- Email: [email protected]
Port-Hamiltonian Representation and Discretization of Undamped Wave Equation System
2nd IFAC Workshop on Control of Systems Governed 2nd IFAC Workshop Control of Systems Governed by Differentialon Equations 2ndPartial IFAC Workshop Workshop on Control of of Systems Systems Governed Governed 2nd IFAC on Control by Partial Differential Equations Available online at www.sciencedirect.com June 13-15, 2016. Bertinoro, Italy by Partial Differential Equations by Partial Differential Equations June 13-15, 2016. Bertinoro, Italy June 13-15, 13-15, 2016. 2016. Bertinoro, Bertinoro, Italy Italy June
ScienceDirect
IFAC-PapersOnLine 49-8 (2016) 309–314
Port-Hamiltonian Representation and Port-Hamiltonian Port-Hamiltonian Representation Representation and and Discretization of Undamped Wave Discretization of Undamped Wave Discretization of Undamped Wave Equation System Equation Equation System System Qingqing Xu ∗∗ , Stevan Dubljevic ∗∗ Qingqing Xu ∗∗ ,, Stevan Dubljevic ∗∗ Qingqing Qingqing Xu Xu , Stevan Stevan Dubljevic Dubljevic ∗ ∗ Department of Chemical and Materials Engineering, University of ∗ Department of Chemical and Materials Engineering, University of ∗ Department ofEdmonton, Chemical and and Materials Engineering, University of of Alberta,of Alberta Canada T6G 2V4(e-mail: Department Chemical Materials Engineering, University Alberta, Edmonton, Alberta Canada T6G 2V4(e-mail: Alberta, Edmonton, Alberta Canada T6G 2V4(e-mail: [email protected];[email protected].) Alberta, Edmonton, Alberta Canada T6G 2V4(e-mail: [email protected];[email protected].) [email protected];[email protected].) [email protected];[email protected].) Abstract: The port-Hamiltonian representation of undamped wave equation system captures Abstract: The port-Hamiltonian representation of undamped wave equation system captures Abstract: port-Hamiltonian of wave system captures the essentialThe energy and dynamical representation features in an unified framework. Inequation this paper, an undamped Abstract: The port-Hamiltonian representation of undamped undamped waveIn equation system captures the essential energy and dynamical features in an unified framework. this paper, an undamped the essential energy andisdynamical dynamical features in an an unified unified framework. framework. In this this paper, an undamped undamped waveessential equation system given in features the port-Hamiltonian formulation. Thepaper, important issue of the energy and in In an wave equation system is in the formulation. The important issue of wave equation system is given given in the port-Hamiltonian port-Hamiltonian formulation. The important issue of obtaining discrete version of the in port-Hamiltonian system formulation. representation is explored. The exact wave equation system is given the port-Hamiltonian The important issue of obtaining discrete version of the port-Hamiltonian system representation is explored. The exact obtaining discrete version of the port-Hamiltonian system representation is explored. The exact discretization method is applied to the port-Hamiltonian system which transforms the system obtaining discrete version of the port-Hamiltonian system representation is explored. The exact discretization method is applied to the port-Hamiltonian system which transforms the system discretization method applied the port-Hamiltonian which transforms the from a continuous to ais discrete state space setting. The system development of the discretized port discretization method is discrete applied to to thespace port-Hamiltonian system which of transforms the system system from a continuous to a state setting. The development the discretized port from a continuous continuous toformulation a discrete discrete state state space setting. setting. The for development of and the computer discretizedbased port Hamiltonian systemto is necessary foundation the discrete from a a space The development of the discretized port Hamiltonian system formulation is necessary foundation for the discrete and Hamiltonian system formulation is necessary foundation forby thenumerical discrete simulations. and computer computer based based regulator andsystem model formulation realizations. is The results are illustrated Hamiltonian necessary foundation for the discrete and computer based regulator and model realizations. The results are illustrated by numerical simulations. regulator and model realizations. The results are illustrated by numerical simulations. regulator and model realizations. The results are illustrated by numerical simulations. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Port-Hamiltonian, Exact Discretization, Undamped Wave Equation System. Keywords: Keywords: Port-Hamiltonian, Port-Hamiltonian, Exact Exact Discretization, Discretization, Undamped Undamped Wave Wave Equation Equation System. System. Keywords: Port-Hamiltonian, Exact Discretization, Undamped Wave Equation System. 1. INTRODUCTION However, most of the studies of the port-Hamiltonian 1. INTRODUCTION INTRODUCTION However, most most of of the the studies studies of of the the port-Hamiltonian 1. However, formulation are of developed for continuous finite and/or 1. INTRODUCTION However, most the studies of the port-Hamiltonian port-Hamiltonian formulation are developed for continuous finite and/or and/or formulation are developed for continuous finite infinite dimensional systems. For example, M. Seslija et formulation are developed for continuous finite and/or The port-Hamiltonian framework is an energy-based rep- infinite dimensional systems. For example, M. Seslija et The port-Hamiltonian port-Hamiltonian framework framework is is an an energy-based energy-based reprep- infinite dimensional systems. For example, M. Seslija et al. research on discrete exterior geometry approach for infinite dimensional systems. For example, M. Seslija et The resentation of physical systems and captures the energy The port-Hamiltonian is an energy-based rep- al. research on discrete exterior geometry approach for resentation of physical physical framework systems and and captures the energy energy al. research exterior approach for time-continuous spatially-discrete port-Hamiltonian sysal. research on on discrete discrete exterior geometry geometry approach sysfor resentation of captures and dynamical features systems in an unified setting. the Nowadays, resentation of physical systems and captures the energy time-continuous spatially-discrete port-Hamiltonian and dynamical features in an unified setting. Nowadays, time-continuous spatially-discrete port-Hamiltonian system (Seslija et al. (2012)). In particular, the discretization time-continuous spatially-discrete port-Hamiltonian sysand dynamical features in an unified setting. Nowadays, the application of the port-Hamiltonian formulation has tem (Seslija et al. (2012)). In particular, the discretization and dynamical features in an unified setting. Nowadays, the application application of of the the port-Hamiltonian port-Hamiltonian formulation formulation has has tem (Seslija (2012)). In the of distributed parameter system is a difficult and chal(Seslija et et al. al. (2012)). system In particular, particular, the discretization discretization the beenapplication widely and of successfully explored within lump and disthe the port-Hamiltonian formulation has tem of distributed distributed parameter is aa difficult difficult and chalchalbeen widely and successfully explored within lump and disof parameter system is and lenging problem. Traditionally, the approximate represenof distributed parameter system is a difficult and chalbeen widely and successfully explored within lump and distributed parameter systems setting (Maschke and van der been widely and successfully explored within lump and dislenging problem. Traditionally, the approximate representributed parameter parameter systems systems setting setting (Maschke (Maschke and and van van der der lenging Traditionally, the approximate tation ofproblem. the original continuous time system is represenobtained lenging problem. Traditionally, the approximate representributed Schaft (2005); Maschke and Van der(Maschke Schaft (1991); Ortega tributed parameter systems setting and van der tation of of the the original original continuous continuous time time system system is is obtained obtained Schaft (2005); (2005); Maschke Maschke and and Van Van der der Schaft Schaft (1991); (1991); Ortega Ortega tation by using numerical simulation techniques such as Euler, of the originalsimulation continuoustechniques time system obtained Schaft et al. (2002); Van der Schaft and der Maschke (2002)). In par- tation Schaft (2005);Van Maschke and Van Schaft(2002)). (1991); Ortega by using using numerical suchis as as Euler, et al. (2002); der Schaft and Maschke In parby numerical simulation techniques such Euler, Runge-Kutta, etc (Kazantzis and Kravaris (1999); Golo by using numerical simulation techniques such as Euler, et al. (2002); Van der Schaft and Maschke (2002)). In particular, the application of theand port-Hamiltonian formalism et al. (2002); Van der Schaft Maschke (2002)). In par- Runge-Kutta, etc (Kazantzis and Kravaris (1999); Golo ticular, the application application of the the port-Hamiltonian port-Hamiltonian formalism Runge-Kutta, etc (Kazantzis and Kravaris (1999); Golo et al. (2004); Bassi et al. (2007)). These methods have Runge-Kutta, etc (Kazantzis and Kravaris (1999); Golo ticular, the of formalism to wave equation system captures the energy balance featicular, the application of captures the port-Hamiltonian formalism et al. al. (2004); (2004); Bassi Bassi et et al. al. (2007)). (2007)). These These methods methods have have to wave equation system the energy balance feaet the disadvantage that the accuracy of the approximate et al. (2004); Bassi et al. (2007)). These methods have to wave equation system the balance tures of dynamic systems and provides the interconnection to wave equation systems system captures captures the energy energy balance feafea- the disadvantage that the accuracy of the approximate tures of dynamic dynamic and provides provides the interconnection interconnection disadvantage that the of approximate discrete time system rapidly deteriorates as the sampling the disadvantage thatrapidly the accuracy accuracy of the the approximate tures of structure related systems with theand exchange ofthe energy among the the tures of dynamic systems and provides the interconnection discrete time system system deteriorates as the the sampling structure related related with with the the exchange exchange of of energy energy among among the the discrete time rapidly deteriorates as sampling period increases. Thus, an accurate and exact representadiscrete time system rapidly deteriorates as the sampling structure system andrelated the environment through ports (Van der Schaft structure with the exchange of energy the period increases. Thus, an accurate and exact representasystem and and the the environment environment through ports ports (Vanamong der Schaft Schaft period Thus, an and representation of increases. the original continuous time system is required for period Thus, an accurate accurate and exact exact representasystem through (Van der and Maschke (2002); Le Gorrec et ports al. (2005); Duindam system and the environment through (Van der Schaft tion of of increases. the original original continuous time system system is required required for and Maschke (2002); Le Gorrec et al. (2005); Duindam tion the continuous time is for accurate discrete representation of the distributed paramtion of the original continuous time system is required for and et al.Maschke (2009)). (2002); Le Gorrec et al. (2005); Duindam accurate discrete representation of the distributed paramand et al. al.Maschke (2009)). (2002); Le Gorrec et al. (2005); Duindam accurate discrete eter system. accurate discrete representation representation of of the the distributed distributed paramparamet (2009)). et al. (2009)). eter system. system. In engineering applications, there are three common phys- eter eter system. In engineering engineering applications, applications, there there are are three three common common physphys- The contribution of this paper is the exploration of a In ical systems that carry waves: the electrical transmisIn applications, there the are three common phys- The contribution of this paper is the exploration of a icalengineering systems that that carry waves: waves: electrical transmiscontribution of paper the proper discretization method in is order to preserve of thea The contribution of this this paperin isorder the exploration exploration ical systems carry the electrical transmission systems line, the that flexible string, andthe the electrical compressible fluid. The ical carry waves: transmisproper discretization method to preserve preserve ofthe thea sion line, line, the the flexible flexible string, string, and and the the compressible compressible fluid. fluid. proper discretization method in order to intrinsic energy and dynamical characteristics of the disproper discretization method in order to preserve the sion The wave equation of the above systems is described by sion line, the flexible and systems the compressible fluid. intrinsic energy energy and and dynamical dynamical characteristics characteristics of of the the disdisThe wave wave equation of string, the above above is described described by intrinsic tributed energy parameter system. The novelty of the design intrinsic and dynamical characteristics of the disThe equation of the systems is by the hyperbolic partial differential equation (PDE) derived The wave equation of the above systems is described by tributed parameter system. The novelty of the design the hyperbolic hyperbolic partial partial differential differential equation equation (PDE) (PDE) derived derived tributed system. The novelty of the design procedureparameter lies in the fact that spatial discretization and/or tributed parameter system. The novelty of the design the fromhyperbolic the mass partial and/or differential energy balance. It is(PDE) a distributed the equation derived procedure lies lies in in the fact fact that that spatial spatial discretization discretization and/or and/or from the the mass mass and/or and/or energy energy balance. balance. It It is is aa distributed distributed procedure any other lies type ofthe spatial approximation of the system is fact that spatial discretization and/or from parameter system and energy due tobalance. its spatial nature it rep- procedure from the mass and/or It isnature a distributed any other other type typeinof ofthe spatial approximation of the the system system is parameter system and due to its spatial it repany spatial approximation of is not considered and the undamped wave equation system any other type of spatial approximation of the system is parameter system and due to its spatial nature it represents one of interesting systems realizations to be adparameter and duesystems to its realizations spatial nature not considered considered and and the the undamped undamped wave wave equation equation system system resents one onesystem of interesting interesting to it be repad- not is completely captured with the proposed transformations not consideredcaptured and thewith undamped wave equation system resents of systems realizations to be addressed one within the port-Hamiltonian setting (Duindam resents of interesting systems realizations to be adis completely the proposed transformations dressed within within the the port-Hamiltonian port-Hamiltonian setting setting (Duindam (Duindam is completely captured the transformations from a continuous to awith discrete state space setting. The completely captured with the proposed proposed transformations dressed et al. (2009)). An important issue of the distributed pa- is dressed within the port-Hamiltonian setting (Duindam from a continuous to a discrete state space setting. The et al. al. (2009)). (2009)). An An important important issue issue of of the the distributed distributed papa- from aa continuous to space discretization is realized by the state Cayley-Tustin time The disfrom continuous to aa discrete discrete state space setting. setting. The et rameter systemAn presented by the port-Hamiltonian frameet al. (2009)). important issue of the distributed padiscretization is realized by the Cayley-Tustin time disrameter system system presented presented by by the the port-Hamiltonian port-Hamiltonian frameframe- discretization is realized by the Cayley-Tustin time discretization transformation and is applied to the portdiscretization is realized by the Cayley-Tustin time disrameter work is that boundary controlled formulations are mainly rameter system presented by the port-Hamiltonian frame- cretization transformation and is applied to the portwork is is that that boundary controlled formulations are are mainly mainly cretization and is to the Hamiltoniantransformation framework of the wave equation system in cretization and wave is applied applied to system the portportwork boundary controlled formulations presented as realizations since actuation is usually realized work is that boundary controlled formulations are mainly Hamiltoniantransformation framework of of the the equation in presented as realizations since actuation is usually realized Hamiltonian framework wave equation system in this work (Havu and Malinen (2007)). Hamiltonian framework of the wave equation system in presented as actuation is at the boundary of the since system and rarely as in realized domain this work (Havu and Malinen (2007)). presented as realizations realizations actuation is usually usually at the the boundary boundary of the the since system and rarely rarely as in in realized domain this work (Havu and (2007)). this and Malinen Malinen (2007)). at of system and as domain actuation (Le Gorrec et al. (2005)). The standard method at the boundary of the system and rarely as in domain The work paper(Havu is organized as follows. In Section 2, we inactuation (Le Gorrec et al. (2005)). The standard method The paper is organized as follows. In Section Section 2, 2, we we inactuation (Le Gorrec et al. (2005)). The standard method to resolve the issue of boundary/point actuation is to apply actuation (Le Gorrec et al. (2005)). The standard method The paper is organized as follows. In troduce and present the port-Hamiltonian formulation of The paper is organized as follows. In Section 2, we ininto resolve resolve the the issue issue of of boundary/point boundary/point actuation actuation is is to to apply apply troduce and present the port-Hamiltonian formulation of to boundary transformation which maps actuation boundaryiscontrol to troduce to resolve the issue of boundary/point to apply and present the port-Hamiltonian formulation of the undamped wave equation system. In section 3, we deal troduce and present the port-Hamiltonian formulation of boundary transformation which maps boundary control to the undamped undamped wave wave equation equation system. system. In In section 3, 3, we we deal boundary which control to in-domain transformation control for the port-Hamiltonian system (Zwart boundary which maps maps boundary boundary control to the withundamped the boundary controlled port-Hamiltonian In the wavecontrolled equation port-Hamiltonian system. In section section system. 3, we deal deal in-domain transformation control for for the the port-Hamiltonian port-Hamiltonian system (Zwart with the boundary system. In in-domain control system (Zwart and Curtain (1995), Moghadam et al. (2013)). in-domain control for the port-Hamiltonian system (Zwart with the boundary controlled port-Hamiltonian system. In with the boundary controlled port-Hamiltonian system. In and Curtain Curtain (1995), (1995), Moghadam Moghadam et et al. al. (2013)). (2013)). and and Curtain (1995), Moghadam et al. (2013)). Copyright 2016 International Federation of 311 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Copyright 2016 responsibility International of Federation of Federation of Automatic 311Control. Peer review© under International Automatic Control Copyright © 2016 International Federation of 311 Copyright © 2016 International Federation of 311 Automatic Control 10.1016/j.ifacol.2016.07.459 Automatic Control Control Automatic
311 311 311 311
IFAC CPDE 2016 310 June 13-15, 2016. Bertinoro, Italy
Qingqing Xu et al. / IFAC-PapersOnLine 49-8 (2016) 309–314
section 4, we present the Cayley-Tustin time discretization of the port-Hamiltonian system. Finally, simulation results of the undamped wave equation system are shown in Section 5. 2. INFINITE DIMENSIONAL PORT-HAMILTONIAN REPRESENTATION OF WAVE EQUATION SYSTEM In this section, the port-Hamiltonian representation of the distributed parameter system with boundary control is introduced and the wave equation described by the hyperbolic PDEs is considered. 2.1 Infinite Dimensional Port-Hamiltonian System
The operator given by ∂ (Hx) + P0 (Hx) ∂t
The the displacement w(ζ, t) is the output y(ζ, t) = w(ζ, t) of the above transformed system and is obtained by the following equation: ζ x1 (η, t)dη (12) y(ζ, t) = 0
The general port-Hamiltonian system representation is given in the following form (Jacob and Zwart (2012)): ∂ ∂x (ζ, t) = P1 [H(ζ)x(ζ, t)] + P0 [H(ζ)x(ζ, t)] (1) ∂t ∂ζ with boundary condition: f (t) = u(t) (2) WB ∂ e∂ (t) where 1 P1 −P1 H(L)x(L, t) f∂ (t) √ = (3) e∂ (t) H(0)x(0, t) 2 I I P1 ∈ K n×n is invertible and self-adjoint; P0 ∈ K n×n is skew-adjoint; H ∈ C 1 ([0, L]; K n×n ), H(ζ) is self-adjoint for all ζ ∈ [0, L] and mI ≤ H(ζ) ≤ M I for all ζ ∈ [0, L] n×2n and some m, M > 0 independent of ζ; W has B ∈ K 0 I ∗ . full rank and WB ΣWB ≥ 0, where Σ = I 0 Ax := P1
where the the displacement w(ζ, t) is state variable. After some basic manipulation and transformation of the system, the above set of equations is transformed to the following form by defining state variables as x1 (ζ, t) = ∂w(ζ,t) ∂w(ζ,t) , so that the system becomes: ∂ζ , x2 (ζ, t) = ∂t ∂ x1 (ζ, t) 0 1 ∂ x1 (ζ, t) = (11) 1 0 ∂ζ x2 (ζ, t) ∂t x2 (ζ, t)
(4)
with domain D(A) = {x ∈ L2 ([0, L]; K n )|Hx ∈ H 1 ([0, L]; K n )} (5) generates a contraction semigroup on the state space (6) X = L2 ([0, L]; K n ) equipped with the inner product 1 L ∗ g (ζ)H(ζ)f (ζ)dζ (7) < f, g >X = 2 0 associated norm is denoted by|| · ||X .
The associated Hamiltonian E : [0, L] → R which represents the energy of the system is given by 1 L ∗ E(t) = x (ζ, t)H(ζ)x(ζ, t)dζ (8) 2 0 2.2 Undamped Wave Equation System Let us consider the case of undamped wave equation which takes the following PDE form (Zwart and Curtain (1995)): ∂ 2 w(ζ, t) ∂ 2 w(ζ, t) = (9) ∂t2 ∂ζ 2 where w ∈ [0, L]⊕R1 . The initial conditions and boundary conditions are as follows: ∂w (ζ, 0) = ψ(ζ) (10) t = 0, w(ζ, 0) = φ(ζ), ∂t w(0, t) = 0 = w(1, t) 312
The system described by Eq.11 can be rewritten in the port-Hamiltonian form: ∂x ∂ (ζ, t) = P1 [H(ζ)x(ζ, t)] + P0 [H(ζ)x(ζ, t)] (13) ∂t ∂ζ y(t) = Cx(ζ, t) 10 0 1 x1 (ζ, t) , P1 = , , P0 = 0, H = where x = x2 (ζ, t) 01 1 0 ζ C = [C1 C2 ], here C1 = 0 (·)dη and C2 = 0. The associated Hamiltonian is given by: 1 L E(t) = x(ζ, t)∗ H(ζ)x(ζ, t)dζ (14) 2 0 1 L = [x1 (ζ, t)2 + x2 (ζ, t)2 ]dζ 2 0 and the following balance equation holds: 1 dE(t) = [(H(ζ)x(ζ, t))∗ P1 (H(ζ)x(ζ, t))]L (15) 0 dt 2 = x1 (L)x2 (L) − x1 (0)x2 (0)
Remark 1: If the boundary condition x2 (0, t) = x2 (L, t) = 0 holds, the derivative of the energy along solutions is zero dE(t) dt = 0, which implies that the system does not dissipate energy. Remark 2: From Eq15, it can be seen that the changes of the overall energy content only occur via the boundary. One can consider the boundary control system with one boundary input: (16) x2 (0, t) = 0 x2 (L, t) = u(t) The boundary effort and boundary flow are given as follows: 1 e∂ (t) = √ [Hx(L, t) + Hx(0, t)] (17) 2 1 x1 (L, t) + x1 (0, t) =√ 2 x2 (L, t) + x2 (0, t) 1 f∂ (t) = √ [P1 Hx(L, t) − P1 Hx(0, t)] (18) 2 1 x2 (L, t) − x2 (0, t) =√ 2 x1 (L, t) − x1 (0, t) then the boundary condition becomes: 1 −1 0 0 1 f∂ (t) f∂ (t) u(t) = √ = WB e∂ (t) e∂ (t) 2 1 0 0 1 −1 0 0 1 . with WB = √12 1 0 0 1
IFAC CPDE 2016 June 13-15, 2016. Bertinoro, Italy
Qingqing Xu et al. / IFAC-PapersOnLine 49-8 (2016) 309–314
The proposed representation in the port-Hamiltonian framework can be easily utilized in the distributed parameter system described by hyperbolic PDEs with boundary control. 3. BOUNDARY CONTROL OF PORT-HAMILTONIAN SYSTEM In this section, we introduce the transformation of boundary control to in-domain control and apply the control to port-Hamiltonian system (Zwart and Curtain (1995); Jacob and Zwart (2012)). The port-Hamiltonian described in Eq.13 with the boundary condition in Eq.16 can be extended to the abstract control system of the form: x(ζ, ˙ t) = Ax(ζ, t), x(ζ, 0) = x0 (19) y(t) = Cx(ζ, t) Bx(ζ, t) = u(t) where A : D(A) ⊂ X → X is linear, the control function u takes values in the Hilbert space U , the boundary operator B : D(B) ⊂ X → U is linear and satisfies D(A) ⊂ D(B) and the operator C : D(C) → X. The above control system is a boundary control system if the following holds: 1.The operator A : D(A) → X with D(A) = D(A) ∩ ker(B) and Ax = Ax x ∈ D(A) (20) is the infinitesimal generator of a C0 -semigroup on X; 2.There exists an operator Bb ∈ L(U, X) such that for all u ∈ U we have Bb u ∈ D(A), ABb ∈ L(U, X) and BBb u = u u ∈ U (21) Defining the following state transformation: v(ζ, t) = x(ζ, t) + Bb (ζ)u(t), v(ζ, 0) = v0
311
according to the boundary control in Eq.16, one can obtain: Bb2 (0) = 0 (28) Bb2 (L) = −1
b2 (ζ) Thus, one can have ∂B∂ζ = −1. Then, according to the extended system described in Eq.25, one can obtain: ∂v2 (ζ, t) v˙ 1 (ζ, t) = + u(t) (29) ∂ζ ∂v1 (ζ, t) + Bb2 (ζ)¯ u(t) v˙ 2 (ζ, t) = ∂ζ u(t) Here, let us define the input as u ¯(t) = and u ˜(t) 1 0 , then, the extended system described B = 0 Bb2 (ζ) in Eq.25 becomes: v(ζ, ˙ t) = Av(ζ, t) + B u ¯(t), v(ζ, 0) = v0 (30) y(t) = Cv(ζ, t) + D¯ u(t)
here the operator D = 0. The above extended system described in Eq.30 maps boundary control to in-domain control. 4. EXACT DISCRETIZATION OF PORT-HAMILTONIAN SYSTEM In this section, we propose an exact time discretization transformation from a continuous to a discrete state space setting. The Cayley-Tustin time discretization transformation is applied to the port-Hamiltonian framework of the undamped wave equation system. 4.1 Exact System Discretization
(22)
here Bb (ζ) = [Bb1 (ζ)Bb2 (ζ)]T . Then, the boundary control system described in Eq.19 can be transferred to the abstract differential equation: v(ζ, ˙ t) = Av(ζ, t) − ABb u(t) + Bb u(t) ˙ (23) where the operator A is given by: ∂ (24) Ax := P1 (Hx) + P0 (Hx) ∂ζ Assuming u ˜(t) = u(t), ˙ the extension of the system is in the following form: u(t) I u(t) ˙ 0 0 + u ˜(t) = v(ζ, t) Bb (ζ) v(ζ, ˙ t) −ABb (ζ) A(ζ) u(t) y(t) = [0 C(ζ)] (25) v(ζ, t)
For the undamped wave equation system described in Eq.13, we define: (26) v1 (ζ, t) = x1 (ζ, t) v2 (ζ, t) = x2 (ζ, t) + Bb2 (ζ)u(t) thus, Bb1 (ζ) = 0. The obtained in-domain control system is designed with homogeneous boundary condition as below: v2 (0, t) = x2 (0, t) + Bb2 (0)u(t) = 0 (27) v2 (L, t) = x2 (L, t) + Bb2 (L)u(t) = 0
313
The continuous system considered in this section is the indomain port-Hamiltonian system described in Eq.30 and let us rewrite the continuous system as follows: v(ζ, ˙ t) = Av(ζ, t) + B u ¯(t), v(ζ, 0) = v0 (31) y(t) = Cv(ζ, t) + D¯ u(t) a slightly non-standard time discretization type is given by (Havu and Malinen (2007)): v(jh) − v((j − 1)h) v(jh) + v((j − 1)h) ≈A + Bu ¯(jh) h 2 v(jh) + v((j − 1)h) y(jh) ≈ C + D¯ u(jh) (32) 2 √ √ ¯(jh) and yjh / h be let u ¯hj / h be the approximation of u the approximation of y(jh), the above equation becomes: h h vjh − vj−1 vjh + vj−1 u ¯hj =A + B√ h 2 h h h h yj u ¯hj vj + vj−1 √ =C + D√ 2 h h
(33)
Applying the Cayley transform, the discrete system is given in the following form: v(ζ, k) = Ad v(ζ, k − 1) + Bd u ¯(k) (34) y(ζ, k) = Cd v(ζ, k − 1) + Dd u ¯(k)
IFAC CPDE 2016 312 June 13-15, 2016. Bertinoro, Italy
Qingqing Xu et al. / IFAC-PapersOnLine 49-8 (2016) 309–314
where √ −1 −1 Ad Bd [δ + A][δ − A] 2δ[δ − A ] B −1 = √ Cd D d 2δC[δ − A]−1 G(δ)
(2005)). The discrete port-Hamiltonian system captures the energy and dynamical features of the system.
G(δ) is the transfer function:
G(δ) = C[δ − A−1 ]−1 B + D here σ = 2/h and operator A−1 is the Yosida extension of operator A.
2 0 -2 -4 0 0.2 0.4 0.6
ζ
0.8 1
0
10
20
30
40
50
k
Fig. 1. Evolution of discrete state x1 (ζ, t) in the undamped wave equation system given by Eq.40.
0
4.2 Resolvent Operator Let us consider the port-Hamiltonian system described in Eq.31: v(ζ, ˙ t) = Av(ζ, t) ∂ where Av := P1 ∂ζ (Hv) + P0 (Hv) and the domain of the operator A is D(A) = {v ∈ L2 (0, 1)|v is absolutely continuous, dv dζ ∈ L2 (0, 1), v(0) = 0}. The resolvent of the operator A can be obtained from Laplace transform: v(ζ, s) = [sI − A]−1 v(ζ, 0) (35) here the resolvent of operator A is R(s, A) = [sI − A]−1 .
One can also obtain the Laplace transform of the system as follows: ∂ sv(ζ, s) − v(ζ, 0) = P1 [Hv(ζ, s)] + P0 [Hv(ζ, s)] (36) ∂ζ then ∂ P1 [Hv(ζ, s)] = [sI − P0 ][Hv(ζ, s)] − v(ζ, 0) (37) ∂ζ the above equation is a set of ordinary differential equations: ∂v1 (ζ, s) = sv2 (ζ, s) − v2 (ζ, 0) ∂ζ ∂v2 (ζ, s) = sv1 (ζ, s) − v1 (ζ, 0) ∂ζ Remark 3: According Theorem 12.3.11 in O. Staffans’s book, the discrete system obtained from Cayley-Tustin transformation is energy preserving system (Staffans 314
3 2 x2(ζ,k)
x1(ζ,k)
4
The resolvent of the operator A operate on v(ζ, 0) is obtained from the solution of the above set of equations when the boundary condition v2 (0, s) = v2 (L, s) = 0, and it is in the following form: (38) R(s, A)v(ζ, 0) = [sI − A]−1 v(ζ, 0) R11 (ζ) R12 (ζ) v1 (ζ, 0) = R21 (ζ) R22 (ζ) v2 (ζ, 0) where cosh(sζ) 1 cosh[s(1 − η)](·)dη R11 (·) = sinh(s) 0 ζ − sinh[s(ζ − η)](·)dη 0 cosh(sζ) 1 R12 (·) = sinh[s(1 − η)](·)dη sinh(s) 0 ζ − cosh[s(ζ − η)](·)dη 0 sinh(sζ) 1 R21 (·) = cosh[s(1 − η)](·)dη sinh(s) 0 ζ cosh[s(ζ − η)](·)dη − 0 sinh(sζ) 1 R22 (·) = sinh[s(1 − η)](·)dη sinh(s) 0 ζ − sinh[s(ζ − η)](·)dη
1 0 -1 -2 -3 0 0.2 0.4 0.6
ζ
0.8 1
0
10
20
30
40
50
k
Fig. 2. Evolution of discrete state v2 (ζ, t) in the undamped wave equation system given by Eq.40. 4.3 Discrete Operator Let us rewrite the continuous system described in Eq.31: v(ζ, ˙ t) = Av(ζ, t) + B u ¯(t), v(ζ, 0) = v0 (39) y(t) = Cv(ζ, t) + D¯ u(t) 1 0 ∂ , where Av := P1 ∂ζ (Hv) + P0 (Hv), B = 0 Bb2 (ζ) ζ C = [C1 C2 ], here C1 = 0 (·)dη and C2 = 0, and D = 0.
IFAC CPDE 2016 June 13-15, 2016. Bertinoro, Italy
Qingqing Xu et al. / IFAC-PapersOnLine 49-8 (2016) 309–314
The discrete system obtained by the exact discretization approach is given in Eq.34. With the resolvent of the operator A obtained Eq.38, one can obtain the discrete system as follows: v(ζ, k) = Ad v(ζ, k − 1) + Bd u ¯(k), y(ζ, k) = Cd v(ζ, k − 1) + Dd u ¯(k)
v(0) = v0
(40)
Here, discrete operator Ad operate on state v(ζ, k − 1) is given as follows: Ad v(ζ, k − 1) = [δ + A][δ − A]−1 v(ζ, k − 1) (41) −1 = [−I + 2δ[δ − A] ]v(ζ, k − 1) Ad11 (ζ) Ad12 (ζ) = Ad21 (ζ) Ad22 (ζ) where cosh(sζ) 1 Ad11 (·) = − (·) + 2δ cosh[s(1 − η)](·)dη sinh(s) 0 ζ − sinh[s(ζ − η)](·)dη 0 cosh(sζ) 1 Ad12 (·) =2δ sinh[s(1 − η)](·)dη sinh(s) 0 ζ − cosh[s(ζ − η)](·)dη 0 sinh(sζ) 1 Ad21 (·) =2δ cosh[s(1 − η)](·)dη sinh(s) 0 ζ cosh[s(ζ − η)](·)dη − 0 sinh(sζ) 1 Ad22 (·) = − (·) + 2δ sinh[s(1 − η)](·)dη sinh(s) 0 ζ − sinh[s(ζ − η)](·)dη
√ ζ cosh(sφ) 1 Cd11 (·) = 2δ cosh[s(1 − η)](·)dη sinh(s) 0 0 φ sinh[s(φ − η)](·)dη dφ − 0 √ ζ cosh(sφ) 1 Cd12 (·) = 2δ sinh[s(1 − η)](·)dη sinh(s) 0 0 φ cosh[s(φ − η)](·)dη dφ − 0
Finally, discrete operator Dd is given as follows: Dd = C[δ − A−1 ]−1 B where ζ cosh(sφ) 1 sinh[s(1 − η)](·)dη Dd = − sinh(s) 0 0 φ − cosh[s(φ − η)](·)dη dφ
The resolvent operator and discrete operators of the portHamiltonian system are presented in this section. The exact time discretization transformation from the continuous to the discrete state space setting is achieved by the Cayley-Tustin time discretization transformation. 5. SIMULATION RESULTS In the simulation study, we choose discrete time and space parameters as dt = 0.5 and dζ = 0.01. The initial conditions of the discrete system described in Eq.40 are v01 (ζ) = 0 and v02 (ζ) = 2 sin(2πζ). The inputs are chosen as u(k) = 0.314 cos(0.314k) and u ˜(k) = u(k) ˙ = 0.099 sin(0.314k). Fig.1-4 gives the simulation results of discrete undamped wave equation system. The states x1 (ζ, k), x2 (ζ, k) and v2 (ζ, k) with one boundary input which operates on x2 (L, k) are given in Fig.1-3. And the output y(ζ, k), which is the displacement w(ζ, k) of the undamped wave equation system is given in Fig.4.
(42)
2
where
−
Bd2
√
2δ ζ
cosh(sζ) sinh(s)
0
1
v2(ζ,k)
Bd1 = −
(44)
0
0
Discrete operator Bd is given as follows: √ Bd = 2δ[δ − A−1 ]−1 B Bd1 (ζ) = Bd2 (ζ)
313
sinh[s(1 − η)](·)dη
1 0 -1
cosh[s(ζ − η)](·)dη √ sinh(sζ) 1 = − 2δ sinh[s(1 − η)](·)dη sinh(s) 0 ζ sinh[s(ζ − η)](·)dη − 0
-2 0
50 40
0.2
30
0.4
20
0.6
ζ
10
0.8 1
0
Similarly, discrete operator Cd operate on v(ζ, k − 1) is given as follows: √ Cd v(ζ, k − 1) = 2δC[δ − A]−1 v(ζ, k − 1) (43) = [ Cd11 (ζ) Cd12 (ζ) ] where 315
k
0
Fig. 3. Evolution of discrete state x2 (ζ, t) in the undamped wave equation system given by Eq.40. Since the boundary control is transferred to in-domain control, it can be seen from Fig.2 that the state x2 (ζ, k) is with one boundary control u(k) = 0.314 cos(0.314k) at
IFAC CPDE 2016 314 June 13-15, 2016. Bertinoro, Italy
Qingqing Xu et al. / IFAC-PapersOnLine 49-8 (2016) 309–314
ζ = L. In addition, the transformed state v2 (ζ, k) is with homogeneous boundaries, see Fig.3. Finally, Fig.4 t gives the output y(ζ, k) with one boundary control 0 u(τ )dτ at ζ = L, which is y(L, k) = sin(0.314k).
1
y(ζ,k)
0.5 0 -0.5 -1 -1.5 0
50 0.2
40 0.4
30 0.6
ζ
20 0.8
10 1
k
0
Fig. 4. Evolution of discrete output y(ζ, t) in the undamped wave equation system given by Eq.40. It can be seen from the simulation results that the exact discretization captures the distributed nature of the hyperbolic PDEs system obtained from the undamped wave equation system. 6. CONCLUSION The application of the port-Hamiltonian framework to the undamped wave equation system captures the energy balance features of dynamical systems. The undamped wave equation system derived from mass and energy balances is described by the port-Hamiltonian framework in this paper. Then, the boundary transformation method which maps boundary control to in-domain control for the port-Hamiltonian system is presented. In this work, the Cayley-Tustin time discretization is applied to portHamiltonian system and the simulation results are given. The obtained discrete formulation provides a basis for the development of more advanced discrete controller formulations and discrete system modelling realizations which intrinsically capture and preserve energy of the underlying system dynamics. More advanced control and optimization schemes, such as model predictive control can be pursued in order to help the undamped wave equation system operate more efficiently. REFERENCES Bassi, L., Macchelli, A., and Melchiorri, C. (2007). An algorithm to discretize one-dimensional distributed port hamiltonian systems. In Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, 61–73. Springer. Duindam, V., Macchelli, A., Stramigioli, S., and Bruyninckx, H. (2009). Modeling and control of complex physical systems: the port-Hamiltonian approach. Springer Science & Business Media. Golo, G., Talasila, V., Van Der Schaft, A., and Maschke, B. (2004). Hamiltonian discretization of boundary control systems. Automatica, 40(5), 757–771. 316
Havu, V. and Malinen, J. (2007). The Cayley transform as a time discretization scheme. Numerical Functional Analysis and Optimization, 28(7-8), 825–851. Jacob, B. and Zwart, H. (2012). Linear port-Hamiltonian systems on infinite-dimensional spaces, volume 223. Springer Science & Business Media. Kazantzis, N. and Kravaris, C. (1999). Time-discretization of nonlinear control systems via Taylor methods. Computers & chemical engineering, 23(6), 763–784. Le Gorrec, Y., Zwart, H., and Maschke, B. (2005). Dirac structures and boundary control systems associated with skew-symmetric differential operators. SIAM Journal on Control and Optimization, 44(5), 1864–1892. Maschke, B. and van der Schaft, A. (2005). 4 compositional modelling of distributed-parameter systems. In Advanced topics in control systems theory, 115–154. Springer. Maschke, B. and Van der Schaft, A. (1991). Portcontrolled hamiltonian systems: modelling origins and systemtheoretic properties. Moghadam, A.A., Aksikas, I., Dubljevic, S., and Forbes, J.F. (2013). Boundary optimal (lq) control of coupled hyperbolic PDEs and ODEs. Automatica, 49(2), 526– 533. Ortega, R., Van Der Schaft, A., Maschke, B., and Escobar, G. (2002). Interconnection and damping assignment passivity-based control of port-controlled hamiltonian systems. Automatica, 38(4), 585–596. Seslija, M., van der Schaft, A., and Scherpen, J.M. (2012). Discrete exterior geometry approach to structurepreserving discretization of distributed-parameter porthamiltonian systems. Journal of Geometry and Physics, 62(6), 1509–1531. Staffans, O. (2005). Well-posed linear systems. 103. Cambridge University Press. Van der Schaft, A. and Maschke, B. (2002). Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics, 42(1), 166–194. Zwart, H. and Curtain, R. (1995). An introduction to infinite dimensional linear systems theory. Texts in Applied Mathematics, Springer-Verlag, New York.
ScienceDirect
IFAC-PapersOnLine 49-8 (2016) 309–314
Port-Hamiltonian Representation and Port-Hamiltonian Port-Hamiltonian Representation Representation and and Discretization of Undamped Wave Discretization of Undamped Wave Discretization of Undamped Wave Equation System Equation Equation System System Qingqing Xu ∗∗ , Stevan Dubljevic ∗∗ Qingqing Xu ∗∗ ,, Stevan Dubljevic ∗∗ Qingqing Qingqing Xu Xu , Stevan Stevan Dubljevic Dubljevic ∗ ∗ Department of Chemical and Materials Engineering, University of ∗ Department of Chemical and Materials Engineering, University of ∗ Department ofEdmonton, Chemical and and Materials Engineering, University of of Alberta,of Alberta Canada T6G 2V4(e-mail: Department Chemical Materials Engineering, University Alberta, Edmonton, Alberta Canada T6G 2V4(e-mail: Alberta, Edmonton, Alberta Canada T6G 2V4(e-mail: [email protected];[email protected].) Alberta, Edmonton, Alberta Canada T6G 2V4(e-mail: [email protected];[email protected].) [email protected];[email protected].) [email protected];[email protected].) Abstract: The port-Hamiltonian representation of undamped wave equation system captures Abstract: The port-Hamiltonian representation of undamped wave equation system captures Abstract: port-Hamiltonian of wave system captures the essentialThe energy and dynamical representation features in an unified framework. Inequation this paper, an undamped Abstract: The port-Hamiltonian representation of undamped undamped waveIn equation system captures the essential energy and dynamical features in an unified framework. this paper, an undamped the essential energy andisdynamical dynamical features in an an unified unified framework. framework. In this this paper, an undamped undamped waveessential equation system given in features the port-Hamiltonian formulation. Thepaper, important issue of the energy and in In an wave equation system is in the formulation. The important issue of wave equation system is given given in the port-Hamiltonian port-Hamiltonian formulation. The important issue of obtaining discrete version of the in port-Hamiltonian system formulation. representation is explored. The exact wave equation system is given the port-Hamiltonian The important issue of obtaining discrete version of the port-Hamiltonian system representation is explored. The exact obtaining discrete version of the port-Hamiltonian system representation is explored. The exact discretization method is applied to the port-Hamiltonian system which transforms the system obtaining discrete version of the port-Hamiltonian system representation is explored. The exact discretization method is applied to the port-Hamiltonian system which transforms the system discretization method applied the port-Hamiltonian which transforms the from a continuous to ais discrete state space setting. The system development of the discretized port discretization method is discrete applied to to thespace port-Hamiltonian system which of transforms the system system from a continuous to a state setting. The development the discretized port from a continuous continuous toformulation a discrete discrete state state space setting. setting. The for development of and the computer discretizedbased port Hamiltonian systemto is necessary foundation the discrete from a a space The development of the discretized port Hamiltonian system formulation is necessary foundation for the discrete and Hamiltonian system formulation is necessary foundation forby thenumerical discrete simulations. and computer computer based based regulator andsystem model formulation realizations. is The results are illustrated Hamiltonian necessary foundation for the discrete and computer based regulator and model realizations. The results are illustrated by numerical simulations. regulator and model realizations. The results are illustrated by numerical simulations. regulator and model realizations. The results are illustrated by numerical simulations. © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Port-Hamiltonian, Exact Discretization, Undamped Wave Equation System. Keywords: Keywords: Port-Hamiltonian, Port-Hamiltonian, Exact Exact Discretization, Discretization, Undamped Undamped Wave Wave Equation Equation System. System. Keywords: Port-Hamiltonian, Exact Discretization, Undamped Wave Equation System. 1. INTRODUCTION However, most of the studies of the port-Hamiltonian 1. INTRODUCTION INTRODUCTION However, most most of of the the studies studies of of the the port-Hamiltonian 1. However, formulation are of developed for continuous finite and/or 1. INTRODUCTION However, most the studies of the port-Hamiltonian port-Hamiltonian formulation are developed for continuous finite and/or and/or formulation are developed for continuous finite infinite dimensional systems. For example, M. Seslija et formulation are developed for continuous finite and/or The port-Hamiltonian framework is an energy-based rep- infinite dimensional systems. For example, M. Seslija et The port-Hamiltonian port-Hamiltonian framework framework is is an an energy-based energy-based reprep- infinite dimensional systems. For example, M. Seslija et al. research on discrete exterior geometry approach for infinite dimensional systems. For example, M. Seslija et The resentation of physical systems and captures the energy The port-Hamiltonian is an energy-based rep- al. research on discrete exterior geometry approach for resentation of physical physical framework systems and and captures the energy energy al. research exterior approach for time-continuous spatially-discrete port-Hamiltonian sysal. research on on discrete discrete exterior geometry geometry approach sysfor resentation of captures and dynamical features systems in an unified setting. the Nowadays, resentation of physical systems and captures the energy time-continuous spatially-discrete port-Hamiltonian and dynamical features in an unified setting. Nowadays, time-continuous spatially-discrete port-Hamiltonian system (Seslija et al. (2012)). In particular, the discretization time-continuous spatially-discrete port-Hamiltonian sysand dynamical features in an unified setting. Nowadays, the application of the port-Hamiltonian formulation has tem (Seslija et al. (2012)). In particular, the discretization and dynamical features in an unified setting. Nowadays, the application application of of the the port-Hamiltonian port-Hamiltonian formulation formulation has has tem (Seslija (2012)). In the of distributed parameter system is a difficult and chal(Seslija et et al. al. (2012)). system In particular, particular, the discretization discretization the beenapplication widely and of successfully explored within lump and disthe the port-Hamiltonian formulation has tem of distributed distributed parameter is aa difficult difficult and chalchalbeen widely and successfully explored within lump and disof parameter system is and lenging problem. Traditionally, the approximate represenof distributed parameter system is a difficult and chalbeen widely and successfully explored within lump and distributed parameter systems setting (Maschke and van der been widely and successfully explored within lump and dislenging problem. Traditionally, the approximate representributed parameter parameter systems systems setting setting (Maschke (Maschke and and van van der der lenging Traditionally, the approximate tation ofproblem. the original continuous time system is represenobtained lenging problem. Traditionally, the approximate representributed Schaft (2005); Maschke and Van der(Maschke Schaft (1991); Ortega tributed parameter systems setting and van der tation of of the the original original continuous continuous time time system system is is obtained obtained Schaft (2005); (2005); Maschke Maschke and and Van Van der der Schaft Schaft (1991); (1991); Ortega Ortega tation by using numerical simulation techniques such as Euler, of the originalsimulation continuoustechniques time system obtained Schaft et al. (2002); Van der Schaft and der Maschke (2002)). In par- tation Schaft (2005);Van Maschke and Van Schaft(2002)). (1991); Ortega by using using numerical suchis as as Euler, et al. (2002); der Schaft and Maschke In parby numerical simulation techniques such Euler, Runge-Kutta, etc (Kazantzis and Kravaris (1999); Golo by using numerical simulation techniques such as Euler, et al. (2002); Van der Schaft and Maschke (2002)). In particular, the application of theand port-Hamiltonian formalism et al. (2002); Van der Schaft Maschke (2002)). In par- Runge-Kutta, etc (Kazantzis and Kravaris (1999); Golo ticular, the application application of the the port-Hamiltonian port-Hamiltonian formalism Runge-Kutta, etc (Kazantzis and Kravaris (1999); Golo et al. (2004); Bassi et al. (2007)). These methods have Runge-Kutta, etc (Kazantzis and Kravaris (1999); Golo ticular, the of formalism to wave equation system captures the energy balance featicular, the application of captures the port-Hamiltonian formalism et al. al. (2004); (2004); Bassi Bassi et et al. al. (2007)). (2007)). These These methods methods have have to wave equation system the energy balance feaet the disadvantage that the accuracy of the approximate et al. (2004); Bassi et al. (2007)). These methods have to wave equation system the balance tures of dynamic systems and provides the interconnection to wave equation systems system captures captures the energy energy balance feafea- the disadvantage that the accuracy of the approximate tures of dynamic dynamic and provides provides the interconnection interconnection disadvantage that the of approximate discrete time system rapidly deteriorates as the sampling the disadvantage thatrapidly the accuracy accuracy of the the approximate tures of structure related systems with theand exchange ofthe energy among the the tures of dynamic systems and provides the interconnection discrete time system system deteriorates as the the sampling structure related related with with the the exchange exchange of of energy energy among among the the discrete time rapidly deteriorates as sampling period increases. Thus, an accurate and exact representadiscrete time system rapidly deteriorates as the sampling structure system andrelated the environment through ports (Van der Schaft structure with the exchange of energy the period increases. Thus, an accurate and exact representasystem and and the the environment environment through ports ports (Vanamong der Schaft Schaft period Thus, an and representation of increases. the original continuous time system is required for period Thus, an accurate accurate and exact exact representasystem through (Van der and Maschke (2002); Le Gorrec et ports al. (2005); Duindam system and the environment through (Van der Schaft tion of of increases. the original original continuous time system system is required required for and Maschke (2002); Le Gorrec et al. (2005); Duindam tion the continuous time is for accurate discrete representation of the distributed paramtion of the original continuous time system is required for and et al.Maschke (2009)). (2002); Le Gorrec et al. (2005); Duindam accurate discrete representation of the distributed paramand et al. al.Maschke (2009)). (2002); Le Gorrec et al. (2005); Duindam accurate discrete eter system. accurate discrete representation representation of of the the distributed distributed paramparamet (2009)). et al. (2009)). eter system. system. In engineering applications, there are three common phys- eter eter system. In engineering engineering applications, applications, there there are are three three common common physphys- The contribution of this paper is the exploration of a In ical systems that carry waves: the electrical transmisIn applications, there the are three common phys- The contribution of this paper is the exploration of a icalengineering systems that that carry waves: waves: electrical transmiscontribution of paper the proper discretization method in is order to preserve of thea The contribution of this this paperin isorder the exploration exploration ical systems carry the electrical transmission systems line, the that flexible string, andthe the electrical compressible fluid. The ical carry waves: transmisproper discretization method to preserve preserve ofthe thea sion line, line, the the flexible flexible string, string, and and the the compressible compressible fluid. fluid. proper discretization method in order to intrinsic energy and dynamical characteristics of the disproper discretization method in order to preserve the sion The wave equation of the above systems is described by sion line, the flexible and systems the compressible fluid. intrinsic energy energy and and dynamical dynamical characteristics characteristics of of the the disdisThe wave wave equation of string, the above above is described described by intrinsic tributed energy parameter system. The novelty of the design intrinsic and dynamical characteristics of the disThe equation of the systems is by the hyperbolic partial differential equation (PDE) derived The wave equation of the above systems is described by tributed parameter system. The novelty of the design the hyperbolic hyperbolic partial partial differential differential equation equation (PDE) (PDE) derived derived tributed system. The novelty of the design procedureparameter lies in the fact that spatial discretization and/or tributed parameter system. The novelty of the design the fromhyperbolic the mass partial and/or differential energy balance. It is(PDE) a distributed the equation derived procedure lies lies in in the fact fact that that spatial spatial discretization discretization and/or and/or from the the mass mass and/or and/or energy energy balance. balance. It It is is aa distributed distributed procedure any other lies type ofthe spatial approximation of the system is fact that spatial discretization and/or from parameter system and energy due tobalance. its spatial nature it rep- procedure from the mass and/or It isnature a distributed any other other type typeinof ofthe spatial approximation of the the system system is parameter system and due to its spatial it repany spatial approximation of is not considered and the undamped wave equation system any other type of spatial approximation of the system is parameter system and due to its spatial nature it represents one of interesting systems realizations to be adparameter and duesystems to its realizations spatial nature not considered considered and and the the undamped undamped wave wave equation equation system system resents one onesystem of interesting interesting to it be repad- not is completely captured with the proposed transformations not consideredcaptured and thewith undamped wave equation system resents of systems realizations to be addressed one within the port-Hamiltonian setting (Duindam resents of interesting systems realizations to be adis completely the proposed transformations dressed within within the the port-Hamiltonian port-Hamiltonian setting setting (Duindam (Duindam is completely captured the transformations from a continuous to awith discrete state space setting. The completely captured with the proposed proposed transformations dressed et al. (2009)). An important issue of the distributed pa- is dressed within the port-Hamiltonian setting (Duindam from a continuous to a discrete state space setting. The et al. al. (2009)). (2009)). An An important important issue issue of of the the distributed distributed papa- from aa continuous to space discretization is realized by the state Cayley-Tustin time The disfrom continuous to aa discrete discrete state space setting. setting. The et rameter systemAn presented by the port-Hamiltonian frameet al. (2009)). important issue of the distributed padiscretization is realized by the Cayley-Tustin time disrameter system system presented presented by by the the port-Hamiltonian port-Hamiltonian frameframe- discretization is realized by the Cayley-Tustin time discretization transformation and is applied to the portdiscretization is realized by the Cayley-Tustin time disrameter work is that boundary controlled formulations are mainly rameter system presented by the port-Hamiltonian frame- cretization transformation and is applied to the portwork is is that that boundary controlled formulations are are mainly mainly cretization and is to the Hamiltoniantransformation framework of the wave equation system in cretization and wave is applied applied to system the portportwork boundary controlled formulations presented as realizations since actuation is usually realized work is that boundary controlled formulations are mainly Hamiltoniantransformation framework of of the the equation in presented as realizations since actuation is usually realized Hamiltonian framework wave equation system in this work (Havu and Malinen (2007)). Hamiltonian framework of the wave equation system in presented as actuation is at the boundary of the since system and rarely as in realized domain this work (Havu and Malinen (2007)). presented as realizations realizations actuation is usually usually at the the boundary boundary of the the since system and rarely rarely as in in realized domain this work (Havu and (2007)). this and Malinen Malinen (2007)). at of system and as domain actuation (Le Gorrec et al. (2005)). The standard method at the boundary of the system and rarely as in domain The work paper(Havu is organized as follows. In Section 2, we inactuation (Le Gorrec et al. (2005)). The standard method The paper is organized as follows. In Section Section 2, 2, we we inactuation (Le Gorrec et al. (2005)). The standard method to resolve the issue of boundary/point actuation is to apply actuation (Le Gorrec et al. (2005)). The standard method The paper is organized as follows. In troduce and present the port-Hamiltonian formulation of The paper is organized as follows. In Section 2, we ininto resolve resolve the the issue issue of of boundary/point boundary/point actuation actuation is is to to apply apply troduce and present the port-Hamiltonian formulation of to boundary transformation which maps actuation boundaryiscontrol to troduce to resolve the issue of boundary/point to apply and present the port-Hamiltonian formulation of the undamped wave equation system. In section 3, we deal troduce and present the port-Hamiltonian formulation of boundary transformation which maps boundary control to the undamped undamped wave wave equation equation system. system. In In section 3, 3, we we deal boundary which control to in-domain transformation control for the port-Hamiltonian system (Zwart boundary which maps maps boundary boundary control to the withundamped the boundary controlled port-Hamiltonian In the wavecontrolled equation port-Hamiltonian system. In section section system. 3, we deal deal in-domain transformation control for for the the port-Hamiltonian port-Hamiltonian system (Zwart with the boundary system. In in-domain control system (Zwart and Curtain (1995), Moghadam et al. (2013)). in-domain control for the port-Hamiltonian system (Zwart with the boundary controlled port-Hamiltonian system. In with the boundary controlled port-Hamiltonian system. In and Curtain Curtain (1995), (1995), Moghadam Moghadam et et al. al. (2013)). (2013)). and and Curtain (1995), Moghadam et al. (2013)). Copyright 2016 International Federation of 311 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Copyright 2016 responsibility International of Federation of Federation of Automatic 311Control. Peer review© under International Automatic Control Copyright © 2016 International Federation of 311 Copyright © 2016 International Federation of 311 Automatic Control 10.1016/j.ifacol.2016.07.459 Automatic Control Control Automatic
311 311 311 311
IFAC CPDE 2016 310 June 13-15, 2016. Bertinoro, Italy
Qingqing Xu et al. / IFAC-PapersOnLine 49-8 (2016) 309–314
section 4, we present the Cayley-Tustin time discretization of the port-Hamiltonian system. Finally, simulation results of the undamped wave equation system are shown in Section 5. 2. INFINITE DIMENSIONAL PORT-HAMILTONIAN REPRESENTATION OF WAVE EQUATION SYSTEM In this section, the port-Hamiltonian representation of the distributed parameter system with boundary control is introduced and the wave equation described by the hyperbolic PDEs is considered. 2.1 Infinite Dimensional Port-Hamiltonian System
The operator given by ∂ (Hx) + P0 (Hx) ∂t
The the displacement w(ζ, t) is the output y(ζ, t) = w(ζ, t) of the above transformed system and is obtained by the following equation: ζ x1 (η, t)dη (12) y(ζ, t) = 0
The general port-Hamiltonian system representation is given in the following form (Jacob and Zwart (2012)): ∂ ∂x (ζ, t) = P1 [H(ζ)x(ζ, t)] + P0 [H(ζ)x(ζ, t)] (1) ∂t ∂ζ with boundary condition: f (t) = u(t) (2) WB ∂ e∂ (t) where 1 P1 −P1 H(L)x(L, t) f∂ (t) √ = (3) e∂ (t) H(0)x(0, t) 2 I I P1 ∈ K n×n is invertible and self-adjoint; P0 ∈ K n×n is skew-adjoint; H ∈ C 1 ([0, L]; K n×n ), H(ζ) is self-adjoint for all ζ ∈ [0, L] and mI ≤ H(ζ) ≤ M I for all ζ ∈ [0, L] n×2n and some m, M > 0 independent of ζ; W has B ∈ K 0 I ∗ . full rank and WB ΣWB ≥ 0, where Σ = I 0 Ax := P1
where the the displacement w(ζ, t) is state variable. After some basic manipulation and transformation of the system, the above set of equations is transformed to the following form by defining state variables as x1 (ζ, t) = ∂w(ζ,t) ∂w(ζ,t) , so that the system becomes: ∂ζ , x2 (ζ, t) = ∂t ∂ x1 (ζ, t) 0 1 ∂ x1 (ζ, t) = (11) 1 0 ∂ζ x2 (ζ, t) ∂t x2 (ζ, t)
(4)
with domain D(A) = {x ∈ L2 ([0, L]; K n )|Hx ∈ H 1 ([0, L]; K n )} (5) generates a contraction semigroup on the state space (6) X = L2 ([0, L]; K n ) equipped with the inner product 1 L ∗ g (ζ)H(ζ)f (ζ)dζ (7) < f, g >X = 2 0 associated norm is denoted by|| · ||X .
The associated Hamiltonian E : [0, L] → R which represents the energy of the system is given by 1 L ∗ E(t) = x (ζ, t)H(ζ)x(ζ, t)dζ (8) 2 0 2.2 Undamped Wave Equation System Let us consider the case of undamped wave equation which takes the following PDE form (Zwart and Curtain (1995)): ∂ 2 w(ζ, t) ∂ 2 w(ζ, t) = (9) ∂t2 ∂ζ 2 where w ∈ [0, L]⊕R1 . The initial conditions and boundary conditions are as follows: ∂w (ζ, 0) = ψ(ζ) (10) t = 0, w(ζ, 0) = φ(ζ), ∂t w(0, t) = 0 = w(1, t) 312
The system described by Eq.11 can be rewritten in the port-Hamiltonian form: ∂x ∂ (ζ, t) = P1 [H(ζ)x(ζ, t)] + P0 [H(ζ)x(ζ, t)] (13) ∂t ∂ζ y(t) = Cx(ζ, t) 10 0 1 x1 (ζ, t) , P1 = , , P0 = 0, H = where x = x2 (ζ, t) 01 1 0 ζ C = [C1 C2 ], here C1 = 0 (·)dη and C2 = 0. The associated Hamiltonian is given by: 1 L E(t) = x(ζ, t)∗ H(ζ)x(ζ, t)dζ (14) 2 0 1 L = [x1 (ζ, t)2 + x2 (ζ, t)2 ]dζ 2 0 and the following balance equation holds: 1 dE(t) = [(H(ζ)x(ζ, t))∗ P1 (H(ζ)x(ζ, t))]L (15) 0 dt 2 = x1 (L)x2 (L) − x1 (0)x2 (0)
Remark 1: If the boundary condition x2 (0, t) = x2 (L, t) = 0 holds, the derivative of the energy along solutions is zero dE(t) dt = 0, which implies that the system does not dissipate energy. Remark 2: From Eq15, it can be seen that the changes of the overall energy content only occur via the boundary. One can consider the boundary control system with one boundary input: (16) x2 (0, t) = 0 x2 (L, t) = u(t) The boundary effort and boundary flow are given as follows: 1 e∂ (t) = √ [Hx(L, t) + Hx(0, t)] (17) 2 1 x1 (L, t) + x1 (0, t) =√ 2 x2 (L, t) + x2 (0, t) 1 f∂ (t) = √ [P1 Hx(L, t) − P1 Hx(0, t)] (18) 2 1 x2 (L, t) − x2 (0, t) =√ 2 x1 (L, t) − x1 (0, t) then the boundary condition becomes: 1 −1 0 0 1 f∂ (t) f∂ (t) u(t) = √ = WB e∂ (t) e∂ (t) 2 1 0 0 1 −1 0 0 1 . with WB = √12 1 0 0 1
IFAC CPDE 2016 June 13-15, 2016. Bertinoro, Italy
Qingqing Xu et al. / IFAC-PapersOnLine 49-8 (2016) 309–314
The proposed representation in the port-Hamiltonian framework can be easily utilized in the distributed parameter system described by hyperbolic PDEs with boundary control. 3. BOUNDARY CONTROL OF PORT-HAMILTONIAN SYSTEM In this section, we introduce the transformation of boundary control to in-domain control and apply the control to port-Hamiltonian system (Zwart and Curtain (1995); Jacob and Zwart (2012)). The port-Hamiltonian described in Eq.13 with the boundary condition in Eq.16 can be extended to the abstract control system of the form: x(ζ, ˙ t) = Ax(ζ, t), x(ζ, 0) = x0 (19) y(t) = Cx(ζ, t) Bx(ζ, t) = u(t) where A : D(A) ⊂ X → X is linear, the control function u takes values in the Hilbert space U , the boundary operator B : D(B) ⊂ X → U is linear and satisfies D(A) ⊂ D(B) and the operator C : D(C) → X. The above control system is a boundary control system if the following holds: 1.The operator A : D(A) → X with D(A) = D(A) ∩ ker(B) and Ax = Ax x ∈ D(A) (20) is the infinitesimal generator of a C0 -semigroup on X; 2.There exists an operator Bb ∈ L(U, X) such that for all u ∈ U we have Bb u ∈ D(A), ABb ∈ L(U, X) and BBb u = u u ∈ U (21) Defining the following state transformation: v(ζ, t) = x(ζ, t) + Bb (ζ)u(t), v(ζ, 0) = v0
311
according to the boundary control in Eq.16, one can obtain: Bb2 (0) = 0 (28) Bb2 (L) = −1
b2 (ζ) Thus, one can have ∂B∂ζ = −1. Then, according to the extended system described in Eq.25, one can obtain: ∂v2 (ζ, t) v˙ 1 (ζ, t) = + u(t) (29) ∂ζ ∂v1 (ζ, t) + Bb2 (ζ)¯ u(t) v˙ 2 (ζ, t) = ∂ζ u(t) Here, let us define the input as u ¯(t) = and u ˜(t) 1 0 , then, the extended system described B = 0 Bb2 (ζ) in Eq.25 becomes: v(ζ, ˙ t) = Av(ζ, t) + B u ¯(t), v(ζ, 0) = v0 (30) y(t) = Cv(ζ, t) + D¯ u(t)
here the operator D = 0. The above extended system described in Eq.30 maps boundary control to in-domain control. 4. EXACT DISCRETIZATION OF PORT-HAMILTONIAN SYSTEM In this section, we propose an exact time discretization transformation from a continuous to a discrete state space setting. The Cayley-Tustin time discretization transformation is applied to the port-Hamiltonian framework of the undamped wave equation system. 4.1 Exact System Discretization
(22)
here Bb (ζ) = [Bb1 (ζ)Bb2 (ζ)]T . Then, the boundary control system described in Eq.19 can be transferred to the abstract differential equation: v(ζ, ˙ t) = Av(ζ, t) − ABb u(t) + Bb u(t) ˙ (23) where the operator A is given by: ∂ (24) Ax := P1 (Hx) + P0 (Hx) ∂ζ Assuming u ˜(t) = u(t), ˙ the extension of the system is in the following form: u(t) I u(t) ˙ 0 0 + u ˜(t) = v(ζ, t) Bb (ζ) v(ζ, ˙ t) −ABb (ζ) A(ζ) u(t) y(t) = [0 C(ζ)] (25) v(ζ, t)
For the undamped wave equation system described in Eq.13, we define: (26) v1 (ζ, t) = x1 (ζ, t) v2 (ζ, t) = x2 (ζ, t) + Bb2 (ζ)u(t) thus, Bb1 (ζ) = 0. The obtained in-domain control system is designed with homogeneous boundary condition as below: v2 (0, t) = x2 (0, t) + Bb2 (0)u(t) = 0 (27) v2 (L, t) = x2 (L, t) + Bb2 (L)u(t) = 0
313
The continuous system considered in this section is the indomain port-Hamiltonian system described in Eq.30 and let us rewrite the continuous system as follows: v(ζ, ˙ t) = Av(ζ, t) + B u ¯(t), v(ζ, 0) = v0 (31) y(t) = Cv(ζ, t) + D¯ u(t) a slightly non-standard time discretization type is given by (Havu and Malinen (2007)): v(jh) − v((j − 1)h) v(jh) + v((j − 1)h) ≈A + Bu ¯(jh) h 2 v(jh) + v((j − 1)h) y(jh) ≈ C + D¯ u(jh) (32) 2 √ √ ¯(jh) and yjh / h be let u ¯hj / h be the approximation of u the approximation of y(jh), the above equation becomes: h h vjh − vj−1 vjh + vj−1 u ¯hj =A + B√ h 2 h h h h yj u ¯hj vj + vj−1 √ =C + D√ 2 h h
(33)
Applying the Cayley transform, the discrete system is given in the following form: v(ζ, k) = Ad v(ζ, k − 1) + Bd u ¯(k) (34) y(ζ, k) = Cd v(ζ, k − 1) + Dd u ¯(k)
IFAC CPDE 2016 312 June 13-15, 2016. Bertinoro, Italy
Qingqing Xu et al. / IFAC-PapersOnLine 49-8 (2016) 309–314
where √ −1 −1 Ad Bd [δ + A][δ − A] 2δ[δ − A ] B −1 = √ Cd D d 2δC[δ − A]−1 G(δ)
(2005)). The discrete port-Hamiltonian system captures the energy and dynamical features of the system.
G(δ) is the transfer function:
G(δ) = C[δ − A−1 ]−1 B + D here σ = 2/h and operator A−1 is the Yosida extension of operator A.
2 0 -2 -4 0 0.2 0.4 0.6
ζ
0.8 1
0
10
20
30
40
50
k
Fig. 1. Evolution of discrete state x1 (ζ, t) in the undamped wave equation system given by Eq.40.
0
4.2 Resolvent Operator Let us consider the port-Hamiltonian system described in Eq.31: v(ζ, ˙ t) = Av(ζ, t) ∂ where Av := P1 ∂ζ (Hv) + P0 (Hv) and the domain of the operator A is D(A) = {v ∈ L2 (0, 1)|v is absolutely continuous, dv dζ ∈ L2 (0, 1), v(0) = 0}. The resolvent of the operator A can be obtained from Laplace transform: v(ζ, s) = [sI − A]−1 v(ζ, 0) (35) here the resolvent of operator A is R(s, A) = [sI − A]−1 .
One can also obtain the Laplace transform of the system as follows: ∂ sv(ζ, s) − v(ζ, 0) = P1 [Hv(ζ, s)] + P0 [Hv(ζ, s)] (36) ∂ζ then ∂ P1 [Hv(ζ, s)] = [sI − P0 ][Hv(ζ, s)] − v(ζ, 0) (37) ∂ζ the above equation is a set of ordinary differential equations: ∂v1 (ζ, s) = sv2 (ζ, s) − v2 (ζ, 0) ∂ζ ∂v2 (ζ, s) = sv1 (ζ, s) − v1 (ζ, 0) ∂ζ Remark 3: According Theorem 12.3.11 in O. Staffans’s book, the discrete system obtained from Cayley-Tustin transformation is energy preserving system (Staffans 314
3 2 x2(ζ,k)
x1(ζ,k)
4
The resolvent of the operator A operate on v(ζ, 0) is obtained from the solution of the above set of equations when the boundary condition v2 (0, s) = v2 (L, s) = 0, and it is in the following form: (38) R(s, A)v(ζ, 0) = [sI − A]−1 v(ζ, 0) R11 (ζ) R12 (ζ) v1 (ζ, 0) = R21 (ζ) R22 (ζ) v2 (ζ, 0) where cosh(sζ) 1 cosh[s(1 − η)](·)dη R11 (·) = sinh(s) 0 ζ − sinh[s(ζ − η)](·)dη 0 cosh(sζ) 1 R12 (·) = sinh[s(1 − η)](·)dη sinh(s) 0 ζ − cosh[s(ζ − η)](·)dη 0 sinh(sζ) 1 R21 (·) = cosh[s(1 − η)](·)dη sinh(s) 0 ζ cosh[s(ζ − η)](·)dη − 0 sinh(sζ) 1 R22 (·) = sinh[s(1 − η)](·)dη sinh(s) 0 ζ − sinh[s(ζ − η)](·)dη
1 0 -1 -2 -3 0 0.2 0.4 0.6
ζ
0.8 1
0
10
20
30
40
50
k
Fig. 2. Evolution of discrete state v2 (ζ, t) in the undamped wave equation system given by Eq.40. 4.3 Discrete Operator Let us rewrite the continuous system described in Eq.31: v(ζ, ˙ t) = Av(ζ, t) + B u ¯(t), v(ζ, 0) = v0 (39) y(t) = Cv(ζ, t) + D¯ u(t) 1 0 ∂ , where Av := P1 ∂ζ (Hv) + P0 (Hv), B = 0 Bb2 (ζ) ζ C = [C1 C2 ], here C1 = 0 (·)dη and C2 = 0, and D = 0.
IFAC CPDE 2016 June 13-15, 2016. Bertinoro, Italy
Qingqing Xu et al. / IFAC-PapersOnLine 49-8 (2016) 309–314
The discrete system obtained by the exact discretization approach is given in Eq.34. With the resolvent of the operator A obtained Eq.38, one can obtain the discrete system as follows: v(ζ, k) = Ad v(ζ, k − 1) + Bd u ¯(k), y(ζ, k) = Cd v(ζ, k − 1) + Dd u ¯(k)
v(0) = v0
(40)
Here, discrete operator Ad operate on state v(ζ, k − 1) is given as follows: Ad v(ζ, k − 1) = [δ + A][δ − A]−1 v(ζ, k − 1) (41) −1 = [−I + 2δ[δ − A] ]v(ζ, k − 1) Ad11 (ζ) Ad12 (ζ) = Ad21 (ζ) Ad22 (ζ) where cosh(sζ) 1 Ad11 (·) = − (·) + 2δ cosh[s(1 − η)](·)dη sinh(s) 0 ζ − sinh[s(ζ − η)](·)dη 0 cosh(sζ) 1 Ad12 (·) =2δ sinh[s(1 − η)](·)dη sinh(s) 0 ζ − cosh[s(ζ − η)](·)dη 0 sinh(sζ) 1 Ad21 (·) =2δ cosh[s(1 − η)](·)dη sinh(s) 0 ζ cosh[s(ζ − η)](·)dη − 0 sinh(sζ) 1 Ad22 (·) = − (·) + 2δ sinh[s(1 − η)](·)dη sinh(s) 0 ζ − sinh[s(ζ − η)](·)dη
√ ζ cosh(sφ) 1 Cd11 (·) = 2δ cosh[s(1 − η)](·)dη sinh(s) 0 0 φ sinh[s(φ − η)](·)dη dφ − 0 √ ζ cosh(sφ) 1 Cd12 (·) = 2δ sinh[s(1 − η)](·)dη sinh(s) 0 0 φ cosh[s(φ − η)](·)dη dφ − 0
Finally, discrete operator Dd is given as follows: Dd = C[δ − A−1 ]−1 B where ζ cosh(sφ) 1 sinh[s(1 − η)](·)dη Dd = − sinh(s) 0 0 φ − cosh[s(φ − η)](·)dη dφ
The resolvent operator and discrete operators of the portHamiltonian system are presented in this section. The exact time discretization transformation from the continuous to the discrete state space setting is achieved by the Cayley-Tustin time discretization transformation. 5. SIMULATION RESULTS In the simulation study, we choose discrete time and space parameters as dt = 0.5 and dζ = 0.01. The initial conditions of the discrete system described in Eq.40 are v01 (ζ) = 0 and v02 (ζ) = 2 sin(2πζ). The inputs are chosen as u(k) = 0.314 cos(0.314k) and u ˜(k) = u(k) ˙ = 0.099 sin(0.314k). Fig.1-4 gives the simulation results of discrete undamped wave equation system. The states x1 (ζ, k), x2 (ζ, k) and v2 (ζ, k) with one boundary input which operates on x2 (L, k) are given in Fig.1-3. And the output y(ζ, k), which is the displacement w(ζ, k) of the undamped wave equation system is given in Fig.4.
(42)
2
where
−
Bd2
√
2δ ζ
cosh(sζ) sinh(s)
0
1
v2(ζ,k)
Bd1 = −
(44)
0
0
Discrete operator Bd is given as follows: √ Bd = 2δ[δ − A−1 ]−1 B Bd1 (ζ) = Bd2 (ζ)
313
sinh[s(1 − η)](·)dη
1 0 -1
cosh[s(ζ − η)](·)dη √ sinh(sζ) 1 = − 2δ sinh[s(1 − η)](·)dη sinh(s) 0 ζ sinh[s(ζ − η)](·)dη − 0
-2 0
50 40
0.2
30
0.4
20
0.6
ζ
10
0.8 1
0
Similarly, discrete operator Cd operate on v(ζ, k − 1) is given as follows: √ Cd v(ζ, k − 1) = 2δC[δ − A]−1 v(ζ, k − 1) (43) = [ Cd11 (ζ) Cd12 (ζ) ] where 315
k
0
Fig. 3. Evolution of discrete state x2 (ζ, t) in the undamped wave equation system given by Eq.40. Since the boundary control is transferred to in-domain control, it can be seen from Fig.2 that the state x2 (ζ, k) is with one boundary control u(k) = 0.314 cos(0.314k) at
IFAC CPDE 2016 314 June 13-15, 2016. Bertinoro, Italy
Qingqing Xu et al. / IFAC-PapersOnLine 49-8 (2016) 309–314
ζ = L. In addition, the transformed state v2 (ζ, k) is with homogeneous boundaries, see Fig.3. Finally, Fig.4 t gives the output y(ζ, k) with one boundary control 0 u(τ )dτ at ζ = L, which is y(L, k) = sin(0.314k).
1
y(ζ,k)
0.5 0 -0.5 -1 -1.5 0
50 0.2
40 0.4
30 0.6
ζ
20 0.8
10 1
k
0
Fig. 4. Evolution of discrete output y(ζ, t) in the undamped wave equation system given by Eq.40. It can be seen from the simulation results that the exact discretization captures the distributed nature of the hyperbolic PDEs system obtained from the undamped wave equation system. 6. CONCLUSION The application of the port-Hamiltonian framework to the undamped wave equation system captures the energy balance features of dynamical systems. The undamped wave equation system derived from mass and energy balances is described by the port-Hamiltonian framework in this paper. Then, the boundary transformation method which maps boundary control to in-domain control for the port-Hamiltonian system is presented. In this work, the Cayley-Tustin time discretization is applied to portHamiltonian system and the simulation results are given. The obtained discrete formulation provides a basis for the development of more advanced discrete controller formulations and discrete system modelling realizations which intrinsically capture and preserve energy of the underlying system dynamics. More advanced control and optimization schemes, such as model predictive control can be pursued in order to help the undamped wave equation system operate more efficiently. REFERENCES Bassi, L., Macchelli, A., and Melchiorri, C. (2007). An algorithm to discretize one-dimensional distributed port hamiltonian systems. In Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, 61–73. Springer. Duindam, V., Macchelli, A., Stramigioli, S., and Bruyninckx, H. (2009). Modeling and control of complex physical systems: the port-Hamiltonian approach. Springer Science & Business Media. Golo, G., Talasila, V., Van Der Schaft, A., and Maschke, B. (2004). Hamiltonian discretization of boundary control systems. Automatica, 40(5), 757–771. 316
Havu, V. and Malinen, J. (2007). The Cayley transform as a time discretization scheme. Numerical Functional Analysis and Optimization, 28(7-8), 825–851. Jacob, B. and Zwart, H. (2012). Linear port-Hamiltonian systems on infinite-dimensional spaces, volume 223. Springer Science & Business Media. Kazantzis, N. and Kravaris, C. (1999). Time-discretization of nonlinear control systems via Taylor methods. Computers & chemical engineering, 23(6), 763–784. Le Gorrec, Y., Zwart, H., and Maschke, B. (2005). Dirac structures and boundary control systems associated with skew-symmetric differential operators. SIAM Journal on Control and Optimization, 44(5), 1864–1892. Maschke, B. and van der Schaft, A. (2005). 4 compositional modelling of distributed-parameter systems. In Advanced topics in control systems theory, 115–154. Springer. Maschke, B. and Van der Schaft, A. (1991). Portcontrolled hamiltonian systems: modelling origins and systemtheoretic properties. Moghadam, A.A., Aksikas, I., Dubljevic, S., and Forbes, J.F. (2013). Boundary optimal (lq) control of coupled hyperbolic PDEs and ODEs. Automatica, 49(2), 526– 533. Ortega, R., Van Der Schaft, A., Maschke, B., and Escobar, G. (2002). Interconnection and damping assignment passivity-based control of port-controlled hamiltonian systems. Automatica, 38(4), 585–596. Seslija, M., van der Schaft, A., and Scherpen, J.M. (2012). Discrete exterior geometry approach to structurepreserving discretization of distributed-parameter porthamiltonian systems. Journal of Geometry and Physics, 62(6), 1509–1531. Staffans, O. (2005). Well-posed linear systems. 103. Cambridge University Press. Van der Schaft, A. and Maschke, B. (2002). Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics, 42(1), 166–194. Zwart, H. and Curtain, R. (1995). An introduction to infinite dimensional linear systems theory. Texts in Applied Mathematics, Springer-Verlag, New York.