Dirac structure obtained from an arbitrary number of component Dirac structures coupled by means of another interconnecting Dirac structure Our work generalizes the results in [1] in two aspects First, the interconnecting structure is not limited to the simple feedback case considered there, and this opens new possibilities for designing control systems Second, the number of simultaneously interconnected systems is not limited to two, which allows for extra flexibility in modeling, particularly in the case of electrical networks Several relevant particular cases are presented, and the application to the interconnection of port- Hamiltonian systems is discussed by means of an example I INRODUCION Dirac structures [2][3][4][5] provide a very elegant encoding of energy conservation for interconnected systems In particular, they allow the formulation of generalized port-hamiltonian systems PHS, which have been shown to describe a wide variety of both lumped parameter and distributed parameter systems [6][7][5] PHS are not only useful for modeling and system analysis, but also for control, since passivity-based control techniques with a natural interpretation can be applied to them [8][9][1] Dirac structures and PHS are also the rigorous mathematical framework underlying bond graph theory [11][5] PHS yield themselves to the description of complex systems made of many open subsystems, since it can be shown that the appropriate essentially, power preserving interconnection of PHS is again a PHS However, even if the individual PHS subsystems are given as a set of differential equations, the resulting PHS constitutes, in general, a set of differential-algebraic equations DAE he practical issue of obtaining the actual representation of the resulting PHS, that is, of its Dirac structure, was addressed in [1] for a simple case, and the generalization of the results presented there is the main objective of this work he paper is organized as follows Section II presents the basic definitions concerning Dirac structures and their representations, and how a PHS can be defined from a Dirac structure, and also reviews one of the fundamental results in [1], which provides explicit representations for the Dirac structure resulting from the interconnection of two of them by means of a simple feedback law Sections III, Work partially supported by Spanish CICY project DPI Carles Batlle is with Department of Applied Mathematics 4 and Institute of Industrial and Control Engineering, Universitat Politècnica de Catalunya, EPSEVG, Av V Balaguer s/n 88 Vilanova i la Geltrú, Spain carlesbatlle@upcedu Imma Massana and Ester Simó are with Department of Applied Mathematics 4, Universitat Politècnica de Catalunya, EPSEVG, Av V Balaguer s/n 88 Vilanova i la Geltrú, Spain imma@ma4upcedu, ester@ma4upcedu IV and V contain the original contributions of the paper In Section III a general interconnection of an arbitrary number of Dirac structures by means of another one is introduced, and the resulting object is shown to be a Dirac structure, using the same technique as in [1] In Section IV, explicit kernel and image representations of the resulting structure are constructed his is the main result of the paper, generalizing the special case treated in [1] Section V provides several particular examples of general interest, and the application of the results of the paper to the interconnection of several port- Hamiltonian systems is presented by means of an example in Section VI Finally, Section VII summarizes the results and discusses some of the avenues for future work hroughout the paper vectors are understood as column vectors, and denotes the transpose A R m n means that A is a real matrix with m rows and n columns If A and B are matrices with the same number of rows, expressions like A B denote the matrix obtained by putting them together along the columns N A and RA stand for the kernel and column spaces of matrix A An identity matrix of appropriate dimension is denoted by I, but zero matrices are written simply as x H is a row vector containing the partial derivatives of Hx, while individual partial derivatives are written as xi H II DIRAC SRUCURES Consider F, a finite-dimensional vector space, called the space of flows, and F, its dual space, called the space of efforts We denote by e f the action of the form e F on the vector f F e and f are known as power variables, since e f has dimensions of power 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 1 A constant Dirac structure on F F is a subspace D F F such that D = D, 2 with the orthogonal complement with respect to If the components of vectors and forms are given with respect to dual basis, then e f = e f, and we can write D = { f, ẽ F F ẽ f + e f = f, e D } 3 If follows immediately from the definition of D that, for any f, e D e f =, 4

2 which encodes the energy conservation property of Dirac structures, also known as power continuity In this sense, Dirac structures are a generalization of Kirchoff s laws of circuit theory, which allow to deduce that conforming currents i and voltages v satisfy ellegen s theorem v i = Every Dirac structure D admits a kernel representation D = {f, e F F F f + Ee = }, 5 for linear maps F : F V and E : F V satisfying EF + F E = 6 rank F + E = dim F, 7 where V is a vector space with the same dimension as F, and the adjoint linear maps F : V F and E : V F are defined by F u f = u F f, and e E u = Ee u for all f F, e F, u V Dirac structures can also be given by means of an image representation D = {f, e F F u V st f = E u, e = F u} 8 with the same maps and spaces of the kernel representation If dim V > dim F one has a relaxed kernel representation or a relaxed image representation Given a basis {f 1,, f n } in F, the corresponding dual basis {e 1,, e n } in F, and any basis in V, with dim V = m n, the linear maps F and E are represented by m n matrices which we denote by the same symbol as the maps satisfying EF + F E =, 9 rank F E = n 1 Kernel and image representations are then given by 5 and 8, with the adjoint maps replaced by the corresponding transpose matrices A port-hamiltonian system can be defined from a Dirac structure if some of the flows and efforts are identified with time derivatives and generalized forces of the states of a system Consider a lumped-parameter physical system defined on a manifold M, with local coordinates x R n he total energy of the system is given by the Hamiltonian function Hx, and the system is assumed to have m open ports For each x we consider F x = x M R m and Fx = x M R m, and define a Dirac structure at each x M, Dx F x Fx It is assumed that the Dirac structure varies smoothly over M, which means, essentially, that the matrices of any given representation are smooth functions of x see [3] for more details We will write the elements of F x Fx 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, ie ports connected to energy storing elements A port-hamiltonian system on M can be defined, pointwise in M, by 1 ẋ, fb, x H, e b Dx x M 11 1 See [5] for a global formulation in terms of Whitney sums he minus sign in f x = ẋ is consistent with the common convention that power flows from the boundary ports into the system and from the internal network into the energy storing elements Indeed, using 4, = e f = e x f x + e b f b = x Hẋ + e b f b, 12 from which Ḣ = e b f b Equation 11 is, in general, a set of differential and algebraic equations DAE, and may include the definition of some boundary variables as inputs or outputs Source or dissipative terms can be added to the system through the boundary ports In [3] it was shown that if some of the efforts and flows of two Dirac structures are connected by means of a relationship that is itself a Dirac structure, the resulting subspace is again a Dirac structure However the proof was not constructive and no general explicit representations of the resulting structure were provided In [1] the interconnection of two Dirac structures by means of a simple feedback relationship is considered, and it is shown, using a different approach, that the result is again a Dirac structure, for which explicit kernel/image representations are computed he remaining of this Section provides a summary of the relevant results in [1] Given Dirac structures D A on the flow space F A = F A F AI and D B on F B = F B F BI, with dim F AI = dim F BI, a feedback interconnection is established by means of f AI = f BI, f AI F AI, f BI F BI, e AI = e BI, e AI F AI, e BI F BI 13 his induces a feedback composition of D A and D B given by D A D B = {f A, e A, f B, e B F A F A F B F B f I, e I such that f A, e A, f I, e I D A and f B, e B, f I, e I D B } 14 he following result is then proven [1], heorem 3: heorem 21: D A D B is a Dirac structure he proof takes advantage of the following linear algebra result, which will be also used in this paper Lemma 22: For any matrix A R m n and vector b R m one has λ R n such that Aλ = b α R m such that A α = α b = 15 he non-trivial part of Lemma 22 follows from the orthogonal decomposition RA N A = R m, 16 valid for any matrix A R m n Furthermore, explicit kernel/image representations for D = D A D B can be constructed If D A and D B have

3 kernel/image representations given by F A, E A = F A F AI, E A E AI, 17 F B, E B = F B F BI, E B E BI, 18 respectively, then [1], theorem 4 a kernel/image representation of D is given by the matrices F = L A F A L B F B, E = L A E A L B E B, 19 where L = L A L B, with L any matrix of maximum rank such that N L = RM and 2 EAI M = E BI F AI F BI 2 III GENERAL COMPOSIION OF DIRAC SRUCURES Consider N Dirac structures D i given in relaxed image representation f i Ei e i f ii = F i EiI λ i, i = 1,, N 21 e ii FiI where each structure has internal or maybe open port flow-effort pairs f i, e i, f i, e i R ni, and a number of interconnected port flow-effort pairs f ii, e ii, f ii, e ii R mi, and where λ i R ri with r i n i + m i he matrices E i, F i R ri ni and E ii, F ii R ri mi satisfy and rang F i F ii E i E ii = n i + m i, 22 E i E ii F i F ii + F i F ii E i E ii = 23 Furthermore, we consider an interconnecting Dirac structure D C given in relaxed kernel form as f 1I F C1 F CN +E C1 E CN = f NI with matrices F Ci, E Ci R r C m i satisfying e NI 24 rang F C1 F CN E C1 E CN = m, 25 where m = N i=1 m i, and E C1 E CN F C1 F CN + F C1 F CN E C1 E CN = 26 Associated to this kernel representation of D C there is also an image one, given by f 1I E C1 F C1 = γ, 27 f NI ECN e NI FCN 2 his M differs from that in [1], but it spans the same column space, which is all that matters with γ R r C Definition he 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 1I,,, f NI, e NI D C such that f i, f ii, e i, e ii D i, i = 1,, N heorem 31: D C N i=1 D i is a Dirac structure Proof: By combining the individual image representations 21 and the kernel representation for the interconnection 24, and reordering the resulting matrices, one gets FC1 E1I + E C1F1I F CN ENI + E CN FNI λ =, 28 where λ = λ 1 λn R r, and where we have defined r = N i=1 r i his can be combined with the equations from 21 for the non-interconnecting variables to get that, for any f 1, e 1,, f N, e N D C N i=1 D i, there exist λ 1,, λ N such that where f 1 e 1 = Kλ 29 f N e N E1 F1 K = EN FN F C1 E1I + E C1F1I F CN ENI + E CN FNI 3 Using Lemma 22, this is equivalent to the implication that, for any β 1, α 1,, β N, α N, γ such that one has K β 1 α 1 β N α N γ = 31 β 1 f 1 + α 1 e β Nf N + α Ne N = 32 aking into account the image representation 27 for the interconnecting Dirac structure, the set of equations 31 can be rewritten as E 1 β 1 + F 1 α 1 + E 1I + F 1I f 1I =, E N β N + F N α N + E NI e NI + F NI f NI =

4 But these are the kernel representations of the D i, with β i the non-interconnecting efforts and α i the non-interconnecting flows, and with the f 1I,,, f NI, e NI belonging to D C, and hence α 1, β 1,, α N, β N D C N i=1 D i hus, we conclude that, if f 1, e 1,, f N, e N D C N i=1 D i, then, for any α 1, β 1,, α N, β N D C N i=1 D i one has β1 f 1 + α1 e βn f N + αn e N =, ie any element of D C N i=1 D i belongs to D C N i=1 i D he chain of arguments can be easily reversed to show also that any element of D C N i=1 D i is also of DC N i=1 D i, and this concludes the proof IV KERNEL AND IMAGE REPRESENAION OF HE COMPOSED SRUCURE with We rewrite 28 as and where M λ =, 33 M = M 1 M i M N R r C r, 34 M i = F Ci E ii + E Ci F ii, i = 1,, N 35 Equation 33 implies that λ N M which, according to 16, is equivalent to λ RM 36 Now we choose an r L r matrix L such that or, equivalently, RM = N L 37 RL = N M 38 his implies that L is such that LM = or, equivalently, M L = Combining 36 and 37 one gets that λ N L or λ RL 39 his in turn implies that there exists µ R r L such that λ = L µ 4 Notice that, since the r L columns of L must expand the kernel of M, one must have r L dim N M 41 Plugging in 4 into the internal part of the relations 21 one gets, with a common µ, fi E = i E e i Fi λ i = i Fi L i µ E = i L i µ, i = 1,, N, 42 F i L i where L i contains the rows of L corresponding to λ i Putting together all the relations in 42 yields f 1 e 1 f N e N = E 1 L 1 F 1 L 1 E N L N F N L N µ 43 From this an image representation of the interconnected structure can be read off, with E = L 1 E 1 L N E N, 44 F = L 1 F 1 L N F N 45 Equations 34, 35, 38, 44 and 45 constitute an algorithm to compute a representation of the composed Dirac structure, and it can be easily implemented by using, for instance, standard linear algebra MALAB functions V SOME BASIC INERCONNECIONS A General feedback interconnection of two structures Consider two Dirac structures D 1 and D 2 with external ports f 1I, R m1, f 2I, e 2I R m2, which are interconnected by means of a general static feedback Figure 1 = Ke 2I, 46 f 2I = K f 1I, 47 where K R m1 m2 hese relations can be put as a Dirac structure in kernel form K I f1i e1i + =, 48 f 2I I K e 2I with r C = m 1 + m 2, from which F C1, F C2, E C1 and E C2 can be read K I F C1 =, F C2 =, E C1 =, E I C2 = K 49 Using then 35 and 34 the matrix M whose kernel is spanned by the columns of L can be obtained, and one finally gets E1I K F M = 1I F 2I K, 5 E 2I which coincides with 2 for K = I B -junction parallel interconnection of 3 structures Consider 3 Dirac structures D 1, D 2, D 3, with a common number m 1 of ports connected in parallel f 1I + f 2I + f 3I =, = e 2I = e 3I, 51 with e ii, f ii R m1 i = 1, 2, 3 A kernel form is given by f 1I f 2I e 2I = f 3I 1 1 e 3I 52

5 K f 2I f 1I 2 1 e 2I K for subsystem 1, ie F 1R = L 1 F 1, E 1R = L 1 E 1 Equation 58, giving L 2, is irrelevant in our case since D 2 has no remaining Dirac structure after the interconnection VI COMPOSIION OF PHS: AN EXAMPLE Consider the three circuits displayed in Figure 2 e 2I e 2I 1 F 2 f 1I K f 2I f 2I Fig 1 Block diagram above and bond graph below of a general feedback interconnection i C1 v C1 i L1 v L1 i 1 v 1 Matrices F Ci and E Ci can be then read off and M 1 = and finally E 1I F 1I, M2 = E 2I F 2I F 2I, M 3 = E 3I, F3I 53 M = E 1I F 1I E 2I F 2I F 2I 54 E 3I F 3I C Effort-constraint reduction of PHS Several methods to truncate the states of a PHS so that the reduced system is again a PHS are presented in [12] One of them, called effort-constraint reduction, sets to zero the generalized forces associated to the states that are going to be neglected, so that power does not flow due to the variation of those states, and, in order to avoid an inconsistency, the corresponding efforts of the underlying Dirac structure are also set to zero, which is the situation that we will study in our framework Consider again the interconnection in Figure 1 but with K = I, so that m 2 = m 1, and with system 2 having only interconnecting ports with port variables f 2I R m1, e 2I R m1 satisfying the trivial Dirac structure D 2 = {f 2I, e 2I e 2I = }, 55 which, in electrical terms, corresponds to a shorted circuit he kernel representation of D 2 is given by F 2I f 2I + E 2I e 2I =, with F 2I = and E 2I = I Using 5 with K = I, one immediately gets see also [5] M = E1I F 1I I 56 Solving LM = with L = L 1 L 2 yields L1 E 1I + L 2 =, L 1 F 1I =, or L 1 F 1I =, 57 L 2 = L 1 E 1I 58 Equation 57, which corresponds to equation 313 in [12], is the key condition that must be satisfied by the matrix L 1 which allows the computation of the reduced Dirac structure i 2 v 2 a A circuit with 2 state ports and one boundary port i C2 i 3 v C2 v 3 b A circuit with one state port and two boundary ports i 4 v 4 i 5 v 5 c A system with just two boundary ports Fig 2 hree circuits to be connected in parallel through ports 1, 2 and 4, leaving 3 and 5 open he third circuit is just a stub, introduced in order to get an open port at the point of connection of the other two circuits Kernel representations they are not unique are given by i C1 1 1 v C1 1 v L1 + 1 i L1 =, 1 1 i 1 1 v i C2 v C2 i v 2 =, i v i4 v4 + = 61 i v 5 Connecting the three circuits in parallel at the ports i 1, v 1, i 2, v 2, i 4, v 4 separates the ports into internal or open ports on one side and interconnected ports on the other, and from the above kernel representations one immediately reads F 1, F 1I, E 1, E 1I, F 2, F 2I, E 2, E 2I, F 3, F 3I, E 3 and E 3I For instance F 1I =, E 2 = 1 1, E 3 = From these and 54 M can be computed M =

6 Using, for instance, the MALAB command null with the r option, a matrix L such that M L = can be obtained L = From L and E i, F i, i = 1, 2, 3, one can compute the matrices of the kernel representation of the composed structure 1 F = , E = he kernel representation F f + Ee =, with f = i C1 v L1 i C2 i 3 i 5 and e = v C1 i L1 v C2 v 3 v 5, yields the network equations of the composed structure v C1 + v L1 =, v C2 v 3 =, v C1 + v C2 =, i L1 i C1 i C2 + i 3 + i 5 =, v C1 v 5 = 67 Finally, putting i C1 = q 1, i C2 = q 2, v L1 = λ, v C1 = q1 H, v C2 = q2 H, and i L1 = λ H, with H = Hq 1, q 2, λ, yields the DAE of the interconnected system, with i 3 and i 5 as inputs and v 3 and v 5 as outputs: q 1 + q 2 = λ H + i 3 + i 5, λ = q1 H, q1 H = q2 H, v 3 = q2 H, v 5 = q1 H 68 VII CONCLUSIONS An algorithm to compute kernel/image representations for the Dirac structure resulting from the interconnection of several Dirac structures has been provided, generalizing a simple special case previously published in the literature As a byproduct, an new proof of the basic result concerning the interconnection of Dirac structures has been obtained Several particular but relevant cases have been worked out in detail, and an example of application to the interconnection of PHS has been presented Besides the immediate application to the modeling of complex systems, the results in the present paper can be useful in several related fields, such as optimal control and model order reduction of PHS systems [12], using the techniques developed for general nonlinear DAE systems [13][14][15][16] Also, from the point of view of the theory of Dirac structures and control by interconnection, some of the results in [1][5] about the achievability of a closedloop Dirac structure could be further investigated using the generalized framework presented here REFERENCES [1] J Cervera, A van der Schaft, and A Baños, Interconnection of port- Hamiltonian systems and composition of Dirac structures, Automatica, vol 43, no 2, pp , 27 [2] B Maschke and A van der Schaft, Port-controlled hamiltonian systems: modelling origins and system theoretic properties, in Proceedings 2nd IFAC Symposium on Nonlinear Control Systems NOLCOS 1992, M Fliess, Ed, Bordeaux, France, 1992, pp [3] M Dalsmo and A van der Schaft, On representations and integrability of mathematical structures in energy-conserving physical systems, SIAM Journal on Control and Optimization, vol 37, no 1, pp 54 91, 1998 [4] H Yoshimura and J E Marsden, Dirac structures in Lagrangian mechanics Part I: Implicit Lagrangian systems, J of Geometry and Physics, vol 57, no 1, pp , Dec 26 [5] V Duindam, A Macchelli, S Stramigioli, and H Bruyninckx, Modeling and Control of Complex Physical Systems: he Port-Hamiltonian Approach, V Duindam, A Macchelli, S Stramigioli, and H Bruyninckx, Eds Springer, 29 [6] A van der Schaft and B Maschke, Hamiltonian formulation of distributed-parameter systems with boundary energy flow, Journal of Geometry and Physics, vol 42, no 1-2, pp , May 22 [7] G Golo, V alasila, A van der Schaft, and B Maschke, Hamiltonian discretization of boundary control systems, Automatica, vol 4, no 5, pp , May 24 [8] R Ortega, A van der Schaft, I Mareels, and B Maschke, Putting energy back in control, Control Systems Magazine, IEEE, vol 21, no 2, pp 18 33, Apr 21 [9] R Ortega, A van der Schaft, B Maschke, and G Escobar, Interconnection and damping assignment passivity-based control of portcontrolled hamiltonian systems, Automatica, vol 38, no 4, pp , 22 [1] C Batlle, A Dòria-Cerezo, G Espinosa-Pérez, and R Ortega, Simultaneous interconnection and damping assignment passivity-based control: the induction machine case study, International Journal of Control, vol 82, no 2, pp , 29 [11] B Maschke, A van der Schaft, and P Breedveld, An intrinsic Hamiltonian formulation of the dynamics of LC-circuits, Circuits and Systems I: Fundamental heory and Applications, IEEE ransactions on, vol 42, no 2, pp 73 82, Feb 1995 [12] R Polyuga, Model reduction of port-hamiltonian systems, PhD dissertation, University of Groningen, he Netherlands, 21 [Online] Available: [13] J Sjöberg and Glad, Computing the controllability function for nonlinear descriptor systems, in Proceedings of the 26 American Control Conference, 26, pp [14] J Sjöberg, K Fujimoto, and Glad, Model reduction of nonlinear differential-algebraic equations, in Proceedings of the 7th IFAC Symposium on Nonlinear Control Systems NOLCOS IFAC, 27, pp [15] J Sjöberg, Optimal control and model reduction of nonlinear DAE models, PhD dissertation, Linköping University, Sweden, 28 [Online] Available: johans/files/phdpdf [16] K Fujimoto and J M A Scherpen, Balanced realization and model order reduction for nonlinear systems based on singular value analysis, SIAM Journal on Control and Optimization, vol 48, no 7, pp , 21