Network Modeling and Control of Physical Systems, DISC

Theory of Port-Hamiltonian systems

Chapter 2: Control of Port-Hamiltonian systems

Arjan van der Schaft∗

April 4, 2005

Abstract

Port-based network modeling of physical systems directly leads to their representation as port-Hamiltonian systems. Key feature of port-Hamiltonian systems is that the power-conservinginterconnection of port-Hamiltonian systems results in another port-Hamiltonian system, withtotal state space the product of the state spaces of the components, total Hamiltonian beingthe sum of the Hamiltonian functions, and with Dirac structure defined by the compositionof the Dirac structures of the subsystems. In this way port-Hamiltonian systems serve as aunified framework for the modeling and analysis of complex physical systems consisting ofmany components from different physical domains, finite- or infinite-dimensional.

We discuss in this chapter a number of approaches to exploit the model structure ofport-Hamiltonian systems for control purposes. Actually, the formulation of physical controlsystems as port-Hamiltonian systems may lead in some cases to a re-thinking of standardcontrol paradigms. Indeed, it opens up the way to formulate control problems in a way that isdifferent and perhaps broader than usual. For example, formulating physical systems as port-Hamiltonian systems naturally leads to the consideration of ’impedance’ control problems,where the behavior of the system at the interaction port is sought to be shaped by theaddition of a controller system, and it suggests ’energy-transfer ’ strategies, where the energyis transferred from one part the system to another. Furthermore, it leads to the investigationof a particular type of dynamic controllers, namely those that can be also represented asport-Hamiltonian systems and that are attached to the given plant system in the same wayas a physical system is interconnected to another physical system. As an application of thisstrategy of ’control by interconnection’ within the port-Hamiltonian setting we consider theproblem of (asymptotic) stabilization of a desired equilibrium by shaping the Hamiltonianinto a Lyapunov function for this equilibrium.

From a mathematical point of view we will show that the mathematical formalism ofport-Hamiltonian systems provides various useful techniques, ranging from Casimir functions,

∗Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Nether-

lands, Phone +31-53-4893449, Fax +31-53-4893800,E-mail a.j.vanderschaft@math.utwente.nl

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Lyapunov function generation, shaping of the Dirac structure by composition. Finally, wealso like to mention the possibility to combine finite-dimensional and infinite-dimensionalsystems within the analysis and control of port-Hamiltonian systems. Infinite-dimensionalport-Hamiltonian systems will be discussed in Chapter 3.

Contents of Chapter 2

1. Introduction

2. Recall of port-Hamiltonian systems theory (see Chapter 1)

3. Interconnection of port-Hamiltonian systems

4. Control by port interconnection

5. The role of energy dissipation

6. Stabilization and Casimir generation for input-state-output port-Hamiltonian systems.

Appendix: Composition of Dirac structures and resistive structures

1 Introduction

Network modeling of complex physical systems (with components from different physicaldomains) leads to a class of nonlinear systems, called port-Hamiltonian systems, see e.g.([24, 48, 22, 21, 7, 44, 47]). Port-Hamiltonian systems are defined by a Dirac structure (formal-izing the power-conserving interconnection structure of the system), an energy function (theHamiltonian), and a resistive structure. Key property of Dirac structures is that the power-conserving interconnection or composition of Dirac structures again defines a Dirac structure,see [21, 45]. This implies that any power-conserving interconnection of port-Hamiltoniansystems is again a port-Hamiltonian system, with Dirac structure being the composition ofthe Dirac structures of its constituent parts, Hamiltonian being the sum of the Hamiltonians,and total resistive structure determined by the resistive structures of the components takentogether.

In this chapter we describe how this framework may be exploited for control purposes. InSection 4 we review the framework of control by interconnection within the port-Hamiltoniansetting in a way that is more general than previous expositions. We discuss three sets ofcontrol problems which may be naturally addressed within this framework. First, we recallthe approach of asymptotic stabilization by Casimir generation for the closed-loop systemand how this relies on the composition of Dirac structures. Secondly, we pose the problemof Port Control, where by the addition of a port-Hamiltonian controller system we seek toshape the behavior of the system at the interaction port. Thirdly, we discuss the problem oftransferring the energy of one part of the system towards another part (Energy Control), anddescribe one strategy to accomplish this by modulating the Dirac structure.

In Section 5 we focus on the precise role of energy dissipation in control strategies forport-Hamiltonian systems. We show in general that for a function to be a conserved quantity(Casimir) for one non-degenerate resistive relation at the resistive port it actually needs to bea Casimir for all resistive relations. Finally, in the last Section we deal with Casimir generationand stabilization in the special case of input-state-output port-Hamiltonian systems.

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Figure 1: Port-Hamiltonian system

2 Recall of port-Hamiltonian systems theory

A port-Hamiltonian system can be represented as in Figure 1. Central in the definition of aport-Hamiltonian system is the notion of a Dirac structure, depicted in the figure by D. Sometheory regarding finite-dimensional Dirac structures is summarized in the Appendix. Basicproperty of any Dirac structure is power conservation: the Dirac structure links the variousport variables in such a way that the total power associated with the port-variables is zero.

In Figure (1) there are two internal ports. One, denoted by S, is corresponding to energy-storage and the other one, denoted by R, is corresponding to internal energy-dissipation. Theport variables associated with the internal storage port will be denoted by (fS , eS). They areinterconnected to the energy storage of the system which is defined by a finite-dimensionalstate space manifold X with coordinates x, together with a Hamiltonian function H : X → R

denoting the energy. The interconnection to the energy storage is accomplished by setting

fS = −x

eS = ∂H∂x

(x)(1)

corresponding to the internal power-balance

d

dtH =

∂T H(x)

∂xx = −eT

SfS (2)

The second internal port corresponds to internal energy dissipation (due to friction, resis-tance, etc.), and its port variables are denoted by (fR, eR). These port variables are termi-nated on a static resistive relation R, which for the rest of this paper we assume to be linear.(Note that a large class of nonlinearities can be already captured in the description of theresistive port. For extensions we refer to [44].) This means that the resistive port variables(fR, eR) satisfy relations of the form

RffR + ReeR = 0 (3)

where the square matrices Rf and Re satisfy the symmetry and semi-positive definitenesscondition

RfRTe = ReR

Tf ≥ 0 (4)

together with the dimensionality condition

rank [Rf |Re] = dim fR (5)

3

By the dimensionality condition (5) and the symmetry (4) we can equivalently rewrite thekernel representation (3) of R into an image representation

fR = RTe λ

eR = −RTf λ

(6)

By the semi-definiteness condition this implies that for all fR, eR satisfying the resistiverelation the following holds

eTRfR = −(RT

f λ)T RTe λ = −λT RfRT

e λ ≤ 0 (7)

A typical example of a resistive relation given in input-output form is

fR = −ReR, R = RT ≥ 0 (8)

Without the presence of additional external ports, the Dirac (interconnection) structure ofthe port-Hamiltonian system satisfies the power-balance

eTSfS + eT

RfR = 0 (9)

which leads by substitution of the equations (2) and (7) to

d

dtH = −eT

SfS = eTRfR ≤ 0 (10)

Now, let us consider external ports to the system. We shall distinguish between two typesof external ports. One is the control port C, with port-variables (fC , eC), which are theport-variables which are accessible for controller action. Other type of external port is theinteraction port I, which denotes the interaction of the port-Hamiltonian system with itsenvironment. The port-variables corresponding to the interaction port are denoted by (fI , eI).By taking both the external ports into account the power-balance (9) extends to

eTSfS + eT

RfR + eTCfC + eT

I fI = 0 (11)

whereby (10) extends tod

dtH = eT

RfR + eTCfC + eT

I fI (12)

The port-Hamiltonian system with state space X , Hamiltonian H corresponding to the energystorage port S, resistive port R, interconnection port I, and total Dirac structure D will besuccinctly denoted by Σ = (X ,H,R, C, I,D). The dynamics of the port-Hamiltonian systemis defined by considering the constraints on the various port variables imposed by the Diracstructure, that is,

(fS, eS , fR, eR, fC , eC , fI , eI) ∈ D

and to substitute fS = −x, eS = ∂H∂x

(x), so as to obtain the implicitly defined dynamics

(−x(t),∂H

∂x(x(t)), fR(t), eR(t), fC(t), eC (t), fI(t), eI(t)) ∈ D (13)

with fR(t), eR(t) satisfying for all t the resistive relation

RffR(t) + ReeR(t) = 0 (14)

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Note that in general (13) consists of a mixed set of differential and algebraic equations (DAEs).An important special case of port-Hamiltonian systems as defined above is the class of

input-state-output port-Hamiltonian systems, where there are no algebraic constraints on thestate space variables, and the flow and effort variables at the resistive, control and interac-tion port are split into conjugated input-output pairs. (Note that the absence of algebraicconstraints on the state variables can be alternatively formulated by requiring that the effortvariables at the energy-storage port, – which are given by the components of the gradientof the Hamiltonian –, are free input variables.) Such input-state-output port-Hamiltoniansystems are of the form

P :

x = [J(x) − R(x)] ∂H∂x

(x) + g(x)u + k(x)d

y = gT (x)∂H∂x

(x)

z = kT (x)∂H∂x

(x)

x ∈ X (15)

where u, y are the input-output pairs corresponding to the control port C, while d, z denotethe input-output pairs of the interaction port I. Here the matrix J(x) is skew-symmetric,that is J(x) = −JT (x). The matrix R(x) = RT (x) ≥ 0 specifies the resistive structure.From a resistive port point of view, it is given as R(x) = gT

R(x)RgR(x) for some resistiverelation fR = −ReR, R = RT ≥ 0, with gR representing the input matrix corresponding tothe resistive port. The total Dirac structure of the system is then given by the graph of theskew-symmetric linear map

−J(x) −g(x) −gR(x)gT (x) 0 0gTR(x) 0 0

(16)

Note that yT u and zT d still denote the power corresponding to the control, respectively,interaction port. For more details, in particular extensions of (15) to feedthrough terms, werefer to [44, 11].

3 Interconnection of port-Hamiltonian systems

The basic property of port-Hamiltonian systems is that the power-conserving interconnectionof any number of port-Hamiltonian systems is again a port-Hamiltonian system.

To be explicit, consider two port-Hamiltonian systems ΣA and ΣB with Dirac structuresDA and DB and Hamiltonians HA and HB, defined on state spaces XA, respectively XB. Forconvenience, split the ports of the Dirac structures DA and DB into the internal energy storageports and all remaining external ports whose port-variables are denoted respectively by fA, eA

and fB, eB . Now, consider any interconnection Dirac structure DI involving the port-variablesfA, eA, fB , eB possibly together with additional port-variables fI , eI . Then it can be shownthat the interconnection of the systems ΣA and ΣB via DI is again a port-Hamiltonian systemwith respect to the composed Dirac structure

DA ‖ DI ‖ DB

involving as port-variables the internal storage port-variables of DA and DB together withthe additional port-variables FI , eI . The main theorem about the composition of (finite-dimensional) Dirac structures is recalled in Appendix A. Furthermore, the state space of the

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interconnected port-Hamiltonian system is the product of the two state spaces XA×XB , whileits Hamiltonian is simply the sum HA + HB of the two Hamiltonians.

This basic statement naturally extends to the interconnection of any number of port-Hamiltonian systems via an interconnection Dirac structure.

4 Control by Port-Interconnection

Control by port-interconnection is based on designing a controller system which is intercon-nected to the control port with port-variables (fC , eC). In principle this implies that we onlyconsider collocated control, where the controller will only use the information about the plantport-Hamiltonian system that is contained in the conjugated pairs (fC , eC) of port variablesof the control port, without using additional information about the plant (e.g. correspond-ing to observation on other parts of the plant system). In the second place, we will restrictattention to controller systems which are themselves also port-Hamiltonian systems. Thereare two main reasons for this. One is that by doing so the closed-loop system is again aport-Hamiltonian system, allowing to easily ensure some desired properties. Furthermore, itwill turn out that the port-Hamiltonian framework suggests useful ways to construct port-Hamiltonian controller systems. Second reason is that port-Hamiltonian controller systemsallow in principle for a physical system realization (thus linking to passive control and systemsdesign) and physical interpretation of the controller action. Of course, this raises the realiza-tion problem which (controller) systems can be represented as port-Hamiltonian systems, forwhich some partial answers are available [44].

Since we do not know the environment (or only have very limited information about it),but on the other hand, the system will interact with this unknown environment, the task ofthe controller is often two-fold: 1) to achieve a desired control goal (e.g. set-point regulationor tracking) if the interaction with the environment is marginal or can be compensated, 2) tomake sure that the controlled system has a desired interaction behavior with its environment.It is fair to say that up to now the development of the theory of control of port-Hamiltoniansystems has mostly concentrated on the second aspect (which at the same time, is oftenunderdeveloped in other control theories).

Most successful approaches to deal with the second aspect of the control goal are thosebased on the concept of ’passivity”, such as dissipativity theory, impedance control and Intrin-sically Passive Control (IPC). In fact, the port-Hamiltonian control theory can be regardedas an enhancement to the theory of passivity, making a much closer link with complex phys-ical systems modeling at one hand and with the theory of dynamical systems (in particular,Hamiltonian dynamics) at the other hand.

As said above, we will throughout consider controller systems which are again port-Hamiltonian systems, in the same way as the plant system is a port-Hamiltonian system.We will use the same symbols as above for the internal and external ports and port-variablesof the controller port-Hamiltonian system, with an added overbar¯or a superscript c in orderto distinguish it from the plant system. (The interaction port of the controller system maybe thought of as an extra possibility for additional controller action (outer-loop control).) Inorder to further distinguish the plant system and the controller we denote the state space ofthe plant system by Xp with coordinates xp, the Dirac structure by Dp and its Hamiltonianby Hp, while we will denote the state space manifold of the controller system by Xc withcoordinates xc, its Dirac structure by Dc and its Hamiltonian by Hc : Xc → R. The intercon-

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nection of the plant port-Hamiltonian system with the controller port-Hamiltonian system isobtained by equalizing the port variables at the control port by:

fC = −fC

eC = eC(17)

where fC , eC denote the control port variables of the controller system. Here, the minus signis inserted to have a uniform notion of direction of power flow. Clearly, this ’synchronizing’interconnection is power-conserving, that is

eTCfC + eT

C fC = 0 (18)

Remark 4.1. A sometimes useful alternative for the power-conserving interconnection (17)is the gyrating power-conserving interconnection

fC = −eC

eC = fC(19)

In fact, the standard feedback interconnection can be regarded to be of this type. Considera plant input-state-output port-Hamiltonian system as in (15)

P :

xp = [J(xp) − R(xp)]∂Hp

∂xp(xp) + gp(xp)u + kp(xp)d

y = gTp (xp)

∂Hp

∂xp(xp)

z = kTp (xp)

∂Hp

∂xp(xp)

xp ∈ Xp (20)

together with a controller input-state-output port-Hamiltonian system

P :

xc = [J(xc) − R(xc)]∂Hc

∂xc(xc) + gc(xc)u + kc(xc)d

y = gTc (xc)

∂Hc

∂xc(xc)

z = kTc (xc)

∂Hc

∂xc(xc)

xc ∈ Xc (21)

Then the standard feedback interconnection

u = −y

y = u(22)

can be seen to be equal to the gyrating interconnection (19).

For both interconnection constraints it directly follows from the theory of compositionof Dirac structures that the interconnected (closed-loop) system is again a port-Hamiltoniansystem with Dirac structure determined by the Dirac structures of the plant PH system andthe controller PH system. (The main theorem about the composition of Dirac structures isrecalled in the Appendix.)

The resulting interconnected PH system has state space Xp × Xc, Hamiltonian Hp +Hc, resistive ports (fR, eR, fR, eR) and interaction ports (fI , eI , fI , eI), satisfying the power-balance

d

dt(Hp + Hc) = eT

RfR + eTRfR + eT

I fI + eTI fI ≤ eT

I fI + eTI fI (23)

7

since both eTRfR ≤ 0 and eT

RfR ≤ 0. Hence we immediately recover the state space formulationof the passivity theorem, see e.g. [44], if Hp and Hc are both non-negative, implying that theplant and the controller system are passive (with respect to their controller and interactionports and storage functions Hp and Hc), then also the closed -loop system is passive (withrespect to the interaction ports and storage function Hp + Hc.)

Nevertheless, we will show in the next sections that, due to the Hamiltonian structure, wecan go beyond the passivity theorem, and that we can derive conditions which ensure that wecan passify and/or stabilize plant port-Hamiltonian systems for which the Hamiltonian Hp isnot non-negative (or bounded from below).

4.1 Stabilization by Casimir generation

As we have seen above, the interconnection of a plant port-Hamiltonian system with a con-troller port-Hamiltonian system leads to a closed-loop port-Hamiltonian system, with closed-loop Dirac structure being the composition of the plant and the controller Dirac structure.Furthermore, we immediately obtain the power-balance (23)

d

dt(Hp + Hc) = eT

RfR + eTRfR + eT

I fI + eTI fI ≤ eT

I fI + eTI fI

What does this mean about the stability properties of the closed-loop system, and how canwe design the controller port-Hamiltonian system in such a way that the closed-loop systemhas desired stability properties? Let us therefore first consider the stability of an arbitraryport-Hamiltonian system Σ = (X ,H,R, C, I,D) without control or interaction ports, that is,an autonomous port-Hamiltonian system Σ = (X ,H,R,D). Clearly, the power-balance (23)reduces to (10), that is

d

dtH = eT

RfR ≤ 0 (24)

Hence we immediately infer by standard Lyapunov theory that if x∗ is a minimum of theHamiltonian H then it will be a stable equilibrium of the autonomous port-Hamiltoniansystem Σ = (X ,H,R,D), which is actually asymptotically stable if the dissipation termeTRfR is negative definite outside x∗, or alternatively if some sort of detectability condition is

satisfied, guaranteeing asymptotic stability by the use of LaSalle’s Invariance principle (seefor details e.g. [44]).

However, what can we say if x∗ is not a minimum of H, and thus we cannot directly useH as a Lyapunov function ?

A well-known method in Hamiltonian systems, sometimes called the Energy-Casimirmethod, is to use in the Lyapunov analysis next to the Hamiltonian other conserved quanti-ties (dynamical invariants) which may be present in the system. Indeed, if we may find otherconserved quantities then candidate Lyapunov functions can be sought within the class ofcombinations of the Hamiltonian H and those conserved quantities. In particular, if we canfind a conserved quantity C : X → R such that V := H + C has a minimum at the desiredequilibrium x∗ then we can still infer stability or asymptotic stability by replacing (24) by

d

dtV = eT

RfR ≤ 0 (25)

and thus using V as a Lyapunov function.

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For the application of the Energy-Casimir method one may distinguish between two maincases. First situation occurs if the desired equilibrium x∗ is not a stationary point of H, andone looks for a conserved quantity C such that H +C has a minimum at x∗. This for examplehappens in the case that the deisred set-point x∗ is not an equilibrium of the uncontrolledsystem, but only a controlled equilibrium of the system. Second situation occurs when x∗ isa stationary point of H, but not a minimum. Typical example of this situation will arise inthe context of stabilization of an unstable plant system; see later on.

Functions that are conserved quantities of the system for every Hamiltonian are calledCasimir functions or simply Casimirs. Casimirs are completely characterized by the Diracstructure of the port-Hamiltonian system. Indeed, a function C : X → R is a Casimir functionof the autonomous port-Hamiltonian system (without energy dissipation) Σ = (X ,H,D) if

and only if the gradient vector e = ∂T C∂x

satisfies

eT fS = 0, for all fS for which ∃eS s.t. (fS , eS) ∈ D (26)

Indeed, (26) is equivalent to

d

dtC =

∂T C

∂x(x(t))x(t) =

∂T C

∂x(x(t))fS = eT fS = 0 (27)

for every port-Hamiltonian system (X ,H,D) with the same Dirac structure D. By the gen-eralized skew-symmetry of the Dirac structure (26) is equivalent to the requirement that

e = ∂T C∂x

satisfies(0, e) ∈ D

Similarly, we define a Casimir function for a port-Hamiltonian system with dissipation Σ =(X ,H,R,D) to be any function C : X → R satisfying

(0, e, 0, 0) ∈ D (28)

Indeed, this will imply that

d

dtC =

∂T C

∂x(x(t))x(t) =

∂T C

∂x(x(t))fp = eT fp = 0 (29)

for every port-Hamiltonian system (X ,H,R,D) with the same Dirac structure D. (In Section5.1 we shall see that in fact by definiteness of the resistive structures the satisfaction of (29)for a particular resistive structure R implies the satisfaction for all resistive structures R.)

Now let us come back to the design of a controller port-Hamiltonian system such that theclosed-loop system has desired stability properties. Suppose we want to stabilize the plantport-Hamiltonian system (Xp,Hp,R, C,Dp) around a desired equilibrium x∗

p. We know thatfor every controller port-Hamiltonian system the closed-loop system satisfies

d

dt(Hp + Hc) = eT

RfR + eTRfR ≤ 0 (30)

What if x∗ is not a minimum for Hp ? A possible strategy is to generate Casimir functionsC(xp, xc) for the closed-loop system by choosing the controller port-Hamiltonian system inan appropriate way. Thereby we generate candidate Lyapunov functions for the closed-loopsystem of the form

V (xp, xc) := Hp(xp) + Hc(xc) + C(xp, xc)

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where the controller Hamiltonian function Hc : Xc → R still has to be designed. The goalis thus to construct a function V as above in such a way that V has a minimum at (x∗

p, x∗c)

where x∗c still remains to be chosen. This strategy thus is based on finding all the achievable

closed-loop Casimirs. Furthermore, since the closed-loop Casimirs are based on the closed-loop Dirac structures, this reduces to finding all the achievable closed-loop Dirac structuresD ‖ D. This will be discussed later on.

Another way to interpret the generation of Casimirs for the closed-loop system is tolook at the level sets of the Casimirs as invariant submanifolds of the combined plant andcontroller state space Xp × Xc. Restricted to every such invariant submanifold (part of) thecontroller state can be expressed as a function of the plant state, whence the closed-loopHamiltonian restricted to such an invariant manifold can be seen as a shaped version of theplant Hamiltonian. To be explicit (see e.g. [44, 32, 35] for details) suppose that we havefound Casimirs of the form

xci − Fi(xp), i = 1, · · · , np

where np is the dimension of the controller state space, then on every invariant manifoldxci − Fi(xp) = αi, i = 1, · · · , np, where α = (α1, · · · , αnp) is a vector of constants dependingon the initial plant and controller state, the closed-loop Hamiltonian can be written as

Hs(xp) := Hp(xp) + Hc(F (xp) + α)

where, as before, the controller Hamiltonian Hc still can be assigned. This can be regardedas shaping the original plant Hamiltonian Hp to a new Hamiltonian Hs.

For the special case of input-state-output port-Hamiltonian systems also involving energydissipation stabilization by Casimir generation will be worked out in detail in a later Section.

4.2 Port Control

In broad terms, the Port Control problem is to design, given the plant port-Hamiltoniansystem, a controller port-Hamiltonian system such that the behavior at the interaction portof the plant port-Hamiltonian system is a desired one, or close to a desired one. This meansthat by adding the controller system we seek to shape the external behavior at the interactionport of the plant system. If the desired external behavior at this interaction port is given ininput-output form as a desired (dynamic) impedance, then this amounts to the ImpedanceControl problem as introduced and studied by Hogan and co-workers [17]; see also [39] forsubsequent developments.

The Port Control problem, as stated in this generality, immediately leads to two fun-damental questions: 1). Given the plant PH system, and the controller PH system to bearbitrarily designed, what are the achievable behaviors of the closed-loop system at the in-teraction port of the plant? 2). If the desired behavior at the interaction port of the plant isnot achievable, then what is the closest achievable behavior? Of course, the second questionleaves much room for interpretation, since there is no obvious interpretation of what we meanby ’closest behavior’. Also the first question in its full generality is not easy to answer, andwe shall only address an important subproblem.

An obvious observation is that the desired behavior, in order to be achievable, needs tobe the port behavior of a PH system. This leads already to the problem of characterizingthose external behaviors which are port behaviors of port-Hamiltonian systems. Secondly, thePort Control problem can be split into a number of subproblems. Indeed, we know that the

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closed-loop system arising from interconnection of the plant PH system with the controllerPH system is specified by a Hamiltonian which is just the sum of the plant Hamiltonian andthe controller Hamiltonian, and a resistive structure which is the ”product” of the resistivestructure of the plant and of the controller system, together with a Dirac structure which isthe composition of the plant Dirac structure and the controller Dirac structure. Thereforean important subproblem is again to characterize the achievable closed-loop Dirac structures.On the other hand, a fundamental problem in addressing the Port Control problem in generaltheoretical terms is the lack of a systematic way to specify ’desired behavior’.

Example 4.2. Consider the plant system (in input-state-output port-Hamiltonian form)

[

q

p

]

=

[

0 1−1 0

]

∂H∂q

∂H∂p

+

[

01

]

u

y =[

0 1]

∂H∂q

∂H∂p

(31)

with q the position and p being the momentum of the mass m, in feedback interconnectionu = −y + fI , u = y = eI , with the controller system (see Figure 2)

me

b

kc

mc

k

Figure 2: Controlled mass

∆qc

pc

∆q

=

0 1 0−1 −b 10 −1 0

∂Hc

∂∆qc

∂Hc

∂pc

∂Hc

∂∆qc

+

001

u

y = ∂Hc

∂∆qc

(32)

where ∆qc is the displacement of the spring kc, ∆q is the displacement of the spring k, and pc

is the momentum of the mass mc. The plant Hamiltonian is H(p) = 12m

p2, and the controller

Hamiltonian is given as Hc (∆qc, pc,∆q) = 12( p2

c

mc+ k(∆q)2 + kc(∆qc)

2). The variable b > 0is the damping constant, and fI is an external force, with eI denoting the correspondingvelocity.The closed-loop system possesses the Casimir function

C(q,∆qc,∆q) = ∆q − (q − ∆qc), (33)

implying that along the solutions of the closed-loop system

∆q = q − ∆qc + c (34)

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with c a constant depending on the initial conditions.With the help of LaSalle’s Invariance principle it can be shown that restricted to the

invariant manifolds (34) the system is asymptotically stable for the equilibria q = ∆qc = p =pc = 0.

The problem of Port Control is to determine the controller system given by (32), or bya more general expression, in such a way that the port behavior in the port variables fI , eI

is a desired one. In this particular (simple and linear) example the desired behavior can bequantified e.g. in terms of a desired stiffness and damping of the closed-loop system, which iseasily expressed in terms of the closed-loop transfer function from fI to eI . Of course, on top ofthe requirements on the closed-loop transfer function we would also require internal stabilityof the closed-loop system. For an appealing example of port control of port-Hamiltoniansystems within a context of hydraulic systems we refer to [19].

4.3 Energy Control

Consider two port-Hamiltonian systems Σi (without internal dissipation) in input-state-output form

xi = Ji(xi)∂Hi

∂xi+ gi(xi)ui

yi = gTi (xi)

∂Hi

∂xi, i = 1, 2

(35)

both satisfying the power-balanced

dtHi = yT

i ui (36)

Suppose now that we want to transfer the energy from the port-Hamiltonian system Σ1 tothe port-Hamiltonian system Σ2, while keeping the total energy H1 + H2 constant. This canbe done by using the following output feedback

(

u1

u2

)

=

(

0 −y1yT2

y2yT1 0

)(

y1

y2

)

(37)

Since the matrix in (37) is skew-symmetric it immediately follows that the closed-loop systemcomposed of systems Σ1 and Σ2 linked by the power-conserving feedback is energy-preserving,that is d

dt(H1 + H2) = 0. However, if we consider the individual energies then we notice that

d

dtH1 = −yT

1 y1yT2 y2 = −||y1||

2||y2||2 ≤ 0 (38)

implying that H1 is decreasing as long as ||y1|| and ||y2|| are different from 0. Conversely, asexpected since the total energy is constant,

d

dtH2 = yT

2 y2yT1 y1 = ||y2||

2||y1||2 ≥ 0 (39)

implying that H2 is increasing at the same rate. In particular, if H1 has a minimum at the zeroequilibrium, and Σ1 is zero-state observable, then all the energy H1 of Σ1 will be transferredto Σ2, provided that ||y2|| is not identically zero (which again can be guaranteed by assumingthat H2 has a minimum at the zero equilibrium, and that Σ2 is zero-state observable).

12

If there is internal energy dissipation, then this energy transfer mechanism still works.However, the fact that H2 grows or not will depend on the balance between the energydelivered by Σ1 to Σ2 and the internal loss of energy in Σ2 due to dissipation.

We conclude that this particular scheme of power-conserving energy transfer is accom-plished by a skew-symmetric output feedback, which is modulated by the values of the outputvectors of both systems. Of course this raises, among others, the question of the efficiency ofthe proposed energy-transfer scheme, and the need for a systematic quest of similar power-conserving energy-transfer schemes. We refer to [9] for a similar but different energy-transferscheme directly motivated by the structure of the example (control of a snakeboard).

4.4 Achievable closed-loop Dirac structures

In all the control problems discussed above (stabilization by Casimir generation, port-controland energy-control) the basic question comes up what are the achievable closed-loop Diracstructures based on a given plant Dirac structure and a controller Dirac structure, whichstill is to be determined. In the sequel we will show how in fact the set of achievable Diracstructures can be completely characterized.

We recall briefly the composition of two Dirac structures with partially shared variables.That is we consider a Dirac structure D1 on a product space F1 × F2, and another Diracstructure D2 on the product space F2 × F3, F1,F2,F3 being linear spaces. The space F2 isthe space of shared flow variables and F ∗

2 the space of shared effort variables between D1 andD2. By following proper sign conventions the interconnection D1 ‖ D2 of the Dirac structuresD1 and D2 is defined as

D1 ‖ D2 := {(f1, e1, f3, e3) ∈ F1 ×F∗

1 ×F3 ×F∗

3 | ∃(f2, e2) ∈ F2 ×F∗

2 s.t.

(f1, e1, f2, e2) ∈ D1 and (−f2, e2, f3, e3) ∈ D2}

Theorem 4.3. [46] Let D1,D2 be Dirac structures (defined with respect to F1×F∗1 ×F2×F∗

2 ,

respectively F2 ×F∗2 ×F3 ×F∗

3 and their bilinear forms). Then D1 ‖ D2 is a Dirac structurewith respect to the bilinear form on F1 ×F∗

1 ×F3 ×F∗3 .

We also know which port-Hamiltonian systems can be achieved by interconnecting a givenplant port-Hamiltonian system P with a controller port-Hamiltonian system C. That is, giventhe plant Dirac structure Dp and the to-be-designed Dirac structure Dc of the controllersystem, what are the achievable Dirac structures Dp ‖ Dc, with ‖ denoting the interconnectionbetween Dp and Dc as above.

Theorem 4.4. [46] Given any plant Dirac structure Dp, a certain interconnected D = Dp ‖Dc can be achieved by a proper choice of the controller Dirac structure Dc if and only if thefollowing two conditions are satisfied

D0p ⊂ D0

Dπ ⊂ Dπp

where

D0p := {f1, e1) | (f1, e1, 0, 0) ∈ Dp}

Dπp := {(f1, e1) | ∃(fP

2 , eP2 ) : (f1, e1, f

P2 , eP

2 ) ∈ Dp}

D0 := {(f1, e1) | (f1, e1, 0, 0) ∈ D}Dπ := {(f1, e1) | ∃(f3, e3) : (f1, e1, f3, e3) ∈ D}

13

4.5 Achievable Casimirs

An important application of the above theorem concerns the characterization of Casimirfunctions which can be achieved by interconnecting a given plant port-Hamiltonian systemwith a controller port-Hamiltonian system. This constitutes a cornerstone for passivity basedcontrol of port-Hamiltonian systems.

We thus consider the question of characterizing the set of achievable Casimirs for theclosed-loop system DP ‖ DC , where DP is the given Dirac structure of the plant port-Hamiltonian system with Hamiltonian H, and DC is the (to-be-designed) controller Diracstructure. In this case, the Casimirs will depend on the plant state x as well as on the con-troller state ξ. Since the controller Hamiltonian HC(ξ) is at our own disposal we will beprimarily interested in the dependency of the Casimirs on the plant state x. (Since we wantto use the Casimirs for shaping the total Hamiltonian H + HC to a Lyapunov function, cf.[32, 35].)

Consider first a port-Hamiltonian system without external (controller or interaction) ports.Also we will assume for simplicity that there is no resistive port. Thus we consider a statespace X with Dirac structure D ⊂ X × X ∗. Then the following subspaces of X , respectivelyX ∗, are of importance

G1 := {fx ∈ X | ∃ex ∈ X ∗ such that (fx, ex) ∈ D}

P1 := {ex ∈ X ∗ | ∃fx ∈ X such that (fx, ex) ∈ D}(40)

The subspace G1 expresses the set of admissible flows, and P1 the set of admissible efforts.In the image representation (102)

G1 = imETx

P1 = imF Tx

(41)

A Casimir function K : X → R of the port-Hamiltonian system is defined to be a functionwhich is constant along all trajectories of the port-Hamiltonian system, irrespectively of theHamiltonian H and the resistive structure. It follows from the above consideration of theadmissible flows that the Casimirs are determined by the subspace G1. Indeed, necessarilyfx = −x(t) ∈ G1 = imET

x , and thus

x(t) ∈ imETx , t ∈ R. (42)

Therefore K : X → R is a Casimir function if dKdt

(x(t)) = ∂T K∂x

(x(t))x(t) = 0 for all x(t) ∈imET

x . Hence K : X → R is a Casimir of the port-Hamiltonian system if it satisfies the setof partial differential equations

∂K

∂x(x) ∈ ker Ex (43)

Geometrically, this can be formulated by defining the following subspace of the dual space ofefforts

P0 = {ex ∈ X ∗ | (0, ex) ∈ D} (44)

Indeed, it can be easily seen that G1 = P⊥0 where ⊥ denotes orthogonal complement with

respect to the duality product < |>. Hence K is a Casimir function iff ∂K∂x

(x) ∈ P0.

14

Pf2

Pe2

DP

DC

1f 3

f

1e 3

e

Cf2

-

Ce2

Figure 3: DP ‖ DC

Remark 4.5. In the case of a non-constant Dirac structure the matrix Ex will depend on x,and kerEx will define a co-distribution on the manifold X . Then the issue arises of integrabilityof this co-distribution, see [7].

Consider the notation given in Figure 3, and assume the ports in (f1, e1) are connectedto the (given) energy storing elements of the plant port-Hamiltonian system (that is, f1 =−x, e1 = ∂H

∂x), while (f3, e3) are connected to the (to-be-designed) energy storing elements of

a controller port-Hamiltonian system (that is, f3 = −ξ, e3 = ∂HC

∂ξ). Note that the number

of ports (f3, e3) can be freely chosen. In this situation the achievable Casimir functions arefunctions K(x, ξ) such that ∂K

∂x(x, ξ) belongs to the space

PCas = {e1 | ∃DC s.t. ∃e3 : (0, e1, 0, e3) ∈ DP ‖ DC} (45)

Thus the question of characterizing the achievable Casimirs of the closed-loop system, re-garded as functions of the plant state x, is translated to finding a characterization of thespace PCas. This is answered by the following theorem.

Theorem 4.6. The space PCas defined in (45) is equal to the linear space

P = {e1 | ∃(f2, e2) : (0, e1, f2, e2) ∈ DP } (46)

.

Proof PCas ⊂ P trivially. By using the controller Dirac structure DC = D∗

P , we immediatelyobtain P ⊂ PCas.

Remark 4.7. For a non-constant Dirac structure on a manifold X PCas defines a co-distribution on X .

In a completely dual way we may consider the achievable constraints of the closed-loopsystem, characterized by the space

GAlg = {f1 | ∃DC s.t. ∃f3 : (f1, 0, f3, 0) ∈ DP ‖ DC} (47)

Theorem 4.8. The space GAlg defined in (47) is equal to the linear space

{f1 | ∃(f2, e2) : (f1, 0, f2, e2) ∈ DP} (48)

Remark 4.9. For a non-constant Dirac structure GAlg defines a distribution on the manifoldX .

15

Example 4.10. Consider the input-state-output port-Hamiltonian plant system with inputsf2 and outputs e2

x = J(x)∂H∂x

(x) + g(x)f2, x ∈ X , f2 ∈ Rm

e2 = gT (x)∂H∂x

(x), e2 ∈ Rm

(49)

where J(x) is a skew-symmetric n× n matrix. The corresponding Dirac structure is given asthe graph of the map

[

f1

e2

]

=

[

−J(x) −g(x)

gT (x) 0

][

e1

f2

]

(50)

It is easily seen that

PCas = {e1 | ∃f2 such that 0 = J(x)e1 + g(x)f2}, (51)

implying that the achievable Casimirs K(x, ξ) are such that e1 = ∂K∂x

(x) satisfies J(x) ∂K∂x

(x) ∈im g, that is, K as a function of x is a Hamiltonian function corresponding to a Hamiltonianvector field contained in the distribution spanned by the input vector fields given by thecolumns of g(x).

5 The role of energy dissipation

In the modeling process of physical systems the precise specification of the resistive relationsis one of the most difficult parts. Furthermore, precisely these resistive relations are oftensubject to time-variation. Therefore, from a robust control perspective it may be desirableto base the construction of the controller in a way that is independent of the precise resistiverelations, and to make sure that the controller behavior meets the required specifications fora sufficiently large range of resistive effects. This raises a number of questions, some of whichare addressed below.

5.1 Casimirs in the case of energy dissipation; the dissipation obstacle

Recall that we define a Casimir function for a port-Hamiltonian system with dissipationΣ = (X ,H,R,D) to be any function C : X → R such that e = ∂C

∂x(x) satisfies

(0, e, 0, 0) ∈ D (52)

Indeed, as derived above, this will imply that

d

dtC =

∂T C

∂x(x(t))x(t) =

∂T C

∂x(x(t))fS = eT fS = 0 (53)

for every port-Hamiltonian system (X ,H,R,D) with the same Dirac structure D.Now, at this point one may think that the definition of Casimir function may be relaxed

by requiring that (53) only holds for a specific resistive relation

RffR + ReeR = 0 (54)

16

where the square matrices Rf and Re satisfy the symmetry and semi-positive definitenesscondition (see (4))

RfRTe = ReR

Tf ≥ 0 (55)

together with the dimensionality condition

rank [Rf |Re] = dim fR (56)

We will show that actually this is not a relaxation, if the required semi-definite positivenessof the resistive relation is strengthened to positive-definiteness

RfRTe = ReR

Tf > 0 (57)

In this case, the condition for a function to be a conserved quantity for one resistive relationwill actually imply that it is a conserved quantity for all resistive relations.

Indeed, let C : X → R be a function satisfying (53) for a specific resistive port R specifiedby matrices Rf and Re as above. This means that e = ∂C

∂x(x) satisfies

eT fS = 0, for all fS for which ∃eS , fR, eR s.t. (fS , eS , fR, eR) ∈ D and RffR + ReeR = 0

However, this implies that (0, e) ∈ (D ‖ R)⊥. Now by Proposition 7.5 in the Appendix(D ‖ R)⊥ = D ‖ (−R), and thus there exists fR, eR such that Rf fR − ReeR = 0 and

(0, e, fR, eR) ∈ D

Hence0 = eT · 0 + eT

RfR = eTRfR

By writing the pseudo-resistive structure −R in image representation (cf. the Appendix)fR = RT

e λ, eR = RTf λ, it follows that

λT RfRTe λ = 0

and by the positive-definiteness condition RfRTe = ReR

Tf > 0 this implies that λ = 0, whence

fR = eR = 0. Hence not only (0, e, fR, eR) ∈ D but actually (0, e, 0, 0) ∈ D, implying that e

is the gradient of a Casimir function as defined before.Of course, the above argument does not fully carry through if the resistive relations are

only semi-positive definite. In particular, this is the case if RfRTe = 0 (implying zero energy

dissipation), corresponding to the presence of ideal power-conserving constraints. In fact, ifRfRT

e = 0 then the resistive structure reduces to a particular type of Dirac structure (see thedefinition of the kernel representation (101) of a Dirac structure in the Appendix.)

The fact that a conserved quantity for one resistive relation is actually a conserved quantityfor all resistive relations (and thus a Casimir) is closely related to the so-called dissipation ob-stacle for the existence of Casimir functions in the case of input-state-output port-Hamiltoniansystems, cf. [32, 35, 44].

In fact, let us consider a conserved quantity C : X → R for an input-state-output port-Hamiltonian system without external (control and interaction) ports for a specific resistivestructure, that is

∂T C

∂x(x) (J(x) − R(x)) = 0

17

Post-multiplication by ∂C∂x

(x) immediately implies

∂T C

∂x(x) (J(x) − R(x))

∂C

∂x(x) = 0

Then by transposition we obtain the second equation

∂T C

∂x(x) (−J(x) − R(x))

∂C

∂x(x) = 0

Adding these two equations one derives

∂T C

∂x(x)R(x)

∂C

∂x(x) = 0

which by positive semi-definiteness of R(x) implies

∂T C

∂x(x)R(x) = 0

(and then also ∂T C∂x

(x)J(x) = 0). This so-called dissipation obstacle implies that Casimirs arenecessarily independent of those state space coordinates that are affected by physical damping.Thus we conclude that the dissipation obstacle for finding Casimirs directly generalizes toport-Hamiltonian systems defined with respect to a general Dirac structure, and is equivalentto the observation that a function that is a conserved quantity with respect to one positivedefinite resistive relation is necessarily a Casimir function (and thus conserved for all resistiverelations).

An obvious way to overcome the dissipation obstacle in particular cases is to apply a pre-liminary feedback (for simplicity we restrict ourselves to input-state-output port-Hamiltoniansystems)

u = K(x)y + v, K(x) = KT (x) ≥ 0

(negative damping injection), which leads to the closed-loop system

x = (J(x) − R(x) + K(x))∂H

∂x+ g(x)v

Hence if we are able to design K(x) in such a way that R(x)−K(x) becomes zero, or at leastof rank less than rankR(x) (exact compensation of energy dissipation) then the dissipationobstacle disappears, respectively is mitigated. An obvious drawback of this strategy is thatit relies on exact compensation, and hence may be highly sensitive with respect to parameteruncertainty in the resistive relations. Another strategy to overcome the dissipation obstaclein some cases is discussed in [28].

6 Stabilization by Casimir generation for input-state-output

port-Hamiltonian system

In this section we investigate the existence of Casimir functions for the feedback interconnec-tion of a plant pH system with a controller pH system, both given in input-state-output formincluding energy dissipation. Thus we consider Casimirs depending both on the plant state as

18

well as the controller state and stabilize an equilibrium in the extended plant-controller states.The resulting Lyapunov function is then the sum of the plant and the controller Hamiltoniansand the corresponding Casimir functions.

Consider a plant port-Hamiltonian system, in input-state-output form

{

x = [J(x) − R(x)] ∂H∂x

+ g(x)u

y = gT (x)∂H∂x

(58)

where x ∈ Rn is the state vector, u ∈ R

m , m < n is the control action, H : Rn → R is the

total stored energy, and J(x) = −JT (x), R(x) = RT (x) ≥ 0 are the natural interconnectionand damping matrices, respectively, and a controller port-Hamiltonian system also in aninput-output form

{

ξ = [JC(ξ) − RC(ξ)]∂HC

∂ξ+ gC(ξ)uC

yC = gT (ξ)∂H∂ξ

(59)

via the standard feedback interconnection

u = −yc

uc = y

By the modularity property of pH systems, the composed system plant–controller is clearlystill pH and can be written as

[

x

ξ

]

=

[

J(x) − R(x) −g(x)gTc (ξ)

gc(ξ)gT (x) Jc(ξ) − Rc(ξ)

]

[

∂H∂x

(x)∂Hc

∂ξ(ξ)

]

[

y

yc

]

=

[

g(x) 00 gc(ξ)

]

[

∂H∂x

(x)∂Hc

∂ξ(ξ)

]

(60)

which again is a port-Hamiltonian system with dissipation, with state space given by theproduct space X × X c, the total Hamiltonian Hcl = H(x) + Hc(ξ). The control objective isto stabilize a desired equilibrium (x∗, ξ∗). To this end we investigate more general Casimirfunctions of the closed-loop system (of the form C(x, ξ)), which means that we are lookingfor the solutions of the p.d.e’s

[

∂T C∂x

(x, ξ) ∂C∂ξ

(x, ξ)]

[

J(x) − R(x) −g(x)gTc (ξ)

gc(ξ)gT (x) Jc(ξ) − Rc(ξ)

]

= 0

or

∂T C

∂x(x, ξ)[J(x) − R(x)] +

∂C

∂ξ(x, ξ)gc(ξ)g

T (x) = 0

∂T C

∂x(x, ξ)g(x)gT

c (ξ) −∂C

∂ξ(x, ξ)[Jc(ξ) − Rc(ξ)] = 0 (61)

post multiplying first equation of (61) by ∂C∂x

(x, ξ) and second equation by ∂C∂ξ

(x, ξ) yields

∂T C

∂x(x, ξ)J(x)

∂C

∂x(x, ξ) =

∂T C

∂ξ(x, ξ)Jc(ξ)

∂C

∂ξ(x, ξ) (62)

19

−∂T C

∂x(x, ξ)R(x)

∂C

∂x(x, ξ) =

∂T C

∂ξ(x, ξ)Rc(ξ)

∂C

∂ξ(x, ξ) (63)

Since by assumption R(x) ≥ 0, Rc(ξ) ≥ 0, (63) is equivalent to

∂T C

∂x(x, ξ)R(x)

∂C

∂x(x, ξ) =

∂T C

∂ξ(x, ξ)Rc(ξ)

∂C

∂ξ(x, ξ) = 0

Since R(x) ≥ 0 the above equation is equivalent to

R(x)∂C

∂x(x, ξ) = Rc(ξ)

∂C

∂ξ(x, ξ) = 0 (64)

Summarizing we have obtained

Proposition 6.1. The functions Ci(x, ξ) satisfy (61) (and thus are Casimir functions for theclosed-loop port-Hamiltonian system (59)) if and only if C satisfies

∂T C

∂x(x, ξ)J(x)

∂C

∂x(x, ξ) =

∂T C

∂ξ(x, ξ)Jc(ξ)

∂C

∂ξ(x, ξ)

R(x)∂C

∂x(x, ξ) = Rc(ξ)

∂C

∂ξ(x, ξ) = 0

∂T C

∂x(x, ξ)J(x) = −

∂T C

∂ξgc(ξ)g

T (x)

∂T C

∂ξ(x, ξ)Jc(ξ) =

∂T C

∂x(x, ξ)g(x)gT

c (ξ) (65)

Proof. Only the last two equations need to be shown, which are easily obtained by substi-tuting (64) into (61).

Remark 6.2. The second equation represents the so-called ”dissipation obstacle”.

Example 6.3. Consider a mechanical system with damping and actuated by external forcesu described as a port-Hamiltonian system with dissipation

[

q

p

]

= (

[

0 Ik

−Ik 0

]

−

[

0 00 D(q)

]

)

[

∂H∂q∂H∂p

]

+

[

0B(q)

]

u

y = BT (q)∂H

∂p(66)

with x =

[

q

p

]

, where q ∈ Rk are the generalized configuration coordinates, p ∈ R

k the

generalized momenta, and D(q) = DT (q) ≥ 0 is the damping matrix. If D(q) > 0, then itis said that the system is fully damped. The outputs y ∈ R

m are the generalized velocitiescorresponding to the generalized external forces u ∈ R

m. In most cases the HamiltonianH(q, p) takes the form

H(q, p) =1

2pTM−1(q)p + P (q) (67)

20

where M(q) = MT (q) > 0 is the generalized inertia matrix, 12pT M−1(q)p = 1

2 qT M(q)q is thekinetic energy, and P (q) is the potential energy of the system.Consider now a controller port-Hamiltonian system

ξ = [Jc(ξ) − Rc(ξ)]∂Hc

∂ξ(ξ) + gc(ξ)uc

yc = gTc (ξ)

∂Hc

∂ξ(ξ)

Then the equations (65) for C = (C1(x, ξ), ..., Cm(x, ξ))T take the form

∂T C

∂p

∂C

∂q−

∂T C

∂q

∂C

∂p=

∂T C

∂ξJc(ξ)

∂C

∂ξ

D(q)∂C

∂p= 0 = Rc(ξ)

∂C

∂ξ

∂T C

∂p= 0, and

∂T C

∂q= −

∂T C

∂ξgc(ξ)B

T (q)

or equivalently∂T C

∂ξJc = 0,

∂C

∂p= 0,

∂T C

∂q+

∂T C

∂ξgc(ξ)B(q) = 0 (68)

Hence if we can solve the PDE in the above equation, then the closed-loop port-Hamiltoniansystem with Jc = 0 admits Casimirs Ci(x, ξ), i = 1, ...,m, leading to a closed loop system

q

p

ξ

=

0 Ik 0−Ik 0 −B(q)gT

c (ξ)0 gc(ξ)B(q) 0

∂Hc

∂q∂Hc

∂p∂Hc

∂ξ

y = BT (q)∂Hc

∂x

yc = gc(ξ)∂Hc

∂ξ(69)

for the shaped systemHs(q, p, ξ) = H(q, p) + Hc(ξ)

If H(q, p) is given as in (67), then

Hs(q, p, ξ) =1

2pTM(q)p + P (q) + Hc(ξ)

which is the closed-loop the energy of the system.

6.1 Stabilization by Casimir generation

Consider the plant system (58) and suppose that there exist Casimirs for the plant controllerinterconnection satisfying (65). On the basis of the Hamiltonian of the plant, the Hamiltonianof the controller and the corresponding Casimir function a Lyapunov function candidate isbuilt as the sum of the plant and controller Hamiltonians and the Casimir function as

V (x, ξ) = H(x) + Hc(ξ) + C(x, ξ) (70)

21

We haved

dtV (x, ξ) = −

∂T H

∂x(x)R(x)

∂T H

∂x(x) −

∂T Hc

∂ξ(ξ)Rc(ξ)

∂T Hc

∂ξ(ξ) ≤ 0

and hence V (x, ξ) qualifies as a Lyapunov function for the closed-loop dynamics.The next step would be to shape the closed-loop energy in the extended state space (x, ξ) insuch a way that it has a minimum at (x∗, ξ∗), therefore we require that the gradient of (70)has an extremum at (x∗, ξ∗) and that the Hessian at (x∗, ξ∗) is positive definite, that is

[

∂∂x

[H(x) + C(x, ξ)] |(x∗,ξ∗)∂∂ξ

[Hc(ξ) + C(x, ξ)] |(x∗,ξ∗)

]

= 0

and[

∂2

∂x2 [H(x) + C(x, ξ)] ∂2

∂ξ∂xC(x, ξ)]

∂2

∂x∂ξC(x, ξ)] ∂2

∂ξ2 [Hc(ξ) + C(x, ξ)]

]∣

∣

∣

∣

∣

(x∗,ξ∗)

≥ 0

Example 6.4. Consider the case of a normalized pendulum

q + sin q + dq = u

with d a positive damping constant, and the total energy function given as H(q, p) = 12p2 +

(1 − cos q). The solution to (68) should be a function of the form f(q − ξ + c)Let (q∗, ξ∗) be the desired equilibrium and we shape the potential energy P (q) in such a waythat it has a minimum at q = q∗, ξ = ξ∗. This can be achieved by choosing a controllerHamiltonian of the form

Hc(ξ) =1

2β(ξ − ξ∗ −

1

βsin q∗)

2

and a Casimir function which is of the form

f(q − ξ + c) =1

2k(q − q∗ − (ξ − ξ∗) −

1

ksin q∗)

2

where β and k are chosen so as to have a minimum at the desired equilibrium. Simplecomputations show that β and k should be such that

cos q∗ + k > 0, β cos q∗ + k cos q∗ + kβ > 0

The resulting passivity based input u is then given by

u = −∂Hc

∂ξ(ξ) = −β(ξ − ξ∗ −

1

βsin q∗)

Remark 6.5. In the same way we can stabilize a fully actuated system of n degrees offreedom, in which case we have to solve n p.d.e.’s of the form (68), in order to find thecorresponding Casimir function.

Next we investigate as to how a port-Hamiltonian system with dissipation can be asymp-totically stabilized around a desired equilibrium point. Suppose that V (x, ξ) has a strictlocal minimum at (x∗, ξ∗), and that there exists an open neighborhood B of (x∗, ξ∗) such that

22

V (x, ξ) > V (x∗, ξ∗) for all (x, ξ) ∈ B. Furthermore assume that the largest invariant set underthe dynamics (60) contained in

{(x, ξ) ∈ B |∂T H

∂x(x)R(x)

∂T H

∂x(x) = 0,

∂T Hc

∂ξ(ξ)Rc(ξ)

∂T Hc

∂ξ(ξ) = 0}

equals (x∗, ξ∗). Then (x∗, ξ∗) is a locally asymptotically stable equilibrium of (58).

6.2 Passivity-based control of input-state-output port-Hamiltonian sys-

tems

In this subsection we investigate Casimir functions for the closed-loop system that are directlyrelating the state variables ξ of the controller system to the state variables x of the plantsystem. In particular, let us consider Casimir functions of the form

ξi − Gi(x) , i = 1, . . . ,dimXC = nC (71)

That means that we are looking for solutions of the p.d.e.’s (with ei denoting the i-th basisvector)

[

−∂T Gi

∂x(x) eT

i

]

J(x) − R(x) −g(x)gTC(ξ)

gC(ξ)gT (x) JC(ξ) − RC(ξ)

= 0

or written out∂T Gi

∂x(x) [J(x) − R(x)] − gi

C(ξ)gT (x) = 0

∂T Gi

∂x(x)g(x)gT

C (ξ) + J iC(ξ) − Ri

C(ξ) = 0

(72)

with ∂T Gi

∂xdenoting as before the gradient vector

(

∂Gi

∂x1, . . . , ∂Gi

∂xn

)

, and giC , J i

C , RiC denoting

the i-th row of gC , JC , respectively RC .

Suppose we want to solve (72) for i = 1, . . . , n , with n ≤ nC (possibly after permutationof ξ1, . . . , ξnC

). with JC(ξ), RC(ξ) the n × n left-upper submatrices of JC , respectively RC .

The following proposition follows from Proposition 6.1.

Proposition 6.6. The functions ξi − Gi(x), i = 1, . . . , n ≤ nC , satisfy (72) (and thusare Casimirs of the closed-loop port-controlled Hamiltonian system (60) if and only if G =(G1, . . . , Gn)T satisfies

∂T G∂x

(x)J(x)∂G∂x

(x) = JC (ξ)

R(x)∂G∂x

(x) = 0

RC(ξ) = 0

∂T G∂x

(x)J(x) = gC (ξ) gT (x)

(73)

with JC(ξ), RC(ξ) the n × n left-upper submatrices of JC , respectively RC .

23

In particular, we conclude that the functions ξi−Gi(x), i = 1, . . . n, are Casimirs if and onlyif they are Casimirs for both the internal interconnection structure Jcl(x, ξ) as well as for thedissipation structure Rcl(x, ξ). Hence it follows directly how the closed-loop port-Hamiltoniansystem reduces to a system any multi-level set {(x, ξ) |ξi = Gi(x) + ci, i = 1, . . . , n}, by re-stricting both Jcl and Rcl to this multi-level set.

Let us next consider the special case n = nC , in which case we wish to relate all thecontroller state variables ξ1, . . . , ξnC

to the plant state variables x via Casimir functions ξ1 −G1(x), . . . , ξnC

−GnC(x). Denoting G = (G1, . . . , GnC

)T this means that G should satisfy (see(73))

∂T G∂x

(x)J(x)∂G∂x

(x) = JC (ξ)

R(x)∂G∂x

(x) = 0 = RC(ξ)

∂T G∂x

(x)J(x) = gC (ξ) gT (x)

(74)

In this case the reduced dynamics on any multi-level set

LC = {(x, ξ)|ξi = Gi(x) + ci, i = 1, . . . nC} (75)

can be immediately recognized. Indeed, the x-coordinates also serve as coordinates for LC .

Furthermore, the x-dynamics is given as

x = [J(x) − R(x)]∂H

∂x(x) − g(x)gT

C (ξ)∂HC

∂ξ(ξ). (76)

Using the second and the third equality of (74) this can be rewritten as

x = [J(x) − R(x)]

(

∂H

∂x(x) +

∂G

∂x(x)

∂HC

∂ξ(ξ)

)

. (77)

and by the chain-rule property for differentiation this reduces to the port-Hamiltonian system

x = [J(x) − R(x)]∂Hs

∂x(x), (78)

with the same interconnection and dissipation structure as before, but with shaped Hamilto-nian Hs given by

Hs(x) = H(x) + HC(G(x) + c). (79)

An interpretation of the shaped Hamiltonian Hs in terms of energy-balancing is the following.Since RC(ξ) = 0 by (74) the controller Hamiltonian HC satisfies dHC

dt= uT

CyC . Hence alongany multi-level set LC given by (75), invariant for the closed loop port-Hamiltonian system

dHs

dt=

dH

dt+

dHC

dt=

dH

dt− uT y (80)

since u = −yC and uC = y. Therefore, up to a constant,

Hs(x(t)) = H(x(t)) −

∫ t

0uT (τ)y(τ)dτ, (81)

and the shaped Hamiltonian Hs is the original Hamiltonian H minus the energy suppliedto the plant system (15) by the controller system (59) (modulo a constant; depending on theinitial states of the plant and controller).

24

Remark 6.7. Note that from a stability analysis point of view (81) can be regarded asan effective way of generating candidate Lyapunov functions Hs from the Hamiltonian H.(Compare with the classical construction of Lur’e functions.)

Example 6.8. A mechanical system with damping and actuated by external forces u ∈ Rm

is described as a port-Hamiltonian system

[

q

p

]

=(

[

0 Ik

−Ik 0

]

−

[

0 00 D(q)

]

)

[

∂H∂q

∂H∂p

]

+

[

0B(q)

]

u

y = BT (q)∂H∂p

(82)

with x = [ qp ], where q ∈ R

k are the generalized configuration coordinates, p ∈ Rk the

generalized momenta, and D(q) = DT (q) ≥ 0 is the damping matrix. In most cases theHamiltonian H(q, p) takes the form

H(q, p) =1

2pTM−1(q)p + P (q) (83)

where M(q) = MT (q) > 0 is the generalized inertia matrix, 12pTM−1(q)p = 1

2 qT M(q)q isthe kinetic energy, and P (q) is the potential energy of the system. Now consider a generalport-Hamiltonian controller system (59) with state space R

m. Then the equations (74) forG = (G1(q, p), . . . , Gm(q, p))T take the form

∂T G∂q

∂G∂p

− ∂T G∂p

∂G∂q

= JC (ξ)

D(q)∂G∂p

= 0

∂T G∂p

= 0, ∂T G∂q

= gC(ξ)BT (q)

(84)

or equivalently

JC = 0,∂G

∂p= 0, gT

C(ξ)B(q) =∂G

∂q(q) (85)

Now let gC(ξ) be the m×m identity matrix. Then there exists a solution G = (G1(q), . . . , Gm(q))to (85) if and only if the columns of the input force matrix B(q) satisfy the integrability con-ditions

∂Bil

∂qj

(q) =∂Bjl

∂qi

(q), i, j = 1, . . . k, l = 1, . . . m (86)

Hence, if B(q) satisfies (86), then the closed-loop port-Hamiltonian system (60) for the con-troller system (59) with JC = 0 admits Casimirs ξi−Gi(q), i = 1, . . . ,m, leading to a reducedport-Hamiltonian system

[

q

p

]

= (

[

0 Ik

−Ik 0

]

−

[

0 00 D(q)

]

)

∂Hs

∂q

∂Hs

∂p

+

[

0B(q)

]

e

y = BT (q)∂Hs

∂p

(87)

25

for the shaped Hamiltonian

Hs(q, p) = H(q, p) + HC(G1(q) + c1, . . . , Gm(q) + cm) (88)

If H(q, p) is as given in (83), then

Hs(q, p) =1

2pTM−1(q)p + [P (q) + HC(G1(q) + c1, · · · , Gm(q) + cm)] (89)

and the control amounts to shaping the potential energy of the system. �

Thus the feedback interconnection of the port-Hamiltonian system having Hamiltonian H

(the “plant”) with another port-Hamiltonian system with Hamiltonian HC (the “controller”)leads to a reduced dynamics given by

x = [J(x) − R(x)]∂Hs

∂x(x) (90)

for the shaped Hamiltonian Hs(x) = H(x) + HC(G(x) + c), with G(x) a solution of (74).From a state feedback point of view the dynamics (90) could have been directly obtained bya state feedback u = α(x) such that

g(x)α(x) = [J(x) − R(x)]∂HC

∂x(G(x) + c) (91)

Indeed, such an α(x) is given in explicit form as

α(x) = −gTC(G(x) + c)

∂HC

∂ξ(G(x) + c) (92)

A state feedback u = α(x) satisfying (91) is customarily called a passivity-based control law,since it is based on the passivity properties of the original plant system (15) and transforms(15) into another passive system with shaped storage function (in this case Hs).Seen from this perspective we have shown in the previous section that the passivity-basedstate feedback u = α(x) satisfying (91) can be derived from the interconnection of the port-Hamiltonian plant system (15) with a port-Hamiltonian controller system (59). This fact hassome favorable consequences. Indeed, it implies that the passivity-based control law definedby (91) can be equivalently generated as the feedback interconnection of the passive system(15) with another passive system (59). In particular, this implies an inherent invarianceproperty of the controlled system: the plant system (90), the controller system (79), as wellas any other passive system interconnected to (90) in a power-conserving fashion, may changein any way as long as they remain passive, and for any perturbation of this kind the controlledsystem will remain stable.The implementation of the resulting passivity-based control u = α(x) is a somewhat complexissue. In cases of analog controller design the interconnection of the plant port-Hamiltoniansystem (15) with the port-Hamiltonian controller system (

In the rest of this section we concentrate on the passivity-based (state feedback) controlu = α(x). The purpose is to more systematically indicate how a port-Hamiltonian systemwith dissipation (15) may be asymptotically stabilized around a desired equilibrium x∗ in twosteps:

26

I Shape by passivity-based control the Hamiltonian in such a way that it has a strictminimum at x = x∗. Then x∗ is a (marginally) stable equilibrium of the controlled system.

II Add damping to the system in such a way that x∗ becomes an asymptotically sta-ble equilibrium of the controlled system.

As before, we shall concentrate on Step I. Therefore, let us consider a port-Hamiltoniansystem with dissipation (15) with X the n-dimensional state space manifold. Suppose wewish to stabilize the system around a desired equilibrium x∗, assigning a closed-loop energyfunction Hd(x) to the system which has a strict minimum at x∗ (that is, Hd(x) > Hd(x

∗) forall x 6= x∗ in a neighbourhood of x∗). Denote

Hd(x) = H(x) + Ha(x), (93)

where the to be defined function Ha is the energy added to the system (by the control action).We have the following

Proposition 6.9. [32, 27] Assume we can find a feedback u = α(x) and a vector functionK(x) satisfying

[J(x) − R(x)] K(x) = g(x)α(x) (94)

such that(i) ∂Ki

∂xj(x) =

∂Kj

∂xi(x), i, j = 1, . . . , n

(ii) K(x∗) = −∂H∂x

(x∗)

(iii) ∂K∂x

(x∗) > −∂2H∂x2 (x∗)

(95)

with ∂K∂x

the n×n matrix with i-th column given by ∂Ki

∂x(x), and ∂2H

∂x2 (x∗) denoting the Hessianmatrix of H at x∗. Then the closed-loop system is a Hamiltonian system with dissipation

x = [J(x) − R(x)]∂Hd

∂x(x) (96)

where Hd is given by (93), with Ha such that

K(x) =∂Ha

∂x(x) (97)

Furthermore, x∗ is a stable equilibrium of (96).

A further generalization is to use state feedback in order to change the interconnectionstructure and the resistive structure of the plant system, and thereby to create more flex-ibility to shape the storage function for the (modified) port-controlled Hamiltonian systemto a desired form. This methodology has been called Interconnection-Damping AssignmentPassivity-Based Control (IDA-PBC) in [32, 35], and has been succesfully applied to a numberof applications. The method is especially attractive if the newly assigned interconnectionand resistive structures are judiciously chosen on the basis of physical considerations, andrepresent some “ideal” interconnection and resistive structures for the physical plant. For anextensive treatment of IDA-PBC we refer to [32, 35].

27

7 Appendix: Composition of Dirac structures and resistive

structures

7.1 Dirac structures

Let us briefly recall the definition of a Dirac structure. We start with a space of powervariables F × F∗, for some linear space F , with power defined by

P = < f∗ | f >, (f, f ∗) ∈ F × F∗, (98)

where < f ∗ | f > denotes the duality product, that is, the linear functional f ∗ ∈ F∗ actingon f ∈ F . We call F the space of flows f , and F ∗ the space of efforts e = f ∗, with the powerof a signal (f, e) ∈ F × F ∗ denoted as < e | f >.

Closely related to the definition of power there exists a canonically defined bilinear form�,� on the space of power variables F × F ∗, defined as

� (fa, ea),(f b, eb) �:=

< ea | f b > + < eb | fa >, (fa, ea), (f b, eb) ∈ F × F∗. (99)

Definition 7.1. [6, 8] A (constant) Dirac structure on F × F ∗ is a subspace

D ⊂ F ×F∗

such that D = D⊥, where ⊥ denotes orthogonal complement with respect to the bilinear form�,�.

If F is a finite-dimensional linear space then it is easily seen that necessarily dimD =dimF for any Dirac structure D. Moreover, in this case a Dirac structure can be equivalentlycharacterized as a subspace D of F × F ∗ such that

(i) < e | f >= 0, for all (f, e) ∈ D,

(ii) dimD = dimF .

Remark 7.2. For many systems, especially those with 3 − D mechanical components, theinterconnection structure is actually modulated by the energy or geometric variables. Thisleads to the notion of non-constant Dirac structures on manifolds, see e.g. [6, 8, 7, 44, 43].For simplicity of exposition we focus here on the constant case, although everything can beappropriately extended to the case of Dirac structures on manifolds.

Dirac structures on finite-dimensional linear spaces admit different representations. Herewe just list a number of them; see e.g. [44] for more information.

1. (Kernel and Image representation) Every Dirac structure D ⊂ F × F ∗, can be repre-sented in kernel representation as

D = {(f, e) ∈ F × F∗ | Ff + Ee = 0} (100)

for linear maps F : F → V and E : F ∗ → V satisfying

(i) EF ∗ + FE∗ = 0,

(ii) rank F + E = dimF ,

(101)

28

where V is a linear space with the same dimension as F , and where F ∗ : V∗ → F∗ andE∗ : V∗ → F∗∗ = F are the adjoint maps of F and E, respectively.

It follows that D can be also written in image representation as

D = {(f, e) ∈ F × F∗ | f = E∗λ, e = F ∗λ, λ ∈ V∗} (102)

Matrix kernel and image representations are obtained by choosing linear coordinatesfor F , F∗ and V. Indeed, take any basis f1, · · · , fn for F and the dual basis e1 =f∗

1 , · · · , en = f∗n for F∗, where dim F = n. Furthermore, take any set of linear coordi-

nates for V. Then the linear maps F and E are represented by n×n matrices F and E

satisfying(i) EF T + FET = 0,

(ii) rank [F |E] = dimF ,

(103)

(Constrained input-output representation) D = {(f, e) ∈ F×F ∗ | f = Je+Gλ,GT e = 0}for a skew-symmetric mapping J : F → F ∗ and a linear mapping G such that ImG = {f | (f, 0) ∈ D}. Furthermore, Ker J = {e | (0, e) ∈ D}.

2. (Hybrid input-output representation, cf. [3]) Let D be given in matrix kernel repre-sentation by square matrices E and F as in 1. Suppose rank F = m(≤ n). Select m

independent columns of F , and group them into a matrix F1. Write (possibly after

permutations) F = [F1...F2], and correspondingly E = [E1

...E2], f =

[

f1

f2

]

, e =

[

e1

e2

]

.

Then the matrix [F1...E2] is invertible, and

D =

{[

f1

f2

]

,

[

e1

e2

] ∣

∣

∣

∣

[

f1

e2

]

= J

[

e1

f2

]}

(104)

with J := −[F1...E2]

−1[F2...E1] skew-symmetric.

3. (Canonical coordinate representation), cf. [6]. There exists a basis for F with dualbasis for F∗, such that with respect to this basis the partioned vector (f, e) =(fq, fp, fr, fs, eq, ep, er, es) is in D iff

fq = ep, fp = −eq

fr = 0, es = 0(105)

Equational representations of a port-Hamiltonian system are now obtained by choosing aspecific representation of the Dirac structure D. For example, if D is given in matrix kernelrepresentation

D = {(fx, ex, f, e) ∈ X × X ∗ ×F ×F∗ | Fxfx + Exex + Ff + Ee = 0}, (106)

with(i) ExF T

x + FxETx + EF T + FET = 0,

(ii) rank [Fx

...Ex

...F...E] = dim(X × F).

(107)

29

Af2

Ae2

DA

DB

1f 3

f

1e 3

e

Bf2

Be2

Figure 4: Notation for DA, DB connection split for composition, power variables.

then the port-Hamiltonian system is given by the set of DAE’s

Fxx(t) = Ex∂H

∂x(x(t)) + Ff(t) + Ee(t), (108)

Remark 7.3. In case of a Dirac structure modulated by the energy variables x and the statespace X being a manifold, the flows fx = −x are elements of the tangent space TxX at thestate x ∈ X , and the efforts ex are elements of the co-tangent space T ∗

xX . Still locally on Xwe obtain the kernel representation (108) for the resulting port-Hamiltonian system, but nowthe matrices Fx, Ex, F,E will depend on x. See for an extensive treatment [7, 44, 43].

7.2 Composition of Dirac structures

First we consider the composition of two Dirac structures with partially shared variables.That is, we consider a Dirac structure DA on a product space F1 × F2 of two linear spacesF1 and F2, and another Dirac structure DB on a product space F2 ×F3, with also F3 beinga linear space. The linear space F2 is the space of shared flow variables, and F ∗

2 the space ofshared effort variables; see Figure(4).

The standard composition of two Dirac structures is to define the interconnection con-straints as

fA = −fB ∈ F2

eA = eB ∈ F∗

2

(109)

Here the minus sign has been included in order to have a consistent power flow convention.The gyrating interconnection (19) can be easily transformed to this case. Therefore thecomposition DA ‖ DB of the Dirac structures DA and DB is defined as

DA ‖ DB := {(f1, e1, f3, e3) ∈ F1 ×F∗

1 ×F3 ×F∗

3 | ∃(f2, e2) ∈ F2 ×F∗

2 s.t.

(f1, e1, f2, e2) ∈ DA and (−f2, e2, f3, e3) ∈ DB}(110)

The composition of two Dirac structures is again a Dirac structure, as already shown in[45](§ 3). In [46] a simpler alternative proof is given, inspired by [29]. Furthermore, this proofallows us to easily extend the result to the composition of a Dirac structure D and a resistivestructure R later on.

Theorem 7.4. Let DA, DB be Dirac structures as in Definition 7.1 (defined with respect toF1 ×F∗

1 ×F2 ×F∗

2 , respectively F2 ×F∗

2 ×F3 ×F∗

3 and their bilinear forms). Then DA ‖ DB

is a Dirac structure with respect to the bilinear form on F1 ×F∗

1 ×F3 ×F∗

3 .

30

Proof. Consider DA, DB defined in matrix kernel representation by

DA = {(f1, e1, fA, eA) ∈ F1 ×F∗

1 ×F2 ×F∗

2 | F1f1 + E1e1 + F2AfA + E2AeA = 0} (111)

DB = {(fB , eB , f3, e3) ∈ F2 ×F∗

2 ×F3 ×F∗

3 | F2BfB + E2BeB + F3f3 + E3e3 = 0} (112)

In the following we shall make use of the following basic fact from linear algebra:

(∃λ s.t. Aλ = b) ⇔ [∀α s.t. αT A = 0 ⇒ αT b = 0] (113)

Note that DA, DB are alternatively given in matrix image representation as

DA = im

ET1

F T1

ET2A

F T2A

00

DB = im

00

ET2B

F T2B

ET3

F T3

(114)

Hence,

(f1, e1, f3, e3) ∈ DA||DB ⇔

⇔ ∃λA, λB s.t.

f1

e1

00f3

e3

=

ET1 0

F T1 0

ET2A ET

2B

F T2A −F T

2B

0 F T3

0 ET3

[

λA

λB

]

⇔

⇔ ∀(β1, α1, β2, α2, β3, α3) s.t. (βT1 αT

1 βT2 αT

2 βT3 αT

3 )

ET1 0

F T1 0

ET2A ET

2B

F T2A −F T

2B

0 F T3

0 ET3

= 0,

βT1 f1 + αT

1 e1 + βT3 f3 + αT

3 e3 = 0 ⇔

⇔ ∀(α1, β1, α2, β2, α3, β3) s.t.

[

F1 E1 F2A E2A 0 00 0 −F2B E2B F3 E3

]

α1

β1

α2

β2

α3

β3

= 0,

βT1 f1 + αT

1 e1 + βT3 f3 + αT

3 e3 = 0 ⇔

⇔ ∀(α1, β1, α2, β2, α3, β3) ∈ DA||DB , βT1 f1 + αT

1 e1 + βT3 f3 + αT

3 e3 = 0 ⇔

⇔ (f1, e1, f3, e3) ∈ (DA||DB)⊥

Thus DA ‖ DB = (DA ‖ DB)⊥, and so it is a Dirac structure.

31

7.3 Composition of Dirac structures and resistive structures

The proof of composition of two Dirac structures (Theorem 7.4) immediately extends to thecomposition of a Dirac structure and a resistive structure.

Proposition 7.5. Let D be a Dirac structure defined with respect to FS × F∗

S × FR × F∗

R.Furthermore, let R be a resistive structure defined with respect to FR ×F∗

R given by

RffR + ReeR = 0 (115)

where the square matrices Rf and Re satisfy the symmetry and semi-positive definitenesscondition

RfRTe = ReR

Tf ≥ 0 (116)

Define the composition D ‖ R of the Dirac structure and the resistive structure in the sameway as the composition of two Dirac structures. Then

(D ‖ R)⊥ = D ‖ (−R) (117)

where −R denotes the pseudo-resistive structure given by

RffR − ReeR = 0 (118)

( −R is called a pseudo-resistive structure since it corresponds to a negative instead of apositive resistance.)

Proof. We follow the same steps as in the above proof that the composition of two Diracstructures is again a Dirac structure (where we take F1 = FS , F2 = FR, and F3 void).Because of the sign difference in the definition of a resistive structure as compared with thedefinition of a Dirac structure we immediately obtain the stated proposition.

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