Stabilization of a class of underactuated mechanical ...chemori/Temp/Islem/ortega2002.pdf · 1218...

1218 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 47, NO. 8, AUGUST 2002

Stabilization of a Class of Underactuated MechanicalSystems Via Interconnection and Damping

AssignmentRomeo Ortega, Fellow, IEEE, Mark W. Spong, Fellow, IEEE, Fabio Gómez-Estern, and Guido Blankenstein

Abstract—In this paper, we consider the application of a newformulation of passivity-based control (PBC), known as intercon-nection and damping (IDA) assignment to the problem of stabi-lization of underactuated mechanical systems, which requires themodification of both the potential and the kinetic energies. Ourmain contribution is the characterization of a class of systems forwhich IDA-PBC yields a smooth asymptotically stabilizing con-troller with a guaranteed domain of attraction. The class is givenin terms of solvability of certain partial differential equations. Oneimportant feature of IDA-PBC, stemming from its Hamiltonian (asopposed to the more classical Lagrangian) formulation, is that itprovides new degrees of freedom for the solution of these equations.Using this additional freedom, we are able to show that the methodof “controlled Lagrangians”—in its original formulation—may beviewed as a special case of our approach. As illustrations we de-sign asymptotically stabilizing IDA-PBCs for the classical ball andbeam system and a novel inertia wheel pendulum. For the former,we prove that for all initial conditions (except a set of zero mea-sure) we drive the beam to the right orientation. Also, we define adomain of attraction for the zero equilibrium that ensures that theball remains within the bar. For the inertia wheel, we prove that it ispossible to swing up and balance the pendulum without switchingbetween separately derived swing up and balance controllers andwithout measurement of velocities.

Index Terms—Energy shaping, Hamiltonian systems, nonlinearcontrol, passivity, underactuated mechanical systems.

I. INTRODUCTION

PASSIVITY-BASED control (PBC) is a design method-ology for control of nonlinear systems which is

well-known in mechanical applications. In some regula-tion problems it provides a natural procedure to shape thepotentialenergy preserving in closed-loop the Euler–Lagrange

Manuscript received November 28, 2000; revised May 24, 2001 and Jan-uary 9, 2002. Recommended by Associate Editor M. Reyhanoglu. This workwas supported in part by the National Science Foundation under Grants ECS-0122412 and INT-0128656. The work of F. Gómez-Estern and G. Blankensteinwas supported in part by the Research Network NACO-2 of the European Union.

R. Ortega is with Laboratoire des Signaux et Systèmes, CNRS-SUPELEC,Gif-sur-Yvette 91192, France (e-mail: [email protected]).

M. W. Spong is with the Coordinated Science Laboratory, Univer-sity of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail:[email protected]).

F. Gómez-Estern is with the Departamento de Ingenieria de Sistemas y Auto-matica, Escuela Superior de Ingenieros, 41092 Seville, Spain (e-mail: [email protected]).

G. Blankenstein is with EPFL/DMA MA, Ecublens CH-1015, Lausanne,Switzerland (e-mail: [email protected]).

Publisher Item Identifier 10.1109/TAC.2002.800770.

(EL) structure of the system. As thoroughly discussed in[16], PBC is also applicable to a broad class of systemsdescribed by the EL equations of motion, including electricaland electromechanical systems. In [17] we proposed theutilization of dynamic EL controllers, coupled with the plantvia power-preserving interconnections (hence preserving theEL structure), to shape the potential energy of a class ofunderactuated EL systems via partial state measurements. Itis well known, however, that to stabilize some underactuatedmechanical devices, as well as most electrical and electro-mechanical systems, it is necessary to modify thetotal energyfunction. Unfortunately, total energy shaping with the classicalprocedure of PBC, where we firstselectthe storage functionto be assigned and then design the controller that enforces thedissipation inequality, destroys the EL structure. That is, inthese cases, the closed-loop—although still defining a passiveoperator—is no longer an EL system, and the storage functionof the passive map (which is typically quadratic in the errors)is not an energy function in any meaningful physical sense. Asexplained in [16, Sec. 10.3.1], this situation stems from the factthat these designs carry out an inversion of the system along thereference trajectories, inheriting the poor robustness propertiesof feedback linearizing designs and imposing an unnaturalstable invertibility requirement to the system.

To overcome this problem we have developed in [18] and[30] (see also [19] and [28]) a new PBC design methodologycalled interconnection and damping assignment (IDA). Themain distinguishing features of IDA-PBC are that: 1) it isformulated for systems described by so-called port-controlledHamiltonian models, which is a class thatstrictly containsELmodels and 2) the closed-loop energy function is obtained—viathe solution of a partial differential equation (PDE)—as aresultof our choice of desired subsystems interconnections anddamping. It is well known that solving PDEs is, in general, noteasy. However, the particular PDE that appears in IDA-PBCis parameterized in terms of the desired interconnection anddamping matrices, which can be judiciously chosen invokingphysical considerations to solve it. This point has been illus-trated in several practical applications including mass-balancesystems, electrical motors, magnetic levitation systems, powersystems, power converters, satellite control and underwatervehicles. See [18] and [30] for a list of references.

0018-9286/02$17.00 © 2002 IEEE

ORTEGAet al.: STABILIZATION OF CLASS OF UNDERACTUATED MECHANICAL SYSTEMS 1219

The present paper, which is an expanded version of [21]and [11], is concerned with the application of IDA-PBC to theproblem of stabilization ofunderactuated mechanicalsystems.1

Our main contribution is the characterization of a class ofsystems for which IDA-PBC yields a smooth stabilizing con-troller. The class is given in terms of solvability of two PDEsthat correspond to the potential and kinetic energy shapingstages of the design. As illustrations we design asymptoticallystabilizing IDA-PBCs for the well-known ball and beamsystem and for a novel inverted pendulum. For the ball andbeam we prove that for all initial conditions (except a set ofzero measure) we drive the beam to the right orientation anddefine a domain of attraction that ensures that the ball remainswithin the bar. For the inertia wheel we present a dynamicnonlinear output feedback which, again for “almost” all initialconditions, stabilizes its upward position. That is, we show thatit is possible to swing up and balance this pendulum withouteither switching or measurement of velocities.

Another concern of our paper is to place the new Hamil-tonian approach in perspective with the method of “controlledLagrangians” for stabilization of simple mechanical systemsdeveloped in a series of papers by Blochet al. (e.g., [5] and[6]), and followed up by [1] and [3].2 IDA-PBC and the con-trolled Lagrangians method are procedures to generate state-feedbacks that transform a given Hamiltonian (respectively, EL)system into another Hamiltonian (respectively, EL) system. Acentral difference between the methods is that while the targetEL dynamics in the controlled Lagrangian method is obtainedmodifying only the generalized inertia matrix and the potentialenergy function, in IDA-PBC we havealso the possibility ofchanging the interconnection matrix, i.e., the Poisson structureof the system. In this respect, the following questions naturallyarise.

1) What is the specific choice of this free parameter thatyields the same controllers for both methods? Althoughthe answer to this question has already been reported inour previous papers, e.g., [19], we use here the more re-cent results of [4] to sharpen the statement.

2) Can we use this additional degree of freedom to sim-plify the task of solving the aforementioned PDEs? Toanswer this question we write the PDEs in a form wherethe free parameters appear explicitly as designer-chosen“control” inputs, and work out two classical examples toillustrate how to select these “inputs.”

3) Is the set ofEL closed-loop models achievable viaIDA-PBC “larger” than the one achievable with La-grangian methods? And if so, can we characterize thegap? We provide here an answer to both questions,showing that with IDA-PBC we can,still preserving the

1We should point out that, as presented here, IDA-PBC applies onlyto smoothly stabilizable systems, ruling out the large class considered in[23]. However, the version of IDA-PBC of [9], applies as well to problemsinvolving time-varying Hamiltonians, allowing to consider stabilization withtime-varying controllers.

2See [12] for an elegant extension to general Lagrangian systems.

EL structure, add gyroscopic terms to the closed-loopsystem, a feature that is not included in the reportedliterature with the Lagrangian approach.3

To further clarify the connections between IDA-PBC andcontrolled Lagrangians we, again invoking [4], prove that thematching conditions of [5] characterize a class of mechanicalsystems such that its kinetic energy functions can be shaped (toa take a particular form) without solving the aforementionedPDEs. Obviating the need of solving the PDEs, which is themain stumbling block in both approaches, clearly enlarges theapplicability of these design methods.

The remainder of this paper is organized as follows. In Sec-tion II, we present the application of IDA-PBC to underactu-ated mechanical systems. Section III is devoted to the com-parison between IDA-PBC and the method of controlled La-grangians. Section IV summarizes some results available onsolvability of the PDEs. Sections V and VI contain the deriva-tions of IDA-PBCs for the inertia wheel pendulum and the balland beam system, respectively. We wrap up this paper with someconcluding remarks in Section VII.

II. STABILIZATION OF UNDERACTUATED MECHANICAL

SYSTEMS

In this section, we apply the IDA-PBC approach to regulatethe position of underactuated mechanical systems with total en-ergy

(2.1)

where , are the generalized position and mo-menta, respectively, is the inertia matrix,and is the potential energy. If we assume that the systemhas no natural damping, then the equations of motion can bewritten as4

(2.2)

The matrix is determined by the manner in whichthe control enters into the system and is invertible incase the system is fully actuated, i.e., . We consider herethe more difficult case where the system is underactuated, andassume .

In the IDA-PBC method we follow the two basic steps ofPBC [20]: 1)energy shaping, where we modify the total energyfunction of the system to assign the desired equilibrium ;and 2)damping injectionto achieve asymptotic stability. As ex-plained later, to preserve the energy interpretation of the stabi-lization mechanism we also require the closed-loop system tobe in port-controlled Hamiltonian form [28].

3Recently, Blochet al.[7] have extended the controlled Lagrangian approachto obtain a method equivalent to the IDA-PBC. See Section III for more infor-mation.

4Throughout the paper, we view all vectors, including the gradient, as columnvectors; except in the Hessian matrix where they appear as rows.


A. Target Dynamics

Motivated by (2.1) we propose the following form for thedesired (closed-loop) energy function:

(2.3)

where and represent the (to be defined)closed-loop inertia matrix and potential energy function, respec-tively. We will require that have an isolated minimum at ,that is

(2.4)

In PBC, the control input is naturally decomposed into twoterms

(2.5)

where the first term is designed to achieve the energy shapingand the second one injects the damping. The desired port-con-trolled Hamiltonian dynamics are taken of the form5

(2.6)

where the terms

represent the desired interconnection and damping structures.The following observations are in order.

— From (2.1) and (2.2), we have that . Sincethis is a nonactuated coordinate, this relationshipshould hold also in closed loop. Fixing (2.3) and (2.6)determines the -block of .

— The matrix is included to add damping into thesystem. As is well-known, this is achieved via negativefeedback of the (new) passive output (also calledcontrol), which in this case is . That is, wewill select the second term of (2.5) as

(2.7)

where we take . This explains the-block of .

— We will show below that the skew-symmetric matrix(and some of the elements of ) can be used

as free parametersin order to achieve the kinetic en-ergy shaping. Providing these degrees of freedom isthe essence of IDA-PBC.6

5See [18], [30], and [28] for the physical and analytical justification of thischoice.

6In Section III, we will show that, for aparticular choiceof skew-symmetricJ , IDA-PBC reduces to the controlled Lagrangian schemes of [1], [12], and[6]. It is reasonable to expect that the possibility of choosing among the wholeclass of skew-symmetric matrices enlarges the class of stabilizable systems.

B. Stability

For the desired closed-loop dynamics, we have the followingproposition, which reveals the stabilization properties of our ap-proach.

Proposition 1: The system (2.6), with (2.3) and (2.4),has a stable equilibrium point at . This equilibrium isasymptotically stable if it is locally detectable from the output7

. An estimate of the domain of attraction is

given by where and

is bounded (2.8)

Proof: From (2.3) and (2.4), we have that is a posi-tive definite function in a neighborhood of . A straight-forward calculation shows that, along trajectories of (2.6),satisfies

since is skew-symmetric and is positive–definite. Hence,is a stable equilibrium. Furthermore, since (by defini-

tion) is proper on its sublevel set , all trajectories startingin are bounded. Asymptotic stability, under the detectabilityassumption, is established invoking Barbashin–Krasovskii’stheorem and the arguments used in the proof of [8, Th. 3.2].Finally, the estimate of the domain of attraction follows fromthe fact that is the largest bounded sublevel set of.

C. Energy Shaping

To obtain the energy shaping term, , of the controller wereplace (2.5) and (2.7) in (2.2) and equate it with (2.6)8

While the first row of the aforementioned equations is clearlysatisfied, the second set of equations can be expressed as

Now, it is clear that if is invertible, i.e., if the system is fullyactuated, then we may uniquely solve for the control inputgiven any and . In the underactuated case,is not invert-ible but only full column rank, and can only influence theterms in the range space of. This leads to the following set ofconstraint equations, which must be satisfied for any choice of

:

(2.9)

7That is, for any solution(q(t); p(t)) of the closed-loop system which be-longs to some open neighborhood of the equilibrium for allt � 0, the fol-lowing implication is true:G (q(t))r H (q(t); p(t)) � 0, 8 t � 0 )lim (q(t); p(t)) = (q ; 0).

8Notice that the damping matrix cancels withu .


where is a full rank left annihilator of , i.e., .Equation (2.9), with given by (2.3), is a set of nonlinearPDEs with unknowns and , with a free parameter,and an independent coordinate. If a solution for this PDE isobtained, the resulting control law is given as

(2.10)The PDEs (2.9) can be naturally separated into the terms that

depend on and terms which are independent of, i.e., thosecorresponding to the kinetic and the potential energies, respec-tively. Thus, (2.9) can be equivalently written as

(2.11)

(2.12)

The first equation is a nonlinear PDE that has to be solved for theunknown elements of the closed-loop inertia matrix. Given

, (2.12) is a simple linear PDE, hence the main difficulty isin the solution of (2.11). Equation (2.11) can be expressed in amore explicit form with the following derivations. First, we usethe fact that , for alland all symmetric , to write (2.11) as

Then, we apply the identity

which holds for all and all , wheredenotes the th column of the matrix , reparametrize interms of the matrices as

(2.13)

and equate terms in to obtain PDEs, expressed only in termsof the independent variable, of the form

(2.14)

where aredesigner chosenmatrices.The key idea is then to chose the free parametersin such a

way that (2.14) admits a solution with symmetric and pos-itive definite. Then, we replace it into (2.12), which is a linearPDE, and look for a solution which satisfies (2.4). The ad-ditional degree of freedom provided by is the main featurethat distinguishes IDA-PBC from the Lagrangian methods thatwe review in the next section.

The following remarks are in order.

— The derivations above characterize a class of under-actuated mechanical systems for which the newlydeveloped IDA-PBC design methodology yieldssmooth stabilization. The class is given in terms ofsolvability of the nonlinear PDE (2.11), [or the moreexplicit (2.14)], and the linear PDE (2.12). Althoughit is well known that solving PDEs is generallyhard, it is our contention that the added degree offreedom—the closed-loop interconnection (equiv-alently, )—simplifies this task. We will elaboratefurther on this point in Section III and in the followingexamples.

— There are two “extreme” particular cases of our pro-cedure. First, if we do not modify the interconnectionmatrix then we recover the well-known potential en-ergy shaping procedure of PBC. Indeed, ifand , then the controller equation (2.10) reducesto

which is the familiar potential energy shaping control.On the other extreme, if we do not change the potentialenergy, but only modify the kinetic energy then, as wewill show in the next section, for a particular choiceof , i.e., (3.6), we recover the controlled-Lagrangianmethod of [5]. Now, if we shape both kinetic and po-tential energies, but fix to (3.6), then IDA-PBC co-incides with the method proposed in [1] and [13].

III. COMPARISONWITH THE LAGRANGIAN APPROACH

The IDA-PBC method may be interpreted as a procedureto generate state-feedback controllers that transform a givenport-controlled Hamiltonian system into another port-controlledHamiltonian system with some desired stability properties, e.g.,in the case of mechanical systems to transform (2.1), (2.2) into(2.3), (2.6) for some positive definite inertia matrix , a po-tential function satisfying (2.4), and an arbitrary skew-sym-metric matrix . Viewed from this perspective, the PDEs (2.11)and (2.12) constitute some matching conditions [that ensure thatthe solutions of both systems are the same]. A sim-ilar approach can be taken proceeding from a Lagrangian per-spective. That is, starting from an EL system

where is the Lagrangian, defined as thedifferencebetween thekinetic and the potential energies, we want to find a static statefeedback such that the behavior of the closed-loop is describedby the controlled Lagrangian system

(3.1)

where is the desired Lagrangian. This problem has beenstudied in great generality in [12]; see also [4]. For the case ofinterest here, that is, restricted to Lagrangians of the form

(3.2)


it has been shown in [1] and [13] that the set of achievable La-grangians is characterized by the solvability of the PDEs9

(3.3)

(3.4)

which, similarly to (2.11), (2.12) of IDA-PBC, match the kineticand the potential energy terms.10 Recalling that anddefining a matrix

(3.5)

which is clearly symmetric and positive definite, we see that(3.4) exactly coincides with (2.12)—setting . Further-more, (2.11)reducesto (3.3) if and only if

(3.6)

Although the previous expression can be verified via direct sub-stitution, a more elegant proof is given in [4] checking thatthe (energy conserving part of the) port-controlled Hamiltoniansystem (2.3), (2.6) is equivalent to the EL (mechanical) system

where and .An alternative form for can be obtained as

with the standard Lie bracket and denotes theterm of a matrix. (This expression was first reported in [19],although with swapped subindexes due to an unfortunate typo).

It has also been shown in [4] that we can add to (3.6) a(skew-symmetric) matrix of the form

, with an arbitrary function of , stillpreserving the closed loop EL structure (3.1). This correspondsto the closed-loop Lagrangian

that includes the gyroscopic terms , which are proven to beintrinsic—roughly speaking, this means that they cannot be re-moved with a change of coordinates. In other words, IDA-PBCnaturally generates a “richer” class of matchable EL systems ofthe form (3.1), which have not been considered in the literatureof the Lagrangian approach.

Recently, Blochet al. [7] have shown that a small adjustmentin the controlled Lagrangian approach yields a method which isfully equivalent with IDA-PBC as described in this paper. Es-sentially, instead of restricting to systems of the form (3.1), theyalso allow to include some external forces into the closed-loop

9In the cited references, the PDE (3.3) is expressed using the Christoffel sym-bols of the second kind, which leads to a more elegant and compact notation.To avoid introducing additional notation, we prefer to use the form given here

10In [17], [18], and [30], we derive matching conditions—expressed interms of algebraic constraints—forpotentialenergy shaping of EL systems inclosed-loop withdynamicEL controllers.

EL system (i.e., the right hand side of (3.1) is not necessarilyequal to zero, but can be any external force). In this way, it is pos-sible to write any mechanical Hamiltonian system in EL formatby including the nonintegrable part of the Hamiltonian systemas an external (gyroscopic) force into the EL system. Notice thatthis method only works for the class of simple mechanical sys-tems, as presented in this paper. Considering this larger class ofclosed-loop EL systems [7] establishes that the controlled La-grangian method isequivalentto the IDA-PBC method.

IV. M ETHODS FORSOLVING THE MATCHING PDEs

Our previous derivations showed that in both approaches,Lagrangian or Hamiltonian, it is necessary to solve some PDEs,a task that is, in general, difficult. We recall now some resultsreported in the literature that allows us to simplify this problem.

First, it has been shown in [10] that if (i.e., thesystem is underactuated only by one degree of freedom), and thekinetic energy matrix depends only on the unactuated coor-dinate, then the nonlinear PDE (2.11) can be transformed, witha suitable choice of and , into a set ofordinary differen-tial equations(ODEs), hence, easier to solve. More precisely, ifwe let be the index of the nonactuated coordinate, and assumethat

Then, restricting to be only a function of , it is possibleto show that (2.11) reduces to the ODEs

(4.1)

The class of systems that satisfies these assumptions is quitecommon in the control literature including the two examplesconsidered here, namely, the inertial wheel pendulum and theclassical ball and beam, as well as the cart and pendulum whichis studied in [10]. For the first example, the interia matrixisconstant and we can take also independent of and, obviating the solution of (2.11). On the other hand, we will

show later that for the second example the reduction to ordinarydifferential equations is instrumental to solve the problems.

Second, we have shown in Section III that if we fix as(3.6), which ensures that the closed-loop system admits an ELrepresentation of the form (3.1), (3.2), then the nonlinear PDE(2.11) reduces to (3.3). An important contribution of [1], see also[3] and [2], is the proof that all of the solutions of these PDEsmay be obtained by sequentially solving a set of three first orderlinear PDEs. It has been shown in [4] that the techniques usedin [1] can be extended to the study of the PDE (2.11), whichincorporate the free parameter. In fact, this leads to a set ofone quadratic and two linear first order PDEs. The first PDEis quadratic in the sense that it contains terms quadratic in theto-be-solved-variables, the derivatives, however, still appear lin-early in the equation.


Third, in a series of interesting papers, e.g., [5], [6], Blochet al. have proven that, in some cases which includes somewell-known examples, the solution of these PDEs can actuallybeobviated. In [4], we have interpreted these conditions usingthe notation employed in this paper. We now briefly summa-rize these results. Toward this end, we find it convenient topartition the generalized coordinates as , with

. This induces a natural partition of theopen-loop inertia matrix as

As in [5], we now introduce the following assumptions.

i) The Lagrangian is independent of thecoordinates, i.e.,they arecyclic variables. In this case, the Lagrangiantakes the form .

ii) The -coordinates are fully actuated, that is .iii) The matrix has the block constant, and satisfies11

Now, let us take the controlled Lagrangian as. (Notice that, as in [5], we only aim

at kinetic energy shaping.) A first immediate observation isthat in this case, under Assumptions i) and ii), the matchingcondition (3.4) reduces to analgebraicequation

(4.2)

A central contribution of [5] is the proof that, for the followingparticular class of , satisfying the so-called, simplifiedmatching assumptions

with , the PDEs (3.4), (3.3) are automatically satisfied.In [4] we have shown that, for the class of considered

in [5], their first matching condition M-1 is equivalent to thealgebraic condition

which in the light of (4.2), clearly obviates the potential energyPDE (3.4). Furthermore, it is also established that the matchingconditions M-2 and M-3 of [5] exactly coincide with the PDE(3.3).

We should recall that solving the PDEs (3.3), (3.4) is just thefirst step in the design procedure. Indeed, to establish stabilityusing Proposition 1 the matrix should be positive definite(at least in a neighborhood of the equilibrium), further,should have an isolated local minimum in. In [5, Th. 3.4]

11These are the simplified matching [5, ass. 2 and 4], respectively.

Fig. 1. Inertia wheel pendulum.

it has been shown that relative equilibria, i.e., equilibria of theform , of conservativesystems arestable if the Hessian of the total energy isdefinite(either posi-tive or negative). Although hard to justify from a physical view-point, we can in this way use these methods to locally stabilize(relative equilibria of) conservative mechanical systems with anegative–definiteclosed-loop inertia matrix. This feature is es-sential in [5] where the potential energy is not modified by thecontrol, and will typically have a maximum at the desired equi-librium, hence the kinetic energy should also have a maximumat this point. The qualifier “conservative” is also important be-cause it is not clear to these authors how to handle the presenceof physical damping in this framework.

V. INERTIA WHEEL PENDULUM

In this section, we apply the preceding design methodologyto the problem of stabilizing the inverted position of the inertiawheel pendulum shown in Fig. 1, which consists of a physicalpendulum with a balanced rotor at the end. The motor torqueproduces an angular acceleration of the end mass which gener-ates a coupling torque at the pendulum axis. We show that theIDA-PBC provides an affirmative answer to the question of ex-istence of acontinuouscontrol law that, for all initial conditionsexcept a set of zero measure, swings up and balances in the up-ward position the pendulum.12 We also show that, as usual inPBC [18], [30], the passivity property allows us to replace thestate-feedback control by a dynamicoutputfeedback that doesnot require the measurements of velocities.

A. Model

The dynamic equations of the device can be written in stan-dard form using the EL formulation [27] as

where , and , are the respective angles and momentsof inertia of the pendulum and disk, is the pendulum mass,

its length, the gravity constant, and is the control input

12This means that the basin of attraction of our controller is an open denseset in the state space, which is the best one can hope for using a continuousfeedback [26].


torque acting between the disk and pendulum. The change ofcoordinates

leads to the simplified description

(5.1)

where . The system can be written in Hamiltonianform (2.2) with , the total energy function(2.1) and

The equilibrium to be stabilized is the upward position with theinertia disk aligned, which corresponds to .13

Caveat: The model of the inertia wheel pendulum can beseen either as a system defined overor, taking modulo ,over , with the unit circle. To allow for feedback lawsthat are not necessarily -periodic on we adopt the formerone for the controller design. However, to determine the domainof attraction we look at the system in the cylinder.

B. Controller Design

We will design our IDA-PBC in three steps. First, we will ob-tain a state-feedback that shapes the energy to globally stabilizethe upward position, then we add the damping for asymptoticstability by feedback of the passive output. Finally, we showthat it is possible to replace the velocity feedback by its dirtyderivative preserving asymptotic stability.

1) Energy Shaping:First, notice that the inertia matrixis independent of, hence, we can take and to be aconstantmatrix too, which we denote by

(5.2)

The inequalities for the coefficients are imposed to ensureispositive definite. The only PDE to be solved is then the potentialenergy PDE given by (2.12), that is

This is a trivial linear PDE whose general solution is of the form

(5.3)

(5.4)

where is an arbitrary differentiable function that we mustchoose to satisfy the condition (2.4) for . We have also

13Since the mass distribution of the inertia disk is symmetric, it may be arguedthat there is no particular reason for aligning it. In spite of that, we impose thisobjective to illustrate the generality of the approach.

defined . Some simple cal-culations show that the necessary condition issatisfied if and only if , while the sufficient con-dition will hold if the Hessian of at the originis positive, and

(5.5)

These conditions on are satisfied with the choice14

, where represents an adjustable gain.The energy shaping term of the control input (2.10) is given

as

(5.6)

where we have defined

We recall that should satisfy the inequalities (5.2)and (5.5), some simple calculations show that these conditionstranslate into

(5.7)

which, together with , define theadmissible regionforthe tuning gains.

C. Damping Injection and Stability Analysis

The control law results in the system

for which we have . Clearly, withoutdamping injection the origin is a stable equilibrium. To makethis equilibrium asymptotically stable, we propose to adddamping feeding back the new passive output , whichcan be computed as

(5.8)

where we defined

with positivity following from (5.2) and (5.5).We are in position to present our first stabilization result,

which establishes that for all initial conditions (except a set ofzero measure) the IDA-PBC drives the pendulum to its upwardposition with the disk aligned.

14For simplicity we have taken here a quadratic function�. Clearly otheroptions, that may be selected to improve the transient performance, are possible,e.g., a saturated function.


(a) (b)

Fig. 2. Evolution of (a)q (t) and (b)q (t) for different values ofk , lettingk = 10.

(a) (b)

Fig. 3. Evolution of (a)q (t) and (b)q (t) for different values ofk , andk = 0:1.

Proposition 2: The inertia wheel pendulum (5.1) in closedloop with the static state-feedback IDA-PBC

(5.9)

where satisfy (5.7), is a proportional gain, andis a damping injection gain, has an asymptotically stable

equilibrium at zero, with domain of attraction the whole statespace minus a set of Lebesgue measure zero.

Proof: We first observe that the closed-loop system hasequilibria , where , are the solu-tions of . We recall from Fig. 1 that the pointswith even correspond to the desired upward position, whilethe ones with odd are with the pendulum hanging.

From Proposition 1 and the aforementioned derivations,we know that the equilibria corresponding to the upwardposition of the pendulum are stable. Further, with some basicsignal chasing, it is possible to show that (in the neighborhoodof zero)the trajectories of the closed-loop system satisfy the(stronger) observability condition: .Therefore, Proposition 1 insures that the desired equilibriumis asymptotically stable. Although Proposition 1 provides alsoan estimate of its domain of attraction, we will show belowthat asymptotic stability is “almost” global, in the sense ofProposition 2.

Toward this end, we make the important observation that theintroduction in the control (5.9) of a nonperiodic function offorces us to consider the system in instead of , andthen is not a proper function of and we cannot insureboundedness of trajectories (starting outside.) Nevertheless,in the coordinates , with , the

closed-loop system is indeed defined over , and the en-ergy function

is positive definite and proper throughout . Then, since, we have thatall solutionsare bounded

in . From the previous analysis, we know that the zeroequilibrium is asymptotically stable. We will now show thatthe other equilibria are unstable. Indeed, the linearization ofthe closed-loop system at these equilibria has eigenvalues withstrictly positive real part and at least one eigenvalue with strictlynegative real part. Associated to the latter there is a stable man-ifold, and trajectories starting in this manifold will converge tothe downward position. However, it is well-known that an-di-mensional invariant manifold of an-dimensional system hasLebesgue measure zero if ; see, e.g., [29]. Consequently,the set of initial conditions that converges to the “bad” equilib-rium has zero measure.

To complete the proof we establish now that trajectories arealso bounded in .15For, we see that our derivations above haveproven that, in , all trajectories are bounded and tendto one of the equilibria or . The firstequilibrium is asymptotically stable and the second one is hy-perbolic. We have furthermore shown that almost any solutionconverges to the stable equilibrium, being trapped in finite timein a sufficiently small neighborhood of it, say . Let us nowimmerse the cylinder in the Euclidean space, and consider

15We thank anonymous reviewer 5 for this proof of global boundedness.


Fig. 4. Control signal fork = 10, andk = 0:1.

the sets of which correspond to on the cylinder. Ifis chosen sufficiently small, then these sets indo not inter-sect. Since the solutions on the cylinder do not leavetheimmersion of the trajectories will not leave the correspondingneighborhood in . This completes the proof.

Output Feedback:It is well known that in PBC designs itis sometimes possible to obviate velocity measurement feedingback instead the dirty derivative of positions [15]. This featurestems from the fact that for feedback interconnection of passivemaps we can replace aconstantfeedback by a feedback throughany positive-real transfer function preserving stability. In partic-ular, to implement the damping injection we can use the feed-back

with , and some filter parameters. The im-portant point is that is implementable without velocity feed-back. This consideration leads us to our final result contained inProposition 3. The proof follows along the lines of the previousproposition, and is only outlined for brevity.

Proposition 3: Consider the inertia wheel pendulum (5.1) inclosed-loop with the dynamic output feedback IDA-PBC

where satisfy (5.7), is a proportional gain; isa damping injection term generated from the dirty derivative ofthe positions as

(5.10)

with . Then, for all initial conditions—except a set ofzero measure—the pendulum converges to its upward positionwith all internal signals uniformly bounded.

Proof: First, notice that from (5.8) and (5.10) we get

Consequently, if we define the new energy function, we can write the

closed-loop system in port-controlled Hamiltonian form as

We have . The proof is completed pro-ceeding as in Proposition 2.

D. Simulation Results

We simulated the response of the inertia wheel pendulumusing the system parametersand the full-state feedback controller with tuning gains

. The following plots show the response of thesystem starting at rest with initial configuration .Thus, the pendulum is hanging nearly vertically downward.

In order to illustrate the influence of the choice ofandon the transient behavior, we have put together several plots of

and , first only changing with kept constantat a value of 10 as seen in Fig. 2. A simple observation showsthat larger values of slow down the convergence toward theequilibrium point. Leaving constant and changing from 10to 100, the plots in Fig. 3 are obtained. These plots show that,counter intuitively but not totally unexpected, the oscillationsvanish faster for lower values of ! Finally, Fig. 4 shows theapplied control torque when and .

VI. BALL AND BEAM SYSTEM

In this section, we will design an IDA-PBC for the well-known ball and beam system depicted in Fig. 5. First, we willprove that for all initial conditions (except a set of zero measure)we drive the beam to the right orientation. Then, we define a do-main of attraction for the zero equilibrium that ensures that theball remains within the bar.

A. Model

The dynamic behavior of the ball and beam system, undertime scaling and some assumptions on mass constants for sim-plicity, is described by the EL equations

(6.1)


Fig. 5. Ball and beam system.

where are the ball position and the bar angle, respec-tively,16 and is the length of the bar. Since we are interestedin ensuring that the ball remains in the bar, we have explicitlyincluded in the model. We refer the reader to [14] for furtherdetails on the model. The Hamiltonian model (2.1), (2.2) is ob-tained defining the matrices

and the potential energy function . The con-trol objective is to stabilize the ball and beam in its rest positionwith .

B. Controller Design

For pedagogical reasons, we will split now the design of theIDA-PBC into kinetic energy shaping, potential energy shaping,asymptotic stability and transient performance analysis.

Kinetic Energy Shaping:First, notice that is a function ofonly, hence it is reasonable to propose of the form (5.2),

but with the coefficients functions(to be defined) ofalso. We will denote

with the function also to be defined.Since this is the scenario of [10] discussed in Section IV we

apply directly the formula (4.1), with , to derive theODEs for . This leads to the system of ODEs

that has to be solved for , and we view as a “free”parameter. To obtain two equations with two unknowns, we fix

, hence, we can easily get explicit solutions of theODEs as

(6.2)

with a free integration constant that must be chosen to ensurethat is positive–definite. It is clear that for all ,furthermore the determinant of results

16Thecaveatof Section V-A applies here as well. Indeed, the model can beseen either as a system defined overor over � S � .

which is positive, as desired, for all . For simplicity,we will take in the sequel. The resulting is then

(6.3)

The kinetic energy shaping is completed evaluatingfrom(2.11) and the calculated above

(6.4)

Potential Energy Shaping:Once we have determined the de-sired inertia and interconnection matrices we proceed now to de-fine the closed-loop potential energy from the solution of (2.12)which in this case is expressed as

(6.5)

Substituting the solutions obtained for and in (6.2) yields

We will solve this PDE using the symbolic language Maple.To help Maple find a suitable solution we introduce the changeof coordinates , the PDE can then berewritten as

where we have defined and . The requiredMaple commands are

which produces (after transforming back the coordinates)

with an arbitrary differentiable function of. This functionmust be chosen to ensure the equilibrium assignment, i.e., tosatisfy (2.4) with . Toward this end, let us evaluate thegradient

Clearly, a necessary and sufficient condition to assign the zeroequilibrium is . Remark that, independently ofthe choice of , (or the integration constant), the closed-loophas other equilibrium points. Indeed, since andis an increasing first–third quadrant function, hasa countable number of roots given by ,with . On the other hand, we have that the only equilibriumwith is the zero equilibrium. Of course, this property ispractically meaningful only if we can show that the trajectories

for all , which will be done later.


(a) (b)

Fig. 6. Level curves ofV (q) around the origin for (a)k = 0:05 and (b)k = 0:01.

It is important to recall that, the equilibria with evencorrespond to the bar in its original orientation, while foroddthe bar is rotated 180. We will prove now that, with a suitablechoice of , we can make the former equilibria stable and thelatter ones unstable.

To study the stability of the equilibria we check positivity ofthe Hessian of , evaluated a, which yields

Taking into account that , depending on whetherthe bar is at its original position or rotated 180, the determi-nant of this matrix is , and sta-bility (instability) of the equilibrium is determined by the signof . We then choose , with .In this way, the equilibria corresponding to the bar in its originalorientation will be stable, while the ones with the rotated bar areunstable.

The new potential energy takes the final form

(6.6)

where we have added a constant to shift the minimum to zero.See Fig. 6.

Asymptotic Stability Analysis:To compute the final controllaw we first determine the energy-shaping termfrom (2.10),which, in this case, takes the form

Replacing the functions derived above for and , and aftersome straightforward calculations, we obtain the expression

(6.7)

where

The controller design is completed with the damping injectionterm (2.7), which yields

(6.8)

It is important to underscore that, in spite of its apparent com-plexity, the controller is globally defined and its highest de-gree is quadratic. This is an important property of the control,since saturation should be avoided in all practical applications.Also, the role of the tuning parameters has a clear interpreta-tion, namely, is abona fideproportional gain in position, as itmultiplies terms that grow linearly in, and injects dampingalong a specified direction of velocities. Commissioning of thecontroller is simplified by this feature, as will be illustrated laterin the simulations.

We give now a first result on asymptotic stability, similar tothe one obtained for the inertia wheel pendulum, but with thefundamental difference that convergence to zero of theball isonly local. A more practical result, that takes into account thefinite length of the bar, will be reported in the next section.

Proposition 4: Consider the ball and beam model (6.1) inclosed-loop with the static state feedback IDA-PBC

, with (6.7) and (6.8), and . Then, the origin isan asymptotically stable equilibrium with domain of attraction

where , with of theform (2.3), given by (6.3), and and defined by (6.6),(2.8), respectively. Further, for all initial conditions, except aset of zero measure, the beam will asymptotically converge toits zero orientation position, i.e., for “almost” all trajectories

.Proof: Stability of the zero equilibrium follows verifying

the conditions of Proposition 1, however, to establish asymp-totic stability, instead of checking detectability, we invoke Ma-trosov’s theorem; see, e.g., [25, Th. 5.5], for which we pick anauxiliary function whose derivative along thetrajectories of the closed loop is


Now, if and only if

see (6.8), which [as ranges in ] defines a triangularsector inside the first–third quadrant of the plane– . On theother hand, the sector where lives in the second–fourthquadrant, consequently, we have thatis a nonvanishing def-inite function at the set , and the conditions ofthe theorem are satisfied (in a neighborhood of zero) withthe required Lyapunov function.

Similarly to the proof of Proposition 2, to prove the “almost”convergence property stated above we look at the system in

to establish boundedness of all trajectories. To immersethe system in this cylinder we change the first coordinate to

(6.9)

and define accordingly the energy function

(6.10)

(6.11)

which is proper and positive–definite in , and withnegative semidefinite derivative. This proves boundedness ofalltrajectories in . The remaining of the proof mimicsthe one done for the inertia wheel pendulum, noting that for allstable equilibria we have the desired asymptotic behavior for,while the other equilibria are unstable.

Before closing this section we note that in this example it isnot possible to replace the measurements of velocities by itsdirty derivative approximation as done for the inertia wheel.This stems from the fact that in the ball and beam controller,besides the damping injection term, the energy shaping term de-pends explicitly on .

Transient Performance:It has been stated in Proposition 4that in closed loop the origin is an asymptotically stable equi-librium with Lyapunov function the desired total energy. In thissection, we will refine this analysis, studying the effect of thetuning parameter on the size of the domain of attraction, andexplicitly quantifying a set of initial conditions such that the ballremains all the time in the bar, that is, for all .

First, we note that as decreases and the kinetic energy isnonnegative, we have that , hencethe sublevel sets of are invariant sets for . Further,if we can show that the kinetic energy is bounded, then thebounded sets provide an estimate of the domain of attraction.To study these sets we find convenient to work in the coor-dinates , where was introduced in (6.9).17

In these coordinates the potential energy function becomes(6.11)—which has the same analytical expression as the totalenergy of the simple pendulum—and the associated sublevelsets, i.e., , are of the form shownin Fig. 7. We are interested here in the bounded connectedcomponents that contain the origin, that we will denote.

17Note that the coordinate mapping(q ; q ) 7! (~q ; q ) defines a globaldiffeomorphism, and recall that boundedness of sub-level sets is invariant underthe action of diffeomorphisms.

The following basic lemma will be instrumental in the sequel.Lemma 1: The set is bounded if and only if .

Proof: The fact that all elements in are bounded forall finite is obvious, as . Hence,we can concentrate only on boundedness of.

We have the following implication:

where we have used the positivity of in the first right handside inequality and to obtain the second one. Note thatthe strict inequality excludes an interval around and

from . This proves that ,and consequently, is bounded.

Necessity will be proved by contradiction. For, we sup-pose that . Then, it is clear that ,which is an unbounded set, and consequentlyis unbounded.

We are in position to present the main result of this section.To simplify the notation we will use to denote the value ofthe functions at .

Proposition 5: Consider the ball and beam model (6.1) inclosed-loop with the static state feedback IDA-PBC

, with (6.7) and (6.8), and .

i) We can compute a constant , function of the initialconditions , such that for all , the set

(6.12)

is an estimate of the domain of attraction of the zero equi-librium. In particular, all trajectories starting with zero ve-locity, and will asymptotically converge tothe origin.

ii) Fix and assume . Then

(6.13)

is a domain of attraction of the zero equilibrium, such thatall trajectories starting in this set satisfy forall .

Proof: We have shown above that the sublevel sets ofare invariant sets for . Further, Lemma 1 establishes

that the connected component of the sublevel sets ofcontaining the origin is bounded if and only if . Hence,setting in Lemma 1 it follows that this set is boundedif and only if

It is clear that, if the first two terms are strictly smaller than, wecan always find and upperbound onsuch that the inequalityholds. To complete the proof of point i) of the proposition weremark that, for all trajectories starting in the set (6.12), isbounded. Hence, from (5.2), we conclude thatfrom some constant . This, together with the fact that

, establishes that the corre-sponding is also bounded and the set (6.12) is an estimateof the domain of attraction.


Fig. 7. Level curves of~V (~q ; q ).

The proof of point ii) proceeds as follows. From (6.9), wehave that

(6.14)

where we have used the fact that is odd and mono-tonic. The region defined by (6.14) is depicted in Fig. 8 togetherwith two sets . Our problem is then to compute the largest

such that the set does not intersect the lines. To simplify the expressions we note that

, and check the intersection with the“closer” lines in the band .For, we substitute in the boundary equation of

, and use the bound , which is valid inthe aforementioned band, to get the first inequality as follows:

The second inequality holds for alland all in the band. This proves that the boundary ofdoesnot intersect the limit lines within the band. They cannot inter-sect outside the interval either becauseimplies that, in , , and this bound on is less strictthan . This completes the proof.

C. Simulations

A set of simulations of the ball and beam system with, has been made. The results are shown in Fig. 9.

The graphs on the upper row depict the ball positionand thebeam angle for zero initial velocity, and varying initial po-sitions and parameters. Under each of these, the correspondinggraphs with the desired Hamiltonian and potential energyare shown. Each column in the graph array belongs to a singlesimulation. From the first two simulations, we see the effect ofincreasing the damping constantstarting with the bar in ver-tical position. Note that the convergence is not always accel-erated with higher values of , as new oscillations come intoplay. The third simulation starts at rest with the bar in horizontalposition and the ball on the edge of the bar. To ensure that theinitial condition is within the domain of attraction, has beenchosen smaller than according to Proposition 5. Fig. 9 also

Fig. 8. Graphical interpretation ofjq j < L, withm= (1=p2)arcsinh(1).

illustrates the monotonic nature of together with the factthat for all .

The effect of the limited bar length and the use of the lastpart of Proposition 5 is illustrated by simulation in Fig. 10. Theparameters are , , , and . Hence,the condition for keeping the ball within the limits of the baris . The first simulation starts at

with . Due to the initial velocitythe controller is unable to slow down the ball before it exceedsthe limit of the bar ( ). In the second simulation, we have

and , thus the boundis guaranteed and the ball remains within the limits of

the bar.

VII. CONCLUSION

We have characterized in this paper a class of underactuatedmechanical systems for which the newly developed IDA-PBCdesign methodology yields smooth stabilization. The class isgiven in terms of solvability of two PDEs (2.11), (2.12). Al-though it is well known that solving PDEs is in general hard,we have added some degrees of freedom, the closed-loop inter-connection , to simplify this task.

As an illustration we have presented a dynamic nonlinearoutput feedback IDA-PBC which stabilizes for all initial con-ditions (except a set of zero measure) the upward position of anovel inverted pendulum. We believe this is the first result ofswinging up and balancing an underactuated pendulum withoutswitching nor measurement of velocities. See also [22] for analternative, full-state feedback, solution using a variation of for-warding. We have also derived a static state-feedback IDA-PBCfor the well-known ball-and-beam system, which asymptoti-cally stabilizes the zero equilibrium for a well-defined set of ini-tial conditions, maintaining the ball inside the length of the bar.Again, to the best of the authors’ knowledge, no similar tran-sient performance result is available in the literature. Currentresearch is under way to test these controllers experimentally.

There are many possible extensions of the work reported here.First, to by-pass the need to solve the (infamous) PDEs we candualize the problem, fixing the energy function to some de-sired form, and trying to solve the resulting algebraic equationfor . This is, in essence, the approach adopted in the recent


Fig. 9. Simulations of the ball and beam starting from rest.

Fig. 10. Ball and beam starting with nonzero initial velocity. Effect of the finite bar length.

paper [9], see also [24]. Second, we discussed in Section III thatIDA-PBC allows the consideration of gyroscopic forces in theenergy function. Although it is not clear at this stage how canthis be profitably used in mechanical applications, they play afundamental role in electromechanical applications, where thedesired equilibrium does not occur at zero kinetic energy. Somepreliminary results along these lines for magnetic levitated sys-tems and electric motors are very encouraging [24]. Third, itwould be, of course, desirable to have a better understanding ofthe PDEs appearing in IDA-PBC, on one hand, to obtain a moresystematic design procedure, and on the other hand, to come toterms with the intrinsic limitations of the methodology [e.g., de-riving conditions for (non)solvability of the PDEs]. Some avail-able results, and open research lines, in this direction are dis-cussed in Section IV. A particularly important aspect is that, forstability purposes, it is not enough to find any solution and

of the PDEs, but we need one that satisfies some kind of“boundary conditions.” The study of this additional constraintis essentially open.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers formany helpful remarks. They are also grateful to the authors of[2] and [7] for providing them an advanced copy of their papers.Part of this work was carried out while R. Ortega was visitingthe University of Illinois at Urbana-Champaign. The hospitalityof this institution is gratefully acknowledged. The first authorwould also like to thank L. Praly of Ecoles des Mines, Paris,France, and D. Alonso and E. Paolini of the Universidad delSur, Bahia Blanca, Argentina, for many helpful discussions.

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[30] R. Ortega, A. van der Schaft, B. Maschke, and G. Escobar,Stabilityand Stabilization of Nonlinear Systems, D. Aeyels, F. Lamnabhi-La-garrigue, and A. van der Schaft, Eds. London, U.K.: Springer-Verlag,1999, LNCIS vol. 246.

Romeo Ortega (S’76–M’76–SM’98–F’99) wasborn in Mexico. He received the B.Sc. degreein electrical and mechanical engineering fromthe National University of Mexico, Mexico City,Mexico, the Master of Engineering degree fromPolytechnical Institute of Leningrad, Russia, andthe Docteur D‘Etat degree from the PolitechnicalInstitute of Grenoble, Grenoble, France, in 1974,1978, and 1984, respectively.

Until 1989, he was with the National University ofMexico. He was a Visiting Professor at the Univer-

sity of Illinois, Urbana-Champaign, in 1987–1988 and at McGill University,Montreal, QC, Canada, in 1991–1992. He is currently with the Laboratoire deSignaux et Systemes (SUPELEC), Paris, France. His research interests are inthe fields of nonlinear and adaptive control, with special emphasis on applica-tions.

Dr. Ortega has been a member of the French National Researcher Council(CNRS) since June 1992, and was a Fellow of the Japan Society for Promo-tion of Science from 1990 to 1991. He was the Chairman of the IEEE WorkingGroup on Adaptive Control and Systems Identification, of the IFAC TechnicalCommittee on Adaptive Control and Tuning, and of the Automatica Paper PrizeAward Committee. He is currently a member of the IFAC Technical Board andchairman of the IFAC Coordinating Committee on Systems and Signals. He isan Associate Editor ofSystems and Control LettersandInternational Journalof Adaptive Control and Signal Processing.

Mark W. Spong (S’81–M’81–SM’89–F’96)received the D.Sc. degree in systems science andmathematics in from Washington University, St.Louis, MO, in 1981.

Since 1984, he has been at the University ofIllinois at Urbana-Champaign, where he is currentlyProfessor of General Engineering, and ResearchProfessor in the Coordinated Science Laboratory.He is the Director of the College of EngineeringRobotics and Automation Laboratory, which hefounded in 1987, and the Director of the John Deere

Mechatronics Laboratory, which he founded in 1995. He has held visitingpositions at the University of Waterloo, Canada, the CINVESTAV del IPN,Mexico City, Mexico, The Lund Institute of Technology, Sweden, The Labo-ratoire d’Automatique de Grenoble, France, The Université de Tecnologie deCompiegne, France, the Katholiek Universitet, Leuven, Belgium, the NationalUniversity of Singapore, and the Technical University of Munich, Germany.He has also served as a consultant to industry and government. His mainresearch interests are in robotics, mechatronics, and nonlinear control theory.He is President of Mechatronic Systems, Inc., Champaign, IL, a companywhich he founded in 1996. He has published over 150 technical articles incontrol and robotics and coauthoredRobot Dynamics and Control(New York:Wiley, 1989) andRobot Control: Dynamics, Motion Planning, and Analysis(Piscataway, NJ: IEEE Press, 1992).

Dr. Spong has served as Associate Editor of the IEEE TRANSACTIONS ON

CONTROL SYSTEMS TECHNOLOGY, the IEEE TRANSACTIONS ON ROBOTICS

AND AUTOMATION, the IEEE Control Systems Magazine, and the IEEETRANSACTIONS ONAUTOMATIC CONTROL. He is a past Editor-in-Chief of theIEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY. He served as amember of the Board of Governors of the IEEE Control System Society from1994 to 2002, and as Vice President for Publication activities from 2000 to2002. In addition, he received an Alexander von Humboldt Foundation SeniorScientist Research Award in 1999.


Fabio Gómez-Esternwas born in Seville, Spain,in 1972. He received the Ingeniero de Telecomu-nicación degree from the Escuela de Ingenieros,Seville, Spain, in 1996.

He joined communications and electronics com-panies in Seville, Spain (Abengoa), and Paris, France(France Telecom) as an R&D Engineer. Since 1999,he has held a teaching position at the Department ofSystems Engineering, University of Seville, Seville,Spain. He has been Visitor in the Laboratoire des Sig-naux et Systemes (CNRS, France) repeatedly. His re-

search fields are communication networks traffic control, nonlinear control sys-tems, Hamiltonian and Lagrangian systems, and robot motion control.

Guido Blankensteinreceived the M.S. and Ph.D. de-grees in applied mathematics from the University ofTwente, Twente, The Netherlands, in 1996 and 2000,respectively.

From January 2001 to July 2001, he held aPost-Doctoral position at Supelec, France. He iscurrently a Postdoctoral Fellow in the Department ofMathematics, Swiss Federal Institute of TechnologyLausanne (EPFL), Lausanne, Switzerland. Hisresearch interests are in mathematical systems andcontrol theory, mainly directed toward Hamiltonian

dynamics, mechanical systems, systems with symmetry, interconnectedsystems, and stabilization.

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Stabilization of a class of underactuated mechanical ...chemori/Temp/Islem/ortega2002.pdf · 1218...

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