University of Groningen
Multidomain Modeling of Nonlinear Networks and Systems ENERGY- AND POWER-BASED PERSPECTIVES Jeltsema, Dimitri; Scherpen, Jacquelien M. A.
Published in: IEEE Control Systems Magazine
DOI: 10.1109/MCS.2009.932927
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record
Publication date: 2009
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA): Jeltsema, D., & Scherpen, J. M. A. (2009). Multidomain Modeling of Nonlinear Networks and Systems ENERGY- AND POWER-BASED PERSPECTIVES: Energy- and power-based perspectives. IEEE Control Systems Magazine, 29(4), 28-59. https://doi.org/10.1109/MCS.2009.932927
Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license. More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne- amendment.
Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Multidomain Modeling of Nonlinear Networks and Systems
ENERGY- AND POWER-BASED PERSPECTIVES
any physical systems, includ- ing mechanical, electrical, electromechanical, fluid, and thermal systems, can be modeled by the Lagrangian M – and Hamiltonian equations of motion [1] [5]. A key aspect of the Lagrangian and Hamiltonian frameworks is the role of energy storage. Apart from the fact that energy is a fundamental concept in physics, there are several motivations for adopting an energy- based perspective in modeling physical sys- tems. First, since a physical system can be viewed as a set of simpler subsystems that exchange energy among themselves and the environment, it is common to view dynami- cal systems as energy-transformation devices. Second, energy is neither allied to a particular physical domain nor restricted to linear ele- ments and systems. In fact, energies from dif- ferent domains can be combined simply by adding up the individual energy contribu- tions. Third, energy can serve as a lingua franca to facilitate communication among sci- entists and engineers from different fields. Lastly, the role of energy and the interconnec- tions between subsystems provide the basis for various control strategies [4], [6]–[8]. In multidomain Lagrangian and Hamil- tonian modeling it is necessary to distin- guish between two types of energies, energy and co-energy. Energy is the ability to do work, while co-energy is the complement of
DIMITRI JELTSEMA and JACQUELIEN M.A. SCHERPEN
D.S. BERNSTEIN Digital Object Identifier 10.1109/MCS.2009.932927
28 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009 1066-033X/09/$25.00©2009IEEE
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. energy as defined and used in [3] and [9]–[15]. To elucidate For mechanical systems, the starting point in setting up the distinction between energy and co-energy, consider a the Lagrangian equations of motion is to determine the point mass M > 0movinginthex-direction. In the nonrela- Lagrangian. The Lagrangian is defined as the difference tivistic case, the momentum p is related to the velocity between the total stored kinetic co-energy associated with v ¼ dx=dt by the linear constitutive relationship p ¼ Mv; the masses and moments of inertia and the total potential and Newton’s second law is given by F ¼ dp=dt, where F is energy associated with gravitational forces and stiffness the force acting on the mass. If the mass is moved by the elements. The Lagrangian equations of motion give a force force, then the increment of work done by the force is Fdx, balance in terms of displacement and velocity, explaining which can be written as why, instead of the kinetic energy, the kinetic co-energy is used for the Lagrangian. More details on co-energy and the dp dx p Fdx ¼ dx ¼ dp ¼ vdp ¼ dp: Lagrangian equations of motion are given in the section dt dt M ‘‘Classical Energy-Based Framework.’’ An analogous situation occurs in the case of electrical Since the kinetic energy of the mass is given by the inte- networks. For both linear and nonlinear electrical networks, gral of the work done by the force, integrating v ¼ p=M energy can be stored magnetically in inductors and electri- from zero to p results in cally in capacitors, and these quantities can be used to p2 derive Lagrangian equations of motion. In particular, the T ( p) ¼ : 2M magnetic energy is defined to be the integral of the current with respect to the flux linkage, whereas the magnetic When plotted in the v-versus-p plane as Figure 1(a), T (p)rep- co-energy is defined as the integral of flux linkage with resents the area of the triangular region below the line p ¼ Mv respect to current. For linear networks, the relationship and to the left of the dashed vertical line. On the other hand, between flux linkage and current is linear, and thus the the complementary kinetic energy, that is, the kinetic co-energy, magnetic energy and the magnetic co-energy are equal. T is defined to be the area of the triangular region above the However, for nonlinear networks, these quantities are gen- line p ¼ Mv and below the horizontal dashed line. The kinetic erally different, similar to the case of the kinetic energy and co-energy cannot be obtained directly from work but rather kinetic co-energy of the relativistic mass described above. is defined in a complementary or dual fashion as the integral ThenetworkLagrangianisthedifferencebetweenthetotal of the momentum with respect to the velocity, which, refer- magnetic co-energy and the total electric energy, where ring to Figure 1(a), is tantamount to extracting the triangular the electric energy is the integral of voltage with respect to area defined by T (p) from the total square area defined by the product pv, that is,
M c T (v) ¼ pv T(p) ¼ v2: 2 p − Mv = 0 M v v v p − 0 = 0 √1 − v 2 / c 2 Note that the kinetic energy is quadratic in p,whilethe kinetic co-energy is quadratic in v,and,furthermore, T (p) ¼T (v). Because of this equality, it is traditional to not make a distinction between T (p)andT (v), and as a 0 p 0 p result T (v) is commonly called the kinetic energy rather (a) (b) than the kinetic co-energy. When the momentum p and T the velocity v are not linearly related, however, (p)and FIGURE 1 Kinetic energy versus kinetic co-energy. (a) Constitutive T (v) are generally different. In fact, when relativistic relationship of a nonrelativistic mass M plotted in the velocity v effects are considered, p and v are related by the nonlinear versus momentum p plane. The kinetic energy is represented by the constitutive relationship region below the line defined by the equation p Mv ¼ 0, whereas the region above the line represents the kinetic co-energy. For a constant mass, the areas of the two regions are equal. Hence the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM0v p ¼ , (1) kinetic energy coincides with the kinetic co-energy. Note that the 2= 2 1 v c relationship between momentum and velocity expresses the original version of Newton’s second law, that is, F ¼ dp=dt, with p ¼ Mv. where M0 denotes the rest mass and c is the velocity of For a constant mass, the latter coincides with the version F ¼ Ma. light. Consequently, for a relativistic mass, the area of the An advantage of stating Newton’s second law in terms of the rela- region below the v-versus-p curve is not equal to the area of tionship between momentum and velocity is that it is also valid when the region above the v-versus-p curve, as shown in Figure M changes in time, a situation where F ¼ Ma loses its validity. (b) For a relativistic mass, Newton’s second law can be extended to Ein- 1(b), and thus the kinetic energy T (p) is no longer equal to stein’s relativistic law of motion by replacing the linear constitutive the kinetic co-energy T (v), even though the units of both relationship p ¼ Mv by the nonlinear function given in (1). In this T (p) and T (v) are Joules (J). case, the kinetic energy clearly differs from the kinetic co-energy.
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 29
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. State Functions key aspect in the classical modeling procedures of analyti- the units of power. Many years later, Wells introduced a power A cal mechanics, due to Lagrange in the late 18th century, is function that is applicable to a much wider range of nonconserva- the use of state functions. A state function is a scalar function tive forces [S1], [S2]. In the context of nonlinear network theory, that depends only on the current state of the system irrespective the generalization of energy and power functions is due to Cherry of the way in which the system reached that state. To physically and Millar in the early 1950s. Cherry [9] introduced a function motivate the idea of a state function, kinetic and potential energy dual to the energy called the co-energy, whereas Millar [18] gen- storage, which are in itself state functions, are combined to form eralized Maxwell’s minimum heat theorem to nonlinear networks the Lagrangian. In the case of a conservative system, the equa- by defining the content and co-content functions. The use of co- tions of motion can be derived from the Lagrangian and thus energy (at that time called dualistic or complementary energy) from knowledge of the kinetic and potential energy contributions can be traced to the work of Count Menebr ea [S3], Maxwell [S4], alone. Nonconservative forces, such as forces due to linear and Essenger [13] in the 19th century. viscous friction, can be included by means of a Rayleigh dissipa- The energy, co-energy, content, and co-content are now defined tion function, which was introduced in 1873 by Lord Rayleigh on basis of the element quadrangle of Figure 2. The state functions [16]. Instead of energy, the values of this quadratic function have associated with memristive elements are known as action and co- action [22]. We first discuss one-port (that is, two-terminal) ele- ments. The extension to multiport elements is then discussed.
INDUCTIVE ENERGYAND CO-ENERGY An element that is characterized by a relationship between f e flow f and generalized momentum p is called an inductive or ‘‘I’’ element. Examples of inductive elements include a mechanical mass or an electrical inductor. More specifically, an inductive element is described by either a p-controlled 0 p 0 q constitutive relationship (a) (b) ^ f ¼ f (p)
or an f -controlled constitutive relationship
e p p ¼ p^(f ):
If both relationships are invertible, the element is said to be one to one. Graphically, the constitutive relationships represent a 0 f 0 q curve separating the associated f -p plane into two areas; see (c) (d) Figure S1(a). The area below the curve represents the induc- tive energy FIGURE S1 Constitutive relationships and state functions. The Z p‘ ^ graph in (a) illustrates the constitutive relationship of a nonlinear T (p‘) ¼ f (p)dp, (S1) inductive element. The area T below the curve is the inductive 0 energy, whereas the area T above the curve is the inductive whereas the area above the curve represents the inductive co- co-energy. In the nonlinear case, inductive energy generally dif- fers from inductive co-energy. However, for a linear inductive energy Z element the constitutive relationship is a straight line through f‘ the origin, and thus the inductive energy coincides with the T (f‘) ¼ p^(f )df : (S2) inductive co-energy. The constitutive relationship in (b) corre- 0 sponds to a capacitive element, where the capacitive energy Inductive energy and inductive co-energy are also referred to and capacitive co-energy are the areas V and V below and above the curve, respectively. In a similar fashion, the constitu- as generalized kinetic energy and generalized kinetic co- tive relationship of a resistive element shown in (c) separates energy. Note that in the linear case the constitutive relations are the effort-flow plane into two areas, which are denoted as resis- straight lines through the origin so that inductive energy coin- tive content D and co-content D . Figure (d) shows the state cides with inductive co-energy, that is, T (p‘) ¼T (f‘). functions associated with a memristive element, which are the memristive action A and the memristive co-action A . Note that CAPACITIVE ENERGYAND CO-ENERGY in (c) the sum of the content and co-content defines the total An element that is characterized by a relationship between effort e power of the element, whereas the sum of the energy and co- energy, as well as the sum of the action and co-action, have no and generalized displacement q is called a capacitive or ‘‘C’’ ele- physical meaning. ment. Examples of a capacitive element include a mechanical
30 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. spring or an electrical capacitor. More specifically, a capacitive ele- The area below and above the curve in the p-q plane (see ment is described by either a q-controlled constitutive relationship Figure S1(d)) represents the memristive action Z ^ qm e ¼ e(q), ^ A(qm) ¼ p(q)dq, (S7) or an e-controlled constitutive relationship 0 ^ : and memristive co-action q ¼ q(e) Z pm ^ If both relationships are invertible, the element is said to be one-to- A (pm) ¼ q(p)dp, (S8) 0 one. The constitutive relationships of a capacitive element sepa- rate the associated e-q plane into two areas; see Figure S1(b). respectively. The area below the curve represents the capacitive energy Z SOURCES qc ^ V(qc ) ¼ e(q)dq, (S3) Observe that in Figure 2 the four classifications are not mutually 0 exclusive. For example, a constant effort source, such as whereas the area above the curve represents the capacitive co- gravity, can be regarded as either a resistive or a capacitive ele- energy ment; for instance, in Figure S1(b) or (c), gravity can be repre- Z ec sented by a horizontal line. Similarly, a flow source can be ^ V (ec ) ¼ q(e)de: (S4) 0 regarded as either a resistive or an inductive element; see [S8] for a discussion in the electrical domain. Capacitive energy and co-energy are often also referred to as generalized potential energy and co-energy. MULTIVARIABLE CASE
RESISTIVE CONTENT AND CO-CONTENT Usually a system consists of more than one of each of the avail- able elements. For example, for a system containing n‘ possibly A resistive or ‘‘R’’ element, such as viscous friction, electrical resist- mutually coupled inductive elements, the flow and generalized ance, externally supplied forces or velocities, and voltage and cur- momentum variables f and p are n‘-dimensional vectors. rent sources, relates effort with flow, or vice versa. Again we have ^ Hence, the constitutive relationships f ¼ f (p) and p ¼ p^(f ) are two specific situations, namely, an f -controlled resistive element vector functions, and T and T represent the total inductive e e^(f ), ¼ energy and total inductive co-energy, respectively, which are and an e-controlled resistive element determined by the sum of the individual energy and co-energy ^ contributions, respectively. Thus, f ¼ f (e), Z Xn‘ p‘ k ^ ...... separating the associated e-f plane into two areas. The area T (p‘) ¼ fk ( , pk , )dpk k¼1 0 below the curve of Figure S1(c) represents the resistive content Z p‘ ^ Z ¼ f >(p)dp, fr ^ 0 D(fr ) ¼ e(f )df , (S5) 0 and Z Xn‘ f‘ whereas the area above the curve of Figure S1(c) represents k ^ T (f‘) ¼ pk ( ..., fk , ...)dfk the resistive co-content 0 Zk¼1 Z f‘ er ^ ^> : D (e ) ¼ f (e)de: (S6) ¼ p (f )df r 0 0 Furthermore, T (p‘) and T (f‘) are related through the Legendre Note that the sum D(f ) þD (e ) ¼ e f of the content and co- r r r r transformation content defines the total power supplied to or extracted from the element, whereas the sum of the energy and co-energy, as well > T (f‘) ¼ p‘ f‘ T(p‘), as the sum of the action and co-action defined below, is devoid of physical meaning. where p‘ ¼rf‘ T (f‘) and f‘ ¼rp‘ T (p‘). Similar extensions apply to the remaining state functions. MEMRISTIVE ACTION AND CO-ACTION A memristive or ‘‘M’’ element relates generalized momentum with REFERENCES displacement, or vice versa, and admits a q-controlled relationship [S1] D. A. Wells, Schaum’s Outline of Lagrangian Dynamics. New York: McGraw Hill, 1967. p ¼ p^(q), [S2] F. A. Fuertes, ‘‘On the power function,’’ J. Appl. Phys., vol. 17, p. 712, 1946. or a p-controlled relationship [S3] O. I. Franksen, ‘‘The nature of data: From measurements to sys- tems,’’ BIT Numer. Math., vol. 25, no. 1, pp. 24–50, 1985. ^ q ¼ q(p): [S4] J. C. Maxwell, Matter and Motion. New York: Dover, 1991.
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 31
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. The Brayton-Moser equations rely on the existence of a function called the mixed-potential function.
charge. For fluid systems, the fluid kinetic co-energy and difference between the total electric co-energy and the total the fluid potential energy are defined in the same manner magnetic energy. Consequently, the co-Lagrangian is ex- as for mechanical systems by using the analogy of force pressed in terms of voltages and flux linkages, instead of and pressure, as well as the analogy of mass velocity and currents and charges. Similarly, for mechanical systems, the flow velocity. Therefore, the fluid Lagrangian is the dif- co-Lagrangian is defined to be the difference between the ference between the total fluid kinetic co-energy and the total potential co-energy and the total kinetic energy and total fluid potential energy. For multidomain systems, hence is expressed in terms of forces and momenta instead such as electromechanical systems, the required energy of velocities and displacements. and co-energy functions follow by addition. Definitions, In the context of mechanical systems, the application and details, and some historical facts on the relevant energies usefulness of the co-Lagrangian equations is limited. First, and co-energies for the various engineering domains are most mechanical systems can be described by Lagrangian discussed in ‘‘State Functions.’’ and Hamiltonian equations. Second, the effect of gravity, The Lagrangian equations constitute a system of second- which can be included in the Lagrangian equations, cannot order differential equations, which can be transformed into a be captured by a potential co-energy function since the rela- system of first-order differential equations, called the Hamil- tionship between gravitational force and its associated dis- tonian equations, by performing a coordinate transformation placement is not globally invertible. In other words, we to express the dynamics in terms of alternative physical vari- cannot define the integral of the displacement with respect ables. For instance, in the mechanical domain the velocities to the gravitational force. However, in other engineering are transformed into their associated momenta. The Hamil- domains, the co-Lagrangian formulation is often useful— tonian equations are generated from the Hamiltonian, which, and sometimes even necessary—to describe the dynamics in instead of the difference between the kinetic co-energy and an energy-based manner. Some examples of networks and the potential energy, is defined to be the sum of the total systems that cannot be described by a Lagrangian, but do kinetic energy and the total potential energy. For electrical allow a co-Lagrangian description, are discussed in the sec- networks, the Hamiltonian is the sum of the total stored mag- tion ‘‘Limitations and Generalizations.’’ The dual form of the netic energy and the electric energy. Hamiltonian formulation is given by the co-Hamiltonian Similar to kinetic energy and kinetic co-energy, potential equations. The corresponding co-Hamiltonian equals the total energy and potential co-energy can be defined. The poten- stored co-energy. For the mechanical and electrical domains, tial energy is defined as the integral of the force with respect the form of the Lagrangian and Hamiltonian, as well as their to the displacement, whereas the complementary potential complements, the co-Lagrangian and co-Hamiltonian, are energy, called the potential co-energy, is defined as the inte- summarized in Table 1. gral of the displacement with respect to the force. The same Apart from energy storage, all physical systems are sub- duality holds in the fluid domain, as well as in the electrical ject to energy dissipation and external energy sources. Con- domain, where electric co-energy is defined as the integral sequently, any practical usage of the Lagrangian and of the capacitor charge with respect to the voltage. Other Hamiltonian frameworks, or their dual forms, must include types of duality can be found at the levels of variables, ele- these phenomena. Although independent energy sources ments, and conservation laws. Dual variables and elements can usually be included through an energy function, such include current and voltage, force and velocity, inductance as the gravitational force through the gravitational potential and capacitance, and inertia and stiffness. An example of energy, the constitutive relationships of dissipative ele- dual conservation laws is given by the relationship between ments must be modeled in terms of power. Even though Kirchhoff’s current and voltage laws. energy and power are often used interchangeably in com- Similar to the duality between energy and co-energy, it mon speech, they are of course different quantities; specifi- is also possible to define a dual, or complementary in the cally, power is the change of energy per unit time. For linear sense of the energies, Lagrangian formulation, which gives mechanical dissipation a Rayleigh dissipation function is rise to the co-Lagrangian equations. The main difference defined [16], [17]. The value of this function has the units of between the Lagrangian and co-Lagrangian equations is the power given by the product of force and velocity. For non- type of variables used in the description. For instance, in linear mechanical dissipation as well as dissipation in other the electrical domain, the co-Lagrangian is defined to be the engineering domains, the content and co-content functions
32 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. TABLE 1 Forms of the Lagrangian and Hamiltonian, and their complements, the co-Lagrangian and co-Hamiltonian, for the mechanical and electrical domain.
Domain Lagrangian Hamiltonian Co-Lagrangian Co-Hamiltonian
Mechanical Kinetic co-energy Kinetic energy Potential co-energy Potential co-energy þ þ Potential energy Potential energy Kinetic energy Kinetic co-energy Electrical Magnetic co-energy Magnetic energy Electric co-energy Electric co-energy þ þ Electric energy Electric energy Magnetic energy Magnetic co-energy
are defined [18], where the adjective ‘‘co-’’ refers to the mechanical, fluid, thermal, and electromechanical systems. complementary form of the content function. The values of Systems containing switches, such as electrical power con- these functions have units of power as well. For instance, in verters and mechanical systems with impacts, are also dis- the electrical domain the content and co-content functions cussed. The application to distributed-parameter systems is involve products of voltage and current, whereas, in the illustrated using two examples, namely, a mechanical sys- fluid domain, the content and co-content involve products tem and Maxwell’s equations. Finally, a few applications of of pressure and volume flow. Such content and co-content the power-based framework are highlighted. functions are called power functions. Although the Lagrangian and co-Lagrangian frameworks ENERGY-BASED MODELING allow for a reasonably large class of nonlinear dissipative ele- OF PHYSICAL SYSTEMS ments, their application may be limited since both frameworks To set up the Lagrangian and Hamiltonian frameworks, as reflect dual properties of the system. For instance, in compari- well as the power-based framework, in a sufficiently generic son to the duality of Kirchhoff’s current and voltage laws, the manner, we adopt the signal analogies used in multi- Lagrangian formulation of an electrical network explicitly cap- domain physical modeling disciplines such as bond graph tures Kirchhoff’s voltage laws, while the current laws are hid- modeling [21]. The various signals are then reduced to a set den in the definition of the configuration coordinates. Dually, of four basic variables called the efforts e, the flows f , the gen- in the co-Lagrangian formulation the appearance of the volt- eralized momenta p, and the generalized displacements q, where age and current laws is reversed. While for mechanical sys- Z t tems it is sufficient to describe only the resultant of the forces p(t) ¼ p(t0) þ e(s)ds (2) in terms of generalized displacements and velocities, for an t0 electrical system the dissipative elements or sources can be and such that either a Lagrangian or a co-Lagrangian formulation Z alone may not be sufficient. In these cases it is necessary to t s s establish combinations of the Lagrangian and co-Lagrangian q(t) ¼ q(t0) þ f ( )d , (3) t0 formulations. In the context of electrical networks, this combi- nation gives rise to the Brayton-Moser (BM) equations stem- respectively. The four basic variables within each physical ming from the early 1960s [19], [20]. The BM equations rely on domain are summarized in Table 2. The analogy between the existence of a function called the mixed-potential function. the mechanical and electrical domains is the classical force- This function consists of the difference between the content voltage or mass-inductance analogy; see ‘‘Analogues, Duals, and co-content functions plus an additional term that reflects and Dualogues.’’ the instantaneous power transfer between the subsystems. Con- sequently, the mixed-potential is a power function, which justi- The Four-Element Quadrangle fies referring to the BM equations as a power-based modeling Since the variables e and p are related by (2), or equivalently, framework. Besides providing a compact system description, p_ ¼ e, and the variables f and q are related by (3), or equiva- the mixed-potential function is useful for determining stability lently, q_ ¼ f , there exist four distinct pairwise combinations criteria for nonlinear electrical networks, especially those con- (p, f ), (q, e), (f , e), and (q, p) that lead to the following classifi- taining regions of negative resistance. cation of generalized system elements, inductive elements The purpose of this article is to provide an overview of (including mechanical masses and electrical inductors), capaci- both the energy- and power-based modeling frameworks tive elements (including mechanical springs and electrical and to discuss their mutual relationships. Furthermore, the capacitors), resistive elements (including mechanical dampers BM equations are shown to be applicable to a large class of and electrical resistors), and memristive elements. The memris- nonlinear physical systems, including lumped-parameter tive element has its origin in the electrical domain [22] as the
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 33
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. Analogues, Duals, and Dualogues n science and engineering, the ideas and concepts devel- information about the direction in which energy is exchanged Ioped in one branch of science and engineering are often within the system, whereas neither voltage nor force include transferred to other branches. One approach to transferring this information. For this reason it is natural to consider current these ideas and concepts is by the use of analogies. Histori- and velocity as flow variables. Although the contribution of the cally, the first attempt to relate mechanical and electrical sys- present article does not depend on the type of analogy used, tems was due to James Clerk Maxwell and Lord Kelvin in the the force-voltage analogy is the one considered here. For 19th century by using the similarity between force and voltage, further discussions on the force-voltage versus force-current as is also apparent from the early use of the term electromotive analogy, see [S6], [S7], and [3]. A fairly extensive overview force (emf). This force-voltage (sometimes called classical) regarding the conception and evolution of both analogies is analogy implies that a mechanical mass is analogous to an givenin[S5]. electrical inductor. In addition to the analogy between mechani- Another closely related concept is the principle of duality. cal and electrical systems, it was observed that phenomena Examples of dual variables and elements are voltage and cur- from other physical domains exhibit similar properties, as sum- rent, force and velocity, inductance and capacitance, and mass marized in Table 2. and stiffness. The various properties for mechanical and electri- Once the force-voltage analogy had been established, cal systems are depicted in Figure S2. For a similar diagram in some scientists started to address some of its limitations. the context of the force-current analogy, see [3]. An analogue These limitations led to the alternative force-current analogy, of a dual phenomenon is called a dualogue. Hence, the force- which implies that a mechanical mass is analogous to an voltage analogue is the dualogue of the force-current analogue, electrical capacitor. The force-current (sometimes called and vice versa. mobility) analogy can be traced back to Darriues (1929), In constructing analogies between mechanical and although it appears to have been discovered independently a electrical systems it is important to realize that the kinetic few years later by Hahnle€ (1932) and Firestone (1933) [S5]. energy of a mass is determined relative to an inertial refer- From a mathematical perspective it seems pointless to dis- ence frame [S8]. The force-voltage analogy therefore sug- cuss which analogy—when it exists—is superior, since both gests that the true electrical analogue of a mass is an analogies lead to equally valid and self-consistent descriptions inductor whose energy can be determined relative to a single of physical systems. Arguments in favor of the force-current current. Using the force-current analogy, the true electrical analogy are mainly related to the preservation of the structural analogue of a mass is a grounded capacitor. Hence, to obtain and topological resemblance. However, from a physical a true mechanical analogue of an electrical circuit with induc- perspective, one of the main arguments in favor of the force- tors that allow their currents to be expressed in terms of more voltage analogy is the analogy between force and pressure than one loop current, or with ungrounded capacitors in case (for the force-current analogy, pressure is considered equiva- of the force-current analogy, a mechanical element called lent to velocity). Furthermore, both current and velocity include the inerter is required [S7]. The inerter differs from a conven- tional mass element since it has two independent terminals, which eliminates the need for a reference frame. The inerter constitutes the dual of a mechanical spring, which also has Inductor Analogue Mass two independent terminals, and constitutes the mechanical analogue of an inductor, or a capacitor in case of the force- current analogy. These analogues allow electrical circuits to Dual Dualogue Dual be translated over to mechanical systems in an unambigu- ous manner.
Capacitor Analogue Spring REFERENCES [S5] P. Gardonio and M. J. Brennan, ‘‘On the origins and development FIGURE S2 Electrical versus mechanical. Based on the force- of mobility and impedance methods in structural dynamics,’’ J. Sound voltage analogy, a mechanical mass is the analogue of an Vib., vol. 249, no. 3, pp. 557–573, 2002. electrical inductor. Likewise, a mechanical spring is the ana- [S6] N. J. Hogan and P. C. Breedveld, ‘‘The physical basis of analo- logue of an electrical capacitor. Mass and spring elements are gies in network models of physical system dynamics,’’ in Proc. complementary or dual elements. The same duality holds for 1999 Int.Conf. Bond Graph Modeling and Simulation (ICBGM’99) Western Multiconf., Simulation Series, no. 1, Jan. 1999, vol. 31, other inductive and capacitive elements, or flow-controlled and pp. 96–104. effort-controlled resistive elements. Furthermore, an analogue [S7] M. C. Smith, ‘‘Synthesis of mechanical networks: The inerter,’’ of a dual element is called a dualogue, as illustrated by the diag- IEEE Trans. Automat. Contr., vol. 47, no. 10, pp. 1648–1662, Oct. onal lines. For instance, an electrical inductor is the dualogue of 2002. a mechanical spring. The force-current, or mobility analogy, is [S8] D. S. Bernstein, ‘‘Newton’s frames,’’ IEEE Control. Syst. Mag., the dualogue of the force-voltage analogy. vol. 28, no. 1, pp. 17–18, Feb. 2008.
34 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. TABLE 2 Domains and variables.
Effort e Flow f Generalized Displacement q Generalized Momentum p Electric Voltage V [V] Current I [A] Charge q [C] Flux linkage / [V-s] Translation Force F [N] Velocity m [m/s] Displacement x [m] Momentum p [N-s] Rotation Torque s [N-m] Angular velocity x [rad/s] Angular displacement h [rad] Angular momentum b [N-m-s] Fluid Pressure P [N/m2] Volume flow Q [m3/s] Volume V [m3] Pressure momentum C [N-s/m2] Thermodynamic Temperature T [K] Entropy flow fs [W/K] Entropy S [J/K] — missing element that constitutes a relationship between The set of basic variables can be subdivided further into charge and flux linkage. An electrical passive two-terminal power variables and energy variables. The power variables are the memristive device was not constructed until recently [23]. An efforts and flows since their product equals power [21], that is, example of a mechanical memristor is the tapered dashpot, : which is a mechanical damper whose resistance depends on P(t) ¼ e(t)f (t) (4) the displacement of its terminals; see ‘‘The Memristor.’’ In a The time integral of power equals energy generalized context, the memristive element establishes the Z relationship between q and p, and hence fills the gap in the t four-element quadrangle showninFigure2.Theassociated E(t) ¼ E(t0) þ e(s)f (s)ds, (5) constitutive relationships and related properties of the four t0 basic elements are discussed in ‘‘State Functions.’’ where E(t0) denotes the energy at initial time t0. Substitut- ing dp(s) ¼ e(s)ds or dq(s) ¼ f (s)ds into (5) yields either a line integral that represents the energy stored by an induc- Inductive f p tive element or by a capacitive element, respectively. For
Memristive C FK FK
a 1 – Resistive xK xK C 2 M x K b R e q v1 v2 Capacitive M M
FM M FIGURE 2 The four-element quadrangle. An inductive element cor- v responds to a static relationship between flow f and generalized M momentum p; a capacitive element corresponds to a static relation- ship between effort e and generalized displacement q; and a resis- FIGURE 3 Mass-spring system. This translational mechanical system tive element corresponds to a static relationship between flow and consists of a rigid interconnection of a cart with constant mass M and effort. The fourth relationship, between generalized momentum and an ideal linear spring with compliance C and natural unstretched generalized displacement, defines a memristive element. The length xK . The motion is restricted to be parallel to the horizontal axis dynamic relationships are represented by the dashed diagonal and relative to a point in a reference frame called ground. The end- lines. Examples of inductive elements include a mechanical mass points of the spring are called terminals (or nodes). The relative dis- M described by p ¼ Mv, where p and v denote its momentum and placement of the spring is determined by the difference between the 2 1 velocity, respectively (see Table 2), or a linear electrical inductor, terminal displacements, that is, xK ¼ xK xK xK . The terminal dis- / / 1 2 with inductance L, described by ¼ LI, where and I denote its placements xK and xK are both measured with respect to the same associated flux linkage and current, respectively. The correspond- ground. Since both sides of the mass are moving with the same ing dynamic relationships are given by p_ ¼ F (Newton’s second velocity, one terminal is associated with the velocity of its center of _ law) and / ¼ V (Faraday’s law). Examples of capacitive elements gravity, while the other terminal is the velocity of the datum or ground. 2 1 include a linear mechanical spring with spring constant K , Hence, the relative velocity of the mass equals vM ¼ vM vM ,with 1 described by F ¼ Kx (Hooke’s law), where x and F denote its dis- vM ¼ 0. If the interconnection constraint of the mass and the spring is placement and force, or an electrical capacitor, with capacitance C, determined by equating their positions at point b,asadvocatedin described by q ¼ CV , where q and V denote its associated charge [41], we first need to choose a ground, for instance, the wall on the 1 and voltage, respectively. For these examples the corresponding left (point a ). Consequently, xK ¼ 0 so that the position of the mass _ _ dynamical relationships are x ¼ v and q ¼ I, respectively. Exam- is determined by xM ¼ xK þ xK . The offset of the mass, which is ples of resistive relationships are Ohm’s law V ¼ RI, with resistance determined by only the natural length of the spring, coincides with the R, or its mechanical analog F ¼ Rv, where R is the coefficient of spurious constant discussed in [41]. This offset can be eliminated by friction. The electrical memristor is discussed in ‘‘The Memristor,’’ shifting the reference to point b. Additionally, in a practical situation while the general nonlinear versions of the four generic elements the cart is subject to friction forces acting on the wheels. These are discussed in ‘‘State Functions.’’ effects are often modeled by a resistive element with resistance R.
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 35
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. this reason, the generalized momenta and displacements For a nonlinear spring, these relationships are expressed as ^ ^ are often referred to as energy variables. For further details, either FK ¼ FK(xK)orxK ¼ xK(FK), referring to a displace- see ‘‘State Functions.’’ ment-controlled or a force-controlled spring, respectively. To distinguish between the variables associated with each Note that the mass belongs to the class of inductive ele- of the four elements, the variables e, f , q,andp are given a sub- ments, while the spring belongs to the class of capacitive ele- script from the set fr, ‘, c, mg, referring to resistive, inductive, ments. According to Table 2, the variables pM and vM capacitive, and memristive elements, respectively. correspond to generalized momentum and flow, respec- tively, whereas xK and FK correspond to generalized dis- Example 1: A Mechanical Mass-Spring System placement and effort, respectively. The corresponding _ To illustrate the classification presented above, consider the dynamical relationships are defined by pM ¼ FM and _ translational mechanical system depicted in Figure 3. This xK ¼ vK, where FM and vK are the force and the velocity asso- system consists of the interconnection of a mass M > 0 and ciated with the mass and the spring, respectively. Further- a linear spring with spring constant K ¼ C 1, where C > 0 more, if the mass is subject to viscous friction, with damping is the compliance. The mass defines the relationship coefficient R, then, in addition, FR ¼ RvR (mechanical ver- between its momentum pM and its velocity vM, which can be sion of Ohm’s law), where FR and vR are the force (effort) 1 expressed as either pM ¼ MvM,orvM ¼ M pM. Similarly, and velocity (flow) associated with the friction, respectively. the spring defines the relationship between its elongation xK The effect of nonlinear friction is expressed either in terms of ^ and the associated force FK, which in the linear case can be FR ¼ FR(vR), referring to velocity-controlled friction, or ^ expressed as either FK ¼ KxK (Hooke’s law) or xK ¼ CFK. vR ¼ vR(FR), referring to force-controlled friction. n
The Memristor ince electronics was developed, engineers designed cir- described by a relationship between current and voltage; a Scuits using combinations of three basic two-terminal ele- capacitor by that of voltage and charge; and an inductor by ments,namely,resistors,inductors,andcapacitors.Froma that of current and flux linkage. But what about the relationship mathematical perspective, the behavior of each of these ele- between charge and flux linkage? As pointed out in [22], a ments, whether linear or nonlinear, is described by relation- fourth element must be added to complete the symmetry. This ships between two of the four basic electrical variables, ‘‘missing element’’ is called the memristor, whose name is a namely, voltage, current, charge, and flux linkage. A resistor is contraction of memory and resistance and refers to a resistor with memory. The memory aspect stems from the fact that a memristor ‘‘remembers’’ the amount of current that has passed through it together with the total applied voltage. More V specifically, letting q denote the charge and / denote the flux linkage, a two-terminal charge-controlled memristor is defined by the constitutive relationship
^ / ¼ /(q):
IqSince flux linkage is the time integral of voltage V and charge is _ the time integral of current I, that is, V ¼ / and I ¼ q_ , we obtain
V ¼ M(q)I, (S9)
^ where M(q) :¼ d/(q)=dq is the incremental memristance. Simi- φ larly, a two-terminal flux-controlled memristor (memductor) is defined by FIGURE S3 The four-element quadrangle for the electrical domain. As indicated by the arrows, an inductor corresponds to q ¼ q^(/): a static relationship between current I and flux linkage /,a capacitor corresponds to a static relationship between voltage V and electric charge q, and a resistive element corresponds to a Differentiating the latter yields the dual of (S9), namely, static relationship between current and voltage. The fourth rela- / tionship, between / and q, defines a memristor. In the last case I ¼ W ( )V , (S10) the variables / and q do not necessarily have the interpretation / : ^ / = / of a physical flux or charge and therefore must be considered as where W ( ) ¼ dq( ) d is the incremental memductance. integrated voltage or current, respectively. The four basic electrical elements are summarized in Figure S2.
36 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. Classical Energy-Based Framework TABLE 3 Domains and energy. For ease of presentation, we begin by considering a class of systems Domain Inductive Energy T or Co-Energy T Capacitive Energy V or Co-Energy V consisting solely of intercon- Electric Magnetic (inductor) Electric (capacitor) Translation Kinetic (mass) Potential (spring, gravity) nected inductive and capacitive Rotation Kinetic (inertia) Potential (rotational spring) energy-storage elements; see Fluid Kinetic (tube, pipeline) Potential (tank) Table 3 for the energy stored in Thermodynamic — (Heating) the elements of various engineer- ing domains. The equations of motion can be deduced from the dynamic relationships configuration of the system, and thus of every element. Any associated with the elements together with a set of constraint such set of variables is called a complete set of generalized coor- relationships that define how the elements are intercon- dinates. The number of degrees of freedom of the system is the nected to form the overall system. In ‘‘State Functions,’’ par- number of independent coordinates required to specify the ticular energy functions are associated with the inductive configuration of each element in the system. and capacitive elements. These functions are given in terms Suppose that a system configuration with n degrees of of the individual element variables, but we may equally well freedom can be described by the complete set of generalized employ any other set of variables that uniquely define the displacement coordinates q ¼ col(q1, ..., qn). The Lagrangian
Observe that (S9) and (S10) are generalized versions of Ohm’s law, in which the memristor and memductor are acting as charge- and flux-modulated resistors, respectively. It is F F important to realize that, for the special cases in which the constitutive relations are linear, that is, when the incremental x memristance M(q) or the incremental memductance W (/)is constant, a memristor and a memductor become an ordinary FIGURE S4 Example of a tapered dashpot. The friction coeffi- resistor and conductor. Hence, memristors and memductors cient depends on the displacement x. In practice the taper is are relevant only in nonlinear circuits. achieved by a conical pin passing through an orifice in the piston. To gain intuition for what distinguishes a memristor from a The shape of the pin can be machined to produce any desired resistor as well as from an inductor and a capacitor, let us briefly memristive relation between displacement and momentum. consider the common analogy of a resistor and a pipe that car- ries a fluid. The fluid can be considered analogous to charge, by fabricating a layered platinum-titanium-oxide-platinum nano- the pressure at the input of the pipe is similar to voltage, and the cell device. rate of flow of the fluid through the pipe is like current. As in the It will be interesting to see what applications arise for this case of a resistor, the flow of fluid through the pipe is faster if the device and whether it invokes a revival of network theory. On pipe is shorter or if it has a larger diameter. Now, an analogy for the other hand, as pointed out in [43], in the mechanical domain, a memristor is a flexible pipe that expands or shrinks according the tapered dashpot is a mechanical damper whose resistance to how fluid flows through it. If fluid flows through the pipe in one depends on the displacement of its terminals; see Figure S4. In direction, the diameter of the pipe increases, thus enabling the practice the taper is achieved by a conical pin passing through fluid to flow faster. If fluid flows through the pipe in the opposite an orifice in the piston. The shape of the pin can be machined to direction, however, the diameter of the pipe decreases, thus produce a desired memristive relation between displacement slowing down the flow of the fluid. If the fluid pressure is turned and momentum, which are the mechanical analogies of charge off, the pipe retains its most recent diameter until the fluid pres- and flux linkages. Other examples of engineering systems in sure is turned back on. Unlike a bucket (or a capacitor) a mem- which the memristor phenomena is apparent can be found in ristive pipe does not store the fluid but ‘‘remembers’’ how much [S9], [43], and [45]. An extension of the concept to a much fluid flowed through it. broader class of systems, called memristive systems,is A physical electrical passive two-terminal memristive device presented in [S10]. was constructed only recently when scientists at Hewlett-Pack- ard Laboratories announced its realization. In particular, it is REFERENCES shown in [23] that memristance naturally arises in nanoscale [S9] L. O. Chua, ‘‘Device modeling via basic nonlinear circuit elements,’’ IEEE Trans. Circuits Syst., vol. CAS-27, no. 11, pp. 1014–1044, Nov. 1980. systems when electronic and atomic transport are coupled [S10] L. O. Chua and S. M. Kang, ‘‘Memristive devices and systems," under an external bias voltage. The memristive effect is realized Proc. IEEE, vol. 64, pp. 209–223, 1976.
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 37
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. equations of motion are given by (see ‘‘Notation’’ for the The Hamiltonian equations (7) and (8) constitute both a notation of partial derivatives) flow and effort balance, respectively. Note that, like (6), the system configuration in (7), (8) is still described in terms of d r L(q, f ) r L(q, f ) ¼ 0, (6) the generalized displacement coordinates q. dt f q _ Example 1 Revisited: The Lagrangian where the flow variables f ¼ q, with f ¼ col( f1, ..., fn), are called generalized velocities. Furthermore, the Lagrangian and Hamiltonian Equations L(q, f ) is defined by For the mechanical system of Figure 3, assume that the mass moves without friction. In deriving the Lagrangian equa- L(q, f ) ¼T ( f ) V(q), tions we need only the kinetic (inductive) co-energy associ- ated with the mass and the potential (capacitive) energy where T ( f ) represents the total inductive co-energy and stored in the spring. According to the definitions given in V(q) the total capacitive energy, which are obtained as the ‘‘State Functions,’’ the kinetic co-energy is determined by sums of the inductive co-energies and the capacitive ener- M gies in the constituent elements, respectively. See ‘‘State T (v ) ¼ v2 , M 2 M Functions’’ for details on energy and co-energy. The Lagrangian equations (6), in both the linear and non- whereas the potential energy is given by linear cases, can be derived in various ways. Perhaps the best x2 known methods are the derivations based on d’Alembert’s V(x ) ¼ K : K 2C principle and the principle of virtual work as well as deriva- tions originating in variational methods such as Hamilton’s The next step is to define the system configuration. Since principle [1], [17]. Alternative methods can be found in [24]. the system has one degree of freedom, a natural choice is to Furthermore, the Lagrangian equations (6) define a set of n take the displacement of the point b in Figure 3, that is, second-order differential equations that constitute an effort x ¼ xK ¼ xM. With this choice, the Lagrangian is defined by balance, and the information necessary to describe the sys- M x2 tem’s dynamic behavior is solely contained in the Lagrangian. L(x, v) ¼ v2 , (9) 2 2C The Hamiltonian equations are established by consider- ing the Legendre transformation. We thus define the general- with corresponding velocity v ¼ x_. Substituting (9) into the ... ized momenta p ¼rf L(q, f ), with p ¼ col(p1, , pn). Then, Lagrange equation (6) yields the second-order differential under the assumption that f can be expressed in terms of p, equation the set of n second-order equations (6) can be transformed x Mx€ þ ¼ 0: (10) into 2n first-order equations of the form C _ q ¼rpH(q, p), (7) Concerning the Hamiltonian counterpart, we first define _ p ¼ rqH(q, p), (8) the momentum
p ¼rvL(x, v) ¼ Mv, where the Hamiltonian H is the total stored energy, that is, and introduce the Hamiltonian : H(q, p) ¼T(p) þV(q) 2 2 p x : H(x, p) ¼½pv L(x, v) v¼ p ¼ þ M 2M 2C Notation Hence, according to (7) and (8), the Hamiltonian equations et for the system are given by 2 3 L p x x1 x_ ¼ , p_ ¼ : (11) 6 . 7 n x ¼ 4 . 5 ¼ col(x1, ..., xn) 2 R M C xn Note that the Lagrangian equation (10) equates only the forces of the mass and the spring, whereas the Hamiltonian denote a column vector, and let V(x) denote a scalar function equations (11) equates both the velocities and the forces in V : Rn ! R. The gradient of V(x)withrespecttox is denoted by terms of x and p. Furthermore, if in Figure 3 the point a is @ @ chosen as the point of reference, the potential energy needs V V n rx V(x) ¼ col (x), :::, (x) 2 R : @x1 @xn to be modified as 2 2 Rn 3 n (x xK) Furthermore, the notation rx V(x) 2 denotes the Hes- V(x) ¼ , sian matrix. 2C where xK is the relaxed length of the spring. n
38 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. The Lagrangian equations of motion (6) and the Hamilto- H ( f , e) ¼T ( f ) þV (e), nian equations (7), (8) are represented in Figure 4 (solid lines). The diagram suggests that there exists a dual form of where T ( f ) represents the total inductive co-energy and V (e) the Lagrangian equations of motion (6) if the system can be is the total capacitive co-energy. It is evident from Figure 4 that expressed in terms of a set of generalized momentum coordi- the co-Hamiltonian equations relate in a cross-wise differential nates and their time-derivatives. The dynamics in terms of manner, in the sense that p_ ¼ e and q_ ¼ f ,asvisualizedbythe the generalized momenta and efforts is called a co-Lagrangian diagonal lines, with the Hamiltonian equations. Hence, the co- system. This system is obtained by replacing the Lagrangian Hamiltonian equations are given by L(q, f ) in (6) by its complementary or dual form, called the co- Lagrangian, defined by the difference between the total d d : rf H ( f , e) ¼ e, reH ( f , e) ¼ f (14) capacitive co-energy and the total inductive energy, that is, dt dt : L (p, e) ¼V (e) T(p) The latter set of equations completes the quadrangle Hence, the co-Lagrangian equations of motion take the form in Figure 4.
d Example 1 Revisited: The reL (p, e) rpL (p, e) ¼ 0, (12) dt Co-Hamiltonian Equations _ where the efforts e ¼ p,withe ¼ col(e1, ..., en), are generalized Returning to the mass-spring system of Figure 3, the co- force coordinates. Note that (12) again defines a set of n second- Hamiltonian takes the form order differential equations, but now constitutes a flow balance. C M H (v, F) ¼ F2 þ v2, (15) 2 2 Example 1 Revisited: The Co-Lagrangian Equations which upon substitution into (14) yields the first-order For the mass-spring system of Figure 3, the formulation of differential equations the co-Lagrangian means that instead of taking the displace- _ _ ¼ ¼ : ment of the connection point as the system configuration, a Mv F, CF v (16) suitable momentum variable must be chosen. One possible Taking the system configuration of ‘‘Example 1 Revisited: The choice is p ¼ pM ¼ pK, where the minus sign stems from the Co-Lagrangian Equations’’ as a reference, the spring force is reference direction of the mass and spring forces. Conse- _ quently, F ¼ p ¼ FK, and the co-Lagrangian has the form Lagrangian C p2 L( p, F) ¼ F2 , 2 2M f p which upon substitution into (12) yields the second-order differential equation f p f p _ p CF þ ¼ 0: (13) e q M Note that (13) equates the velocities of the spring and mass f p 1 n with vM ¼ M p. e q e q The three representations considered above describe the Co-Hamiltonian/ Hamiltonian dynamics of a system consisting only of inductive and Brayton-Moser capacitive elements. The underlying principle of the trans- e q formations between the various energy functions is the exis- tence of the associated Legendre transformations. The Co-Lagrangian diagram in Figure 4 shows that there exists a fourth equa- tion set involving the variables e and f . Starting from the FIGURE 4 Relationship between the Lagrangian and Hamiltonian equations and their complementary versions. The Lagrangian equa- Hamiltonian equations, the Legendre transformation of tions are described in terms of generalized displacements and their both q 7! e and p 7! f is considered simultaneously, that is, corresponding flows (generalized velocities). The associated Hamil- tonian equations are obtained by replacing the flows by generalized H ( f , e) ¼ q>e þ p>f H(q, p), momenta. The complementary or dual Lagrangian equations are referred to as the co-Lagrangian equations since they are described where it is assumed that q can be expressed in terms of e,andp in terms of dual variables, namely, a set of generalized momenta and its associated efforts (generalized forces). The complementary Ham- canbeexpressedintermsoff , that is, if the respective relation- iltonian formulation is represented by the co-Hamiltonian equations, ships e ¼rqH(q, p) ¼rqV(q)andf ¼rpH(q, p) ¼rpT (p) which are described in terms of flows and effort. The co-Hamiltonian are invertible. Thus the co-Hamiltonian is given by formulation coincides with the Brayton-Moser equations.
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 39
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. given by FK ¼ F, while the velocity of the mass equals Note that the contents exist only for resistive elements that vM ¼ v. The co-Hamiltonian representation (16) provides both are flow-controlled. Effort-controlled resistive elements can the force and velocity balance, but now in terms of force (effort) be included in the Lagrangian framework if their constitu- and velocity (flow) variables only. The original system configu- tive relationships are one-to-one. ration variables x and p are eliminated from the model. n Similarly, the co-Lagrangian equations can be modified by introducing a resistive co-content function D (e) associ- Including Resistive Elements ated to the effort-controlled resistive elements, so that (12) The frameworks considered above assume that the system is modified as is free from dissipation and externally supplied efforts or flows. Dissipation and supply effects can be included in the d reL (p, e) rpL (p, e) ¼ reD (e): (18) Lagrangian framework by introducing a content function dt D( f ), (see ‘‘State Functions’’) so that (6) is modified as Flow-controlled elements can be included in the co-Lagran- d : gian framework if their constitutive relationships are one- rf L(q, f ) rqL(q, f ) ¼ rf D( f ) (17) dt to-one. As illustrated in the examples below, the possibility of describing a system by either (17) or (18) strongly depends on the system configuration.
R Example 1 Revisited: Adding Dissipation Suppose that the mass of the system of Figure 3 is subject to viscous friction with friction coefficient R. We know that the M C constitutive relationship can be represented by either 1 FR ¼ RvR (flow-controlled with f ¼ vR)orvR ¼ R FR (effort- controlled with e ¼ FR). Since these relationships are linear, and thus one-to-one, the corresponding content and co-content (a) functions exist. However, for the Lagrangian formulation only the content is needed, which, by noting that vR ¼ v,isgivenby R C Z M v R 2 D(v) ¼ RvRdvR ¼ v : (19) 0 2
Hence the Lagrangian equation of motion for the lossless (b) situation (10) is modified as
R1 x Mv_ þ ¼ rD(v) ¼ Rv: (20) C v n R C 2 M Example 2: Velocity- Versus Force-Controlled Damping A similar, but conceptually different, situation occurs when a (c) viscous damper is connected between the wall and the mass as shown in Figure 5(a). The corresponding content function FIGURE 5 Mass-spring-damper systems. (a) A mechanical system con- coincides with (19), and, since the velocity of the wall is con- sisting of a constant mass M connected to a linear spring with compli- sidered to be zero as a reference velocity, we again have ance C, and a linear viscous damper with friction coefficient R.The vR ¼ v. Hence, the system can be described by the same damper can be modeled as a resistive element, which, at its terminals, Lagrangian equation as derived in (20). On the other hand, a exerts an equal and opposite force that is a function of the relative veloc- dual situation occurs if instead the damper is placed between ity between these terminals. Taking the wall on the left as the ground point, the relative velocity of the damper is determined by the velocity of the spring and the mass; see Figure 5(b). In this case the veloc- the mass. This damper configuration is referred to as a velocity- ity of the damper cannot be related to the chosen velocity controlled damper. In the case of (b), the same damper is placed coordinate, but rather shares the force (effort) of the spring, between the mass and the spring. The relative velocity of the damper is that is, FR ¼ F. Hence, a direct way to describe the dynamics now determined by the difference between the velocity of the mass and of this system is by means of the co-Lagrangian equation the spring. However, in deriving the state equations, the constitutive relationship of the damper is described rather as a function of its associ- _ p F ated force. Dual to the configuration of (a), this damper configuration is CF þ ¼ r D (F) ¼ , (21) referred to as force-controlled damper. (c) Combination of (a) and (b). M F R
40 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. where the co-content D (F) is of the form Additional examples that cannot be described in terms of a Z Lagrangian, but do admit a co-Lagrangian formulation, F 2 FR F include electrical networks with varactors, which are nonlin- D (F) ¼ dFR ¼ : (22) 0 R 2R ear voltage-controlled capacitive elements that cannot be described in terms of the charge, and the Josephson junction Note that this configuration of the damper does not contrib- circuit model, which contains a flux-controlled inductive ele- n ute a force to the Lagrangian equation of motion (20). ment for which the Legendre transform does not exist. Example 2 shows that the Lagrangian and co-Lagrangian An additional assumption that is introduced to further frameworks complement each other. Indeed, as shown in the simplify the setup of the four frameworks is that the induc- section ‘‘The Brayton-Moser Equations,’’ both frameworks tive co-energy does not depend on the generalized displace- can be naturally combined into a single system of equations ments. This assumption means that the displacements allowing for both effort- and flow-controlled resistive ele- associated with the individual inductive elements can all be ments, as in the case of the system of Figure 5(c). expressed in terms of a set of independent generalized dis- placement coordinates, that is, q‘ ¼ Uq, with U a constant Limitations and Generalizations matrix of appropriate dimensions. The corresponding flows The Lagrangian and Hamiltonian equations, together with their f‘ ¼ q_‘ and f ¼ q_ are then related by f‘ ¼ Uf , which means dual formulations, the co-Lagrangian and co-Hamiltonian that the total inductive co-energy can be expressed solely in equations, respectively, introduced above are set up to explain terms of f . Although U is typically constant for electrical the relations between the four frameworks in a straightforward networks and translational mechanical systems, the selec- manner. The two main assumptions that all four representa- tion of a set of generalized configuration coordinates for tions exist simultaneously are i) that the system configuration rotating mechanical systems often gives rise to a displace- can be described by either a set of generalized displacement ment-dependent mapping of the form coordinates, or a set of generalized momentum coordinates, or U both,allhavingthesamedimensionn, and ii) that the Legendre q‘ ¼ (q), (23) transformations from energy to co-energy, and vice versa, exist. changing, for example, Cartesian coordinates to polar, While these assumptions are satisfied by some electrical net- cylindrical, or spherical coordinates. Consequently, differ- works that have the same number of inductive and capacitive entiation of (23) with respect to time yields the flow elements, as well as some mechanical systems, such as the mass-spring example treated above, the class of systems that ^ f‘ ¼rU(q)f ¼: U(q, f ), (24) can be modeled by all four representations is restricted. q The main limitation in the mechanical domain concerns which yields a function of both q and f , and thus extends the existence of the potential co-energy function. If the the inductive co-energy from T ( f )toT (q, f ). Another constitutive relationship between force and displacement, ^ instance where the inductive co-energy may become a say for a stiffness element K, is nonlinear, that is, F ¼ F (x ), K K K function of both q and f is the case in which connections are it may be that its inverse x ¼ x^ (F ) does not exist. In the K K K made between different physical domains [4]. An example absence of an inverse constitutive relationship it is not possi- for which it is convenient to perform a change of coordi- ble to evaluate the potential co-energy given by nates is the inverted pendulum on a cart system. Z F ^ V (F) ¼ xK(FK)dFK: 0
An example for which the potential co-energy cannot be Mb evaluated, at least not globally, is in the case of systems that are subject to gravity; see ‘‘Example 3 Revisited.’’ However, θ for mechanical systems, the co-Lagrangian and co-Hamilto- y nian equations are sometimes invoked to model special l problems; see [15] for examples. Besides mechanical systems, the dual formulation can be x insightful and sometimes even necessary for describing sys- Mc tems from other engineering domains. For instance, some networks and systems, such as the tunnel diode circuit dis- cussed in ‘‘History of the Mixed-Potential Function,’’ cannot be described in a classical Lagrangian or Hamiltonian frame- FIGURE 6 Inverted pendulum on a cart. The mass of the cart and work since the tunnel diode characteristic is not one-to-one the point mass (the bob) at the end of the rod are denoted by Mc and thus cannot be expressed in terms of a content function. and Mb , respectively. The rod with length l is considered massless.
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 41
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. History of the Mixed-Potential Function t a June 1958 conference in Brussels, Leo Esaki presented Lyapunov function. As with many physical systems, a Lyapunov A a new type of diode he had developed at Sony and for function candidate is the total stored energy in the circuit. How- which he received a Ph.D. in physics and, later, the 1973 Nobel ever, this function is not useful for investigating stability for the prize in physics. The characteristic curve of the Esaki or tunnel tunnel diode circuit [S11]. diode, named for the quantum-mechanical tunneling effect it exploited, contains a region in which the current decreases as ALTERNATIVE LYAPUNOV ARGUMENT the voltage increases. This region of negative resistance made To circumvent this problem, Moser’s key idea was to introduce it possible to construct bistable circuits, which were used as the scalar function switching or memory elements (flip-flops) in early computers. Z V To gain a better understanding and to guarantee safe operation 1 2 ^ P(I, V ) ¼ RI EI Ig (Vg )dVg þ IV , (S12) of these circuits, Jurgen€ Moser developed a mathematical 2 0 technique for analyzing their stability [S11]. His method was so that the differential equations (S11) can be rewritten as based on the study of a power related scalar function leading to quantitative restrictions on the circuit parameters so as to _ LI ¼rI P(I, V ), ensure stable switchings. We now briefly outline Moser’s origi- _ CV ¼rV P(I, V ): nal idea and motivation. (Here we use an opposite sign convention in comparison with MOTIVATION FOR A NEW THEORY [S11].) Observe that the values of (S12) have the units of power Consider the tunnel diode circuit depicted in Figure S5, where (current 3 voltage) and that its extrema coincide with the equili-
I ¼ IL denotes the current through the inductor L and V ¼ VC bria of (S11). However, (S12) does not qualify as a Lyapunov denotes the voltage across the capacitor C. The differential function since its time derivative is indefinite. For that reason, equations describing the circuit are given by Moser introduced the alternative function
_ _ ^ LI ¼ E RI V , CV ¼ I I (V ), (S11) L 2 C ^ 2 g O(I, V ) ¼ (E RI V ) þ (I I (V )) , (S13) 2 2 g ^ where the function Ig (V ) characterizes the relation between the whose time derivative implies that all trajectories tend to the set voltage V and the current Ig through the branch of the tunnel ^ ^ of stable equilibria if I 0 (V ) :¼ dI (V )=dV > 0, that is, the slope diode. In Figure S5 the characteristic curve is plotted, and the g g of the characteristic curve of the tunnel diode must be positive three equilibrium points of the circuit are shown. Two of the for all V . However, this function still does not take into account equilibria are asymptotically stable, whereas the equilibrium in the region of negative resistance. Finally, the combination of the region of negative resistance is unstable. For the design of the two functions S¼OþkP, with arbitrary constant k, yields a flip-flop circuit it is useful to know when all trajectories tend to the asymptotically stable equilibrium points to exclude bounded k ^0 k _ R þ L 2 I g (V ) C ^ 2 trajectories, such as limit cycles, that never reach the equilib- S(I, V ) ¼ (E RI V ) (I I (V )) : L2 C2 g rium points. The most common way to proceed is to find a _ The derivative S(I, V ) is negative defi- nite if k is chosen in the interval
E VC ^ − + Stable R I 0 (V ) I < k < g , g Unstable L C Ig Stable ^ where it is assumed that I 0 (V ) > L C g 1 ^0 > CRL . The condition I g (V ) I L CRL 1 puts a restriction on the steepness of the slope of the negative 0 V R g resistance region of the tunnel diode and guarantees that all trajectories FIGURE S5 Tunnel diode circuit. The tunnel diode has a nonlinear voltage-controlled consti- converge to the set of stable equilibria. tutive relationship that contains a region of negative resistance. This characteristic made it possible to construct bistable circuits, which were used as switching or memory elements THE BRAYTON-MOSER EQUATIONS (flip-flops) in early computers. Note that the constitutive relationship of the tunnel diode is Four years after [S11] appeared, not invertible since it is not a single-valued function of the voltage. Consequently, the tunnel diode circuit cannot be described using the Lagrangian equations because the associated Moser generalized the theory together content is not globally defined. with Brayton in [19] to the class of
42 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. topologically complete circuits. A circuit is topologically complete external voltage sources. The terms in Wells’s power function if each branch of the circuit can be expressed either in terms of associated with the conservative forces coincide with the the inductor currents I, or the capacitor voltages V , or both, and instantaneous power transfer term of the mixed-potential func- if it can be split up into two subcircuits, say Ra and Rb ,whereRa tion, where N is the identity matrix. contains the current-controlled inductors and current-controlled resistors, and Rb contains the voltage-controlled capacitors and A FAMILYOF BRAYTON-MOSER DESCRIPTIONS voltage-controlled resistors and conductors. The P-function for Motivated by the tunnel diode example, the principal application of such a circuit then takes the form the mixed-potential function concerns its use in determining stabil- ity criteria for possibly nonlinear networks admitting a description P(I, V ) ¼D(I) D (V ) þ V >NI, (S14) of the form (S15), together with (S14) and (S16). Indeed, it is now _ _ where easily seen that P(z) is a quadratic form in z,thatis,
Z _ _ > _ Nr I P(z) ¼ z Q(z)z: (S17) ^ > D(I) ¼ V r (Ir )dIr 0 Hence for circuits without capacitors (RL circuits), we have represents the resistive content capturing the current-controlled z ¼ I, Q(I) ¼ L(I), and P(I) ¼D(I). Under the condition that resistors and sources contained in Ra, that is, the constitutive L(I) is positive definite, as is usually the case, we obtain _ relations between the currents through the resistors Ir and the P(I) 0 , which, by the invariant set theorem, implies that each ^ voltages across the resistors is given by Vr ¼ Vr (Ir ), and bounded I approaches the set of equilibria, as t !1. A similar result pertains to circuits without inductors, that is, RC circuits. Z Ng V ^> However, for RLC circuits the symmetric part of Q(z)is D (V ) ¼ Ig (Vg )dVg 0 indefinite. Brayton and Moser’s key observation is to generate a ~ ~ new pair fQ, Pg~ such that the symmetric part of Q is at least represents the resistive co-content capturing the voltage-con- negative semidefinite. The construction is as follows. For each trolled resistors, conductors, and sources contained in R , that b constant symmetric matrix M and real number k, a new mixed- is, the constitutive relations between the voltages across the potential is obtained from conductors Vg and the currents through the conductors is given ^ > by Ig ¼ Ig (Vg ). The term V NI represents the instantaneous ~ k 1 > P(z) ¼ P(z) þ rz P(z)Mrz P(z), (S18) power delivered from Ra to Rb. Here, the matrices Nr , Ng , and 2
N, with entries 1, 0, stem from applying Kirchhoff’s voltage > ~ yielding P_(z) ¼ z_ Q(z)z_ , with and current laws to the circuit. Thus, the circuit is completely determined by a mix of three different potential functions; hence ~ Q(z) ¼ kQ(z) þr2P(z)MQ(z): (S19) Brayton and Moser call (S14) the mixed-potential function. In z compact notation, the dynamics of a possibly nonlinear RLC cir- The original ideas of [S11] are then generalized into several cuit can be described as theorems [19], each imposing particular restrictions on the cir- cuit parameters or the topology. The first three theorems Q(z)z_ ¼r P(z), (S15) z presented in [19] are summarized below. with z ¼ col(I, V ) and Theorem S1 L(I)0 If R :¼r2D(I) is constant and nonsingular, D (V ) !1 as Q(z) ¼ , (S16) I 0 C(V ) jV j!1, and where L(I) and C(V ) are the incremental inductance and capac- L1=2(I)R 1N>C 1=2(V ) < 1, itance matrices. This system of differential equations is com- monly known as the Brayton-Moser (BM) equations. then each trajectory of (106) tends to the set of equilibria as t !1. Although the BM equations (S15), together with (S16) and The proof of the theorem follows by selecting k ¼ 1 and the mixed-potential of the form (S14), are due to Brayton and M ¼ diag(2R 1, 0) in (S18) and (S19), and by invoking the Moser [19], it is noteworthy to mention that similar ideas were invariant set theorem. Note that for topologically complete cir- € already developed earlier by Wells [S12] and Stohr [S13]. In cuits the theorem requires the resistors in Ra to be linear and to particular, the similarity of the mixed-potential with Wells’s have sufficient damping in each current coordinate. The latter power function is remarkable. In addition to including dissipa- condition is satisfied if each inductor has some series resist- tive forces to describe the behavior of resistors, Wells used the ance, no matter how small. Also note that the conditions of the power function to include conservative forces to describe the theorem are independent of the resistors in Rb . The following behavior of capacitors and externally applied forces to describe result is the complementary or dual version of Theorem S1.
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 43
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. literature. Most of these extensions are based on the topologi- TABLE S1 Assumptions for Brayton and Moser’s stabil- cal structure of the circuit. In [S14]–[S16], and [3], a topological ity theorems. mixed-potential is derived from a variational point of view. These observations are proved more rigorously in [S17]. The Type of Element Theorem S1 Theorem S2 Theorem S3 inclusion of ideal transformers is treated in [S18]. In [S19], the Resistive Linear Nonlinear Nonlinear concept of pseudocontent and pseudohybrid content is intro- Conductive Nonlinear Linear Nonlinear Inductive and Nonlinear Nonlinear Linear duced to carry over the ideas of BM to topologically noncom- capacitive pete circuits. The problem of finding the largest class of circuits for which a mixed-potential function can be constructed is dis- cussed in [S20] and [28]. Furthermore, based on the ideas Theorem S2 presented in [S19], the extension of Theorem S1 and S2 to cir- : 2 cuits having noninvertible R or G matrices, as well as the If G ¼rV D (V ) is constant and nonsingular, D(I) !1 as applicability of the stability theory to noncomplete circuits, is jIj!1, and C1=2(V )G 1NL 1=2(I) < 1, presented in [26]. A generalization of Brayton and Moser’s stability theorems that also includes the analysis of circuits that then each trajectory of (S15) tends to the set of equilibria as t !1. contain nonlinear resistors, conductors, inductors, and capaci- The proof of Theorem S2 is similar to the proof of Theorem tors simultaneously is given in [S21]. Geometrical aspects of S1, except that k ¼ 1 and M ¼ diag(0, 2G 1), which means that the concept of the mixed-potential function can be found in every capacitor must possess some parallel conductance. The [S20], [S22]–[S26], and [11]. third theorem does not require the resistors to be linear, but assumes linear inductors and capacitors. REFERENCES Theorem S3 [S11] J. K. Moser, ‘‘Bistable systems of differential equations with applications to tunnel diode circuits,’’ IBM J. Res. Develop., vol. 5, pp. If L and C are constant, symmetric, and positive definite, and 226–240, 1960. ÈÉÈÉ [S12] D. A. Wells, ‘‘A power function for the determination of l 1=2 1=2 l 1=2 1=2 > 1 L R(I)L þ 2 C G(V )C 0, Lagrangian generalized forces,’’ J. Appl. Phys., vol. 16, pp. 535– 538, 1945. € l [S13] A. Stohr,€ ‘‘Uber gewisse nich notwendig lineare (nþ1)-pole,’’ where f g represents the infimum of the eigenvalues of the € Arch. Electron. Ubertr. Tech., vol. 7, pp. 546–548, 1953. respective matrices, then each bounded trajectory of (S15) [S14] J. O. Flower, ‘‘The topology of the mixed-potential function,’’ tends to the set of equilibria as t !1. Proc. IEEE, vol. 56, pp. 1721–1722, Oct. 1968. k = l l [S15] J. O. Flower, ‘‘The existence of the mixed-potential function,’’ Theorem S3 follows by selecting ¼ð1 2Þ( 2 1) and Proc. IEEE, vol. 56, pp. 1735–1736, Oct. 1968. M diag(L, C) 1. The requirement that M in (S18) and (S19) ¼ [S16] D. L. Jones and F. J. Evans, ‘‘Variational analysis of electrical must be chosen to be constant is precisely the reason for the dif- networks,’’ J. Franklin Inst., vol. 295, no. 1, pp. 9–23, 1973. ferent linearity assumptions on the admissible circuit elements. [S17] F. M. Massimo, H. G. Kwatny, and L. Y. Bahar, ‘‘Derivation of the Brayton-Moser equations from a topological mixed-potential In summary, Table S1 shows the assumptions on the circuit ele- function,’’ J. Franklin Inst., vol. 310, no. 415, pp. 259–269, Oct./Nov. ments regarding the applicability of each of the three theorems. 1980. It is important to realize that the requirements of theorems [S18] E. H. Horneber, ‘‘Application of the mixed-potential function for formulating normal form equations for nonlinear RLC networks,’’ Int. J. S1, S2, and S3 are only sufficient conditions that establish com- Circuit Theory Appl., vol. 6, pp. 345–362, 1978. parisons between different time constants or frequencies. [S19] L. O. Chua, ‘‘Stationary principles and potential functions for non- Indeed, the application of Theorem S1 to the tunnel diode circuit linear networks,’’ J. Franklin Inst., vol. 296, no. 2, pp. 91–114, Aug. pffiffiffiffiffiffiffiffiffi 1973. yields the condition R > L=C, which can be rewritten as [S20] W. Marten, L. O. Chua, and W. Mathis, ‘‘On the geometrical meaning of pseudo hybrid content and mixed-potential,’’ Arch. Elec- L € – RC > , tron. Ubertr., vol. 46, no. 4, pp. 305 309, 1992. R [S21] D. Jeltsema and J. M. A. Scherpen, ‘‘On Brayton and Moser’s missing stability theorem,’’ IEEE Trans. Circuits Syst. II, vol. 52, no. 9, where both sides of the inequality have the units of seconds. A pp. 550–552, 2005. ^ similar discussion holds for the stability condition I0 (V ) > [S22] R. K. Brayton, ‘‘Nonlinear reciprocal networks,’’ in Mathematical g Aspects of Electrical Network Analysis. Providence: American Mathe- 1 CRL discussed above. Note that the same condition follows matical Society, 1971, pp. 1–15. from the application of Theorem S3. Theorem S2 cannot be [S23] S. Smale, ‘‘On the mathematical foundations of electrical circuit theory,’’ J. Differ. Equ., vol. 7, no. 1–2, pp. 193–210, 1972. applied since it requires linearity of the resistors in Rb, which in [S24] W. Mathis, Theorie Nichtlinearer Netzwerke. Berlin: Springer- this case is the tunnel diode. Verlag, 1987. [S25] G. Blankenstein, ‘‘Geometric modeling of nonlinear RLC circuits,’’ STATE OF THE ART IEEE Trans. Circuits Syst. I, vol. 52, no. 2, pp. 396–404, 2005. [S26] D. Jeltsema. (2005) Modeling and control of nonlinear networks: During the last four decades several notable extensions and A power-based perspective, Ph.D. Thesis, Delft Univ. Technol. generalizations of the BM theory have been presented in the [Online]. Available: http://repository.tudelft.nl/file/82874/363811
44 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. R Example 3: Inverted Pendulum on a Cart On the the other hand, consider a system b that consists of Consider the inverted pendulum mounted on a cart in nc capacitive and ng resistive elements, either effort-controlled Figure 6. We assume that the rod with length l is massless, or one-to-one, having the same number of external ports or R and denote the cart mass by Mc and the point mass (the terminals as a and as underlying configuration variables the bob) at the upper end of the inverted pendulum by Mb. generalized momenta associated with the capacitive elements, ... Since the system moves in the x-y plane, its configuration that is, p ¼ col(p1, , pnc ). If the efforts and flows associated can be described in terms of Cartesian coordinates, namely, with these external ports are denoted as eb and fb, respectively, _ 3 the displacement of the cart in the horizontal direction (x,0) and eb ¼ Nbe,wheree ¼ p and Nb an ne nc matrix, then the and the displacement of the bob (xb, yb). The kinetic co- co-Lagrangian equations (18) take the form energy in this case is given by Z Nbe d > : reL (e) ¼ re D (e) f deb (27) M M dt b T (x_, x_ , y_ ) ¼ c x_ 2 þ b (x_ 2 þ y_ 2): 0 b b 2 2 b b In this case, the co-Lagrangian is reduced to the total stored However, since the pendulum has only one degree of freedom, capacitive co-energy, that is, L (e) ¼V (e), and the right- a more convenient choice of configuration coordinates for the hand term inside parentheses represents the total co-content h R bob is its angular displacement with respect to the vertical associated with the system b. h axis. Denoting q ¼ col(x, )andq‘ ¼ col(x, xb, yb), we proceed Suppose now that the two systems of Figure 7 are inter- by applying a coordinate transformation of the form (23), with connected through ea ¼ eb and fb ¼ fa, which in this case 2 3 is tantamount to connecting the respective terminals. Sub- x 4 5 tracting the total content and co-content functions produces U(q) ¼ x þ l sin (h) : the scalar function l cos (h) > _ _ _ P( f , e) ¼D( f ) D (e) þ e Nf , (28) Hence the velocities are given by f‘ ¼ col(x, xb, yb), where 2 3 2 3 whereweusethefactthatthesumofthetwointegralsappear- x_ 10 4 5 4 5 x_ ^ ing at the right-hand sides of (26) and (27) reduce, by means of x_ ¼ 1 lcos(h) ¼: U(q, f ), b h_ > : > _ integration by parts, to e Nf ,withN ¼ N Na.Consequently, yb 0 lsin(h) b (26) and (27) combine into one set of equations given by so that the kinetic co-energy can be expressed as d 2 rf H ( f , e) ¼rf P( f , e), (29) Mc þ Mb _ Mbl _ T (q, f ) ¼ x_ 2 þ M lx_h cos(h) þ h2, (25) dt 2 b 2 d r H ( f , e) ¼rP( f , e), (30) dt e e which now depends on both generalized displacement and n velocity coordinates. where H ( f , e) ¼T ( f ) þV (e) represents the total co-energy. Equations (29) and (30), together with (28), are the BM equa- THE BRAYTON-MOSER EQUATIONS tions, originally derived for nonlinear electrical circuits in [19] Having in mind the modified Lagrangian equations (17) and (18), consider a system that consists of n‘ inductive and nr resistive elements, either flow controlled or one to one, and R ea f denote this system as a. The underlying configuration varia- −1 + b1 fa e eb f bles are the generalized displacements associated with the Σ 1 a2 + 1 b2 a − ¼ ... f e inductive elements, that is, q col(q1, , qn‘ ). Furthermore, a2 b2 assume that the system has ne external ports (or ne þ 1 external terminals) associated with a set of efforts ea and flows fa (see Figure 7). The external flows are related to the inductive flows e f _ a−k + bk Σ f ¼ q through the relationship fa ¼ Naf ,whereNa is an ne 3 n‘ b f eb matrix. If the resistive elements admit a resistive content func- ak k tion D( f ), then the Lagrangian equations (17) take the form Z Naf d > rf L( f ) ¼ rf D( f ) ea dfa , (26) FIGURE 7 Brayton-Moser system. The subsystem Ra contains the dt 0 flow-controlled inductive and resistive elements, whereas Rb con- where the Lagrangian is reduced to the total stored induc- tains the effort-controlled capacitive and resistive elements. Since inductive and capacitive elements, as well as flow-controlled and tive co-energy, that is, L( f ) ¼T ( f ), and the right-hand effort-controlled resistors, are complementary or dual elements term inside parentheses represents the total content associ- (see ‘‘Analogues, Duals, and Dualogues’’), the subsystems Ra and ated with the system Ra. Rb can also be considered as complementary or dual subsystems.
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 45
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. and [20]. The scalar function (28) is termed the mixed-potential elementary dc motor, a nonlinear fluid system, and a heat function. The principle application of the mixed-potential is to exchanger. An example of a nonlinear electrical circuit is derive Lyapunov-type stability theorems; see ‘‘History of the presented in ‘‘History of the Mixed-Potential Function.’’ Mixed-Potential Function’’ for a historical overview. Although More involved examples are discussed in the section ‘‘Rotat- the original construction of the mixed-potential function starts ing Mechanical Systems and Beyond.’’ from a differential version of Tellegen’s theorem, the above derivation starts from the classical energy-based framework. Examples 1 and 2 Revisited: A similar exposition in the context of nonlinear electrical cir- The Brayton-Moser Equations cuits can be found in [25]. From a system-theoretic perspec- As a first example to illustrate the BM equations, let us once tive, the BM equations can be interpreted as a gradient system more consider the mass-spring system depicted in Figure with respect to the mixed-potential function (28) and the 3. Following the line of thought presented in the previous indefinite metric defined by the symmetric matrix section, suppose that the mass and the spring are repre- R R sented by the subsystems a ¼fMg and b ¼fCg, respec- r2H ( f , e)0 tively. The next step is to define the system configuration. ¼ f : Q( f , e) 2 (31) R R 0 re H ( f , e) For system a we select x ¼ xM and for system b we select p ¼ pK, so that, as above, the flow equals v ¼ vM and the Note that the mixed-potential consists of three different effort equals F ¼ FK. The mixed-potential function is deter- potential functions (hence the adjective ‘‘mixed’’), all of mined from the interconnection of the two subsystems Ra R which have values with units of power. Moreover, the term and b.SinceFM ¼ F and vK ¼ v,weobtain > e Nf equals the instantaneous power delivered from Ra to R P(v, F) ¼ Fv, (33) b, where N can be considered as the ‘‘turns ratio’’ matrix of a bank of ideal transformers. A proof of this fact can be while the form of the co-Hamiltonian is given in (15). found in [26]. Evidently, if N equals the identity matrix Hence, substituting (33) into (29) and (30) yields the equa- and the system does not contain any resistive elements, tions of motion then the mixed-potential reduces to P( f , e) ¼ e>f . Hence _ _ : the form of the resulting BM equations coincides with the Mv ¼ F, CF ¼ v (34) co-Hamiltonian equations (14), which suggests that (29) Additionally, if the mass is subject to linear viscous fric- and (30) can be considered as a generalized co-Hamiltonian tion or if a damper is placed between the wall and the mass, description. such as in Figure 3 or Figure 5(a), we have Ra ¼fM, Rg and For ease of notation, the BM equations (29) and (30) can R b ¼fCg, respectively. Hence, the mixed-potential can be be compactly written as modified with the addition of the resistive content function _ Q(z)z ¼rzP(z), (32) R P(v, F) ¼ v2 þ Fv: 2 where z ¼ col( f , e) and Q(z) is given by (31). We proceed by exemplifying the BM equations using the If we also insert a damper between the mass and the spring, R R mechanical mass-spring system in Figure 3, followed by the as in Figure 5(c), we have a ¼fM, R1g and b ¼fC, R2g, and thus
R F2 P(v, F) ¼ 1 v2 þ Fv: 2 2R2 Ra La Note that the difference in sign between (34) and the + a ω co-Hamiltonian equations (16) is due to the chosen refer- r ence directions in the selection of the system configura- n Va km Jr tion variables. τ l Example 4: DC Motor [21] _ Ia b In its simplest practical form the dc motor shown in Figure 8 consists of an armature inductance La,anarmatureresistance FIGURE 8 A dc motor. The electrical energy supplied to the system by Ra,andarotorinertiaJr. The input of the system is the armature s the armature voltage Va is converted into rotational energy driving a load voltage Va, and the load torque is denoted by l. The total stored with torque sl . The conversion of electric energy into mechanical energy x = 2 = x2 co-energy is given by H (Ia, r) ¼ð1 2ÞLaIa þð1 2ÞJr r , is represented by a gyrator whose gyration ratio given by the motor con- where the flows I and x denote the armature current and stant k . The flow variables are represented by the current I through a r m a the angular velocity of the rotor, respectively. Furthermore, the armature inductor La and the angular velocity xr of the rotor with inertia Jr . Since the armature resistor Ra is connected in series with the the armature inductance and the rotor inertia both belong inductor it is considered as a current-controlled resistive element. to the class of inductive elements. However, due to the
46 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. gyrative nature of the system we cannot find a mixed-poten- tial function for this setting. To circumvent this problem, we Qin use the fact that the inertia ‘‘seen’’ from the electrical terminals C1 C2 a and b behaves like a capacitive element with capacitance Rp, Lp Ro : 2 : x Cr ¼ km Jr and associated effort er ¼ km r,whereastheload QL : 1s P p P torque translates into the flow source fl ¼ km l.Conse- C1 C2 quently, we can split the system into a subsystem Ra ¼ R fLa, Ra, Vag and a subsystem b ¼fCr, flg. Setting fa ¼ Ia,the FIGURE 9 Fluid system. The two tanks are considered as the corresponding content, co-content, and co-energy in terms of capacitive elements with capacities C1 and C2. The pipe that con- flowsandeffortsaregivenby nects the two tanks is modeled as an inductive element Lp and a resistance Rp. Furthermore, the fluid that is fed to the system and R the orifice dissipator Ro at the outlet are both modeled as nonlinear D( f ) ¼ a f 2 V f , D (e ) ¼ e f , a 2 a a a r r l effort-controlled resistive elements. The state variables are the pressure drops associated with the two tanks and the flow rate and through the pipe. L C H ( f , e ) ¼ a f 2 þ r e2, Furthermore, the co-Hamiltonian is given by the total a r 2 a 2 r stored fluid co-energy respectively. The coupling between the two subsystems Ra R and b is represented by a unit transformer with N ¼ 1, so C1 2 C2 2 Lp 2 H (QL , PC , PC ) ¼ P þ P þ Q : that the mixed-potential takes the form p 1 2 2 C1 2 C2 2 Lp
Ra 2 : Substituting P(QLp , PC1 , PC2 ) and H (QLp , PC1 , PC2 ) into (29) P( fa, er) ¼ fa Vafa erfl þ erfa 2 and (30) yields the equations of motion Substituting P( f , e ) into (29) and (30) yields the equations a r _ of motion LpQLp ¼ RpQLp þ PC2 PC1 , _ _ C1PC1 ¼ Qin QLp , Lafa ¼ er þ Rafa Va, pffiffiffiffiffiffiffiffiffiffiffi _ : _ C2PC2 ¼ QLp GPC2 Crer ¼ fa fl, R R n or, equivalently, in terms of the electrical and mechanical Note that a ¼fLp, Rpg and b ¼fC1, C2, Ro, Qing. variables, Example 6: A Heat Exchanger Cell [27] _ x LaIa ¼ km r þ RaIa Va, Figure 10 shows a heat exchanger in which energy is exchanged x_ s : Jr r ¼ kmIa l n between a cold stream, with inlet and outlet temperatures Tci
and Tco , and a hot stream, with inlet and outlet temperatures
Example 5: A Fluid System Thi and Tho , respectively. The associated thermal capacities are Consider the fluid system shown in Figure 9, which consists denoted by Cc and Ch, whereas the heat transfer is modeled by of two open tanks with capacities C1 and C2, respectively. The first tank, with pressure drop (effort) PC1 , is fed by a volume flow source Qin andlinkedtothesecondtank,withpressure f , T h hi drop (effort) PC2 , by a long pipe with fluid inertia Lp,resistance
Rp,andflowrateQLp (flow). The second tank discharges at f , T atmospheric pressure through an orifice dissipator Ro that can c ci be described by the nonlinear constitutive relationship 1 2 PR ¼ G Q ,wherePR and QR denote the pressure drop o Ro o o T across and the flow rate through the orifice. Hence, the content co D ¼ð = Þ 2 function is (QLp ) 1 2 RpQLp , the co-content function is Z P C2 pffiffiffiffiffiffiffiffiffiffiffi Th o D (PC1 , PC2 ) ¼ GPRo dPRo QinPC1 , 0 FIGURE 10 A heat exchanger system in which energy is exchanged > and the interconnection matrix is given by N ¼½ 11 . between a hot stream and a cold stream. The system has two efforts
Hence, the mixed-potential for the system has the form as state variables, namely, the temperature Tco of the cold stream and the temperature T of the hot stream. The associated thermal Z ho P pffiffiffiffiffiffiffiffiffiffiffi R C2 capacities are Co and Ch, respectively, whereas the heat transfer is P ¼ p 2 þ (QLp , PC1 , PC2 ) QLp GPRo dPRo QinPC1 modeled by a thermal conductance Ghc . The inlet temperatures Tci 2 0 and T are assumed to be constant. The control variables are the : hi þ QLp (PC2 PC1 ) volumetric flow rates fc and fh.
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 47
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. a thermal conductance Ghc. The system can be described by the associated with the capacitive elements without violating
effort variables Tco and Tho . Under the assumption that the inlet the interconnection constraints applicable to the domain temperatures are constant and the volumetric flow rates fc and under consideration, such as Kirchhoff’s laws or D’Alem- fh are the control inputs, the total co-content is found as bert’s principle. An electrical circuit that satisfies these topological properties is said to be topologically complete; for G c D (T , T ) ¼ hc (T T )2 c (T T )2f details, see ‘‘History of the Mixed-Potential Function.’’ This co ho 2 co ho 2 co ci c c terminology can naturally be administered to the multido- h 2 (Th Th ) fh, main case treated here. 2 o i If a system is not topologically complete, we can try to c c where the constants c and h depend on the density and augment the system topology by adding inductive or specific heat of the respective streams. Since there are no capacitive elements, as described in [19] and [28], so that the inductive elements, D¼0 and N ¼ 0. Then the mixed- augmented system becomes topologically complete. Indeed,
potential consists only of the co-content, that is, P(Tco , Tho ) ¼ consider the linear mechanical system of Figure 11(a). Obvi- D (Tco , Tho ), which, together with the co-Hamiltonian ously, the system is not topologically complete since the 2 2 H (T , T ) ¼ð1=2ÞC T þð1=2ÞC T , yields the nonlinear velocities and forces associated with the dampers R1 and R2 co ho c co h ho equations of motion cannot directly be expressed in terms of the velocity v ¼ vM of the mass M and the force F ¼ FK of the spring with com- _ c 1 CcTco ¼ Ghc(Tco Tho ) þ c(Tco Tci )fc, pliance C ¼ K . On the other hand, suppose that we add an _ 0 0 c : additional mass M as shown in Figure 11(b). Since vR1 ¼ v ChTho ¼ Ghc(Tco Tho ) þ h(Tho Thi )fh 0 0 0 and vR2 ¼ v v, with v ¼ vM , the augmented system is R [ R n Note that a ¼ and b ¼fCc, Ch, Ghc, fc, fhg. now topologically complete, and the associated mixed- potential function is given by Topological Completeness 0 0 R1 0 2 R2 0 2 0 The main assumptions that lead to a mixed-potential func- P (v, v , F) ¼ (v ) þ (v v) Finv þ F(v v) (35) tion of the form (28) are i) that the system under considera- 2 2 R R tion can be split into the two subsystems a and b and ii) and has the form (28). However, to find a mixed-potential R that each element in a can be described by the flow varia- for the original system we need to be able to eliminate the bles associated with the inductive elements, and each ele- additional velocity v0 from (35). Letting M0 ! 0 implies that ment in R can be described by the effort variables 0 0 0 0 b rv0 P (v, v ,F) 0, or, equivalently, R1v þ R2(v v) þ F 0. Consequently, the original topologically noncomplete sys- tem is described implicitly by a set of differential algebraic vM equations (DAEs). Solving the latter constraint for v0 yields R2
R1 0 1 v ¼ (R2v F), (36) C M Fin R
where R :¼ R1 þ R2. Substituting (36) into (35) then pro- vides the mixed-potential function
(a) R R F2 R P(v, F) ¼ 1 2 v2 F v 1 Fv: (37) R in R R vM′ vM 2 2
R2 Although (37) appears to be of the form (28), the content R1 and co-content in (37) are not simply the sums of the content ′ M C M Fin and co-content of the individual resistive elements in the system, respectively. Moreover, R1 and R2 act as a force divider that can be interpreted as a mechanical transformer with transformation ratio N ¼ R1=R. Therefore, the system R R (b) cannot be decomposed into a and b since the interconnec- tion structure depends on both R1 and R2. Thus, even FIGURE 11 Example of a mechanical topologically noncomplete sys- though the mixed-potential for a topologically noncomplete tem. The system shown in (a) is not topologically complete since the system cannot be interpreted as easily as in the topologi- velocities and forces associated with the dampers R1 and R2 cannot cally complete case, the concept per se remains applicable. directly be expressed in terms of the velocity of the mass M and the force of the spring C. As shown in (b), the system can be rendered In [28] algorithms are provided for constructing mixed- topologically complete by adding a small mass M0 to the common potential functions for a wide class of topologically non- 0 connection point of R1, R2, and C. Then let M ! 0. complete circuits. In addition, necessary conditions are
48 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2009
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. given in [28] for the existence of the mixed-potential func- multiple switches is easily accomplished by appropriately tion. Mathematically speaking, the resulting DAE system extending (40) with rj 2f0, 1g,forj ¼ 1, ..., ns,wherens belongs to the subset of index 1 systems. denotes the number of independent switches. Noncontrollable The procedure illustrated above can in many cases be car- switches, such as diodes, can be treated as nonlinear resistors. ried over to other domains. The original topologically non- Switched BM equations are closely related to the aver- complete system is then considered as a limiting case of the aged pulse-width modulation (PWM) models. See [4] for a augmented system. Moreover, the addition of inductive and discussion on this subject in the Lagrangian and Hamiltonian capacitive elements can often be justified on physical grounds framework. A PWM switching function may be specified as since these elements are often present as parasitic elements. 1 for t t < t þ d(t )T, In the section ‘‘Rotating Mechanical Systems and Beyond’’ r(t) ¼ k k k 0 for t þ d(t )T t < t þ T, we provide necessary conditions for the existence of the k k k mixed-potential function for a large class of systems. ... for tkþ1 ¼ tk þ T, k ¼ 0, 1, 2, , where tk represents a sam- pling instant, T is the fixed sampling period (duty cycle), SWITCHED-MODE SYSTEMS and d( ) is the duty ratio function of the switch whose val- Switched-mode systems are systems for which the topology ues are in the closed interval ½0; 1 . For (40) the averaging may change depending on certain discrete parameters. Exam- process means that z is replaced by the average state z, ples include electrical power converters and mechanical sys- representing the average efforts and flows, and the discrete tems with impacts. The switchings may be induced internally, control r is replaced by its duty ratio function d. The consis- asinthecaseofadiodeinapowerconverteroranimpactina tency conditions on the averaged mixed-potential functions mechanical system, or it may be triggered externally as in the are thus given by case of firing a thyristor in a power converter. Since the topol- d 1 ogy of a system is determined by the interconnection structure P (z)jd¼1 ¼P (z), d 0 of the elements, it is not surprising that the BM description P (z)jd¼0 ¼P (z), yields a mixed-potential function that depends on the position 1 of the switches. To illustrate how the mixed-potential is where P (z) is the mixed-potential function for the extreme d 0 altered by switching phenomena, we first extend the BM saturation value ¼ 1, and P (z) is the mixed-potential d equations for systems containing a single independent switch. function for the extreme saturation value ¼ 0. Note that d The switch position, denoted by the scalar function r,is P (z) can be considered as a weighted ratio, with weighting d 1 0 assumed to take values in the discrete set f0, 1g. Further- parameter , between P (z) and P (z). more, we assume that, for each switch position, the associ- ated system admits the construction of a mixed potential. Example 7: A Power Converter Adopting the terminology from [4], each mode can be char- Consider the single switch dc-to-dc boost power converter acterized by a set of BM parameters as follows. When the depicted in Figure 12. Assume that the load resistor Ro, switch position function takes the value r ¼ 1, the associ- ated system, denoted by R1, is characterized by a known set L of BM parameters R1 ¼fQ1, P1g satisfying σ = 1 + 1 _ 1 V C R Q (z)z ¼rzP (z): (38) in − o
L σ VC Similarly, when the switch position function takes the value I 0 r ¼ 0, the associated system is characterized by R0 ¼ + L 1 V − C R fQ0, P0g and satisfies in o L 0 _ 0 Q (z)z ¼rzP (z): (39) + Vin − C Ro Hence, a switched system arising from the systems R1 and σ = 0 R0 defines a switched BM system whenever it is completely characterized by the set of switched BM parameters r r R ¼fQr, P g with switched mixed-potential FIGURE 12 A single switch boost converter topology. This power converter is used to realize an output dc voltage greater than its r 0 1 P (z) ¼ (1 r)P (z) þ rP (z), (40) input dc voltage Vin. A boost converter is also called a step-up converter since it increases the input voltage. The boosting effect is satisfying accomplished by charging the inductor L with magnetic energy (switch in position r ¼ 1). This magnetic energy is then released to r _ r : Q (z)z ¼rzP (z) (41) charge the capacitor C (switch in position r ¼ 0). Additionally, the capacitor operates as a filter element to smooth the switching The Q-matrices are usually not altered by the switch positions, effects in the current and voltage fed to the load Ro . In practice the r in which case Q (z) ¼ Q(z). Furthermore, the inclusion of switch is realized by a transistor and a diode.
AUGUST 2009 « IEEE CONTROL SYSTEMS MAGAZINE 49
Authorized licensed use limited to: University of Groningen. Downloaded on December 23, 2009 at 05:19 from IEEE Xplore. Restrictions apply. capacitor C, and inductor L have linear constitutive rela- which, in turn, provides the switched equations of motion tionships. The active switch r is the external control input _ LI ¼ Vin þ (1 r)V, for the network. The converter has two stages, namely, _ V r ¼ 1 (switch ON) and r ¼ 0 (switch OFF). Letting I ¼ IL CV ¼ (1 r)I : Ro denote the current (flow) through the inductor, and V ¼ VC the voltage (effort) across the capacitor, then the total stored The conditions for the transition from ON to OFF, and vice co-energy equals H (I, V) ¼ð1=2ÞLI2 þð1=2ÞCV2. The versa, are determined externally by controlling the switch. n mixed-potential for the switch in position r ¼ 0 is given by
2 Example 8: A Bouncing Pogo-Stick 0 V P (I, V) ¼ VinI þ VI, Consider the vertically bouncing pogo stick depicted in Fig- 2Ro ure 13. The system consists of a mass M and a massless foot, yielding the differential equations interconnected by a linear spring with compliance C ¼ K 1 and a damper R. The mass can move vertically under the _ LI ¼ Vin þ V, influence of gravity g. The contact situation is described by _ V r ¼ 0 (foot has no ground contact, OFF) and r ¼ 1 (foot has CV ¼ I : Ro ground contact, ON). The co-energy storage in the mass and the spring equals H (v, F) ¼ð1=2ÞMv2 þð1=2ÞCF2,where Similarly, for the switch in position r ¼ 1, we have _ v ¼ xM denotes the velocity of the mass and F ¼ KxK denotes the force associated with the spring. Note that for r ¼ 0the V2 P1(I, V) ¼ V I , mass is ‘‘disconnected’’ from the spring and damper, and the in 2R o system equations can be described with help of the mixed- yielding the differential equations potential function F2 _ P0(v, F) ¼ Mgv , LI ¼ Vin, 2R _ V : CV ¼ yielding the no-contact dynamics Ro _ Substituting P0(I, V) and P1(I, V) into (40), we obtain the Mv ¼ Mg, switched mixed-potential function _ F CF ¼ : R 2 r V r P (I, V) ¼ VinI þ (1 )VI, r 2Ro On the other hand, for ¼ 1, we obtain
R P1(v, F) ¼ v2 þ Mgv þ Fv, 2
g M fromwhichwededucethecontact dynamics