Theory of complex systems

Theory of Complex Systems Varchasva Chapter 1 Port-Based Modeling of Dynamic Systems Varchasva Abstract Many engineering activities, in particular mechatronic design, require that a multi-domain or ‘multi-physics’ system and its control system be designed as an integrated system. This chapter discusses the background and concepts of a portbased approach to integrated modeling and simulation of physical systems and their controllers, with parameters that are directly related to the real-world system, thus improving insight and direct feedback on modeling decisions. It serves as the conceptual motivation from a physical point of view that is elaborated mathematically and applied to particular cases in the remaining chapters. 1.1 Introduction 1.1.1 Modeling of dynamic systems If modeling, design and simulation of (controlled) systems are to be discussed, some initial remarks at the meta-level are required. It should be clear and it probably will be, due to the way it is phrased next, that no global methodology exists that deals with each problem that might emerge. In other words, no theory or model can be constructed independently of some problem context. Nevertheless, in practice, not only well-established theories are treated as some form of absolute ‘truth’, but also (sub-)models of physical components are often considered as constructs that can be independently manipulated, for instance in a so-called model library. Without some reference to a problem context, such a library would be useless, unless there is an implicit agreement about some generic problem context, such that some generic sub-models can be stored for re-use. However, such a foundation is rather weak, as implicit agreements tend to diverge, especially in case of real world problems. Herein, we will focus on the generic problem context of the dynamic behavior of systems that primarily belong to the domain of the (control) engineer, but also of the physicist, the biologist, the physiologist, etc. These systems can be roughly 1 2 1 Port-Based Modeling of Dynamic Systems characterized as systems that can be considered to consist for a large part of subsystems for which it is relevant to their dynamic behavior that they obey the basic principles of macroscopic physics, like the conservation principles for fundamental physical quantities like ‘charge’, ‘momentum’, ‘matter’, ‘energy’ etc. as well as the positive-entropy-production principle, in other words: the ‘laws’ of thermodynamics. The other part is considered to consist of sub-models for which the energy bookkeeping is generally not considered relevant for the dynamic behavior. Such parts are generally addressed as the signal processing part (‘controller’) that is commonly for a large part realized in digital form for the common reasons of flexibility, reproducibility, robustness (error correction), maintainability, etc., even though analogue solutions are in some cases much less costly in terms of material and energy consumption. This chapter focuses on the description of the part for which energy bookkeeping is relevant for the dynamic behavior, while keeping a more than open eye for the connection to the signal part, either in digital or in analogue form. It is argued that so-called ‘port-based modeling’ is ideally suited for the description of the energetic part of a multi-domain system, sometimes also called multi-physics system. This means that the approach by definition deals with multidisciplinary systems like those encountered in mechatronics for example. Port-based physical system modeling aims at providing insight, not only in the behavior of systems that an engineer working on multidisciplinary problems wishes to design, build, troubleshoot or modify, but also in the behavior of the environment of that system. A key aspect of the physical world around us is that ‘nature knows no domains’. In other words, all boundaries between disciplines are man-made, but highly influence the way humans interact with their environment. A key point each modeler should be aware of is that any property of a model that is a result of one of his own choices, should not affect the results of the model. Examples of modeler’s choices are: relevance of time and space scales, references, system boundaries, domain boundaries, coordinates and metric. If a variation in one of these choices leads to completely different conclusions about the problem for which the model is constructed, the model obviously does not serve its purpose as it tells us more about the modeler than about the problem to be solved. Again, when the issue is phrased in this manner, it is hard to disagree, but practice shows that this modeling principle is often violated. 1.1.2 History of physical systems modeling of engineering systems Several attempts to unified or systematic approaches of modeling have been launched in the past. In the upcoming era of the large-scale application of the steam engine over 150 years ago, the optimization of this multi-domain device (thermal, pneumatic, mechanical translation, mechanical rotation, mechanical controls, etc.) created the need for the first attempt to a systems approach. This need for such a ‘mecha-thermics’ approach was then named thermodynamics. Although many will Chapter 2 Port-Hamiltonian Systems A. J. van der Schaft Abstract In this chapter, we will show how the representation of a lumped-parameter physical system as a bond graph naturally leads to a dynamical system endowed with a geometric structure, called a port-Hamiltonian system. The dynamics are determined by the storage elements in the bond graph (cf. Sect. 1.6.3), as well as the resistive elements (cf. Sect. 1.6.4), while the geometric structure arises from the generalized junction structure of the bond graph. The formalization of this geometric structure as a Dirac structure is introduced as the key mathematical concept to unify the description of complex interactions in physical systems. It will also allow to extend the definition of a finite-dimensional port-Hamiltonian systems as given in this chapter to the infinite-dimensional case in Chapter 4, thus dealing with distributed-parameter physical systems. We will show how this port-Hamiltonian formulation offers powerful methods for the analysis of complex multi-physics systems, also paving the way for the results on control of port-Hamiltonian systems in Chapter 5 and in Chapter 6. Furthermore, we describe how the port-Hamiltonian structure relates to the classical Hamiltonian structure of physical systems as being prominent in e.g. classical mechanics, as well as to the Brayton-Moser description of RLC-circuits. 2.1 From junction structures to Dirac structures In the preceding chapter, we have seen how port-based network modeling of lumped-parameter physical systems leads to a representation of the physical system by generalized bond graphs. Generalized bond graphs consist of energy-storing elements, resistive elements and power-continuous elements like transformers, gyrators, 0- and 1-junctions. These elements are linked by bonds, each carrying a pair of flow and effort variables, whose product equals the power through the bond. In order to fix the direction of power flow, a half arrow is attached to each bond, indicating the positive direction of power flow. Thus, a generalized bond graph is an oriented graph with its nodes being decorated by one of the elements indicated above, and ev- 53 54 2 Port-Hamiltonian Systems ery edge (called ‘bond’) labeled by two scalar conjugate variables f ∈ R (flow) and e ∈R (effort). Furthermore, the elements at every node only involve the flow and effort variables associated with the bonds that are incident on that node. An important extension of this definition of a bond graph is obtained by allowing for multi-bonds (cf. Sect. 1.9.2) which are labeled by flow vectors f ∈ Rk and dual effort vectors e ∈ (Rk)∗ (cf. Sect. B.1.1 in Appendix B). Still a further extension (see Sect. 3.2), which comes in naturally for 3-D mechanical systems, is to consider flows f which take value in the Lie algebra se(3) (‘twists’) and efforts e which take value in the dual Lie algebra se∗(3) (‘wrenches’). The key concept in the formulation of port-based network models of physical systems as port-Hamiltonian systems is the geometric notion of a Dirac structure. Loosely speaking, a Dirac structure is a subspace of the space of flows f and efforts e such that for every pair (f,e) in the Dirac structure the power e× f is equal to zero, and, furthermore, the subspace has maximal dimension with respect to this property. This means that it is not possible to extend the subspace to a larger subspace that still has this power-conserving property. 2.1.1 From 0- and 1-junctions to Dirac structures Before mathematically formalizing the notion of a Dirac structure, we will start with showing how the basic bond graph elements of 0-junctions and 1-junctions as encountered in the previous chapter share these properties of power-conservation and maximal dimension. Let us start with the simple 0-junction relating two pairs of flows and efforts (f1,e1) and (f2,e2) by e1 = e2 f1 + f2 = 0 Clearly, the 0-junction is power-conserving, that is, (2.1) e1 f1 +e2 f2 = 0 (2.2) But there is more: the 0-junction is described by two independent equations involving 4 variables, and thus represents a 2-dimensional subspace of the 4-dimensional space of total vectors (f1, f2,e1,e2) of flow and effort variables. Furthermore, it can be seen (this will be shown later on in full generality) that we cannot leave out one of the equations in (2.1) while still retaining the power-conservation property (2.2), that is, the dimension 2 is the maximal achievable dimension with respect to the power-conservation property. The same situation occurs for the simple 1-junction described by the relations f1 = f2 e1 +e2 = 0 (2.3) Chapter 3 Port-Based Modeling in Different Domains C. Batlle, F. Couenne, A. Doria-Cerezo, V. Duindam, E. Fossas, C. Jallut,` L. Lefevre, Y. Le Gorrec, B. M. Maschke, R. Ortega, K. Schlacher,` S. Stramigioli, M. Tayakout Abstract In this Chapter we present some detailed examples of modelling in several domains using port and port-Hamiltonian concepts, as have been presented in the previous chapters. We start with the electromechanical domain in Sect. 3.1, while in Sect. 3.2 it is shown how port-Hamiltonian systems can be fruitfully used for the structured modelling of robotics mechanisms. In Sect. 3.3, it is show how to model simple elastic systems either in the Lagrangian and Hamiltonian framework, while, in Sect. 3.4, an expressions of the models representing momentum, heat and mass transfer as well as chemical reactions within homogeneous fluids in the port-based formalism is proposed. To this end, the entropy balance and the associated source terms are systematically written in accordance with the principle of irreversible thermodynamics. Some insights are also given concerning the constitutive equations and models allowing to calculate transport and thermodynamic properties. As it will be shown, for each physical domain, these port-based models can be translated into bond-graph models, in the case of distributed as well as lumped parameters models. 3.1 Modeling of electrical systems Electromechanical energy conversion has already been discussed in Sect. 1.9.3, and, in particular, the constraints imposed by energy conservation on the constitutive laws of the ports, Maxwell’s relations, have been derived. As the name indicates, electromechanical systems (EMS) bridge the gap between the electrical and mechanical domains. In practice, on the electrical side one has an electric circuit of a very special class, what is called an electronic power converter, which, if the system is working as an electrical motor, takes the electrical energy from some source and provides a suitable voltage to the EMS so that the desired mechanical speed is reached; likewise, if the system acts as a generator, the power converter transforms the raw electrical energy into a form adapted for immediate use, storage or transportation. The main characteristic of electronic power converters is that they are variable structure systems (VSS). They contain a number of switches and diodes, of 131 132 3 Port-Based Modeling in Different Domains Fig. 3.1 A functional descrip- v1 v which the former can be opened or closed in a periodic manner by a suitable control algorithm, in order to effect the necessary electrical energy conversion. Since electronic power converters are so important for EMS, and also for many other applications, such as portable equipment, energy supply systems in the aerospace industry, or uninterruptible power supply systems, we present first an explicit example of modelling of a power converter in the port-Hamiltonian framework. Next we discuss in detail the port-Hamiltonian description of a general EMS, and use it to describe an elementary electromagnet. Finally we couple both systems and display the complete port-Hamiltonian structure. Although modelling of VSS in the port-Hamiltonian framework is straightforward, numerical simulation can be quite complex and time-intensive, due to the abrupt structure changes. Approximate, smooth models can be obtained from a VSS, using suitable averages of the state variables and the control signals. For completeness, we also present the simplest form of this averaging theory, which yields models which can be easily implemented in bond graph theory. 3.1.1 Electronic power converter circuits Fig. 3.1 shows a functional model of the boost (or elevator) converter (the detailed electronics of how the switches are implemented is not shown). The switches s1 and s2 are complementary: when s1 is closed (s1 = 1), s2 is open (s2 = 0), and viceversa. Thus, the different circuit topologies can be described with a single boolean variable S = s2. The port Hamiltonian modeling of electric circuits can be done in a systematic way using tools from graph theory [145], but since we are dealing here with a circuit of very small size we will adopt a more pedestrian approach and concentrate on the problems presented by the switches, using the ideas of [74]. A more in-deep conceptual analysis of the switches can be found in [63,73,82]. The Hamiltonian dynamical variables of the boost converter are the magnetic flux at the coil, φL, and the charge of the capacitor, qC. Hence we have two one-dimensional Hamiltonian Chapter 4 Infinite-Dimensional Port-Hamiltonian Systems A. Macchelli, B. M. Maschke Abstract This chapter presents the formulation of distributed parameter systems in terms of port-Hamiltonian system. In the first part it is shown, for different examples of physical systems defined on one-dimensional spatial domains, how the Dirac structure and the port-Hamiltonian formulation arise from the description of distributed parameter systems as systems of conservation laws. In the second part we consider systems of two conservation laws, describing two physical domains in reversible interaction, and it is shown that they may be formulated as port-Hamiltonian systems defined on a canonical Dirac structure called canonical Stokes-Dirac structure. In the third part, this canonical Stokes-Dirac structure is generalized for the examples of the Timoshenko beam, a nonlinear flexible link, and the ideal compressible fluid in order to encompass geometrically complex configurations and the convection of momentum. 4.1 Modelling origins of boundary port-Hamiltonian systems The aim of this section is to introduce the main concepts and the origin of boundary port Hamiltonian systems that extend the port Hamiltonian formulation from lumped parameter systems defined in Chapter 2 to distributed parameter systems. Dynamic models of distributed parameter systems are defined by considering not only the time but also the space as independent parameters on which the physical quantities are defined. They allow to model objects such as vibrating strings or plates, transmission lines or electromagnetic fields and mass and heat transfer phenomena in tubular reactors or in the heart of fuel cells. In this section we shall use a formulation of distributed parameter systems in terms of systems of conservation laws as they arise in their mathematical analysis [85, 189] or in terms of systems of balance equations as they arise in models of heat and mass transfer phenomena [21]. We shall show how the port Hamiltonian formulation arises from the combination of systems of conservation laws with the axioms of irreversible thermodynamics [42,97,168]. The first subsection recalls briefly the concepts of con- 211 212 4 Infinite-Dimensional Port-Hamiltonian Systems servation law and the axioms of Thermodynamics on the example of a the heat conduction in a cylindrical rod. The second subsection shows how one may define a canonical Dirac structure for reversible physical systems such as the transmission line, the vibrating string, consisting in two coupled conservation laws by applying an unusual thermodynamic perspective to these systems. The third section shows that Stokes-Dirac structures also arise in dissipative phenomena and how one may define dissipative boundary port Hamiltonian systems on the examples of the heat conduction in a cylindrical rod, the lossy transmission line and the vibrating string with structural damping. Finally it should be mentioned that in this section we shall consider for the sake of simplicity, only 1-dimensional spatial domain, i.e. systems defined on some interval in R and postpone the general case to the next section. 4.1.1 Conservation law and irreversible thermodynamics Let us recall briefly on the example of the heat conduction, the main concepts which we shall use to define port Hamiltonian systems for distributed parameter systems. For a detailed lecture on the foundations of irreversible thermodynamics the reader is referred to [42,60,168], [77, chap. 6.3] and for application to modelling of heat and mass transfer phenomena to [21]. In this section we shall consider the heat diffusion in some 1-dimensional medium (for instance a rod with cylindrical symmetry) and denote its spatial domain by the interval Z = [a,b] ⊂ R. The time interval on which the variables of the system are defined is denoted by I ∋ t. We assume the medium to be undeformable (i.e. its deformations are neglected) and consider only one physical domain, the thermal domain and its dynamics. The first step consist in writing a conservation law of the conserved quantity, here the conservation of the density of internal energy, denoted by u(t,z), a extensive thermodynamic variable of the medium: ∂u ∂ ∂t = −∂zJQ (4.1) where JQ(t,z) is the flux variable, here the heat flux across the section at z. The heat flux itself arises from the thermodynamic non-equilibrium and is defined by some phenomenological law, for instance defined according to Fourier’s law by: JQ (4.2) whereλ(T,z) denotes the heat conduction coefficient and T denotes the temperature of the medium, the intensive thermodynamic variable of the thermal domain. Actually the axioms of the Irreversible Thermodynamics near equilibrium, decompose the preceding relation by saying that the flux variable is a function of the thermodynamic driving force F which characterizes the non- Chapter 6 Analysis and Control of Infinite-Dimensional Systems A. Macchelli, C. Melchiorri, R. Pasumarthy, A. J. van der Schaft Abstract Infinite dimensional port Hamiltonian systems have been introduced in Chapter 4 as a novel framework for modeling and control distributed parameter systems. In this chapter, some results regarding control applications are presented. In some sense, it is more correct to speak about preliminary results in control of distributed port Hamiltonian systems, since a general theory, as the one discussed in Chapter 5 for the finite dimensional port Hamiltonian systems, has not been completely developed, yet. We start with a short overview on the stability problem for distributed parameter systems in Sect. 6.2, together with some simple but useful stability theorems. Then, in Sect. 6.3, the control by damping injection is generalized to the infinite dimensional case and an application to the boundary and distributed control of the Timoshenko beam is presented. In Sect. 6.4, a simple generalization of the control by interconnection and energy shaping to the infinite dimensional framework is discussed. In particular, the control scheme is developed in order to cope with a simple mixed finite and infinite dimensional port Hamiltonian system. Then, an application to the dynamical control of a Timoshenko beam is discussed in Sect. 6.5. 6.1 Introduction In the last years, stimulated by the applications arising from space exploration, automated manufacturing and other areas of technological development, the control of distributed parameter systems has been an active field of research for control system people. The problem is quite complex since the systems to be controlled are described by a set of partial differential equations, the study of which is not an easy task. It is well-known that the semi-group theory provides a large number of results on the analysis of systems of PDEs, and, in particular, on the exponential stability of feedback laws for beam, wave and thermoelastic equations. Classical results can be found in [56,157], while in [117] some new contributions concerning the stability and feedback stabilization of infinite dimensional systems are reported. In partic- 319 320 6 Analysis and Control of Infinite-Dimensional Systems ular, second order PDEs, such as the Euler-Bernoulli beam equation which arises from control of several mechanical structures (e.g. flexible robots arms and large space structures), are discussed. As pointed out in Sect. 6.2, when dealing with infinite dimensional systems, the main problem concerns about the intrinsic difficulties related to the proof of stability of an equilibrium point. Moreover, it is important to underscore that this limitation does not depend on the particular approach adopted to study the problem. Even if a distributed parameter systems is described within the port Hamiltonian framework, the stability proof of a certain control scheme will always be a difficult task. So the distributed port-Hamiltonian approach does not simplify the control task. Then, we can ask ourself: from the control point of view, what is the advantage related to a port-Hamiltonian description of a distributed parameter system? It is author’s opinion that two are the main advantages in adopting the distributed portHamiltonian framework. At first, the development of control schemes for infinite dimensional systems is usually based on energy considerations or, equivalently, the stability proof often relies on the properties of an energy-like functional, a generalization of the Lyapunov function to the distributed parameters case. Some examples, related to the stabilization of flexible beams, are in [102,202]. The Hamiltonian description of a distributed parameter system is given in terms of time evolution of energy variables depending on the variation of the total energy of the system. In this way, the energy of the system, which is generally a good Lyapunov function, appears explicitly in the mathematical model of the system itself and, consequently, both the design of the control law and the proof of its stability can be deduced and presented in a more intuitive (in some sense physical) and elegant way. Secondly, the port-Hamiltonian formulation of distributed parameter systems originates from the idea that a system is the result of a network of atomic element, each of them characterized by a particular energetic behavior, as in the finite dimensional case. So, the mathematical models originates from the same set of assumptions. This fact is important and allows us to go further: in particular, it is of great interest to understand if also the control schemes developed of finite dimensional port Hamiltonian systems could be generalized in order to deal with distributed parameter ones. For example, suppose that the total energy (Hamiltonian) of the system is characterized by a minimum at the desired equilibrium configuration, [121]. This happens, for example, in the case of flexible beams, for which the zero-energy configuration corresponds to the undeformed beam. In this situation, the controller can be developed in order to behave as a dissipative element to be connected to the system at the boundary or along the distributed port. The amount of dissipated power can be increased in order to reach quickly the configuration with minimum energy. As in the finite dimensional case, it can happen that the minimum of the energy does not correspond to a desired configuration. Then, it is necessary to shape the energy function so that a new minimum is introduced. In other words, it is interesting to investigate if the control by interconnection and energy shaping discussed in Sect. 5.3 can be generalized to the infinite dimensional case. More details in [118,119,121,171]. Appendix A Model Transformations and Analysis using Bond Graphs P. C. Breedveld Abstract This appendix illustrates the main procedures for model transformations and analysis using the bond graph formalism. A.1 Model transformations A.1.1 Conversion of an ideal physical model into a bond graph A standard procedure for the translation of an ideal physical model (IPM) into a bond graph contains the following steps (specifics between brackets apply to the mechanical domain that is treated in a dual way due to the common choice of variables, i.e. the force-voltage analogy): 1. Identify the domains that are present. 2. Choose a reference effort (velocity reference and direction) for each of the domains (degrees of freedom). 3. Identify and label the other points with common effort (velocity) in the model. 4. Identify, classify and accordingly label the ports of the basic one- and two-port elements: C, I, GY, etc. in the model. A label consists of a node type and a unique identifier, e.g. in the linear case the constitutive parameter connected by a colon. 5. Identify the efforts (velocities) and effort differences (relative velocities) of all ports identified in the previous step. 6. Represent each effort by a 0-junction, (each velocity by a 1-junction). Use a 1-junction (0-junction) and bonds to construct a relation between each effort difference (relative velocity) and the composing efforts (velocities) as follows, taking care that each effort difference (relative velocity) is explicitly represented by a 0-junction (1-junction), see Fig. A.1. 7. Connect all ports identified in steps 4 and 5 to the corresponding junctions. Note that after this step all of the ports identified in step 4 are directly connected to a 0-junction (1-junction) only. 369 Fig. A.1 Construction of effort and flow differences. Table A.1 Equivalence rules for simple junction structures. 8. (optional) Simplify the bond graph where necessary according to the equivalence rules in Table A.1. These steps not only support this translation process, but also give a better insight when dynamic models are directly written in terms of a bond graph as well. In that case steps 1, 3, 4 and 5 should be changed from a translation of already made modeling choices into the modeling choices themselves. This process is merely intended to establish a link between familiar representations and a bond graph representation. It does not suggest that the use of bond graphs to support modeling always takes place on the basis of an existing IPM. On the contrary, the process of making modeling choices is supported best by direct application of the bond graph representation, especially when it is causally augmented Fig. A.2a Sketch of an elevator sys- Fig. A.2b Word bond graph. tem. Fig. A.3 Iconic diagram of the IPM of the elevator system (step 1). as will be shown in section , where the advantage of a bond graph as an alternative model ‘view’ will be illustrated. A.1.2 Example: systematic conversion of a simple electromechanical system model into a bond graph representation. In order to demonstrate the relationship between the domain-independent bond graph representation of a model and a domain dependent representation, e.g. the iconic diagram of the model of the elevator system in Fig. A.2a, a systematic conversion is demonstrated in Figures A.3 through A.10 following the eight steps described in the previous section. This conversion ignores the automatic feedback on modeling decisions from a bond graph, as the modeling decisions have already been made. Fig. A.4 The IPM with references (step 2). Fig. A.5 The IPM with relevant voltagees and (angular velocities indicated (step 3). Fig. A.6 Representation of voltages by 0-junctions and of velocities by 1-junctions (step 4). Fig. A.7 Construction of the difference variables (steps 5 and 6). A.1.3 Conversion of causal bond graphs into block diagrams As a causal bond represents a bi-lateral signal flow with fixed directions, a causal bond graph (e.g. Fig. A.11) can be expanded into a block diagram in three to four steps: 1. All node symbols are encircled and all bonds are expanded into bilateral signal flows according to the assigned causality (Fig. A.12). 2. All constitutive relations of each node are written into block diagram form, according to the assigned causality of each port. 0-junctions are represented by a signal-node for the efforts and a summation for the flows, while 1-junctions Fig. A.8 Complete bond graph (step 7). Fig. A.9 Simplification of the bond graph (step 8a). Fig. A.10 Simplification of the bond graph (step 8b). are represented by a signal-node for the flows and a summation for the efforts (Fig. A.13). 3. All signals entering a summation resulting from a junction are given a sign corresponding to the half-arrow direction: if, while traveling from causal input to causal output, the bond orientation does not change (this does not exclude an orientation opposite to the signal direction!), then a plus sign is added representing a positive contribution to the summation; by contrast if the bond orientation does change, then a minus sign is added representing a negative contribution to the summation (Fig. A.14). In principle, a complete block diagram is obtained at this Fig. A.11 Causal bond graph. Fig. A.12 Expansion of causal bonds into bilateral signals. Fig. A.13 Expansion of the nodes into operational blocks. Fig. A.14 Addition of signs to the summations. point. However, its topology is not common due to the location of the conjugate signals. This may be omitted in the next step. 4. (optional) Redraw the block diagram in such a way that the inputs are at the left-hand side and the outputs (observed variables) are at the right-hand side (Fig. A.15) with the integrators in the forward path. The block diagram may be manipulated according to the standard rules for block diagrams as to obtain a canonical form. The procedure to obtain a signal flow graph is completely analogous to the above procedure as all operations represented by blocks, including the signs of the summations, are combined as much as possible and then written next to an edge, while all summations become nodes, as signal nodes can be distinguished from signal summation points by observing the signal directions (signal node has only one input, summation has only one output). Fig. A.15 Conversion into conventional form. A.1.4 Generation of a set of mixed algebraic and differential equations An arbitrary bond graph with n bonds contains 2n conjugate power variables, 2n ports and 2n corresponding port relations (constitutive relations). If a bond graph is made causal, the order in which the causal strokes are assigned to the bonds can uniquely label the bonds and their corresponding efforts and flows by using the sequence numbers of this process as indices. Next the constitutive relation of each port is written in the form that corresponds to the assigned causality. This results in a mixed set of 2n algebraic and first-order differential equations in an assignment statement form. Note that the differential equations that belong to storage ports in preferred integral causality have a time derivative at the left-hand side of the assignment statements, indicating a ‘postponed’ integration, if it were. During numerical simulation, this integration is performed by the numerical integration method to allow for the next model evaluation step. The switched junctions have the same causal port properties as the regular junctions, but no acausal form of the constitutive relations exists, while it necessarily contains ‘if-then-else’ statements that can only be written after causality has been assigned. The algebraic relations can be used to eliminate all the variables that do not represent the state of a storage port or an input variable, thus resulting in a set of ordinary differential equations (ODE) if all storage ports have preferred causality or in a set of differential and algebraic equations (DAE) if there are dependent storage ports. If the elimination of the algebraic relations is done by hand, the following three intermediate steps are advised: 1. Eliminate all variables that are dependent on the identities of junctions (0, 1) and sources ((M)Se, (M)Sf); 2. Eliminate all variables that are related by the algebraic port relations of all the ports that are not junction ports and not source ports ((M)R(S), (M)TF, (M)GY); 3. Eliminate all variables that are related by the port relations of all the ports that are junction summations (0, 1); 4. If an algebraic loop is present (active arbitrary causality) choose a variable in this loop to write an implicit algebraic relation and solve symbolically if possible. Otherwise the use of an implicit numerical method is required; 5. If present and possible, eliminate a differentiated state variable at the right-hand side of the relations symbolically if possible. Otherwise the use of an implicit numerical method is required. For example, the bond graph in Fig. A.11 contains 10 bonds, 20 equations and 20 variables of which two are state variables, such that 18 variables have to be eliminated. There are 9 identities (2 source and 3 + 2 + 2 = 7 junction ports), 6 multiplications (2×2 transducer +2 R) and 3 summations (3 junctions) resulting in the 18 necessary algebraic relations. The final result, assuming linearity of the elements, is or in matrix form d n R (A.1) dt A.2 Linear analysis A.2.1 Introduction Even though it may support a frequently encountered, persistent misapprehension that the port-based approach and its bond graph representation require the restriction that all constitutive relations of the nodes of a bond graph should be linear, this assumption will be made in the next section, but only to show the link between bond graphs and commonly used linear analysis techniques in system dynamics. However, it should be strongly emphasized that much of the linear analysis directly applied to the bond graph representation can be qualitatively generalized to the nonlinear case, as it still provides an insight. At the least, it gives an impression about small-signal behavior near an operating point of a nonlinear system model. Direct application of the wide range of linear analysis techniques on a bond graph should serve a purpose in the sense that it provides some additional information. If this is not the case, there is no need to change from a conventional model representation already obtained, like a set of linear state equations, into a bond graph representation. If all constitutive relations of the nodes of a bond graph are assumed to be linear, the bond graph represents a linear system model and each elementary node other than the junctions (and the unit gyrator called symplectic gyrator) can be characterized by one parameter (C, I, R, TF, GY) or input(signal) ((M)Se, (M)Sf). In case of (external) modulation, the linear system model becomes time-variant (MR, MTF, MGY). Note that internal modulation causes nonlinearity and cannot occur in the linear case. A.2 Linear analysis Table A.2 Impedance and admittance formulations of 1-port elements and corresponding gains. Table A.3 Gains 2-port elements in various causal forms. Given that a causal, linear bond graph is equivalent with any other linear system representation, it can be used to support all kinds of linear analysis. The conversion of a bond graph into a block diagram, a signal flow graph or a set of differential equations was already discussed. This makes clear that any linear analysis technique that exists for these kinds of models formulations can be directly applied to a causal, linear bond graph as well. In particular transmission matrices and Mason’s loop rule can be used to derive transfer functions between a chosen input and a chosen output in case of tree and chain structures [36]. As the identification of signal loops takes place in a bond graph via the causal paths, there is an immediate connection during modeling between the properties of a transfer function and the physical properties. The advantage of applying these techniques directly to the bond graph is that the relation of certain aspects of the linear analysis or the transfer function in particular with the physical structure can be directly observed and used to create or to adapt to desired behavior. This not only supports modeling decisions, but also allows insight in how physical changes can be made to obtain a required transfer. In particular, an impedance analysis will be discussed, as it provides a means to directly generate port equivalent compositions and decompositions. For linear analysis purposes, it is often useful to write the gain related to a node directly in the bond Table A.4 Composition rules for junctions and 1-ports. Table A.5 Composition rules for 2-ports and 1-ports. Table A.6 (De-)composition rules involving transducers. graph. In order to distinguish this notation from the regular notation of characteristic parameters (:) or generating functions (::) the gains, in which differentiations and integrations are replaced by the Laplace operator s and 1/s respectively, are placed between square brackets ([]). A.2.2 Impedance analysis using bond graphs A port of an element in effort-out causality can be characterized by an impedance, while a port with flow-out causality is best described by an admittance. Tables A.2 and A.3 provide listings of the possible gains that characterize the basic elements, both in impedance and in admittance form. Table A.4 illustrates that (de-)composition operations involving a 1-junction are best performed in impedance A.3 Port-based modeling and simulation: a simple example Fig. A.16 Word bond graph of a simple servo system. Fig. A.17 Bond graph of the dominant behaviors. form, while (de-)composition operations involving a 0-junction are best performed in admittance form as this leads to simple summation operations. Tables A.5 and A.6 list some elementary (de-)composition rules and the results for basic elements respectively. A.3 Port-based modeling and simulation of dynamic behavior of physical systems in terms of bond graphs: a simple example The model structure of a simple servo system is generated in order to give an impression of the port-based approach and the feedback on modeling decisions provided by the causal analysis. First a word bond graph is drawn at the component level, combined with a block diagram representation of setpoint, controller and closed loop (Fig. A.16). This gives an impression of the important domains and the corresponding variables of interest. Next the components are replaced by the nodes of a bond graph that represent the dominant behavior of each of the components (Fig. A.17). The causality shows that, apart from the dynamics of the controller and the integration in the position sensor the drive system model has no dynamics: the imposed voltage directly determines the servo speed (1st order system). In order to add some dynamic behavior, the resistance and inductance of the motor circuit, the friction and inertia of the rotor are added as well as the inertia of the load (Fig. A.18). The causality not only shows that the rigid connection between the rotor inertia and the load inertia makes them dependent, but also that the inductance of the motor circuit forms a second order loop (causal path) with the mechanical inertia (rotor & load) via the gyrator (3rd order system). Fig. A.19 demonstrates that modeling the torsion of the drive system resolves the dependency between rotor and load inertia, but creates a new second order loop (5th order system). Fig. A.20 shows that changing from a voltage control of the motor to current control not only suppresses its electrical time constant, but also the second order loop between inertia and inductance via the gyrator. Figures A.16 through A.20 are all screen dumps of models that can directly be simulated when relevant parameters are selected. The information of all these steps supports the modeling process, depending on the problem context. The introduction to bond graph concepts and notation in previous sections has already discussed many links between the representation and the modeling process. It Fig. A.18 Bond graph of the dominant behaviors and some important dynamics. Fig. A.19 Addition of the torsion of the drive system resolves the dependency between rotor and load inertia. Fig. A.20 Current-control of the motor suppresses the electrical time constant as well as the second order loop via the GY. cannot be sufficiently emphasized that modeling is a decision process that is different each time, but which can be supported by looking for conceptual structure based on universal principles as well as a direct link with computational issues, which provides direct feedback on modeling decisions. The bond graph notation supports this due to its domain independence and its ability to represent conceptual and computational information simultaneously. A true understanding of these features is only obtained by sufficient practicing. Appendix B Mathematical Background B.1 Linear algebra and differential geometry This appendix presents a short intuitive overview of some of the mathematical concepts used in this book. More detailed, precise, and extended treatments can be found in many textbooks, such as [111, 208] for linear algebra and vector spaces, [41,68,186] for differential geometry and manifolds, and [175,187] for Lie groups. B.1.1 Duality in vector spaces We start with two basic definitions of mappings and some possible properties. These properties are illustrated in the figures B.1a-B.1d. Definition B.1 (mappings). A mapping f between two sets A and B associates exactly one element of B to each element of A. We denote it abstractly as f : A → B, and its action on an element a ∈ A as f(a) 7→ b with b ∈ B. The set A is called the domain of f, and the set B its co-domain. The set of all b ∈ B such that there exists an a ∈ A with f(a) 7→ b is called the range of f. Definition B.2 (surjective, injective, bijective). A mapping f : A → B is surjective (or onto), if its range is equal to its co-domain. It is injective (or one-to-one) if for every b in its range, there is exactly one a ∈ A such that f(a) 7→ b. A mapping is bijective (or one-to-one and onto) if it is injective and surjective. In addition, a diffeomorphism is a mapping between Rn and Rn that is injective, continuously differentiable, and has a continuously differentiable inverse. Some examples of different types of mappings f : R → R are the functions f(x) 7→ x2 (not surjective, not injective), f(x) 7→ tan(x) (surjective, not injective), f(x) 7→ ex (not surjective, injective), and f(x) 7→ x3 (bijective). 381 Definition B.3 (vector space). A real vector space V is a set of elements (called vectors), one element called the identity (or zero-vector →−0 ), and two operations ⊕ (addition of two vectors) and · (multiplication of a vector by a scalar), such that • for all two elements v1,v2 ∈V, also v1 ⊕v2 ∈V; •• for all elements v ∈V and x ∈ R, also x·v ∈V; ⊕ →− for all elements v ∈V, there exists a v−1 ∈V such that v v−1 = 0 ; and such that the following properties hold for all v1,v2,v3 ∈V and x1,x2 ∈ R. v1 ⊕→−0 = v1 (v1 ⊕v2)⊕v3 = v1 ⊕(v2 ⊕v3) 1·v1 = v1 (x1 +x2)·v1 = (x1 ·v1)⊕(x2 ·v1) x1 ·(x2 ·v1) = (x1x2)·v1 x1 ·(v1 ⊕v2) = (x1 ·v1)⊕(x1 ·v2) where x1 +x2 and x1x2 are standard addition and multiplication of real numbers. The abstract definition of a vector space includes many different spaces with a linear structure. Not only an obvious example like the space of all velocity vectors of a point mass is a vector space, but, for example, also the space of all 2×3 matrices, if we take element-wise addition as the ⊕ operator and the zero-matrix as the identity element. Since a vector space is closed under addition and scalar multiplication, we can search for the smallest number of elements ei ∈ V such that any element of V can be constructed by addition and scalar multiplication of the elements ei. If we can find n < ∞ elements ei that accomplish this, then the vector space is said to be ndimensional, and the ei elements are called a basis of the vector space. The dimension n of a vector space is unique, but the choice of basis ei is not. By definition, we can express any element v ∈V as a linear combination of the basis elements, i.e. as n v = ∑viei = v1e1 +v2e2 +...+vnen (B.1) i=1 the n numbers vi ∈ R can serve as coordinates for V, as they define a bijective mapping between Rn and V. Definition B.4 (dual vector space). The dual space V∗ of a vector space V is the space of all linear mappings (called co-vectors) from V to R, i.e. all mappings f : V → R such that for all vi ∈V and xi ∈ R. f ((x1 ·v1)⊕...⊕(xk ·vk)) = x1 f(v1)+...+xk f(vk) (B.2) Definition B.5 (dual product). The dual product is the natural pairing of an element v ∈V and an element f ∈V∗ as hf | vi:=f(v) ∈ R If we choose a basis ei for V, we can express any element v ∈ V as (B.1), and hence from (B.2) we see that the mapping of v by an element f ∈V∗ can be written as hf | vi = f(v) = f (B.3) i.e. as a linear combination of the mappings of the basis elements. This shows that a dual element f is fully defined by how it maps the basis elements, and, since each element f(ei) is a single real number, it shows that the dimension of V∗ is equal to the dimension of V, i.e. the number of basis elements ei. It also suggests a basis for the dual space V∗, which we denote by ej and is defined by the condition j j j 1 when i = j (B.4) 0 when i =6 j where δji is the Kronecker delta. Any f ∈ V∗ can then be written as a linear combination of the basis elements ei n f = ∑ fjej = f1e1 + f2e2 +...+ fnen (B.5) j=1 with the numbers fj ∈ R again defining coordinates for V∗ in the basis ej. With these choices of bases, computing the dual product (B.3) becomes n fjviej fivi (B.6) so, for these choices of bases, computing the dual product of a vector and a co-vector simply means summing the pair-wise products of their coordinates. Vector spaces and their duals define an interesting mathematical structure, but they can also be used to represent a physical structure, for example as follows. Consider a robotic mechanism with n joints. The space of velocities ˙q (at a point q) forms a vector space V, and we can choose a basis for example as ei = q˙i, i.e. each basis element describes the unit-velocity of one joint, and zero velocity of the other joints. This vector space V automatically induces a dual space V∗ of abstract linear operators mapping a velocity to a real number. We can just ignore this dual space, but we can also think of it as the space of all collocated joint torques, i.e. the ndimensional space with elements τ and basis elements ei =τi. From the structure of the vector space and its dual, we can pair elements as n hτ| q˙i = ∑τiq˙i (B.7) i=1 such that applying τ to ˙q produces a real number. The reason for choosing this interpretation of a vector space and its dual becomes clear when we interpret also this real number: the dual product represents the mechanical power flowing into the system when it is moving with velocity ˙q and with applied torques τ. Associating the abstract mathematical concept of (dual) vector spaces to the practical physical concept of collocated power variables (force and velocity) can help reasoning about the physical concepts. The mathematical structure constrains computations to make sense. For example, computing the power as (B.7) only makes sense when τ and ˙q are collocated, which is equivalent to V and V∗ having dual bases as defined in (B.4). In this way, keeping the mathematical structure between physical variables in mind can help to avoid mistakes. Definition B.6 (tensor). Given a vector space V and its dual V∗, a tensor T is a mapping of the form T : (B.8) p times q times that is linear in all its arguments. The tensor T is said to have order p+q, order p contra-variant and order q co-variant, and is called a type (p,q) tensor. Tensors are linear operators that map vectors and co-vectors to R, and as such, are generalizations of the concepts of vectors and co-vectors. In fact, a co-vector is a type (0,1) tensor, since it maps a vector (an element of one copy of V) to R. Similarly, a vector is a type (1,0) tensor, since it maps a co-vector (an element of one copy of V∗) to R. Both mappings are defined by the dual product. Basis elements and the corresponding coordinates for tensors can be constructed from coordinates for vectors and co-vectors, simply by taking the appropriate coordinates for each of the arguments. The (i1,...,ip, j1,..., jq)th coordinate of a type (p,q) tensor T, i.e. the result of applying T to the basis vectors e1,...,ep and i1,...,ip co-vectors e1,...,eq, is denoted by Tj1,...,jq . This convention of writing the contravariant indices as superscripts and the co-variant indices as subscripts can be useful to quickly assess the type of tensor from its representation in coordinates. A metric tensor, often denoted by g, is a symmetric positive-definite type (0,2) tensor. It is symmetric in the sense that g(v,w) = g(w,v) for any two vectors v,w, and positive-definite in the sense that g(v,v) > 0 for all vectors v except the zerovector (in which case it returns zero by linearity of tensors). With a metric-tensor, we define the inner product between two vectors v and w as hv,wi:=g(v,w) = g(w,v) = ∑gijviwj i,j the length of a vector v as (B.9) kvk:=phv,vi = pg(v,v) = r∑gijvivj (B.10) i,j and the cosine of the angle between two nonzero vectors v and w as (B.11) When hv,wi = 0, the vectors v and w are said to be orthogonal in the metric g. We can again relate the mathematical concept of a metric to physical variables. In this case, we take e.g. again the space of velocities ˙q as the vector space V, and now the mass matrix M as a metric tensor, since it is indeed symmetric and positive definite. Then, when we apply the tensor M to two copies of ˙q, that is, we multiply the matrix with the vectors as ˙qTMq˙, we obtain a number that represents the physical quantity of twice the kinetic co-energy associated with the velocity ˙q. We can also define a new tensor by applying the metric tensor only to one copy of V. The resulting tensor maps one tangent vector to R, and is hence a tensor of type (0,1) – a co-vector. If we again take the physical example of a robot with velocity ˙q and mass matrix M, the new tensor is Mq˙ – the generalized momentum (co) vector. Hence, we have seen two interpretations of the dual vector space: one as the space of forces, and one as the space of generalized momenta. The distinction between the different types of tensors allow to assess what operations between them are possible. For example, a metric tensor is an operator mapping two vectors to R, and hence it does not make mathematical sense to apply them to co-vectors, even though the coordinates of a metric tensor (represented by an n×n matrix) can be multiplied by the coordinates of a co-vector (represented by an n-dimensional column vector). B.1.2 Manifolds The configuration space of a system is generally represented by an abstract space that is not directly equal to Rn for some suitable n. For example, Sect. 3.2 shows how the configuration of a rigid body is described by an element of SE(3), and how using R6 (six numbers, e.g. including Euler angles) leads to singularities and other numerical problems that are not present in physics. Differential geometry is a field of mathematics that makes exact the global properties of a configuration space such as SE(3), while still allowing to do computations locally using real numbers. In this section, we give a very brief intuitive overview of the idea of differential geometry and how it can help to use some concepts from this field. The central concept in differential geometry is the concept of a manifold. We can think intuitively of a manifold as some kind of abstract space (such as SE(3)) that locally looks like Rn. More precise, if we take a point in the abstract space, then the space around this point can locally be described by coordinates in an open subset of Rn. An example of such a manifold is the surface of the earth, which globally is (more or less) a sphere, and locally (at each point) can be described by coordinates in R2, i.e. a flat chart. These charts, unfortunately, are not global, due to the topology of the sphere. Once it becomes clear that a space is a manifold, i.e. once we have found enough local coordinate charts to Rn to cover the whole space, we can define global objects on the manifold (such as functions) by defining them first locally for each chart, and then checking certain compatibility conditions between the charts. These compatibility conditions ensure, for example, that a certain point in the abstract space, with two different coordinates in two different local charts, still has the same function value. A manifold is called differentiable, if the mappings that change coordinates between different charts are diffeomorphisms. At each point p of a differentiable manifold M, the space of all tangent vectors is called the tangent space, denoted TpM. This space is a linear vector space of dimension n, and describes all possible directions around p. The union of the tangent spaces over all points of M is called the tangent bundle, denoted TM. An element of the tangent bundle consists of a point p ∈ M plus a vector in TpM: the tangent bundle is hence 2n dimensional. The fact that the tangent space is a vector space allows to generalize the linear algebra concepts from Sect. B.1.1 to the setting of differential geometry. Since the tangent vector space at every point p has a dual, denoted by Tp∗M, we can define the cotangent or dual tangent bundle T∗M as the union of all dual tangent spaces. The concept of a tensor can be generalized to a tensor field, which is an object, defined on the manifold, that at each point p maps copies of the tangent space TpM and dual tangent space Tp∗M to a real number, and that varies smoothly over M. Note that tensor fields only operate on vectors and co-vectors that are elements of tangent and co-tangent space at the same point p. An example of a tensor field is a vector field, which is a tensor field of type (1,0) that assigns to each point of the manifold a tangent vector. Fig. B.2a shows an example. It also shows how, from a vector field, we can define its integral curves as the curves with velocity vector at all points equal to the value of the vector field Fig. B.2a Vector field X and one Fig. B.2b A function f and its Lie derivative along X. of its integral curves φ(t). The curve φ(t) is parameterized by t with φ(0)= p. at those points. Integral curves can be interpreted as the trajectories of a particle flowing along the vector field. Given a function f : M → R of the points of the manifold, this function is obviously also defined for the points of the integral curves. If the function is differentiable, we define its Lie derivative along the vector field X at a point p as d (LX f)(p):=(B.12) dt where φ(t) is an integral curve of X with φ(0) = p. It can be shown that this expression is independent of the choice of integral curve. An example of a Lie derivative is shown in Fig. B.2b. From two vector fields X and Y, we can also define a third vector field Z as the unique vector field such that for all functions f LZ f = L[X,Y] f = LX(LY f)−LY(LX f) (B.13) This new vector field is called the Lie bracket of the two vector fields. It roughly represents the velocity when moving a little along X, then a little along Y, then a little along −X, and finally a little along −Y. Another example of a tensor field is a metric tensor field, which assigns to each point of the manifold a metric tensor, i.e. a symmetric positive definite type (0,2) tensor. Such a tensor defines the metric concepts (dot-product, length, and angle) for tangent vectors at all points of the manifold. B.1.3 Geometric structures A geometric structure on a manifold is, loosely speaking, an object (structure) that is defined on all points of the manifold and that does not depend on the chosen coordinate systems for the manifold. The word ‘geometric’ refers to the fact that you can usually picture these structures as geometric entities (subspaces, planes, lines, etcetera) attached in some way to the manifold. Examples of geometric structures were already given in the previous section. For example, the tangent bundle is a geometric structure, as it is defined independently from the coordinates on the manifold as the space of all vectors tangent to the manifold. Similarly, a certain vector field on a manifold is a geometric structure, as the elements of the field attach certain ‘arrows’ to all points of the manifold. Of course, the parameterization of geometric structures does depend on the coordinates, in the sense that the elements that belong to the structure are expressed in different coordinates for different coordinate systems on the manifold, but these expressions are such that the elements of the structure themselves are invariant under coordinate changes. For example, the coordinates of elements of the tangent bundle may change under coordinate changes, but the geometrical ‘arrows’ that they represent remain the same. Another example of a geometric structure on a manifold is a vector bundle, which assigns to each point of the manifold a vector space (not necessarily the tangent space), in such a way that the vector spaces at different points have the same structure, such that the vector spaces at different points can be mapped to each other by isomorphisms. A direct example of a vector bundle is the tangent bundle on a manifold, but another example is the Dirac structure, as defined in Definition 2.2. Finally, the Whitney sum of two vector bundles B1 and B2 is defined as the vector bundle B1 ⊕B2 with vector space at each point equal to the direct sum of the two vector spaces of B1 and B2 at that point. In other words, if two vector bundles Bi assign to each point of the manifold a vector spaceVi, then the Whitney sum B1 ⊕B2 assigns to each point of the manifold a vector space V1 ⊕V2, the direct sum of V1 and V2. B.1.4 Lie groups and algebras Definition B.7 (group). A group G is a set S together with a binary operator • : S×S → S and an element I ∈ S, such that for all s1,s2,s3 ∈ S we have identity element: s1 •I = I •s1 = s1 (B.14) associativity: (s1 •s2)•s3 = s1 •(s2 •s3) (B.15) inverse element: ∃ s−1 1 ∈ S such that s1 •s−1 1 = s−1 1 •s1 = I (B.16) If also s1 •s2 = s2 •s1 (commutativity property), the group is called abelian. A simple example of a group is the set Z = {..., −2, −1, 0, 1, 2, ...} together with the standard addition operator and identity element 0. It can be checked that this is indeed a group: adding zero to any element of the set indeed gives that same element, addition is associative, and for each element, the inverse is simply the negation of that element. Since summation is even commutative, this group is also abelian. Definition B.8 (Lie group). A Lie group G is a manifold that is also a group, i.e. it has a binary operator • : G ×G → G and an identity element I ∈ G that satisfy the group properties. A Lie group is basically a group with a differentiable structure, which allows to talk about curves, velocities, tangent spaces, etcetera. We discuss a few examples of Lie groups that are useful for robotics, i.e. examples that describe positions and orientations in space. Since these groups are generally abstract, we also discuss matrix representations, i.e. sets of matrices with certain properties, which, together with the usual matrix multiplication as binary operator and the identity matrix as identity element, can be related one-to-one to abstract elements of the group. The matrix representations can be used in numerical computations as a type of singularity-free (though redundant) set of coordinates. Example B.1 (Translation). The group of all translations in n dimensions is denoted by T(n), e.g. the group of translations in three dimensions is denoted by T(3). Clearly these groups can be directly identified with Rn, and so a matrix representation of T(n) would be the space of n-dimensional column-vectors pn, together with vector addition as the the binary operator, and the zero vector as the identity element. Another possible representation is as the set of all (n+1×n+1) dimensional matrices structured as (B.17) with pn the translation vector, together with matrix multiplication as binary operator and the identity matrix as identity element. This method looks cumbersome and the matrix representation is highly redundant, but it proves useful when translations are combined with rotations. Example B.2 (Fixed-axis rotation). The space of rotations around a fixed axis forms a group under the binary operator of combining rotations by performing one rotation after the other. The group is denoted by SO(2) and can be identified with the circle S 1. The group is one-dimensional, and in practice, it is often described by a single real number (the angle of rotation). This, however, neglects the fact that a full 360◦ rotation does not change the group element, although it does change the angle; this is the difference between a circle and a straight line. Instead, the group of rotations can be described by the set of special orthogonal 2×2 matrices (whence the name SO(2)), meaning 2×2 orthogonal matrices with determinant +1. These matrices have the form R (B.18) where φ is the angle of rotation. Together with matrix multiplication as the binary operator, and the identity matrix (φ= 0) as the identity element, these matrices form a complete representation of the group of fixed-axis rotations. Example B.3 (Spatial rotation). The space of free rotations around any axis in three dimensions forms a group, and is denoted by SO(3). The group is three-dimensional, and is often represented locally by three angles, called the Euler angles, that describe three consecutive rotations around three (local) axes. Such a parameterization, however, has singularities, which results in non-smooth behavior of the coordinates around singularities. Instead, rotations can be fully and uniquely identified with the set of all special orthogonal 3×3 matrices (whence the name SO(3)), meaning 3×3 orthogonal matrices with determinant +1. Another representation of the group of spatial rotations is by unit quaternions. In this representation, a vector of the form q (B.19) is used to describe rotation around an axis n with angle θ. The axis n is constrained to have unit norm (in the Euclidean sense), which means that also the vector q has unit norm. This representation is singularity free, but it doubly covers SO(3), since the rotation angles α and α+ 360◦ (for some α) define the same rotation, but are represented by different vectors q. Example B.4 (Planar motion). We can combine the group of two-dimensional translations, i.e. translations in a plane, with the group of fixed-axis rotations and take the fixed axis to be orthogonal (in the Euclidean sense) to the translational plane. The resulting object is again a Lie group, and it describes all planar motions, that is, the set of all possible ways that an object can be positioned in a plane. This group is called the special Euclidean group of dimension two, denoted SE(2). As representation of this group, we can simple use a combination of a twodimensional vector p to describe translation and a matrix R of the form (B.18) for the translation. This choice is often made in literature, but computations in this representation are cumbersome, since two consecutive motions need to be combined as R13 = R23R12 p13 = p23 +R23p12 (B.20) which leads to long and tedious equations for multiple consecutive motions. Instead, we combine translation and rotation in one so-called homogeneous matrix of the form H (B.21) where R is the rotation matrix (B.18). Consecutive planar motions can now be represented by simple matrix multiplications of the corresponding homogeneous matrices. The matrix representation of a translation as (B.17) is a special case of (B.21) for zero rotation, R = I. Example B.5 (Three-dimensional motion). Similar to the planar situation, we can combine the group of translations in three dimensions T(3) with the group of freeaxis rotations SO(3). The result is the special Euclidean group in three dimensions, or, SE(3), that describes the space of all possible relative positions and orientations in three-dimensional space. It can also be represented by a matrix of the form (B.21) but now with R a three-dimensional rotation matrix, and p a three-dimensional translation vector. Again, consecutive motions are simply represented by matrix multiplication of the appropriate homogeneous matrices. The examples show how many useful transformations are actually Lie groups, and that these can be represented globally and without singularities by matrices with the appropriate properties. The realization that these transformations are Lie groups allows to perform certain useful operations on them. First, because of the group structure, we can take an element of the group and combine it with another element of the group. This is called (left or right) translation, since effectively it transports one element of the group to another place, by means of group multiplication. In particular, since every element of a group has an inverse, we can transport a group element to the identity of the group. This transport can be done in two ways, either by pre- or post-multiplication with the inverse. Secondly, since a Lie group has a differentiable manifold structure, we can talk about continuous and differentiable curves in the group, which represent smooth consecutive transformations, i.e. smooth motions of an object. The derivatives of these curves represent the (angular, linear, or combined) velocities of the moving objects. Combining these two aspects (the group aspect and the manifold aspect), we can transport a curve γ(t) near an element A ∈G to a curve near the identity by applying A−1 to every element of the curve. We can then take the derivative of the transformed curve to obtain an element of the tangent space TIG at the identity. Depending on whether left or right translation is chosen, different velocity vectors are obtained. Since in this way, velocity vectors at any point A ∈ G can be transported to the tangent space at the identity, this tangent space provides a common vector space which allows to compare and add different velocities. Furthermore, the tangent space at the identity can be given the structure of a Lie algebra, which is defined as follows. Definition B.9 (Lie algebra). A Lie algebra is a vector space together with a binary operator [,] : V ×V → V (called Lie bracket or commutator), that satisfies the following properties for all v1,v2,v3 ∈V and a1,a2 ∈ R ( [a1v1 +a2v2,v3] = a1[v1,v3]+a2[v2,v3] bilinearity:(B.22) [v1,a1v2 +a2v3] = a1[v1,v2]+a2[v1,v3] skew-symmetry: [v1,v2] = −[v2,v1] (B.23) Jacobi’s identity: [v1,[v2,v3]]+[v2,[v3,v1]]+[v3,[v1,v2]] = 0 (B.24) An example of a Lie algebra is the vector space V of all n×n matrices with the Lie bracket defined as [A,B]:=AB−BA for A,B ∈V. In the case of a Lie group, the tangent space at the identity can be given the structue of a Lie group by defining the appropriate Lie bracket on it, and this tangent space is thus usually called the Lie algebra of the group, and is denoted by g:=TIG. Finally, as shown in Sect. 3.2, the tangent vectors at the identity of a group can have a clear physical interpretation, much more than tangent vectors at general points A ∈ G. B.2 Legendre transforms and co-energy A concise overview of homogeneous functions and Legendre transformations is given and applied to energy functions, thus leading to the concept of co-energy. Various properties of energy and co-energy related to storage are shortly discussed and related to a physical interpretation. Finally some domain specific forms of coenergy are discussed. B.2.1 Homogeneous functions and Euler’s theorem A function F(x), with x = x1,...,xk, is homogeneous of order (or degree) n if F(αx) =αnF(x). Define yi(x) = ∂∂xFi , then yi(x) or ∂F(αx) αn ∂F(x) yi(αx) = = =αn−1yi(x) (B.25) ∂αxi α ∂xi is homogeneous of order (n − 1). For a homogeneous function, Euler’s theorem holds: ∑i = 1k ∂ xFi xi = nF(x) or F(x) = n 1 i∑=k1yixi = 1n yT ·x ∂ with y = y1,...,yK. By definition: (B.26) k ∂F k T dF = i∑=1 ∂ xi ·dxi = i∑=1yidxi = y ·dx but also (B.27) dF dx (B.28) Hence: dyT ·x = (n−1)yT ·dx → for n = 1: dyT ·x = 0 (B.29) for n 6= 1: dF x Fig. B.3 Bond graph repre- sentation of a storage elementSf: 0 with n−1 ports constrained. B.2.2 Homogeneous energy functions The energy of a system with k state variables qi is E(q) = E(q1,...,qk). If qi is an extensive state variable, this means that E(αq) =αE(q) =α1E(q). Hence E(q) is first order (i.e., n = 1) homogeneous, so ei ∂ is zero-th order (i.e., n − i 1 = 0) homogeneous, which means that ei(q) is an intensive variable, i.e. ei(αq) = α0ei(q) = ei(q). This also means that in case n = 1 and k = 1, e(q) is constant, i.e. ∂e = de = 0, which changes the behavior of this element into that of a source. In ∂q dq order to enable storage, storage elements = multiports (k > 1). Hence, the common ‘1-port storage element’ = n-port storage element with flows of n−1 ports kept zero, i.e. the corresponding n−1 states remain constant and are not recognized as states. Such a state is often considered a parameter: if E(q1,q2,...,qn)|dqi = 0 ∀i =6 1 = E′(q1), then E′(qi) is not necessarily first-order homogeneous in q1. This can be represented in bond graph form as in Fig. B.3. For n = 1 and k independent extensities there are only k−1 independent intensities, because for n = 1 we find a generalized Gibbs’ fundamental relation: E(q) = eT ·q By definition, already dE = eT · dq, and combining these equations results in the generalized Gibbs-Duhem relation: deT ·q = 0 (B.30) B.2.3 Legendre transform A Legendre transform of a homogeneous function F(x) with respect to xi is defined as L{F(x)}xi = Lxi = F(x)−yixi (B.31) where yi = ∂∂xFi . Moreover, the total Legendre transform of F(x) is k L{F(x} = L = F(x)− ∑yixi (B.32) i=1 Note that for n = 1, L = 0. Now dLxi = dF −d(yixi) = dF −yidxi −didyi = ∑yjdxj −didyi (B.33) j6=i independentor Lxi = Lxi(xvariable or ‘coordinate’! Hence1,...,xi−1,yi,xi+1,...,xk), which means thatL = L(y) and dxi is replaced byL = −∑ki=1 xidyyii as= −dyT ·x. B.2.4 Co-energy function The co-energy Eq∗i of E(q) with respect to qi is by definition: Eq∗i (B.34) Hence E eiqi. The total co-energy E∗(e) of E(e) is E∗ = −L, hence E(q)+E∗(e) = eT ·q. For n = 1, we have that E∗(e) = 0 thus confirming the earlier conclusion that there are only k−1 independent ei. For n = 2, E q and for n = 3, E q E q B.2.5 Relations for co-energy functions dEq∗i = deiqi − ∑ ejdqj j6=1 (B.35) k dE∗ = ∑deiqi = deT ·q = (n−1)eT ·dq = (n−1)dE i=1 (B.36) E∗ q (B.37) Table B.1 Extensities qi and intensities ei(q), with internal energy U. extensities qi intensities ei(q)= ∂∂qEi entropy S temperature T = ∂∂US volume V pressure p =−∂∂UV total mole number N total material potential µtot = ∂∂UN mole number per i-th species Ni chemical potential µi = ∂∂NUi B.2.6 Legendre transforms in simple thermodynamics In thermodynamics, the Legendre transforms appear in the following relations. The meaning of the involved quantities is explained in Table B.1. m−1µiNi +µtot ·N Free energy F: F = LS =U m−1 tot dF = −SdT − pdV + ∑µidNi +µ ·dN i=1 F(T,V,N,Ni) (= Ni f(T,v,c)) Enthalpy H: H = LV =U −(−pV) =U + pV dH = TdS+Vdp+ dN H(S, p,N,Ni) (= Nih(s, p,c)) Gibbs free enthalpy G: − − − tot · m−1µiNi G = LS,V =U TS ( pV) = µ N + ∑ i=1 m−1 tot dG = −SdT +Vdp+ ∑µidNi +µ ·dN i=1 G(T, p,N,Ni) (= Nig(T, p,c)) For m = 1: g = µtot(T, p) B.2.7 Legendre transforms and causality If an effort is forced on a port of a C-element (i.e. with derivative causality or flow causality), this means that the roles of e and q are interchanged in the set of independent variables, which means that the energy has to be Legendre transformed in order to continue to serve as a generating function for the constitutive relations. Se: Tconst C Fig. B.4a Bond graph associated to dF = udq. Fig. B.4b Bond graph associated to P = ddFt Such a transformation is particularly useful when the effort e is constant (e.g. an electrical capacitor in an isothermal environment with T = Tconst): • dF = udq−SdT = udq, or as the bond graph of Fig. B.4a; • P = u·q˙ = ddFt , or as the bond graph of Fig. B.4b. B.2.8 Constitutive relations The function ei(q) is called constitutive relation, also called constitutive equation. constitutive law, state equation, characteristic equation, etcetera. If ei(q) is linear, i.e. first order homogeneous, then E(q) is second order homogeneous, i.e. E(q) is quadratic. In this case, and only in this case: E(αq) =α2E(q) E q deT ·q = eT ·dq = dE B.2.9 Maxwell reciprocity From the principles of energy conservation (first law) can be derived that ∂2E ∂2E = ∂qi∂qj ∂qj∂qi ∂ej ∂ei ∂qi ∂qj = i.e. the Jacobian matrix of the constitutive relation is symmetric. This is called Maxwell reciprocity or Maxwell symmetry. B.2.10 Intrinsic stability Intrinsic stability requires that this Jacobian is positive-definite and that the diagonal elements of the Jacobian are positive ∂ei > 0 i ∂qi ∀ B.2.11 Legendre transforms in mechanics In mechanical systems with kinetic energy T, potential energy V, displacements x, momenta p, velocities v, and forces F, the following holds: Hamiltonian H: E(q) = H(x, p) = T +V Lagrangian L: H H = (T +T∗)−(T +V) = T∗ −V = L(x,v) ∂H with v = ∂p co-Hamiltonian: H H co-Lagrangian or Hertzian: Hx∗ =V −T = −L (F, p) ∂H with F = ∂x B.2.12 Legendre transforms in electrical circuits In electrical circuits with capacitor charger q, voltages u, coil flux linkages Φ, and currents i, we have E(q,Φ) = EC(q)+EL(Φ) E∗(u,i) = uT ·q+iT ·Φ−E Only in the linear case, it follows that E∗ = E. Appendix C Nomenclature and Symbols appearing in Sect. 3.4 Abstract This appendix summarizes the notation (i.e nomenclature, symbols and vector/tensor operators) adopted in Sect. 3.4. Nomenclature A surface area or affinity a activity defined with respect to the state of pure component (AB)∓ activated complex A, B chemical symbol C molar concentration c specific heat capacity d driving force for diffusion D diffusion coefficient e effort variable for diffusion E activation energy of a chemical reaction f flux of scalar quantity per unit of surface area F total flux of a scalar quantity G, g total and specific Gibbs free energy g external body force per unit of mass H, h total and specific enthalpy h¯, h¯′ total energy per mass unit for a thermodynamic system h¯ Planck constant I unit tensor of order 2 k¯ Boltzmann constant k chemical rate constant K equilibrium constant 399 400 C Nomenclature and Symbols appearing in Sect. 3.4 l distance M, M¯ total and molar mass N number of moles or number of components in a mixture p momentum per mass unit P pressure Pr chemical symbol of a product r, r˜ rate of a chemical reaction by unit of volume and its linearized counterpart R number of independent chemical reactions and ideal gas constant Re chemical symbol of a reactant r radial position S, s total and specific entropy T absolute temperature Tr trace of a tensor t time U, u total and specific internal energy V, v total and specific volume v, v velocity and its modulus W, w total or work per unit of volume Y, y total and specific scalar quantity y specific vectorial quantity For a total quantity Y, the specific quantity per mass unit is y, and the specific quantity per mole unit is ¯y. ∆rY is the variation of Y associated to a chemical reaction. Greek symbols α heat transfer coefficient or conductance by unit of surface area θ1, θ2 parameters in an equation of state ϕ equation of state κ constant for a linearized rate equation or volume viscosity Φ tensor of the momentum flux τ shear stress tensor σ source term per unit of volume η viscosity µ chemical potential ρ density or mass concentration χ extent of a chemical reaction or molar fraction ν stoechiometric coefficient γ activity coefficient defined with respect to the state of pure component λ, λ0 heat conductivity and a parameter in the heat conductivity expression C Nomenclature and Symbols appearing in Sect. 3.4 401 ω mass fraction Ψ potential energy per mass unit ∆ difference δ small quantity Σ total source term Subscripts ext surrounding f forward h¯, h¯′ total energy i, j indexes for components k index for chemical reaction l index for pipes m mixtures P constant pressure q heat or thermal energy p momentum r reverse ref origin or reference state s entropy u internal energy v constant volume w non-thermal energy y scalar quantity ∓ activated state Superscripts a anti-symmetric b boundary e excess eq equilibrium f forward k index for chemical reactions R flux expressed with respect to a moving frame Reac due to a chemical reaction Rev reversibly r reverse 402 C Nomenclature and Symbols appearing in Sect. 3.4 s symmetric Sc scalar T thermal diffusion coefficient (Soret effect) Tens tensorial Vect vectorial ∗ pure component in the same state then a mixture id ideal solution ig ideal gas 0 standard state Vectors and tensors notation We give here only the notations of the main quantities that are manipulated in Sect. 3.4 defined in a cartesian system of coordinates (x1,x2,x3). For a more complete presentation of this topic, see for example [5, 21] and Sect. 4.2.1 (and the references therein) for their geometric, coordinate-free interpretation in terms of differential forms. The gradient of a scalar quantity y, i.e. ∇y, is a vector of which the coordinates are By extension, one can also define the gradient of a vector y = [y1 y2 y3]T, i.e. ∇y, by the second order tensor: The divergence of a vector y is a scalar quantity: y3 ∇·y x3 An element of τ, the viscous part of the momentum flux (or shear stress tensor) is defined as follows: τij is the force for unit of surface area exerted in the direction i on a plane perpendicular to direction j. 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