on computing the distance to stability for matrices using linear dissipative hamiltonian systems abstract in this paper we consider the problem of computing the nearest stable matrix to an unstable one we propose new algorithms to solve this problem based on a reformulation using linear dissipative hamiltonian systems we show that a matrix a is stable if and only if it can be written as a j r q where j j t r 0 and q 0 that is r is positive semidefinite and q is positive definite this reformulation results in an equivalent optimization problem with a simple convex feasible set we propose three strategies to solve the problem in variables j r q i a block coordinate descent method ii a projected gradient descent method and iii a fast gradient method inspired from smooth convex optimization these methods require o n 3 operations per iteration where n is the size of a we show the effectiveness of the fast gradient method compared to the other approaches and to several state of the art algorithms

dirac nijenhuis structures in this paper we study the concept of dirac nijenhuis structures we consider them to be a pair where d is a dirac structure defined with respect to a lie bialgebroid a a and is a nijenhuis operator which defines a deformation of the lie algebroid structure of d in a compatible way the transformation can be considered to deform also the double of the lie bialgebroid which leads to the concept of dirac nijenhuis structures of type i or to not affect it leading to dirac nijenhuis structures of type ii we prove that the concept contains poisson nijenhuis structures as a particular case and provides another example for the case of kahler manifolds

brief paper on the addition of integral action to port controlled hamiltonian systems a technique that provides closed loop integral action depending on the passive outputs of port controlled hamiltonian systems is already available this paper addresses a new method that allows us to add integral action also on system variables having relative degree higher than one while still preserving the hamiltonian form and thus closed loop stability the new approach is applied to design speed regulation controllers for the permanent magnet synchronous motor closed loop stability and asymptotic rejection of unknown piecewise constant load torques are formally proved this theoretically predicted control system performance is illustrated via simulation experiments which also show that the properties hold under parameter uncertainties this is in line with the usual practice of including integral action in a controller with the aim of improving its closed loop robustness the fact that the method enhances the range of possible integral actions in the controller enriched with this robustness property allows us to assess it as a practically important complement to the well known interconnection and damping assignment techniques developed in the framework of port controlled hamiltonian systems

contact geometric descriptions of vector fields on dually flat spaces and their applications in electric circuit models and nonequilibrium statistical mechanics contact geometry has been applied to various mathematical sciences and it has been proposed that a contact manifold and a strictly convex function induce a dually flat space that is used in information geometry here such a dually flat space is related to a legendre submanifold in a contact manifold in this paper contact geometric descriptions of vector fields on dually flat spaces are proposed on the basis of the theory of contact hamiltonian vector fields based on these descriptions two ways of lifting vector fields on legendre submanifolds to contact manifolds are given for some classes of these lifted vector fields invariant measures in contact manifolds and stability analysis around legendre submanifolds are explicitly given throughout this paper legendre duality is explicitly stated in addition to show how to apply these general methodologies to applied mathematical disciplines electric circuit models and some examples taken from nonequilibrium statistical mechanics are analyzed

determinants of laplacians and a second limit formula in mathrm gl 3 on evalue le determinant du carre de l operateur de dirac sur des tores a 3 dimensions avec structure de spin ou de facon equivalente l operateur de laplace sur une fonction asymetrique automorphe

interconnection and damping assignment passivity based control of port controlled hamiltonian systems passivity based control pbc is a well established technique that has shown to be very powerful to design robust controllers for physical systems described by euler lagrange el equations of motion for regulation problems of mechanical systems which can be stabilized shaping only the potential energy pbc preserves the el structure and furthermore assigns a closed loop energy function equal to the difference between the energy of the system and the energy supplied by the controller thus we say that stabilization is achieved via energy balancing unfortunately these nice properties of el pbc are lost when used in other applications which require shaping of the total energy for instance in electrical or electromechanical systems or even some underactuated mechanical devices our main objective in this paper is to develop a new pbc theory which extends to a broader class of systems the aforementioned energy balancing stabilization mechanism and the structure invariance towards this end we depart from the el description of the systems and consider instead port controlled hamiltonian models which result from the network modelling of energy conserving lumped parameter physical systems with independent storage elements and strictly contain the class of el models

on the stability of the finite difference time domain method in this paper we give a necessary and sufficient condition for the stability of the finite difference time domain method fdtd method this is an explicit time stepping method that is used for solving transient electromagnetic field problems a necessary but not a sufficient condition for its stability is usually obtained by requiring that discrete fourier modes defined on the fdtd grid remain bounded as time stepping proceeds here we follow a different approach we rewrite the basic fdtd equations in terms of an iteration matrix and study the eigenvalue problem for this matrix from the analysis a necessary and sufficient condition for stability of the fdtd method follows moreover we show that for a particular time step the 2 norm of the fdtd iteration matrix is equal to the golden ratio

some remarks on the stability of time domain electromagnetic simulations aspects of the stability of time domain update schemes for electromagnetic problems are examined based on the consideration of system amplification matrices arising from the discretization of maxwell s equations in the sequence space we discuss time domain stability for a given discretization scale and under three different assumptions of decreasing generality general updates general leapfrog updates and reciprocal leapfrog updates

the matching conditions of controlled lagrangians and ida passivity based control this paper discusses the matching conditions resulting from the controlled lagrangians method and the interconnection and damping assignment passivity based control ida pbc method both methods have been presented recently in the literature as means to stabilize a desired equilibrium point of an euler lagrange respectively hamiltonian system in the context of mechanical systems with symmetry the original controlled lagrangians method is reviewed and an interpretation of the matching assumptions in terms of the matching of kinetic and potential energy is given secondly both methods are applied to the general class of underactuated mechanical systems and it is shown that the controlled lagrangians method is contained in the ida pbc method the lambda method as described in recent papers for the controlled lagrangians method transforming the matching conditions a set of non linear pdes into a set of linear pdes is discussed the method is used to transform the matching conditions obtained in the ida pbc method into a set of quadratic and linear pdes finally the extra freedom obtained in the ida pbc method with respect to the controlled lagrangians method is used to discuss the integrability of the closed loop system explicit conditions are derived under which the closed loop hamiltonian system is integrable leading to the introduction of gyroscopic terms

on the synthesis of boundary control laws for distributed port hamiltonian systems this paper is concerned with the energy shaping of 1 d linear boundary controlled port hamiltonian systems the energy casimir method is first proposed to deal with power preserving systems it is shown how to use finite dimensional dynamic boundary controllers and closed loop structural invariants to partially shape the closed loop energy function and how such controller finally reduces to a state feedback when dissipative port hamiltonian systems are considered the casimir functions do not exist anymore dissipation obstacle and the immersion via a dynamic controller reduction through invariants method cannot be applied the main contribution of this paper is to show how to use the same ideas and state functions to shape the closed loop energy function of dissipative systems through direct state feedback i e without relying on a dynamic controller and a reduction step in both cases the existence of solution and the asymptotic stability by additional damping injection of the closed loop system are proven the general theory and achievable closed loop performances are illustrated with the help of a concluding example the boundary stabilization of a longitudinal beam vibrations

on the relation between fdtd and fibonacci polynomials in this paper we show that the finite difference time domain method fdtd method follows the recurrence relation for fibonacci polynomials more precisely we show that fdtd approximates the electromagnetic field by fibonacci polynomials in ta where t is the time step and a is the first order maxwell system matrix by exploiting the connection between fibonacci polynomials and chebyshev polynomials of the second kind we easily obtain the courant friedrichs lewy cfl stability condition and we show that to match the spectral width of the system matrix the time step should be chosen as large as possible that is as close to the cfl upper bound as possible

parallel simultaneous stabilization of two systems governed by partial differential equations subject to actuator saturation abstract a parallel simultaneous stabilization of two systems governed by partial differential equation pde with conservation law subject to actuator saturation is considered and a method on the control design is proposed based on the orthogonal decomposition for a port controlled hamiltonian system an abstract approach to the parallel simultaneous stabilization of two systems governed by partial differential equation with conservation law subject to actuator saturation is established the study shows that the parallel simultaneous stabilization controller obtained in this paper is valid in the case of the systems governed by the pdes

dirac structures and boundary control systems associated with skew symmetric differential operators associated with a skew symmetric linear operator on the spatial domain a b we define a dirac structure which includes the port variables on the boundary of this spatial domain this dirac structure is a subspace of a hilbert space naturally associated with this dirac structure is an infinite dimensional system we parameterize the boundary port variables for which the c 0 semigroup associated with this system is contractive or unitary furthermore this parameterization is used to split the boundary port variables into inputs and outputs similarly we define a linear port controlled hamiltonian system associated with the previously defined dirac structure and a symmetric positive operator defining the energy of the system we illustrate this theory on the example of the timoshenko beam

gaussian spectral rules for the three point second differences i a two point positive definite problem in a semi infinite domain we suggest an approach to grid optimization for a second order finite difference scheme for elliptic equations a model problem corresponding to the three point finite difference semidiscretization of the laplace equation on a semi infinite strip is considered we relate the approximate boundary neumann to dirichlet map to a rational function and calculate steps of our finite difference grid using the pade chebyshev approximation of the inverse square root it increases the convergence order of the neumann to dirichlet map from second to exponential without increasing the stencil of the finite difference scheme and losing stability

hybrid port hamiltonian systems from parameterized incidence matrices to hybrid automata abstract this paper shows how to formally design a hybrid automaton model for a wide class of dissipative physical systems with sources and switching topology this method is based on a mathematical representation of the dynamic network graph and of its dual graph using the hybrid incidence matrix and on a constructive method for analyzing admissible and constrained configurations the port hamiltonian representation associated with the set of hybrid system configurations parameterized by the discrete state of the switches is synthesized to be part of the hybrid automaton of the system this is a further step towards a generic control synthesis for physical switching systems

hamiltonian systems discrete time approximation losslessness passivity and composability abstract in this paper a passive integrator dedicated to input output hamiltonian systems approximation is presented in a first step a discrete hamiltonian framework endowed with a lie derivative like formula is introduced it is shown that the discrete dynamics encodes energy conservation and passivity additionally the characterization of the discrete dynamics in terms of dirac structure is shown to be invariant by interconnection the class is thus composable networked systems belong to the class in a second step the discrete dynamics is considered as a one step integration method the method is shown to be convergent and provides a discrete time approximation of an input output hamiltonian system accordingly the discrete dynamics inherits intrinsic energetic characteristics storage function and dissipation rate from the original system the method is thus tagged as passive integrator as an illustration the closed loop behavior of interconnected subsystems and the stabilization of a rigid body spinning around its center of mass are presented

approximate dissipative hamiltonian realization and construction of local lyapunov functions the key in applying energy based control approach is to be able to express the system under consideration as a dissipative hamiltonian system i e to obtain dissipative hamiltonian realization dhr for the system in general the precise dhr form is hard to obtain for nonlinear dynamic systems when a precise dhr does not exist for a dynamic system or such a precise realization is difficulty to obtain it is necessary to consider its approximate realization this paper investigates approximate dhr and construction of local lyapunov functions for time invariant nonlinear systems it is shown that every nonlinear affine system has an approximate dhr if linearization of the system is controllable based on the diagonal normal form of nonlinear dynamic systems a new algorithm is established for the approximate dhr finally we present the concept of kth degree approximate lyapunov function and provide a method to construct such a lyapunov function example studies show that the methodology presented in this paper is very effective

networks of hamiltonian systems and feedback abstract motivated by the design of complex networks transmitting pulse like signals we investigate and develop a network theory for hamiltonian systems the main topics of this paper are articulated around the following result under a few natural assumptions a network of hamiltonian systems with energy conserving interactions will itself be hamiltonian for the symplectic form c l g k i c k 1 where is the symplectic form of the individual components of the network k encodes the network structure and c which we call the symplectic gain matrix is related to the non involutivity of the control vector fields a further analysis of this form will reveal how certain interactions result in effectively putting constraints on the system giving a new result in the flavor of the dirac theory of constrained hamiltonian systems but of a control theoretic origin throughout the paper we use the toda lattice to illustrate the results and find a new approach to the description of the periodic toda lattice as a hamiltonian system on the space of jacobi matrices

on an intrinsic formulation of time variant port hamiltonian systems in this contribution we present an intrinsic description of time variant port hamiltonian systems as they appear in modeling and control theory this formulation is based on the splitting of the state bundle and the use of appropriate covariant derivatives which guarantees that the structure of the equations is invariant with respect to time variant coordinate transformations in particular we will interpret our covariant system representation in the context of control theoretic problems typical examples are time variant error systems related to trajectory tracking problems which allow for a hamiltonian formulation furthermore we will analyze the concept of collocation and the balancing interaction of power flows in an intrinsic fashion

energy based lyapunov functions for forced hamiltonian systems with dissipation in this paper we propose a constructive procedure to modify the hamiltonian function of forced hamiltonian systems with dissipation in order to generate lyapunov functions for nonzero equilibria a key step in the procedure which is motivated from energy balance considerations standard in network modeling of physical systems is to embed the system into a larger hamiltonian system for which a series of casimir functions can be easily constructed interestingly enough for linear systems the resulting lyapunov function is the incremental energy thus our derivations provide a physical explanation to it an easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given some examples that illustrate the method are given

simultaneous stabilization of a set of nonlinear port controlled hamiltonian systems this paper investigates simultaneous stabilization of a set of nonlinear port controlled hamiltonian pch systems and proposes a number of results on the design of simultaneous stabilization controllers for the pch systems firstly the case of two pch systems is studied using the dissipative hamiltonian structural properties the two systems are combined to generate an augmented pch system with which some results on the control design are then obtained for the case that there exist parametric uncertainties in the two systems hamiltonian structures an adaptive simultaneous stabilization controller is proposed when there are external disturbances and parametric uncertainties in the two systems two simultaneous stabilization controllers are designed for the systems one is a robust controller and the other is a robust adaptive one secondly the case of more than two pch systems is investigated and a new result is proposed for the simultaneous stabilization of the systems finally two illustrative examples are studied by using the results proposed in this paper simulations show that the simultaneous stabilization controllers obtained in this paper work very well

global output regulation for a class of single input port controlled hamiltonian disturbed systems abstract the problem of global output regulation for a class of single input port controlled hamiltonian disturbed systems is studied a composite control method is developed by combining the damping injection and the nonlinear disturbance observer techniques together a series of coordinate transformation is presented to transform the system with mismatched disturbance into matched disturbance system by means of lyapunov stability theorems global stability analysis for the closed loop hamiltonian system is presented it is shown that the system output is regulated to zero asymptotically in the presence of mismatched disturbance without larger overshoots and longer settling time caused by integral action study on two examples with simulation results demonstrates the effectiveness of the proposed control method

dynamic extension is unnecessary for stabilization via interconnection and damping assignment passivity based control interconnection and damping assignment passivity based control ida pbc is a technique that regulates the behavior of nonlinear systems assigning a desired port hamiltonian structure to the closed loop this basic idea introduced seven years ago has turned out to be very successful and has provided solutions to a wide variety of physical problems although ida pbc is originally formulated as a static state feedback technique a natural question that arises is whether it is possible to extend the realm of applicability of the method by considering dynamic controllers more precisely is the set of plants that is stabilizable with static state feedback ida pbc smaller than the one stabilizable with dynamic ida pbc the main contribution of this paper is to prove that the answer to this question is unfortunately negative

feedback equivalence of input output contact systems abstract control contact systems represent controlled or open irreversible processes which allow us to represent simultaneously the energy conservation and the irreversible creation of entropy such systems systematically arise in models established in chemical engineering the differential geometric of these systems is a contact form in the same manner as the symplectic 2 form is associated to hamiltonian models of mechanics in this paper we study the feedback preserving the geometric structure of controlled contact systems and render the closed loop system again as a contact system it is shown that only a constant control preserves the canonical contact form hence a state feedback necessarily changes the closed loop contact form for strict contact systems arising from the modelling of thermodynamic systems a class of state feedback that shapes the closed loop contact form and contact hamiltonian function is proposed the state feedback is given by the composition of an arbitrary function and the control contact hamiltonian function the similarity with structure preserving feedback of input output hamiltonian systems leads to the definition of input output contact systems and to the characterization of the feedback equivalence of input output contact systems an irreversible thermodynamic process namely the heat exchanger is used to illustrate the results

the local structure of poisson manifolds varietes de poisson et applications decomposition structures de poisson lineaires approximation lineaire systemes hamiltoniens le probleme de linearisation groupes de fonction realisations et applications impulsion paires duales et groupes de jauge existence des realisations unicite des realisations les problemes des 3 corps restreints et autres exemples

pseudo spectral methods for the spatial symplectic reduction of open systems of conservation laws a reduction method is presented for systems of conservation laws with boundary energy flow it is stated as a generalized pseudo spectral method which performs exact differentiation by using simultaneously several approximation spaces generated by polynomials bases and suitable choices of port variables the symplecticity of this spatial reduction method is proved when used for the reduction of both closed and open systems of conservation laws for any choice of collocation points i e for any polynomial bases the symplecticity of some more usual collocation schemes is discussed and finally their accuracy on approximation of the spectrum on the example of the ideal transmission line is discussed in comparison with the suggested reduction scheme

numerical stability of nonorthogonal fdtd methods in this paper a sufficient test for the numerical stability of generalized grid finite difference time domain fdtd schemes is presented it is shown that the projection operators of such schemes must be symmetric positive definite without this property such schemes can exhibit late time instabilities the origin and the characteristics of these late time instabilities are also uncovered based on this study nonorthogonal grid fdtd schemes nfdtd and the generalized yee gy methods are proposed that are numerically stable in the late time for quadrilateral prism elements allowing these methods to be extended to problems requiring very long time simulations the study of numerical stability that is presented is very general and can be applied to most solutions of maxwell s equations based on explicit time domain schemes

a power based description of standard mechanical systems this paper is concerned with the construction of a power based modeling framework for mechanical systems mathematically this is formalized by proving that every standard mechanical system with or without dissipation can be written as a gradient vector field with respect to an indefinite metric the form and existence of the corresponding potential function is shown to be the mechanical analog of brayton and moser s mixed potential function as originally derived for nonlinear electrical networks in the early sixties in this way several recently proposed analysis and control methods that use the mixed potential function as a starting point can also be applied to mechanical systems

stabilization of input disturbed stochastic port hamiltonian systems via passivity it is a universal phenomenon that the inputs of a real stochastic process are disturbed however this phenomenon is often ignored when stochastic port hamiltonian system sphs is used for modeling in which only the disturbance on the states is considered for this reason an extension of sphs named input disturbed sphs id sphs is proposed to describe the inputs disturbance as well as the states one the definition and some properties of id sphs including structure invariance and stochastic passivity are presented one by one to construct the basic framework of id sphs based on this framework a passivity based controller is further developed to stabilize id sphs at its steady state the control strategy is demonstrated in a thermodynamic heat conduction process with the results supporting strongly the proposed id sphs theory

riemannian optimal control and model matching of linear port hamiltonian systems this paper presents a unified controller design method for h 2 optimal control and model matching problems of linear port hamiltonian systems the controller design problems are formulated as optimization problems on the product manifold of the set of skew symmetric matrices the manifold of the symmetric positive definite matrices and euclidean space a riemannian metric is chosen for the manifold in such a manner that the manifold is geodesically complete i e the domain of the exponential map is the whole tangent space for every point on the manifold in order to solve these problems the riemannian gradients of the objective functions are derived and these gradients are used to develop a riemannian steepest descent method on the product manifold the geodesic completeness of the manifold guarantees that all points generated by the steepest descent method are on the manifold numerical experiments illustrate that our method is able to solve the two specified problems

dirac structures on hilbert spaces and boundary control of distributed port hamiltonian systems abstract aim of this paper is to show how the dirac structure properties can be exploited in the development of energy based boundary control laws for distributed port hamiltonian systems usually stabilization of non zero equilibria has been achieved by looking at or generating a set of structural invariants namely casimir functions in closed loop since this approach fails when an infinite amount of energy is required at the equilibrium dissipation obstacle this paper illustrates a novel approach that enlarges the class of stabilizing controllers the starting point is the parametrization of the dynamics provided by the image representation of the dirac structure that is able to show the effects of the boundary inputs on the state evolution in this way energy balancing and control by state modulated source methodologies are extended to the distributed parameter scenario and a geometric interpretation of these control techniques is provided the theoretical results are discussed with the help of a simple but illustrative example i e a transmission line with an rlc load in both serial and parallel configurations in the latter case energy balancing controllers are not able to stabilize non zero equilibria because of the dissipation obstacle the problem is solved thanks to a boundary state modulated source

abstractions of hamiltonian control systems given a control system and a desired property an abstracted system is a reduced system that preserves the property of interest while ignoring modeling detail in previous work abstractions of linear and nonlinear control systems were considered while preserving reachability properties in this paper we consider the abstraction problem for hamiltonian control systems where in addition to the property of interest we also preserve the hamiltonian structure of the control system we show how the hamiltonian structure of control systems can be exploited to simplify the abstraction process we then focus on local accessibility preserving abstractions and provide conditions under which local accessibility properties of the abstracted hamiltonian system are equivalent to the local accessibility properties of the original hamiltonian control system

bounded stabilisation of stochastic port hamiltonian systems this paper proposes a stochastic bounded stabilisation method for a class of stochastic port hamiltonian systems both full actuated and underactuated mechanical systems in the presence of noise are considered in this class the proposed method gives conditions for the controller gain and design parameters under which the state remains bounded in probability the bounded region and achieving probability are both assignable and a stochastic lyapunov function is explicitly provided based on a hamiltonian structure although many conventional stabilisation methods assume that the noise vanishes at the origin the proposed method is applicable to systems under persistent disturbances

reduction principles and the stabilization of closed sets for passive systems in this technical note we explore the stabilization of closed invariant sets for passive systems and present conditions under which a passivity based feedback asymptotically stabilizes the goal set our results rely on novel reduction principles allowing one to extrapolate the properties of stability attractivity and asymptotic stability of a dynamical system from analogous properties of the system on an invariant subset of the state space

on representations and integrability of mathematical structures in energy conserving physical systems in the present paper we elaborate on the underlying hamiltonian structure of interconnected energy conserving physical systems it is shown that a power conserving interconnection of port controlled generalized hamiltonian systems leads to an implicit generalized hamiltonian system and a power conserving partial interconnection to an implicit port controlled hamiltonian system the crucial concept is the notion of a generalized dirac structure defined on the space of energy variables or on the product of the space of energy variables and the space of flow variables in the port controlled case three natural representations of generalized dirac structures are treated necessary and sufficient conditions for closedness or integrability of dirac structures in all three representations are obtained the theory is applied to implicit port controlled generalized hamiltonian systems and it is shown that the closedness condition for the dirac structure leads to strong conditions on the input vector fields

families of moment matching based structure preserving approximations for linear port hamiltonian systems in this paper we propose a solution to the problem of moment matching with preservation of the port hamiltonian structure in the framework of time domain moment matching we characterize several families of parameterized port hamiltonian models that match the moments of a given port hamiltonian system at a set of finite interpolation points we also discuss the problem of markov parameters matching for linear systems as a moment matching problem for descriptor representations associated with the given system at zero interpolation points solving this problem yields families of parameterized reduced order models that achieve markov parameter matching finally we apply these results to the port hamiltonian case resulting in families of parameterized reduced order port hamiltonian approximations

control by interconnection and standard passivity based control of port hamiltonian systems the dynamics of many physical processes can be suitably described by port hamiltonian ph models where the importance of the energy function the interconnection pattern and the dissipation of the system is underscored to regulate the behavior of ph systems it is natural to adopt a passivity based control pbc perspective where the control objectives are achieved shaping the energy function and adding dissipation in this paper we consider the pbc techniques of control by interconnection cbi and standard pbc in cbi the controller is another ph system connected to the plant through a power preserving interconnection to add up their energy functions while in standard pbc energy shaping is achieved via static state feedback in spite of the conceptual appeal of formulating the control problem as the interaction of dynamical systems the current version of cbi imposes a severe restriction on the plant dissipation structure that stymies its practical application on the other hand standard pbc which is usually derived from a uninspiring and non intuitive ldquopassive output generationrdquo viewpoint is one of the most successful controller design techniques the main objectives of this paper are 1 to extend the cbi method to make it more widely applicable in particular to overcome the aforementioned dissipation obstacle 2 to show that various popular variants of standard pbc can be derived proceeding from a unified perspective 3 to establish the connections between cbi and standard pbc proving that the latter is obtained restricting the former to a suitable subset providing a nice geometric interpretation to standard pbc and comparing the size of the set of ph plants for which they are applicable

achievable casimirs and its implications on control of port hamiltonian systems in this paper we extend results on interconnections of port hamiltonian systems to infinite dimensional port hamiltonian systems and to mixed finite and infinite dimensional port hamiltonian systems the problem of achievable dirac structures is now studied for systems with dissipation in the finite dimensional infinite dimensional and the mixed finite and infinite dimensional case we also characterize the set of achievable casimirs and study its application for the control of port hamiltonian systems

power based control canonical coordinate transformations integral and adaptive control power based modeling was originally developed in the early sixties to describe a large class of nonlinear electrical rlc networks in a special gradient form recently this idea has been extended for modeling and control of a larger class of physical systems in this paper first coordinate transformations are introduced for systems described in this framework such that the physical structure is preserved such a transformation can provide new insights for both analysis and control design second power based integral and adaptive control schemes are presented advantages of these schemes are shown by their application on standard mechanical systems

application of the difference gaussian rules to solution of hyperbolic problems two of the authors earlier suggested a method of calculating special grid steps for three point finite difference schemes which yielded exponential superconvergence of the neumann to dirichlet map we apply this approach to solve the two dimensional time domain wave problem and the 2 5 d elasticity system in cylindrical coordinates our numerical experiments exhibit exponential convergence at prescribed points with the cost per grid node close to that of the standard second order finite difference scheme the scheme demonstrates high accuracy with slightly more than two grid points per wavelength the reduction of the grid size by one order compared to the standard scheme with the equidistant grids is observed

hamilton jacobi theory for degenerate lagrangian systems with holonomic and nonholonomic constraints we extend hamilton jacobi theory to lagrange dirac or implicit lagrangian systems a generalized formulation of lagrangian mechanics that can incorporate degenerate lagrangians as well as holonomic and nonholonomic constraints we refer to the generalized hamilton jacobi equation as the dirac hamilton jacobi equation for non degenerate lagrangian systems with nonholonomic constraints the theory specializes to the recently developed nonholonomic hamilton jacobi theory we are particularly interested in applications to a certain class of degenerate nonholonomic lagrangian systems with symmetries which we refer to as weakly degenerate chaplygin systems that arise as simplified models of nonholonomic mechanical systems these systems are shown to reduce to non degenerate almost hamiltonian systems i e generalized hamiltonian systems defined with non closed two forms accordingly the dirac hamilton jacobi equation reduces to a variant of the nonholonomic hamilton jacobi equation associated with the reduced system we illustrate through a few examples how the dirac hamilton jacobi equation can be used to exactly integrate the equations of motion

implicit and explicit representations of continuous time port hamiltonian systems implicit and explicit representations of smooth nite dimensional port hamiltonian systems are studied from the perspective of their use in numerical simulation and control design implicit representations arise when a system is modeled in cartesian coordinates and when the system constraints are applied in the form of additional algebraic equations explicit representations are derived when generalized coordinates are used a relationship between the phase spaces for both system representations is derived in this article justifying the equivalence of the representations in the sense of preserving their hamiltonian functions as well as their hamiltonian symplectic forms ultimately resulting in the same hamiltonian ow

brief paper full order observer design for a class of port hamiltonian systems we consider a special class of port hamiltonian systems for which we propose a design methodology for constructing globally exponentially stable full order observers using a passivity based approach the essential idea is to make the augmented system consisting of the plant and the observer dynamics to become strictly passive with respect to an invariant manifold defined on the extended state space on which the state estimation error is zero we first introduce the concept of passivity of a system with respect to a manifold by defining a new input and output on the extended state space and then perform a partial state feedback passivation which leads to the construction of the observer we then illustrate this observer design procedure on two physical examples the magnetic levitation system and the inverted pendulum on the cart system

a port hamiltonian formulation of physical switching systems with varying constraints this paper extends a generic method to design a port hamiltonian formulation modeling all geometric interconnection structures of a physical switching system with varying constraints a non minimal kernel representation of this family of structures named dirac structures is presented it is derived from the parameterized incidence matrices which are a mathematical representation of the primal and dual dynamic network graphs associated with the system this representation has the advantage of making it possible to model complex physical switching systems with varying constraints and to fall within the framework of passivity based control

modeling and control of the timoshenko beam the distributed port hamiltonian approach the purpose of this paper is to show how the timoshenko beam can be fruitfully described within the framework of distributed port hamiltonian dph systems so that rather simple and elegant considerations can be drawn regarding both the modeling and control of this mechanical system after the dph model of the beam is introduced the control problem is discussed in particular it is shown how control approaches already presented in the literature can be unified and a new control methodology is presented and discussed this control methodology relies on the generalization to infinite dimensions of the concept of structural invariant casimir function and on the extension to distributed systems of the so called control by interconnection methodology in this way finite dimensional passive controllers can stabilize distributed parameter systems by shaping their total energy i e by assigning a new minimum in the desired equilibrium configuration that can be reached if a dissipative effect is introduced

generalized port hamiltonian dae systems abstract motivated by recent work in this area we expand on a generalization of port hamiltonian systems that is obtained by replacing the hamiltonian function representing energy storage by a lagrangian subspace this leads to a new class of algebraic constraints and dae systems in physical systems modeling it is shown how dirac structures and lagrangian subspaces allow for similar representations and how this can be exploited to convert algebraic constraints originating from dirac structures into algebraic constraints corresponding to lagrangian subspaces and conversely

theta functions modular invariance and strings we use quillen s theorem and algebraic geometry to investigate the modular transformation properties of some quantities of interest in string theory in particular we show that the spin structure dependence of the chiral dirac determinant on a riemann surface is given by riemann s theta function we use this result to investigate the modular invariance of multiloop heterotic string amplitudes

stability of fdtd on nonuniform grids for maxwell s equations in lossless media in this paper we present practical stability conditions for the finite difference time domain method on nonuniform tensor product grids yee grids these stability conditions apply to maxwell s equations for inhomogeneous and lossless media rectangular domains are considered and the conditions are expressed in terms of the minimum spatial stepsizes of the grid and the maximum electromagnetic wave speed in the configuration the maximum wave speed is known as soon as the media are specified while the minimum spatial stepsizes are known after the configuration has been discretized for two dimensional configurations we present a number of numerical examples which illustrate the effectiveness of the proposed stability conditions

the hamilton pontryagin principle and multi dirac structures for classical field theoriesa we introduce a variational principle for field theories referred to as the hamilton pontryagin principle and we show that the resulting field equations are the euler lagrange equations in implicit form second we introduce multi dirac structures as a graded analog of standard dirac structures and we show that the graph of a multisymplectic form determines a multi dirac structure we then discuss the role of multi dirac structures in field theory by showing that the implicit euler lagrange equations for fields obtained from the hamilton pontryagin principle can be described intrinsically using multi dirac structures finally we show a number of illustrative examples including time dependent mechanics nonlinear scalar fields maxwell s equations and elastostatics

controlled lagrangians and the stabilization of mechanical systems i the first matching theorem we develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled lagrangians the procedure involves making structured modifications to the lagrangian for the uncontrolled system thereby constructing the controlled lagrangian the euler lagrange equations derived from the controlled lagrangian describe the closed loop system where new terms in these equations are identified with control forces since the controlled system is lagrangian by construction energy methods can be used to find control gains that yield closed loop stability we use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group the procedure is demonstrated for several underactuated balance problems including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane

energy shaping of distributed parameter port hamiltonian systems based on finite element approximation the main contribution of this paper is a procedure for the control by energy shaping via casimir generation of infinite dimensional port hamiltonian systems based on a particular finite element approximation the proposed approach is justified by the fact that the adopted spatial discretization technique is able to preserve casimir functions in the closed loop system when going from the distributed to the approximated lumped parameter system besides the intrinsic difficulties related to the large number of state variables the finite element model is generally given in terms of a dirac structure and is completely a causal which implies that the plant dynamics is not given in standard input state output form but as a set of daes consequently the classical energy casimir method has to be extended in order to deal with dynamical systems with constraints usually appearing in the form of lagrangian multipliers the general methodology is illustrated with the help of an example in which the distributed parameter system is a lossless transmission line

dirac structures and their composition on hilbert spaces dirac structures appear naturally in the study of certain classes of physical models described by partial differential equations and they can be regarded as the underlying power conserving structures we study these structures and their properties from an operator theoretic point of view in particular we find necessary and sufficient conditions for the composition of two dirac structures to be a dirac structure and we show that they can be seen as lagrangian hyper maximal neutral subspaces of krein spaces moreover special emphasis is laid on dirac structures associated with operator colligations it turns out that this class of dirac structures is linked to boundary triplets and that this class is closed under composition

port hamiltonian modeling of a nonlinear timoshenko beam with piezo actuation in this paper we develop a mathematical model for the dynamics of a nonlinear timoshenko beam with piezoelectric actuation this model can then be used to design controllers with the goal of achieving a desired shape of the beam the control scheme can be used for several applications e g vibration control in structures or shape control for high precision structures like inflatable space reflectors the starting point of the control design is modeling for control we do this in the framework of port hamiltonian ph modeling which has favorable properties such as passivity and a hamiltonian representing the energy and serving as a lyapunov function that can be exploited for controller design an important property of the ph modeling framework is that it facilitates modeling multiphysics systems or systems which consist of several subsystems where all parts are modeled separately and then can be interconnected easily this is possible because any interconnection of finite dimensional ph systems yiel

canonical transformation and stabilization of generalized hamiltonian systems abstract this paper introduces generalized canonical transformations for generalized hamiltonian systems which convert a generalized hamiltonian system into another one and preserve the generalized hamiltonian structure of the original as in classical mechanics it is expected that canonical transformations will provide new insights and fundamental tools for both analysis and synthesis of those systems firstly the class of generalized canonical transformations and some of their properties are indicated secondly it is shown how to stabilize the generalized hamiltonian systems using canonical transformations in addition some examples illustrate how such transformations are utilized for control systems design

an lmi approach to stabilization of linear port controlled hamiltonian systems in this paper it is shown that controllers for stabilizing linear port controlled hamiltonian pch systems via interconnection and damping assignment can be obtained by solving a set of linear matrix inequalities lmis two sets of almost equivalent lmis are proposed in the first set the interconnection and damping matrices do not appear explicitly which makes it more difficult to directly manipulate those matrices by requiring the system to have no uncontrollable pole at s 0 the second set of lmis explicitly containing the interconnection and damping matrices can be obtained taking into account the physical properties of the system some prespecified structures can be imposed directly on those matrices

nonholonomic control systems on riemannian manifolds this paper gives a general formulation of the theory of nonholonomic control systems on a riemannian manifold modeled by second order differential equations and using the unique riemannian connection defined by the metric the main concern is to introduce a reduction scheme replacing some of the second order equations by first order equations the authors show how constants of motion together with the nonholonomic constraints may be combined to yield such a reduction the theory is applied to a particular class of nonholonomic control systems that may be thought of as modeling a generalized rolling ball this class reduces to the classical example of a ball rolling without slipping on a horizontal plane

effort and flow constraint reduction methods for structure preserving model reduction of port hamiltonian systems abstract the geometric formulation of general port hamiltonian systems is used in order to obtain two structure preserving reduction methods the main idea is to construct a reduced order dirac structure corresponding to zero power flow in some of the energy storage ports this can be performed in two canonical ways called the effort and the flow constraint methods we show how the effort constraint method can be regarded as a projection based model reduction method both the effort and flow constraint reduction methods preserve the stability and passivity properties of the original system as a consequence of preserving the port hamiltonian structure

partial stabilization of input output contact systems on a legendre submanifold this technical note addresses the structure preserving stabilization by output feedback of conservative input output contact systems a class of input output hamiltonian systems defined on contact manifolds in the first instance achievable contact forms in closed loop and the associated legendre submanifolds are analysed in the second instance the stability properties of a hyperbolic equilibrium point of a strict contact vector field are analysed and it is shown that the stable and unstable manifolds are legendre submanifolds in the third instance the consequences for the design of stable structure preserving output feedback are derived in closed loop one may achieve stability only relatively to some invariant legendre submanifold of the closed loop contact form and furthermore this legendre submanifold may be used as a control design parameter the results are illustrated along the technical note on the example of heat transfer between two compartments and a controlled thermostat

stabilization analysis of time delay hamiltonian systems in the presence of saturation abstract the stabilization of a class of hamiltonian systems with state time delay and input saturation is addressed in this paper sufficient conditions are derived by using lyapunov krasovskii functional theorem to guarantee the systems as well as the resulted closed loop systems by output feedback to be asymptotically stable when input saturation effectively occurs a numerical example is presented to illustrate the effectiveness of the obtained results in this paper

some properties of conservative port contact systems the dynamics of open irreversible thermodynamic systems that is systems including both the balance equation of the energy and the entropy has been formulated as contact vector fields with generating functions depending on some external control variable and called conservative port contact systems in this paper we relate the dynamical properties of these systems equilibrium points asymptotic stability to properties of the generating functions the contact hamiltonian functions we show that the equilibrium points of the system satisfy certain conditions involving the contact hamiltonian function we also consider lyapunov s first theorem to emphasize a stability criterion for the equilibrium points in terms of this contact hamiltonian function and relate it to some thermodynamical properties these results are then related to the physical phenomena that are taking place in the system

a convergence analysis of yee s scheme on nonuniform grids the yee scheme is the principal finite difference method used in computing time domain solutions of maxwell s equations on a uniform grid the method is easily seen to be second order convergent in space this paper shows that the yee scheme is also second order convergent on a nonuniform mesh despite the fact that the local truncation error is nodally only of first order

discrete port hamiltonian systems either from a control theoretic viewpoint or from an analysis viewpoint it is necessary to convert smooth systems to discrete systems which can then be implemented on computers for numerical simulations discrete models can be obtained either by discretizing a smooth model or by directly modeling at the discrete level itself one of the goals of this paper is to model port hamiltonian systems at the discrete level we also show that the dynamics of the discrete models we obtain exactly correspond to the dynamics obtained via a usual discretization procedure in this sense we offer an alternative to the usual procedure of modeling at the smooth level and discretization

hamiltonian discretization of boundary control systems a fundamental problem in the simulation and control of complex physical systems containing distributed parameter components concerns finite dimensional approximation numerical methods for partial differential equations pdes usually assume the boundary conditions to be given while more often than not the interaction of the distributed parameter components with the other components takes place precisely via the boundary on the other hand finite dimensional approximation methods for infinite dimensional input output systems e g in semi group format are not easily relatable to numerical techniques for solving pdes and are mainly confined to linear pdes in this paper we take a new view on this problem by proposing a method for spatial discretization of boundary control systems based on a particular type of mixed finite elements resulting in a finite dimensional input output system the approach is based on formulating the distributed parameter component as an infinite dimensional port hamiltonian system and exploiting the geometric structure of this representation for the choice of appropriate mixed finite elements the spatially discretized system is again a port hamiltonian system which can be treated as an approximating lumped parameter physical system of the same type in the current paper this program is carried out for the case of an ideal transmission line described by the telegrapher s equations and for the two dimensional wave equation

an in depth stability analysis of nonuniform fdtd combined with novel local implicitization techniques this work focuses on efficient full wave solutions of multiscale electromagnetic problems in the time domain three local implicitization techniques are proposed and carefully analyzed in order to relax the traditional time step limit of the finite difference time domain fdtd method on a nonuniform staggered tensor product grid newmark cranknicolson cn and alternating direction implicit adi implicitization all of them are applied in preferable directions alike hybrid implicitexplicit hie methods as to limit the rank of the sparse linear systems both exponential and linear stability are rigorously investigated for arbitrary grid spacings and arbitrary inhomogeneous possibly lossy isotropic media numerical examples confirm the conservation of energy inside a cavity for a million iterations if the time step is chosen below the proposed relaxed limit apart from the theoretical contributions new accomplishments such as the development of the leapfrog alternating direction hybrid implicit explicit adhie fdtd method and a less stringent courant like time step limit for the conventional fully explicit fdtd method on a nonuniform grid have immediate practical applications a novel newmark beta fdtd method is proposed in 3d the leapfrog adhie fdtd method is for the first time proposed a courant like time step limit less stringent than the currently known limits is found for conventional nonuniform fdtd the stability of three local implicitization techniques is rigorously analyzed in the algebraic sense the stability of the proposed implicitization techniques is numerically validated by a cavity example

interconnection of port hamiltonian systems and composition of dirac structures port based network modeling of physical systems leads to a model class of nonlinear systems known as port hamiltonian systems port hamiltonian systems are defined with respect to a geometric structure on the state space called a dirac structure interconnection of port hamiltonian systems results in another port hamiltonian system with dirac structure defined by the composition of the dirac structures of the subsystems in this paper the composition of dirac structures is being studied both in power variables and in wave variables scattering representation this latter case is shown to correspond to the redheffer star product of unitary mappings an equational representation of the composed dirac structure is derived furthermore the regularity of the composition is being studied necessary and sufficient conditions are given for the achievability of a dirac structure arising from the standard feedback interconnection of a plant port hamiltonian system and a controller port hamiltonian system and an explicit description of the class of achievable casimir functions is derived

nonholonomic mechanical systems with symmetry this work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of lagrangian mechanics and with a view to control theoretical applications the basic methodology is that of geometric mechanics applied to the formulation of lagrange d alembert generalizing the use of connections and momentum maps associated with a given symmetry group to this case we begin by formulating the mechanics of nonholonomic systems using an ehresmann connection to model the constraints and show how the curvature of this connection enters into lagrange s equations unlike the situation with standard configuration space constraints the presence of symmetries in the nonholonomic case may or may not lead to conservation laws however the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables this momentum equation which plays an important role in control problems involves parallel transport operators and is computed explicitly in coordinates an alternative description using a body reference frame relates part of the momentum equation to the components of the euler poincar e equations along those symmetry directions consistent with the constraints one of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not an example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard a modified version of the skateboard which uses momentum coupling for locomotion generation we construct a synthesis of the mechanical connection and the ehresmann connection defining the constraints obtaining an important new object we call the nonholonomic connection when the nonholonomic connection is a principal connection for the given symmetry group we show how to perform lagrangian reduction in the presence of nonholonomic constraints generalizing previous results which only held in special cases several detailed examples are given to illustrate the theory september 1994 revised march 1995 revised june 1995

brief paper power shaping control of reaction systems the cstr case power shaping control is a recent approach for the control of nonlinear systems based on the physics of the dynamical system it rests on the formulation of the dynamics in the brayton moser form one of the main obstacles for using the power shaping approach is to write the dynamics in the required form since a partial differential equation system submitted to sign constraints has to be solved this work comes within the framework of control design approaches that could possibly generate a closer link between the notions of energy that are specific to reaction systems as derived from thermodynamics concepts and the dynamic system stability theory the objective of this paper is to address the design of power shaping control to reaction systems and more particularly the step of solving the partial differential equation system in order to illustrate the approach we have selected the classical yet complex continuous stirred tank reactor cstr as a case study we show how using the power shaping approach leads to a global lyapunov function for the unforced exothermic cstr this lyapunov function is then reshaped by means of a controller in order to stabilize the process at a desired temperature

power shaping control writing the system dynamics into the brayton moser form power shaping control is an extension of energy balancing passivity based control that is based on a particular form of the dynamics the brayton moser form one of the main difficulties in this control approach is to write the dynamics in the suitable form since this requires the solution of a partial differential equation pde system with an additional sign constraint here a general methodology is described for solving this partial differential equation system the set of all solutions to the pde system is given as the solution of a linear equation system furthermore a necessary condition is given so that a solution of the linear system which meets the sign condition exists this methodology is illustrated on the case of a chemical reactor where the physical knowledge of the system is used to find a suitable solution

control by interconnection of mixed port hamiltonian systems in this note the regulation problem for mixed finite and infinite dimensional port hamiltonian systems m ph systems is discussed a m ph system results from the power conserving interconnection of finite and infinite dimensional systems in port hamiltonian form in particular the system given by the interconnection of two finite dimensional systems one of which is the controller by means of an infinite dimensional connection is studied the proposed control methodology is a generalization to the infinite dimensional case of a well established passivity based control technique for finite dimensional port hamiltonian systems the control by interconnection and energy shaping according to which the open loop energy function is shaped so that a minimum in the desired configuration is introduced this procedure is possible once the state variable of the controller is related to the state variable of the plant by constraining the state of the closed loop system on a structural invariant defined by a set of casimir functions in this way the energy function of the controller which is freely assignable becomes a function of the configuration of the plant and then it can be easily shaped in order to solve the regulation problem

a physically motivated class of scattering passive linear systems we introduce a class of scattering passive linear systems motivated by examples from mathematical physics the state space of the system is x h oplus e where h and e are hilbert spaces we also have a hilbert space e 0 which is dense in e with continuous embedding and e 0 is the dual of e 0 with respect to the pivot space e the input space is the same as the output space and it is denoted by u the semigroup generator has the structure a big begin smallmatrix 0 l l g frac 1 2 k k end smallmatrix big where l in cal l e 0 h and k in cal l e 0 u are such that big begin smallmatrix l k end smallmatrix big with domain e 0 is closed as an unbounded operator from e to h oplus u the operator g in cal l e 0 e 0 is such that rm re langle gw 0 w 0 rangle leq 0 for all w 0 in e 0 the observation operator is c left begin matrix 0 k end matrix right the control operator is b c and the output equatio

a pod reduced order finite difference time domain extrapolating scheme for the 2d maxwell equations in a lossy medium abstract this paper is concerned with establishing a reduced order finite difference time domain fdtd extrapolating scheme based on proper orthogonal decomposition pod method for the two dimensional 2d maxwell equations in a lossy medium for this purpose we first review the classical fdtd scheme and associated results for the 2d maxwell equations in a lossy medium and then we formulate three sets of pod bases by means of the solutions obtained from the classical fdtd scheme on the very short time interval establish the pod reduced order fdtd extrapolating scheme and furnish the error estimates among the pod reduced order fdtd solutions and the classical fdtd solutions as well as the accuracy solution for the 2d maxwell equations in a lossy medium finally we enumerate two numerical examples to confirm that the pod reduced order fdtd extrapolating scheme is feasible and efficient for simulating the phenomena for the electric and magnetic fields in a lossy medium

on determinants of laplacians on riemann surfaces determinants of laplacians on tensors and spinors of arbitrary weights on compact hyperbolic riemann surfaces are computed in terms of values of selberg zeta functions at half integer points