Article

Bond-graph modeling

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Abstract

The bond-graph method is a graphical approach to modeling in which component energy ports are connected by bonds that specify the transfer of energy between system components. Power, the rate of energy transport between components, is the universal currency of physical systems. Bond graphs are inherently energy based and thus related to other energy-based methods, including dissipative systems and port-Hamiltonians. This article has presented an introduction to bond graphs for control engineers. Although the notation can initially appear daunting, the bond graph method is firmly grounded in the familiar concepts of energy and power. The essential element to be grasped is that bonds represent power transactions between components

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... Bond graphs were originally invented by Henry Paynter for use in hydroelectric systems ( Paynter, 1961 ), but they also naturally represent electrical and mechanical systems ( Borutzky, 2010 ). The reader is referred to the texts by Borutzky (2010) ; Gawthrop and Smith (1996) , and Gawthrop and Bevan (2007) for a comprehensive introduction to bond graph theory. More recently, bond graphs have been extended to chemical ( Thoma and Bouamama, 20 0 0 ), biochemical ( Gawthrop and Crampin, 2014;Oster et al., 1973 ) and electrochemical systems ( Gawthrop, 2017 ), enabling bond graph modelling of membrane transporters such as the sodium-glucose transport protein 1 (SGLT1) . ...
... The chemical energy stored within the species of a biochemical system is used to drive reactions, which convert chemical species into other chemical species, dissipating chemical energy in the process. The rate of a reaction is related to the chemical potentials of its reactants and products using the Marcelin-de Donder equation ( Gawthrop and Bevan, 2007;Oster et al., 1973 ): [J/mol] is the reverse affinity (representing the total chemical potential of the products). For the enzyme cycle example in Fig. 1 B, the reaction rates of the upper and lower reactions are given by the equations ...
... (13) The internal behaviour of systems A and B are captured via the constitutive relationships Φ A and Φ B , which store or dissipate energy. We refer interested readers to the tutorial by Gawthrop and Bevan [19] for an overview of bond graph modelling and for an in depth treatment see [20]. ...
... Bond graphs have the potential to drive theoretical advances in these fields due to their generalised nature, which naturally allows models to span across different physical domains. Connections between ports representing different physical domains are made using the energy-transmitting TF (transformer) component [19,20]. Thus, for example, the chemical and electrical domains are connected using a transformer with modulus F ; the Faraday constant. ...
Preprint
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Like all physical systems, biological systems are constrained by the laws of physics. However, mathematical models of biochemistry frequently neglect the conservation of energy, leading to unrealistic behaviour. Energy-based models that are consistent with conservation of mass, charge and energy have the potential to aid the understanding of complex interactions between biological components, and are becoming easier to develop with recent advances in experimental measurements and databases. In this paper, we motivate the use of bond graphs (a modelling tool from engineering) for energy-based modelling and introduce, BondGraphTools, a Python library for constructing and analysing bond graph models. We use examples from biochemistry to illustrate how BondGraphTools can be used to automate model construction in systems biology while maintaining consistency with the laws of physics.
... Bond graphs were originally invented by Henry Paynter for use in hydroelectric systems (Paynter, 1961), but they also naturally represent electrical and mechanical systems (Borutzky, 2010). The reader is referred to the texts by Gawthrop and Smith (1996), Borutzky (2010) and Gawthrop and Bevan (2007) for a comprehensive introduction to bond graph theory. More recently, bond graphs have been extended to chemical (Thoma and Bouamama, 2000), biochemical (Oster et al., 1973;Gawthrop and Crampin, 2014) and electrochemical systems (Gawthrop, 2017), enabling bond graph modelling of membrane transporters such as the sodium-glucose transport protein 1 (SGLT1) . ...
... The chemical energy stored within the various species of a biochemical system is used to drive reactions, which convert chemical species into other chemical species, dissipating chemical energy in the process. The rate of reaction can be related to the chemical potential of its reactants and products using the Marcelin-de Donder equation (Gawthrop and Bevan, 2007;Oster et al., 1973): ...
Preprint
Membrane transporters contribute to the regulation of the internal environment of cells by translocating substrates across cell membranes. Like all physical systems, the behaviour of membrane transporters is constrained by the laws of thermodynamics. However, many mathematical models of transporters, especially those incorporated into whole-cell models, are not thermodynamically consistent, leading to unrealistic behaviour. In this paper we use a physics-based modelling framework, in which the transfer of energy is explicitly accounted for, to develop thermodynamically consistent models of transporters. We then apply this methodology to model two specific transporters: the cardiac sarcoplasmic/endoplasmic Ca$^{2+}$ ATPase (SERCA) and the cardiac Na$^+$/K$^+$ ATPase.
... Bond graphs were originally invented by Henry Paynter for use in hydroelectric systems ( Paynter, 1961 ), but they also naturally represent electrical and mechanical systems ( Borutzky, 2010 ). The reader is referred to the texts by Borutzky (2010) ; Gawthrop and Smith (1996) , and Gawthrop and Bevan (2007) for a comprehensive introduction to bond graph theory. More recently, bond graphs have been extended to chemical ( Thoma and Bouamama, 20 0 0 ), biochemical ( Gawthrop and Crampin, 2014;Oster et al., 1973 ) and electrochemical systems ( Gawthrop, 2017 ), enabling bond graph modelling of membrane transporters such as the sodium-glucose transport protein 1 (SGLT1) . ...
... The chemical energy stored within the species of a biochemical system is used to drive reactions, which convert chemical species into other chemical species, dissipating chemical energy in the process. The rate of a reaction is related to the chemical potentials of its reactants and products using the Marcelin-de Donder equation ( Gawthrop and Bevan, 2007;Oster et al., 1973 ): [J/mol] is the reverse affinity (representing the total chemical potential of the products). For the enzyme cycle example in Fig. 1 B, the reaction rates of the upper and lower reactions are given by the equations ...
Article
Membrane transporters contribute to the regulation of the internal environment of cells by translocating substrates across cell membranes. Like all physical systems, the behaviour of membrane transporters is constrained by the laws of thermodynamics. However, many mathematical models of transporters, especially those incorporated into whole-cell models, are not thermodynamically consistent, leading to unrealistic behaviour. In this paper we use a physics-based modelling framework, in which the transfer of energy is explicitly accounted for, to develop thermodynamically consistent models of transporters. We then apply this methodology to model two specific transporters: the cardiac sarcoplasmic/endoplasmic Ca2+ ATPase (SERCA) and the cardiac Na+/K+ ATPase.
... This required the modification of several equations (Section 2) and re-parameterisation through fitting to the original data (Section 3). To verify the physical plausibility of the updated model we have also developed a bond graph version (Oster et al., 1971;Gawthrop and Crampin, 2014), and we refer readers to Gawthrop and Smith (1996); Borutzky (2010); and Gawthrop and Bevan (2007) for further information on bond graph theory. Given the thermodynamic consistency of our updated model we believe that it is particularly well-suited for incorporation into community efforts for developing a thermodynamic model of a cardiomyocyte to ultimately study whole-heart cardiac energetics. ...
Article
The Na$^+$/K$^+$ ATPase is an essential component of cardiac electrophysiology, maintaining physiological Na$^+$ and K$^+$ concentrations over successive heart beats. Terkildsen et al. (2007) developed a model of the ventricular myocyte Na$^+$/K$^+$ ATPase to study extracellular potassium accumulation during ischaemia, demonstrating the ability to recapitulate a wide range of experimental data, but unfortunately there was no archived code associated with the original manuscript. Here we detail an updated version of the model and provide CellML and MATLAB code to ensure reproducibility and reusability. We note some errors within the original formulation which have been corrected to ensure that the model is thermodynamically consistent, and although this required some reparameterisation, the resulting model still provides a good fit to experimental measurements that demonstrate the dependence of Na$^+$/K$^+$ ATPase pumping rate upon membrane voltage and metabolite concentrations. To demonstrate thermodynamic consistency we also developed a bond graph version of the model. We hope that these models will be useful for community efforts to assemble a whole-cell cardiomyocyte model which facilitates the investigation of cellular energetics.
... We assumed that the gyroscope has two degrees of freedom with mass (M), spring (K), and damper (B) elements in the X (sense) and Y (drive)-axes [24]. This lumped mass-spring-damper model can be expressed as a bond graph, which is a graphical representation of a physical dynamic system, for better understanding of the mathematical relationship [25], as shown in Figure 2b. The effort source (SE), which is the applied input voltage generated by the electrical circuit, was converted into a force on the Y (drive)-axis by a transformer (TF:G 1 : comb-drive). ...
Article
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Recently, consumer applications have dramatically created the demand for low-cost and compact gyroscopes. Therefore, on the basis of microelectromechanical systems (MEMS) technology, many gyroscopes have been developed and successfully commercialized. A MEMS gyroscope consists of a MEMS device and an electrical circuit for self-oscillation and angular-rate detection. Since the MEMS device and circuit are interactively related, the entire system should be analyzed together to design or test the gyroscope. In this study, a MEMS vibratory gyroscope is analyzed based on the system dynamic modeling; thus, it can be mathematically expressed and integrated into a circuit simulator. A behavioral simulation of the entire system was conducted to prove the self-oscillation and angular-rate detection and to determine the circuit parameters to be optimized. From the simulation, the operating characteristic according to the vacuum pressure and scale factor was obtained, which indicated similar trends compared with those of the experimental results. The simulation method presented in this paper can be generalized to a wide range of MEMS devices.
... Here we briefly outline bond graph components as used in electrophysiological modelling. For a more comprehensive introduction, the texts by Gawthrop and Smith [33] and Borutzky [34] provide detailed descriptions of bond graph theory, and Gawthrop and Bevan [35] [34]. ...
Article
Mathematical models of cardiac action potentials have become increasingly important in the study of heart disease and pharmacology, but concerns linger over their robustness during long periods of simulation, in particular due to issues such as model drift and non-unique steady states. Previous studies have linked these to violation of conservation laws, but only explored those issues with respect to charge conservation in specific models. Here, we propose a general and systematic method of identifying conservation laws hidden in models of cardiac electrophysiology by using bond graphs, and develop a bond graph model of the cardiac action potential to study long-term behaviour. Bond graphs provide an explicit energy-based framework for modelling physical systems, which makes them well-suited for examining conservation within electrophysiological models. We find that the charge conservation laws derived in previous studies are examples of the more general concept of a "conserved moiety". Conserved moieties explain model drift and non-unique steady states, generalising the results from previous studies. The bond graph approach provides a rigorous method to check for drift and non-unique steady states in a wide range of cardiac action potential models, and can be extended to examine behaviours of other excitable systems.
... This power is represented by a half arrow labelling the two power variables ( effort e and flow f ) independently from the physical nature of the modelled part of the system. The advantage of the BG is its causal concept that allows not just the modelling but also the control analysis, sizing, diagnosis etc. BG Theory and its applications can be consulted in the literature [64,130]. ...
Thesis
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This research work constitutes a general contribution towards a simpler modelling and diagnosis of the multidisciplinary hybrid systems. Hybrid renewable energy systems where hydrogen is used to store the surplus of the power fits perfectly under this description. Such system gathers different energetic components that are needed to be connected or disconnected according to different operating conditions. These different switching configurations generate different operating modes and depend on the intermittency of the primary sources, the storage capacities and the operational availability of the different hardwares that constitute the system. The switching behaviour engenders a variable dynamic which is hard to be expressed mathematically without investigating all the operating modes. This modelling difficulty is transmitted to affect all the model-based tasks such as the diagnosis and the operating mode management. To solve this problematic, a new modelling tool, called event-driven hybrid bond graph, is developed. Entirely graphic, this formalism allows a multidisciplinary global modelling for all the operating modes at once. By separating the continuous dynamic driven by the bond graph, from the discrete states handled by an integrated automaton, this approach simplifies the management of the operating modes. The model issued using this methodology is also well-adapted to perform a robust diagnosis which is achievable without referring back to the analytical description of the model. The operating mode management, when associated with the on-line diagnosis, allows the implementation of reconfiguration strategies and protection protocols when faults are detected.
... An example is RLC electric circuits, where components such as capacitors and inductors are associated with first-order linear differential equations while the algebraic constraints impose Kirchhoff's conservation laws for voltages in closed loops and currents at each junction [1]. A more general approach that leads to DAEs as the underlying dynamics is that of bond graphs (e.g., [2]), a unifying model for networked physical systems such as electrical, mechanical, and hydraulic networks. ...
... Each physical domain involves a pair of (e, f ) variables such as voltage and current in electrical systems. Besides the electrical domain, we can mention, not only the mechanical (translation or rotation), magnetic, hydraulic, thermal and chemical domains, but also generalizations toward architecture, micro-economy and macro-economy [3][4][5][6][7]13,14,20,35,36,42]. Systems can be described within the BG formalism by means of sources, one-port elements (energy stores and dissipators), and interconnections with junctions (series and parallel) or two-port elements (transformers and gyrators). ...
Article
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This paper proposes a mechanism for describing DNA information in a global perspective based on the bond graph and the memristor concepts. Present-day schemes rely upon local techniques for comparing distinct portions of DNA and, consequently, they are limited in the scope of their applicability. Systematic modeling procedures may shed an additional light toward efficient encoding algorithms, for the analysis and design of natural and artificial DNA structures.
... Recently, the bond graph approach from engineering [30][31][32][33] has been adapted to biochemistry [34][35][36][37]. Bond graphs are close in spirit and application to TKM in that they produce ordinary differential equations for dynamic simulation [37] and that their parameters satisfy thermodynamic consistency without the need to invoke Wegscheider conditions [34][35][36][37]. ...
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A fault propagates along physical paths until it reaches the boundary of the equipment or system, which shows as a functional failure. Hence, inferring the fault propagation helps to ensure the normal operation of the industrial system. To infer the fault propagation in the steam turbine system, a graph model is developed. Firstly, a process graph topology is constructed according to the system mechanism, whose nodes and edges represent the equipment and mutual relationships. Meanwhile, a fault graph topology is built, in which nodes indicate potential faults and edges are inferred propagation paths. Then, the representations of fault nodes are realized through a graph neural network. Lastly, link prediction methods based on nodes’ representations are conducted, along with the paths inference results. Consequently, the accuracy of fault propagation inference for the steam turbine system is over 86%.
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Interactions among biomolecules, electrons, and protons are essential to many fundamental processes sustaining life. It is therefore of interest to build mathematical models of these bioelectrical processes not only to enhance understanding but also to enable computer models to complement in vitro and in vivo experiments. Such models can never be entirely accurate; it is nevertheless important that the models are compatible with physical principles. Network Thermodynamics, as implemented with bond graphs, provide one approach to creating physically compatible mathematical models of bioelectrical systems. This is illustrated using simple models of ion channels, redox reactions, proton pumps, and electrogenic membrane transporters thus demonstrating that the approach can be used to build mathematical and computer models of a wide range of bioelectrical systems.
Chapter
The boom systems of mobile cranes and aerial platform vehicles can be described as hydraulic actuated long boom manipulators. The purpose of this paper is to develop a complete mathematical model for such a boom system which is a multi-domains system consisting of the boom structure and hydraulic drive system. The hydraulic system and the boom structure are described in the port-Hamiltonian formulation. The port-Hamiltonian systems can be easily interconnected through energy exchanges, thus allowing the description of a complex system as a composition of subsystems. The structure of the long boom manipulator is specified as two main types, telescopic boom, and folding boom. These two boom types are correspondingly simplified as rotational non-homogeneous Timoshenko beam and double rotational Timoshenko beams. A structure-preserving discretization for the Timoshenko beam model is applied to transfer the boom model from infinite into finite. Then the interconnections between the hydraulic model and discretized boom structure model are illustrated and simulations of two types of long boom manipulators are accomplished in MATLAB/Simulink.
Article
Common rail system is a key technology of energy saving and emission reduction for modern diesel engines. Multiple injection, as one of the most interesting features of common rail system, allows both optimal fuel consumption and exhaust emissions. In order to explore the method for controlling the fluctuation of fuel injection quantity during multiple injection, experiments have been carried out in this paper, focusing on pilot-main injection. The high pressure fuel circuits of the system have been equivalent to a spring-mass vibration system. Comparison with the experiment shows that the proposed fluctuation equation can reasonably predict the fluctuation characteristics of main injection quantity with pilot-main injection interval. The correction control strategy for the main injection quantity fluctuation has been proposed, in which the relative damping coefficient, rail pressure, pilot-main injection interval and main injection pulse width are chosen as the input variables. The experimental results with different rail pressure and main injection quantity show that the fluctuation of injection quantity during pilot-main injection can be controlled effectively by the proposed correction strategy. The maximum average fluctuation of main injection quantity decreases by as much as 44.66 %.
Chapter
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Composability and modularity in relation to physics are useful properties in the development of cyber-physical systems that interact with their environment. The bond-graph modeling language offers these properties. When systems structures conform to the bond-graph notation, all interfaces are defined as physical “power ports” which are guaranteed to exchange power. Having a single type of interface is a key feature when aiming for modular, composable systems. Furthermore, the facility to monitor energy flows in the system through power ports allows the definition of system-wide properties based on component properties. In this paper we present a metamodel of the bond-graph language aimed to facilitate the description and deployment of software components for cyber-physical systems. This effort provides a formalized description of standardized interfaces that enable physics-conformal interconnections. We present a use-case showing that the metamodel enables composability, reusability, extensibility, replaceability and independence of control software components.
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The dynamics of a twin-rotor multi-input multi-output system, which is similar to that of a helicopter in many ways, is highly nonlinear in nature. In this article, a detailed dynamical model of twin-rotor multi-input multi-output system is developed and simulated by using bond graph approach. Nonlinear nature of the interface gain, thrust, and drag forces, and the stiffness of cable attached to support column joint are estimated. The rotors are modeled through the Newton–Euler equations. The bond graph model is created by using the generic sub-models and the same set of sub-models can be assembled differently to model many other similar systems such as tricopters and quadcopters. Inertial forces and moments, rotor thrust and drag forces, active and reactive motor torques, and direct current motor dynamics are considered in the model. The responses from the model are compared with the test data for validation.
Conference Paper
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Today, Cyber-Physical Production Systems (CPPS) are controlled by manually written software, therefore the software is not able to adapt to unforeseen faults or external system changes. So even if a fault is diagnosed correctly, the system normally needs to be repaired manually by a human operator. To implement the vision of an autonomous system, besides self-diagnosis a self-reconfiguration or self-repair step is needed. Here reconfiguration is the task of restoring valid system behavior after an invalid system behavior occurred. For complex CPPS, finding such a new valid configuration always requires a system model covering all potential new configurations-only for rather simple systems the possible reconfigurations for a fault can be modeled explicitly. Unfortunately , such models are hardly available for such systems. To solve this challenge, in this paper, a novel approach for the automated reconfiguration of CPPS is presented. It is based on Satisfiabil-ity Modulo Theories and operates on observed system data as well as on information about the system topology. By doing this, the mod-eling efforts are reduced. To evaluate this new approach, a simulation of such CPPS is used.
Article
The Bond Graph approach and the Chemical Reaction Network approach to modelling biomolecular systems developed independently. This paper brings together the two approaches by providing a bond graph interpretation of the chemical reaction network concept of complexes. Both closed and open systems are discussed. The method is illustrated using a simple enzyme-catalysed reaction and a trans-membrane transporter.
Article
One of the challenges of modern engineering design is the amount of data that designers must keep track while performing system analysis and synthesis. This task is particularly important in the design process of complex systems such as novel aerospace systems where Modeling and Simulation play an essential role. The Agile philosophy stems from the field of Software Engineering and describes an approach to development in which requirements and solutions gradually develop through collaboration between self-organising cross-functional teams and end users. Agile Model-Based System Engineering (AMBSE) is the application of the Agile philosophy to Model-Based System Engineering. In this paper, AMBSE is accomplished through the application of the Object-Oriented System Engineering Method (OOSEM). OOSEM employs a top-down scenario-driven process that adopts System Modeling Language (SysML) and leverages the object-oriented paradigm to support the analysis, specification, design, and verification of systems. AMBSE assisted by mathematical modelling and safety assessment techniques is applied to the first design iterations of the main aircraft systems, allowing a comprehensive design exploration. The flight control system was chosen to illustrate the procedure in detail, emphasising the synthesis of a six-degrees-of-freedom model augmented by dynamic inversion control for a hypothetical supersonic transport aircraft satisfying class II MIL-F-8785C handling qualities. It is concluded that AMBSE presents promising properties to support future aircraft development within the current regulatory framework for aircraft design, while enabling a smooth transition from conceptual to preliminary design.
Thesis
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In the energy supply, passive single-carrier energy distribution grids (such as in electricity distribution, natural gas, district heat) are changing towards active multi- carrier energy distribution networks (MEDNs). Key factors explaining this transi- tion are the increased role of renewable energies and decentralized microgeneration (e.g. combined heat and power units). A multi-carrier perspective is advantageous, as synergies of the different energies can be taken into account, e.g. storing excessive electric energy in the gas network. In order to ensure the safe and efficient oper- ation of MEDNs, operational variables need to be either measurable or calculable. This leads to the question of observability for MEDNs, where the system must be observable in order to calculate all important operation variables. As today’s sensor placement is done by rules of thumb, it is generally not the optimal placement. In this work, the approach of a graph-theoretical modeling of MEDNs and the com- puting of a cost-efficient optimal sensor placement ensures that the MEDN is ob- servable. The derivation of the MEDN graph and the mixed-integer linear program (MILP) for the sensor placement will be illustrated in a case study for a subnetwork of the MEDN of Karlsruhe, Germany.
Article
Model-based systems engineering is becoming important and is being more widely used as the computing power is improved, because it can be extensively utilized for evaluating performances of vehicles with less effort and time than real-world tests. In this study, a port-based model technique is reviewed, especially for investigating the model connection between the transmission output and the final-drive input in vehicle powertrain models. Commercialized tools that model vehicle powertrains provide well-organized processes for users to connect conventional component models, but a confliction problem of connecting ports frequently arise when users develop their own models and integrate them into a vehicle system. In this study, several ideas to address the conflict issue have been reviewed, and an idea is implemented to solve the port conflict of the vehicle powertrain, and discussions about the method are provided. The idea is very useful especially when the powertrain model uses effort and flow signals to connect sub-component models. Simulation results show that the method produces accurate results and is fast enough, which requires additional calculation time less than 1 %, compared to an alternative method, and it is proved that it guarantees numerical stability in computations. The idea has been widely used in powertrain models, but the feasibility and the numerical stability have not been carefully investigated in the previous studies, so this study would be helpful for proving a basis to model users who develop new sub-component models.
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Full-text available
Modelica is an open-source, object-oriented equation-based modeling language. It is suitable for describing sophisticated dynamic systems (symmetry/asymmetry) as it uses mathematical acausal equations to express physical characteristics. The hydraulic mechanical units (HMU) of gas turbine engine control systems couple the contents of mechanical, hydraulic, symmetry, and other multidisciplinary fields. This paper focuses on the Modelica description method of those HMU models. The content of this work is threefold: firstly, the division form of basic elements in HMU is defined, and the method for describing these element models with Modelica is proposed; secondly, the organization of the element models is defined by using the inheritance characteristics of Modelica, and a lightweight (small code scale) component model is designed; and finally, the causal/acausal connections are designed according to bond graph theory, and the elements and components are integrated into a prototype modeling library. In this paper, the modeling library is verified by comparing simulation results of five typical HMU subsystem models with commercial modeling and simulation software.
Article
Full-text available
A continuous Galerkin finite element method that allows mixed boundary conditions without the need for Lagrange multipliers or user-defined parameters is developed. A mixed coupling of Lagrange and Raviart-Thomas basis functions are used. The method is proven to have a Hamiltonian-conserving spatial discretisation and a symplectic time discretisation. The energy residual is therefore guaranteed to be bounded for general problems and exactly conserved for linear problems. The linear 2D wave equation is discretised and modelled by making use of a port-Hamiltonian framework. This model is verified against an analytic solution and shown to have standard order of convergence for the temporal and spatial discretisation. The error growth over time is shown to grow linearly for this symplectic method, which agrees with theoretical results. A modal analysis is performed which verifies that the eigenvalues of the model accurately converge to the exact eigenvalues, as the mesh is refined. The port-Hamiltonian framework allows boundary coupling with bond-graph or, more generally, lumped parameter models, therefore unifying the two fields of lumped parameter modelling and continuum modelling of Hamiltonian systems. The wave domain discretisation is shown to be equivalent to a coupling of canonical port-Hamiltonian forms. This feature allows the model to have mixed boundary conditions as well as to have mixed causality interconnections with other port-Hamiltonian models. A model of the 2D wave equation is coupled, in a monolithic manner, with a lumped parameter model of an electromechanical linear actuator. The combined model is also verified to conserve energy exactly.
Chapter
Bond-Graph modeling approach is constructive for the understanding and analysis of system dynamics. It allows a decomposition of a complex system into simpler subsystems. It is a formalism with a unified terminology for all physical domains, based on the analogy between phenomena, highlighting the power exchanges among subsystems and the causality simultaneously. These characteristics make Bond-Graph a powerful tool for structural analysis of the properties of the system and thus for the design of control systems. This article introduces and propose a methodology to construct Bond-Graph of simple systems and a methodology to couple Bond-Graphs with Energetic Macroscopic Representation formalism.
Article
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A complete account is given of the theory of so-called dissipative dynamical systems. The concept of dissipativeness is defined as a general input-output property which includes, as notable special cases, passivity and other properties related to finite-gain. The aim is to treat input-output and state properties side-by-side with emphasis on exploring connections between them. The key connection is that a dissipative system in general possesses a set of energy-like functions of the state. The properties of these functions are studied in some detail. It is demonstrated that this connection represents a direct generalization of the well-known Kalman-Yakubovich lemma to arbitrary dynamical systems. Applications to stability theory and passive system synthesis are briefly discussed for non-linear systems.
Article
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Energy is one of the fundamental concepts in science and engineering practice, where it is common to view dynamical systems as energy-transformation devices. This perspective is particularly useful in studying complex nonlinear systems by decomposing them into simpler subsystems that, upon interconnection, add up their energies to determine the full system's behavior. The action of a controller may also be understood in energy terms as another dynamical system. The control problem can then be recast as finding a dynamical system and an interconnection pattern such that the overall energy function takes the desired form. This energy-shaping approach is the essence of passivity-based control (PBC), a controller design technique that is very well known in mechanical systems. Our objectives in the article are threefold. First, to call attention to the fact that PBC does not rely on some particular structural properties of mechanical systems, but hinges on the more fundamental (and universal) property of energy balancing. Second, to identify the physical obstacles that hamper the use of standard PBC in applications other than mechanical systems. In particular, we show that standard PBC is stymied by the presence of unbounded energy dissipation, hence it is applicable only to systems that are stabilizable with passive controllers. Third, to revisit a PBC theory that has been developed to overcome the dissipation obstacle as well as to make the incorporation of process prior knowledge more systematic. These two important features allow us to design energy-based controllers for a wide range of physical systems
Article
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Bond graphs are primarily used in the network modeling of lumped parameter physical systems, but controller design with this graphical technique is relatively unexplored. It is shown that bond graphs can be used as a tool for certain model matching control designs. Some basic facts on the nonlinear model matching problem are recalled. The model matching problem is then associated with a particular disturbance decoupling problem, and it is demonstrated that bicausal assignment methods for bond graphs can be applied to solve the disturbance decoupling problem as to meet the model matching objective. The adopted bond graph approach is presented through a detailed example, which shows that the obtained controller induces port-Hamiltonian error dynamics. As a result, the closed loop system has an associated standard bond graph representation, thereby rendering energy shaping and damping injection possible from within a graphical context.
Article
Good quality control of dynamical systems relies on good models of the controlled system; it follows that the control engineer must also be a good system modeller and have effective paradigms for supporting system modelling. Much control engineering is based on the block diagram paradigm. It will be argued in this Editorial that such use of the block diagram paradigm is unfortunate and that the bond graph paradigm is preferable for a number of reasons.
Article
A physical interpretation of the inverse dynamics of linear and non-linear systems is given in terms of the bond graph of the inverse system. It is argued that this interpretation yields physical insight to guide the control engineer. Examples are drawn from both robotic and process systems.
Article
This paper introduces into a graphical, computer aided modelling methodology that is particularly suited for the concurrent de-sign of mechatronic systems, viz. of engineering systems with mechan-ical, electrical, hydraulic or pneumatic components including interac-tions of physical effects from various energy domains. Beyond the introduction, bond graph modelling of multibody sys-tems, as an example of an advanced topic, is briefly addressed in order to demonstrate the potential of this powerful approach to modelling mechatronic systems. It is outlined how models of multibody systems including flexible bodies can be build in a systematic manner.
Article
System inversion is a concept that often appears either implicitly or explicitly in the design of control systems (e.g, decoupling, feedback linearization or feedforward control). This paper presents an algorithm for the representation of an inverse systems model using bond graph techniques. Structural invertibility conditions are investigated graphically and the structure of the inverse model is discussed. This graphical inversion method is carried out in parallel with classical inversion algorithms using causality manipulations in bond graphs. The physical approach adopted here is enhanced with applications presented in Part 2 which follows.
Article
This paper proposes to extend the set of causality assignment procedures. The proposed alternative procedures are mainly inspired by formulations developed in the mechanical domain. They enable Lagrange equations, Hamilton equations, and Boltzmann-Hamel equations to be obtained, as well as formulations with the Lagrange multipliers. In the context of system modeling a varied set of mechanical oriented equations are available in a systematic way from the bond graph representation and the proposed corresponding procedures provide an algorithmic frame for programming these mathematical formulations. The graphical features of the bond graph tool and the causality stroke concept enable formulations to be methodically obtained, formulations that can otherwise be very awkward to express. Also these procedures emphasize certain interesting properties of the bond graph tool e.g.: there is a clear distinction between the energy topology of a system and its dynamic equations; it also enables graphic structural analyses to be undertaken; and finally it can play a pedagogical role in engineering education.
Article
The implementation of impedance control is considered. A feedback control algorithm for imposing a desired cartesian impedance on the end-point of a nonlinear manipulator is presented. This algorithm completely eliminates the need to solve the 'inverse kinematics problem' in robot motion control. The modulation of end-point impedance without using feedback control is also considered, and it is shown that apparently 'redundant' actuators and degrees of freedom such as exist in the primate musculoskeletal system may be used to modulate end-point impedance and may play an essential functional role in the control of dynamic interaction.
Book
Modelling and simulation have become endeavours central to all disciplines of science and engineering. they are used in the analysis of physical systems where they help us gain a better understanding of the functioning of our physical world. They are also important to the design of new engineering systems where they enable us to predict the behaviour of a system before it is ever actually built. Modelling and simulation are the only techniques available that allow us to analyze arbitrarily non-linear systems accurately and under varying experimental conditions. The two books "Modeling and Simulation of Continuous Systems," of which this is the first, introduce the student to an important subclass of these techniques. They deal with the analysis of systems described through a set of ordinary or partial differential equations or through a set of difference equations. This first volume introduces concepts of modelling physical systems through a set of differential and/or difference equations. The purpose is twofold: it enhances the scientific understanding of our physical world by codifying (organizing) knowledge about this world, and it supports engineering design by allowing the reader to assess the consequences of a particular design alternative before it is actually built. This text has a flavour of the mathematical discipline of dynamical systems, and is strongly orientated towards Newtonian physical science.
Article
The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper.
Article
This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials.
Conference Paper
Design in the Physical Domain is proposed as a means of integrating control systems design with mechanical systems design. This approach facilitates separation of design issues from implementation issues through high-level abstraction, and provides guidance in selecting the proper physical architecture for a given control task. It is shown, as an example, how this philosophy may lead to alternative robot architectures that are inherently stable and well-suited for high performance endpoint control.
Conference Paper
Manipulation fundamentally requires a manipulator to be mechanically coupled to the object being manipulated. A consideration of the physical constraints imposed by dynamic interaction shows that control of a vector quantity such as position or force is inadequate and that control of the manipulator impedance is also necessary. Techniques for control of manipulator behaviour are presented which result in a unified approach to kinematically constrained motion, dynamic interaction, target acquisition and obstacle avoidance.
Article
Analysis and simulation of non-linear inverse systems are sometimes necessary in the design of control systems particularly when trying to determine an input control required to achieve some predefined output specifications. But unlike physical systems which are proper, the inverse systems are very often improper leading to numerical problems in simulation as their models sometimes have a high index when written in the form of differential-algebraic equations (DAE). This paper provides an alternative approach whereby performance specifications and the physical system are combined within a single bond graph leading to a greatly simplified simulation problem.
Article
A bond graph representation of model-based observer control is introduced and shown to provide a convenient framework for the design of controllers in the physical domain. The approach is illustrated by a series of examples and the robustness of the method is investigated by simulation.
Article
“Design in the Physical Domain” is proposed as a means of integrating control systems design with mechanical systems design. This approach facilitates separation of design issues from implementation issues through high-level abstraction, and provides guidance in selecting the proper physical architecture for a given control task. As an example it is shown how this philosophy may lead to alternative robot architectures that are inherently stable and well-suited for high performance end-point control.
Article
A bond graph approach to hybrid simulation of dynamical systems using numerical-experimental real-time substructuring is presented. The bond graph concepts of a virtual junction and a virtual actuator, hitherto used in the context of physical-model based control, are used to perform the substructuring in an intuitively appealing way. The approach is illustrated by the reworking of a previously-published example. The approach is verified experimentally using a bench-top multiple mass-spring system for the physical substructure and automatically generated real-time code is used to implement the numerical substructure.
Article
The virtual earth concept, well known to designers of active electronic circuits with operational amplifier components, is shown to have a novel bicausal bond graph interpretation. This leads to simplified bond graph modelling of such circuits. Some simple operational amplifier circuits, together with a more complex active filter are used to illustrate the approach. A complex electro-mechanical system shows that the method is useful in creating a unified bond graph model of systems involving both analogue electronic and mechanical systems.
Article
A bond-graph based approach to design in the physical domain is described which uses the concept of virtual actuators and virtual sensors. The approach is illustrated by, and implemented on, an experimental ball and beam system.
Book
This is a book about modelling, analysis, and control of linear time-invariant systems. The book uses what is called the behavioral approach towards mathematical modelling. Thus a system is viewed as a dynamical relation between manifest and latent variables. The emphasis is on dynamical systems that are represented by systems of linear constant coefficient differential equations. In the first part of the book the structure of the set of trajectories that such dynamical systems generate is analyzed. Conditions are obtained for two systems of differential equations to be equivalent in the sense that they define the same behavior. It is further shown that the trajectories of such linear differential systems can be partitioned in free inputs and bound outputs. In addition the memory structure of the system is analyzed through state space models. The second part of the book is devoted to a number of important system properties, notably controllability, observability, and stability. An essential feature of using the behavioral approach is that it allows these and similar concepts to be introduced in a representation free manner. In the third part control problems are considered, more specifically stabilization and pole placement questions. The book is a textbook for advanced undergraduate or beginning graduate students in mathematics and engineering. It contains numerous exercises, including simulation problems, and examples, notably of mechanical systems and electrical circuits.
Article
The purpose of this paper is to study interconnections and control of dynamical systems in a behavioral context. We start with an extensive physical example which serves to illustrate that the familiar input-output feedback loop structure is not as universal as we have been taught to believe. This leads to a formulation of control problems in terms of interconnections. Subsequently, we study interconnections of linear time-invariant systems from this vantage point. Let us mention two of the results obtained. The first one states that any polynomial can be achieved as the characteristic polynomial of the interconnection with a given plant, provided the plant is not autonomous. The second result states that any subsystem of a controllable system can be implemented by means of a singular feedback control law. These results yield pole placement and stabilization of controllable plants as a special case. These ideas are finally applied to the stabilization of a nonlinear system around an operating point
Article
Conventional bond graph theory is predicated on the notion that a bond has a single causal stroke: an effort imposed at one end implies a flow imposed at the other. This notion is implied by components having a known constitutive relationship. This paper discusses bond graphs with two causal strokes - Bicausal bond-graphs. These can, for example, handle systems with unknown parameters. This paper addresses issues arising from the general idea of inverse systems - physical systems with a non-standard input-output pattern; in particular: dynamic inverses, parameter estimation and state estimation. 1. Introduction r v 1 i 1 c v c Fig. 1. An electrical circuit 1 R C S e 0 i 1 v 1 v c i 1 Fig. 2. An electrical circuit: bond graph Consider the linear time invariant electrical circuit of Figure 1 and the associated Bond Graph of Figure 2. In addition to the qualitative description of the system structure implied by the Bond Graph of Figure 2, there is quantitative info...
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