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1 University of Groningen Port Hamiltonian Formulation of Infinite Dimensional Systems II. Boundary Control by Interonnetion Mahelli, Alessandro; van der Shaft, Abraham; Melhiorri, Claudio Published in: EPRINTS-BOOK-TITLE IMPORTANT NOTE: You are advised to onsult the publisher's version (publisher's PDF) if you wish to ite from it. Please hek the doument version below. Doument Version Publisher's PDF, also known as Version of reord Publiation date: 24 Link to publiation in University of Groningen/UMCG researh database Citation for published version (APA): Mahelli, A., Shaft, A. J. V. D., & Melhiorri, C. (24). Port Hamiltonian Formulation of Infinite Dimensional Systems II. Boundary Control by Interonnetion. In EPRINTS-BOOK-TITLE University of Groningen, Johann Bernoulli Institute for Mathematis and Computer Siene. Copyright Other than for stritly personal use, it is not permitted to download or to forward/distribute the text or part of it without the onsent of the author(s) and/or opyright holder(s), unless the work is under an open ontent liense (like Creative Commons). Take-down poliy If you believe that this doument breahes opyright please ontat us providing details, and we will remove aess to the work immediately and investigate your laim. Downloaded from the University of Groningen/UMCG researh database (Pure): For tehnial reasons the number of authors shown on this over page is limited to 1 maximum. Download date:

2 Port Hamiltonian Formulation of Infinite Dimensional Systems II. Boundary Control by Interonnetion Alessandro Mahelli, Arjan J. van der Shaft and Claudio Melhiorri Abstrat In this paper, some new results onerning the boundary ontrol of distributed parameter systems in port Hamiltonian form are presented. The lassial finite dimensional port Hamiltonian formulation of a dynamial system has been generalized to the distributed parameter and multivariable ase by extending the notion of finite dimensional Dira struture in order to deal with an infinite dimensional spae of power variables. Consequently, it seem natural that also finite dimensional ontrol methodologies developed for finite dimensional port Hamiltonian systems an be extended in order to ope with infinite dimensional systems. In this paper, the ontrol by interonnetion and energy shaping methodology is applied to the stabilization problem of a distributed parameter system by means of a finite dimensional ontroller. The key point is the generalization of the definition of Casimir funtion to the hybrid ase, that is the dynamial system to be onsidered results from the power onserving interonnetion of an infinite and a finite dimensional part. A simple appliation onerning the stabilization of the onedimensional heat equation is presented. I. INTRODUCTION The port Hamiltonian representation of a finite dimensional system 1, 2 has been reently generalized to the infinite dimensional ase, 3, 4 by extending the notion of Dira struture in order to ope with an infinite dimensional spae of power variables. From the modeling point of view, the port Hamiltonian formulation of an infinite dimensional system with spatial domain provides a deep insight of the struture of the system, whose dynamis an be interpreted as the result of the interation among (at least) two energy domains within and/or between the system and its environment through. From the ontrol perspetive, one of the main advantages in adopting the port Hamiltonian approah in both the finite either the infinite dimensional ase is that the energy (Hamiltonian) funtion, whih is usually a good Lyapunov funtion, expliitly appears in the dynamis of the system. Given a desired state of equilibrium, if the Hamiltonian of the system assumes its minimum at this onfiguration, then asymptoti stability an be assured by introduing a dissipative effet with the ontroller. In this way, energy dereases until the minimum of energy or, equivalently, the This work has been done in the ontext of the European sponsored projet GeoPlex with referene ode IST Further information is available at A. Mahelli and C. Melhiorri are with CASY DEIS, University of Bologna, viale Risorgimento 2, 4136 Bologna, Italy {amahelli, melhiorri}@deis.unibo.it A. J. van der Shaft is with the Department of Applied Mathematis, University of Twente, 75 AE Enshede, The Netherlands a.j.vandershaft@math.utwente.nl desired equilibrium onfiguration is reahed. This ontrol methodology is alled ontrol by damping injetion, 5, 2. On the other hand, if the Hamiltonian funtion of the system does not assume its minimum in the desired equilibrium state, it is neessary to shape the open-loop energy funtion and to introdue a new minimum in the desired onfiguration. The idea is to interonnet a ontroller to the plant and to hoose the Hamiltonian of the regulator in order to properly shape the total (losed-loop) energy funtion. It is important to note that, in general, there is no a priori relation between the state of the plant and the state of the ontroller, so it is not immediate how the ontroller Hamiltonian funtion an be hosen in order to orretly shape the total energy. This problem an be solved by hoosing the struture of the ontroller, i.e. its interonnetion, damping and input/output matries, in suh a way that the state of the losed-loop system is onstrained on ertain subspae independently of the energy funtion of both the plant either the ontroller. Equivalently, this an be done by introduing a set of Casimir funtions in the system, 6, 7. Under some tehnial hypothesis, then, it is possible to introdue an intrinsi non-linear state feedbak law that will be used in order to hoose the energy funtion of the ontroller so that the losedloop Hamiltonian an be properly shaped. Note that, under these hypothesis, this energy funtion depends on the state variables of the plant. This ontrol methodology is alled invariant funtion method or, within the framework of port Hamiltonian systems, ontrol by interonnetion and energy shaping and it is deeply disussed in 6, 7 and also in 8, 9 for the stabilization of non-linear port Hamiltonian systems. In this paper, the ontrol by interonnetion and energy shaping is extended and applied to the regulation problem of an infinite dimensional system by means of a finite dimensional ontroller that an at on the system by exhanging power through the boundary. Some preliminary results in this diretion have already been presented in 1, 11 where the infinite dimensional system is given by a set of transmission lines, while an appliation to stabilization of the Timoshenko beam has been disussed in 12, 13. The main result onerns the neessary and suffiient onditions for a real-valued funtion defined over the losed-loop state spae to be a strutural invariant (Casimir funtion) for the ontrolled system whih is an hybrid system sine it results from the power onserving interonnetion of an infinite and of a finite dimensional system. One these

3 onditions are dedued, by hoosing a proper family of Casimir funtions, the ontrol by interonnetion and energy shaping methodology an be applied as in the finite dimensional ase. In this way, the open-loop energy funtion an be shaped by introduing a new minimum at the desired equilibrium onfiguration. This paper is organized as follows. In Set. II, a short introdution about the ontrol by interonnetion and energy shaping for finite dimensional port Hamiltonian systems is given and then the boundary ontrol by interonnetion for infinite dimensional systems is disussed in Set. III. Neessary and suffiient onditions for the existene of Casimir funtions in the losed loop system are dedued and their appliations in the energy shaping proedure is desribed. Finally, a simple example onerning the boundary stabilization of the heat equation is disussed in Set. IV, while onlusions are presented in Set. V. II. CONTROL BY INTERCONNECTION IN FINITE DIMENSIONS Denote by X an n-dimensional spae of state (energy) variables and by H : X R a salar energy funtion (Hamiltonian) bounded from below. Denote by U an m- dimensional (linear) spae of input variables and by its dual Y U the spae of output variables. Then, ẋ = J(x) R(x) H + G(x)u y = G T (x) H (1) with J(x) = J T (x), R(x) = R T (x) and G(x) matries of proper dimensions, is a port Hamiltonian system with dissipation. The n n matries J and R are alled interonnetion and damping matrix respetively. Suppose (1) has to be asymptotially stabilized around the onfiguration x X by means of the following dynamial ontroller in port Hamiltonian form: ẋ = J (x ) R (x ) H + G (x )u y = G T (x ) H (2) Denote by X the ontroller state spae, with dim X = n, and by H : X R the Hamiltonian funtion, bounded from below. Moreover, suppose that J (x ) = J T (x ) and R (x ) = R T (x ) and that dim U = dim Y = m. If systems (1) and (2) are interonneted in power onserving way, that is if { u = y y = u (3) the resulting dynamis is given by the following autonomous port Hamiltonian systems, with state spae X X and Hamiltonian H + H : ẋ J(x) R(x) G(x)G T = (x ) H ẋ G (x )G T (4) (x) J (x ) R (x ) x H Given a generi port Hamiltonian system, it is possible to give the following fundamental definition of strutural invariant or, equivalently, of Casimir funtion, 6, 7, 2. Definition 2.1 (Casimir funtion): Consider the port Hamiltonian system (1) with state spae X and Hamiltonian funtion H : X R. A funtion C : X R is a Casimir funtion for (1) if and only if C = for every possible hoie of Hamiltonian H. From Def. 2.1, a salar funtion C : X X R is a Casimir funtion for (4) if and only if the following relations are satisfied: T C (J R) + T C G G T = (5a) T C (J R ) T C GGT = (5b) These onditions are diret onsequene of the interonnetion law (3). The existene of Casimir funtions for the losed-loop system (4) plays an important role in the ontrol by interonnetion and energy shaping methodology. If x X is the desired equilibrium onfiguration for (1), asymptoti stability in x an be ahieved by properly hoosing the Hamiltonian funtion of (2) in order to shape the losedloop energy H + H so that a (possibly) global minimum in the desired equilibrium onfiguration an be introdued. It is important to note that there is no relation between the state of the ontroller and the state of the system to be ontrolled. Then, it is not lear how the ontroller energy, whih is freely assignable, has to be hosen in order to solve the regulation problem. A possible solution an be to onstrain the state of the losed-loop system (4) on a ertain subspae of X X, for example given by: Ω := {(x, x ) X X x = S(x) + } where R n and S : X X is a funtion to be omputed. In other words, we are looking for a set of Casimir funtions C i : X X R, i = 1,..., n for the losed-loop system (4) suh that C i (x, x ) := S i (x) x,i (6) where S 1 (x),..., S n (x) T = S(x). Due to the nature of a Casimir funtion, it is possible to introdue an intrinsi non-linear state feedbak law that will be used in order to hoose the energy funtion of the ontroller so that the losed-loop Hamiltonian an be properly shaped. Note that, under these hypothesis, this energy funtion depends on the state variables of system (1). This ontrol methodology is alled invariant funtion method, 6, 7. From (5), the set of funtions (6) are Casimir funtions for (4) if and only if T S T S GGT = J R (J R) = G G T Then, the following proposition an be proved, 9, 2.

4 Proposition 2.1: The funtions C i, i = 1,..., n, defined in (6) are Casimir funtions for the system (4) if and only if the following onditions are satisfied: T S J(x) S = J (x ) (7a) R(x) S = (7b) R (x ) = (7) T S J(x) = G (x )G T (x) (7d) Suppose that (7) are satisfied. Then, from (6), the state variables of the ontroller are robustly related to the state variable of the system to be stabilized sine x,i = S i (x) + i, i = 1,..., n (8) with i R depending on the initial onditions. Moreover, the losed-loop dynamis (4) evolves on the foliation indued by the level sets L i C i = {(x, x ) X X x,i = S i (x) + i } (9) with i = 1,..., n, whih an be expressed as a funtion of the x oordinate. If onditions (7b) and (7d) are taken into aount, the redued dynamis of (4) on these level sets is given by ẋ = J(x) R(x) H G(x)GT (x ) H ( H = J(x) R(x) + S ) H (1) From (8), we have that H (x ) H (S(x) + ): the ontroller energy funtion is finally dependent from x b through the non-linear feedbak ation S( ). If H d (x) := H(x) + H (S(x) + ) (11) then (1) an be written as ( H ẋ = (J R) + S ) H = (J R) H d (12) In onlusion, the following proposition has been proved, 9, 2. Proposition 2.2: Consider the losed-loop port Hamiltonian system (4) and suppose that the vetor funtion S(x) = S 1 (x),..., S n (x) T satisfies onditions (7). Then, the redued dynamis on the level sets (9) is given by (12), where the losed-loop energy funtion H d is given by (11). By properly hoosing the ontroller energy funtion H, it is possible to shape the losed-loop energy funtion H d defined in (11) so that a new minimum in x is introdued. Then, the desired onfiguration an be reahed with the dynamis given by (12). III. BOUNDARY CONTROL BY INTERCONNECTION OF MDPH SYSTEMS A. Introdution In this setion, the ontrol by interonnetion and energy shaping, disussed in Set. II for the finite dimensional ase, is generalized to distributed parameter systems in port Hamiltonian form. In partiular, it is shown how it is possible to shape the open loop energy funtion of a distributed parameter system by interonneting a finite dimensional ontroller to its boundary. The struture of the ontroller has to be hosen so that a proper set of strutural invariants (Casimir funtions) are introdued in the losed loop system. In this way, the energy variables of the distributed parameter system an be robustly related to the state variables of the ontroller, thus introduing an impliit state feedbak law. Then, same proedure presented for the finite dimensional ase an be applied. B. Existene of Casimir funtions Consider the following multi-variable distributed port Hamiltonian system with spatial domain R d (losed and ompat), 3: = (J R) δ x H t (13) w = B (δ x H) where x X is the onfiguration variable, w W are the boundary terms defined by the boundary operator B, H : X R is the Hamiltonian funtion, J is a skew adjoint differential operator and R is a self-adjoint differential operator taking into aount the dissipative effets. Both X either W are spaes of vetor value smooth funtions of proper dimension. It is possible to prove that the following energy balane relation holds, 3: dh dt 1 2 B {J, R} (w, w) da (14) where the integral over represent the power exhanged with the environment through the boundary and the B {J, R} is a onstant operator depending on the differential operators J and R. See 3 for more details. Suppose that (13) has to be stabilized in the onfiguration x X by means of the finite dimensional ontroller (2) that has to be interonneted to the system (13) in power onserving way. Then, relation (3) has to be generalized in order to deal with a situation in whih the power port of the system to be stabilized is not a finite dimensional vetor spae. A possible solution an be the following. Denote by Ψ u (z) and Ψ y (z) a ouple of matries depending eventually on z and suppose that it is possible to write the boundary terms in (13) as follows: w = Ψ y u Ψ u y (15)

5 The interonnetion law expressed in (15) is power onserving if and only if y T u + 1 B {J, R} (w, w) da = 2 where, from (15), we have that B {J, R} (w, w) da = = m i,j=1 + 2 m i,j=1 B {J, R} (Ψ y,i, Ψ y,j ) da u,i u,j m i,j=1 B {J, R} (Ψ u,i, Ψ u,j ) da y,i y,j B {J, R} (Ψ u,i, Ψ y,j ) da u,i y,j and, then, relation (15) an be satisfied if and only if B {J, R} (Ψ u,i, Ψ u,j ) da = (16a) B {J, R} (Ψ y,i, Ψ y,j ) da = (16b) B {J, R} (Ψ u,i, Ψ y,j ) da = δ ij (16) for every i, j = 1,..., m and where δ is the Kroneker symbol. Note that, given w W u,i = B {J, R} (Ψ u,i, w) da (17a) y,i = B {J, R} (Ψ y,i, w) da (17b) or equivalently that u = B{J, u R} (w) y = B y {J, R} (w) (18) where B u {J, R} : W U and B y {J, R} : W Y are two linear operator whose definition is base on (17). Consider a funtion C : X X R defined over the state spae of the losed loop system resulting from the power onserving interonnetion (15) of (13) and (2). From Def. 2.1, we an say that C is a Casimir funtion if and only if dt = T C (J R ) H + T C G B{J, u R} (w) + (δ x C) T (J R) δ x H dv = for every Hamiltonian funtions H and H, where u is expressed as a funtion of the boundary terms as in (17). Sine J and R are a skew adjoint and a self adjoint differential operator respetively, we have that (see 3, 14): (δ x C) T (J R)δ x H = (δ x H) T (J + R)δ x C + div B {J, R} (B (δ x C), w) and then dt = T H (J + R ) C (δ x H) T (J + R) δ x C dv ) + B {J, R} (w, B (δ x C) + Ψ u G T C da that has to be for every Hamiltonian funtion of the losed loop system and for every w given by (15). This means that (15) has to be satisfied by some u U and y Y, whih are related to the boundary variables w by (18). Consequently, it is possible to verify that the last integral is equal to T B{J, u R} (B (δ x H)) ( ) B y {J, R} (B (δ x C)) + G T C T H G B{J, u R} (B (δ x C)) and then dt = T H (J + R ) C + G B{J, u R} (B (δ x C)) (δ x H) T (J + R) δ x C dv T + B{J, u R} (B (δ x H)) ( ) B y {J, R} (B (δ x C)) + G T C that has to be for every Hamiltonian funtion of the losed loop system. This is true if and only if (J + R ) C + G B{J, u R} (B (δ x C)) = (19a) (J + R) δ x C = (19b) B y {J, R} (B (δ x C)) + G T C = (19) In onlusion, the following proposition has been proved. Proposition 3.1: Consider the losed loop system resulting from the power onserving interonnetion (15) of the infinite dimensional system (13) with the finite dimensional ontroller (2). Denote by X and X the state spae of the distributed parameter system and of the ontroller respetively. Then, a real value funtion C : X X R is a Casimir funtion for the losed loop system with respet to the interonnetion law (15) if and only the set of onditions (19) are satisfied. Note 3.1: The set of neessary and suffiient onditions (19) onerning the existene of strutural invariants in the losed loop system are the generalization of the analogous onditions (5) in the finite dimensional ase. In the hybrid ase desribed in Prop. 3.1, the strutural invariants have to satisfy the PDEs (19a) and (19) in the ontroller/plant variables and the PDE (19b) in the spatial variable of

6 the distributed parameter system providing the variational derivative of the andidate Casimir funtion with respet to the onfiguration variable. Note that the boundary onditions for (19b) have to be hosen in suh a way that (19a) and (19) are satisfied. The Casimir funtions are a onsequene of the interonnetion law (15). C. Energy shaping via strutural invariants As disussed in finite dimensions, the existene of a partiular lass of Casimir funtions in the ontrolled system an be of great interest in the energy shaping proedure. Also in the distributed parameter ase, a possible solution an be to hoose the struture of the ontroller (2) in order to introdue a set of n n strutural invariants in the losed loop system in the form C i (x, x ) = S i (x) x,i (2) where now S i (x) = S i(z, x) dv, with i = 1,..., n. These funtions are Casimir funtion for the losed loop system if and only if the set of onditions (19) are satisfied. In partiular, denote by J, R and Ḡ the sub-matries of the interonnetion, damping and input matries of (2) orresponding to the first n state variables and define S : X X R n as S = S 1 S n T. Then, from (19), we obtain the following set of onditions: G B{J, u R} (δ xs 1 ) B{J, u R} (δ xs n ) = J + R (21a) (J + R) δ x S i = (21b) B y {J, R} (δ xs 1 ) B y {J, R} (δ xs n ) = ḠT (21) with i = 1,..., n and where the operator B has been removed in order to keep a lighter notation. Note that, from (21a) and (21) only ( J + R ) is determined by the set of funtionals S i, while (21) gives the expression of the input sub-matrix Ḡ. Clearly, J, R and Ḡ depend on S i, whih have to be solution of the PDE (21b) whose boundary onditions have to be hosen in suh a way that (21a) and (21) are satisfied. If the set of onditions (21) an be satisfied, then the losed loop Hamiltonian funtion beomes H l (x, x ) = H(x) + H (x,1,..., x,n ) = H(x) + H (S 1 (x),..., S n (x),..., x,n ), thus depending expliitly on the onfiguration variable of the distributed parameter system. If n = n, then (2) are Casimir funtions of the losed loop system if and only if onditions (21) are satisfied and the losed loop Hamiltonian beomes H l (x, x ) = H(x) + H (S 1 (x),..., S n (x)), i.e. only a funtion of the onfiguration variable of the distributed parameter system. By properly hoosing the ontroller energy funtion, it is possible to introdue a minimum at the desired equilibrium onfiguration that an be reahed is some dissipative effet is present in the system. In partiular, if in (13) R =, that is no dissipative/diffusion phenomena are present in the infinite dimensional plant, it is onvenient to hose the ontroller struture in order to have n < n Casimir funtion in the form (2) and then to introdue energy dissipation by ating on the remaining energy variables. IV. EXAMPLE: STABILIATION OF THE HEAT EQUATION Consider the heat equation that an be written in mdph form as follows, 3: t w = = 2 z 2 δ xh δ x H z δ xh {, 1} (22) where =, 1 is the spatial domain, X = L 2 () is the spae of energy variables, H(x) = x2 dz is the Hamiltonian funtion and 1 B {J, R} = 1 is the onstant matrix representing the operator whih gives the power through the boundary as in (14). Note that ( dh dt = x ) ( ) 2 dz x L z z z z (23) Denote by x X a desired equilibrium onfiguration of (22) that the following one dimensional ontroller (with J = R = ) should render asymptotially stable. ẋ = G u y = G T H (24) Suppose that x R and that u, y R 2. From (23), it is easy to verify that (22) and (24) are interonneted in a power onserving way if z δ xh() x() u = z δ xh(1) = z () z (1) y = x(1) (25) The first step in the ontrol by interonnetion and energy shaping is to hoose the ontroller struture in order to have a set of strutural invariants in the form (2). In this ase, sine the ontroller is a dynamial system of order 1, it is neessary to determine G suh that the funtion C(x, x ) = x S(x) = x 1 S(z, x)dz (26) is a Casimir funtion for the losed loop system, that is C = for every H and H. We have that dt = (G z x() + δ x S() δ x S(1) ) z x(1) 1 x 2 z 2 (δ xs)dz z δ x S() z δ x S(1) G T H Consequently, (26) is a Casimir funtion for the ontrolled system if and only if 2 z 2 δ xs = (27a)

7 and { G + δ x S() δ x S(1) = z δ x S() z δ x S(1) G T = (27b) Condition (27a) provides the admissible funtionals S, while (27b) the input matrix G of the ontroller. From (27a), δ x S = az + b, with a, b R, while, in order to satisfy (27b), it is neessary that a = and it is possible to hoose b = 1. Consequently, and G = 1 1 (28) C(x, x ) = x 1 x(z) dz (29) is a Casimir funtion for the losed loop system. From (28), the ontroller (24) beomes ẋ = z x(1) z x() H y = x H and then, from (25), x() = x(1) = H (3) that is the ontroller ats on the system by imposing the same temperature on both the extremities of the infinite dimensional system. Moreover, the ontroller internal energy hanges, that is ẋ, only if there is a differene in the gradient of temperature at the extremities of the domain. As a onsequene, the ontroller an stabilize the distributed parameter system only in the onfigurations for whih the temperature is onstant along the domain, that is x (z) = x for every z. Under the hypothesis that the initial onfiguration of the system is known, from (29) and from the properties of the Casimir funtions, we have that x = x (x) = 1 x(z) dz for the losed loop system. Define x = x (x ) = x. The onfiguration x is asymptotially stable if the ontroller Hamiltonian is H (x ) = x x x x. In fat, if H l (x, x ) = H(x) + H (x ) is the energy funtion of the losed loop system, taking into aount (3), we have that dh l dt = 1 ( z ) 2 dz Then, Ḣ l = if z x = on, that is if x(z) is onstant on. Sine x() = x(1) = x, the only admissible onfiguration is x that results to be asymptotially stable. The asymptoti stability of x an be alternatively proved by looking at the expression of the losed loop Hamiltonian. We have that 1 1 H l (x) = 1 2 (x 2 dz x x ) dz = 1 (x x ) 2 dz + κ 2 whih has a global minimum in x and where κ is a onstant. V. CONCLUSIONS In this paper, a novel tehnique for the boundary ontrol of distributed parameter systems in port Hamiltonian form has been developed by extending the well known ontrol by interonnetion and energy shaping methodology. The basi result is the generalization of the onditions for obtaining a partiular set of Casimir funtion to the hybrid ase, that is the dynamial system to be onsidered results from the power onserving interonnetion of an infinite dimensional system (the plant) and of a finite dimensional one (the ontroller). A simple appliation onerning the stabilization of the one-dimensional heat equation has been presented. ACKNOWLEDGMENT Alessandro Mahelli wishes to thank prof. Bernhard Mashke for the friendly disussions and the nie suggestions for improving the paper. REFERENCES 1 B. M. Mashke and A. 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