PAMM · Proc. Appl. Math. Mech. 10, 623 – 624 (2010) / DOI 10.1002/pamm.201010304

On Passivity-Based Control of Non-Classical Electromechanical Systems

Andreas Müller1,∗, Uwe Jungnickel2,∗∗, Gerald Kielau2,∗∗∗, and Peter Maiser2,†

1 Chair of Mechanics and Robotics, University Duisburg-Essen, Lotharstr. 1, 47057 Duisburg2 Institute of Mechatronics, Reichenhainer Str. 88, 09126 Chemnitz

Passivity-based control has exclusively been pursued for dynamical systems possessing energetic state functions that are

quadratic in the generalized velocities. This assumption does not apply to many dynamical systems, for instance in elec-

tromechanics, thus passivity-based control has not been established for these classes of systems. This contribution presents an

augmented PD control scheme for such systems. To this end the system dynamics is represented in the event space considered

as a Finsler space. It is shown that in this setting the skew symmetry property is retained. A passivity-based augmented PD

controller is designed in the event space, and restricted to the configuration space giving rise to a passivity-base control law.

c© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Passivity has evolved as a design concept for model-based control of Euler-Lagrange systems, i.e. systems whose governing

motion equations arise from energetic state functions via a variational principle. The passivity property ensures the stability

of the controller [4], [6]. Vital for the passivity is a certain skew symmetry property of the motion equations. The theory

developed so far has exclusively focused on energetic state functions that are quadratic in the generalized velocities, like

the kinetic energy of multibody systems. For such systems the required skew symmetry condition is fulfilled. On the other

hand electromechanical systems, for instance, may possess energetic state functions that are not quadratic in the generalized

velocities, e.g. systems that are described by the Maxwell equations for moving media (Minkowski equations). Hence, such

systems cannot be treated with the classical passivity-based control techniques.

In the quadratic case, the skew symmetry property results from the Riemannian geometry of the n-dimensional configu-

ration space, which is lost for non-quadratic state functions. In fact, the passivity property depends on the metric property

induced by the energetic state function that actually allows to define geodesics in configuration space. However, by consid-

ering the system dynamics in the n + 1-dimensional event space the energetic state function can be homogenized and does

in fact define a Finsler metric making the event space a Finsler space, in which one can relate the system’s dynamics to the

Finsler geometry. As reported in [2] this allows for the design of a controller in event space, which can be shown to be passive.

Restriction of this control law to the configuration space gives rise to a passivity-based control law.

2 The Control Problem

It is assumed that there exist two state functions: Λ (q, q, t), the Lagrangian, and D (q, q, t), the dissipation function, where

q = (qa) ∈ V n, a = 1, . . . , n are the generalized coordinates and V n is the n-dimensional configuration space. The system

dynamics is governed by the Euler-Lagrange differential equations

d

dt

(

∂aΛ)

− ∂aΛ + ∂aD = Qa, (1)

with generalized forces Qa (t) not represented by Λ or D, like control forces. The Lagrangian of an electromechanical system

can be expressed as Λ = T + U , with kinetic coenergy and potential energy. The position control problem for (1) consists in

finding a control law Ra (q, q, q, t) steering the systems along a nominal trajectory qa0 (t), i.e. the equations

d

dt

(

∂aΛ)

− ∂aΛ + ∂aD = Ra (2)

should imply uniformly asymptotic convergence of the system’s state.

∗ Corresponding author E-mail: andreas-mueller@uni-due.de, Phone: +49 203 379 3514, Fax: +49 203 379 2494∗∗ E-mail: uwe.jungnickel@ifm-chemnitz.de∗∗∗ E-mail: gerald.kielau@ifm-chemnitz.de† E-mail: peter.maisser@ifm-chemnitz.de

c© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

624 Section 19: Dynamics and Control

3 Proposed Control Law

By considering time as an additional coordinate q0 := t one can use (qα) ≡

(

q0, qa)

as generalized coordinates on the

(n + 1)-dimensional event space V n+1. A curve qα (τ ) in V n+1 is then parameterized in terms of a curve parameter τ . It is

assumed that the kinetic coenergy T (q, q, t) is a positive definite function with respect to q. Then a geometry of V n+1 can be

introduced on the basis of the kinetic coenergy. With q′α := dqα

dτthe function

L (q′α, qα) := T(

q′α/q′0, q, t)

q′0 (3)

is positively homogenous of degree one in q′α. This is a necessary and sufficient condition for the integral

I (C) :=

∫C τ2

τ1

L (q′α, qα) dτ (4)

to be parameter invariant. It is further assumed that L (q′α, qα) is positive definite w.r.t. q′ in a certain domain G ⊂ TV n+1,

and that the quadratic form ∂′α∂′

βL (q′α, qα) ξαξβ is positive definite with respect to ξ in G. Then Fn+1 := V n+1 is the Finsler

space associated with L ( [1], [3], [5]). With DT being the dissipation function in V n+1 the motion equations (conditions for

(4) to be stationary) of the controlled system are, with the Finsler metric gαβ (q′α, qα) :=1

2∂′

α∂′βL2,

gαβq′′β + ∆αβq′β + ∂′αDT = Rα, (5)

where, with the corresponding Christoffel symbols Γαβγ (q′α, qα) := 1

2(∂βgαγ + ∂γgαβ − ∂αgβγ),

∆αβ := Γαβγq′γ +1

2∂′

γgαβq′′β (6)

It can be shown that g′αβ − 2∆αβ is skew symmetric. This suggests using the following augmented PD control law in Fn+1

Rα := gαβq′′β0 + ∆αβq

′β0 + ∂′

αDT − Kαβe′β − Cαβeβ (7)

with positive definite gain matrices Kαβ and Cαβ . The configuration space V n is a constraint manifold in Fn+1 defined by

q′0 = 1

T. Restriction of (7) to the configuration space finally yields the control law

Qa :=1

T

(

gaβ

(

qβ0 − qβ

0

T

T

)

+ γaβγ qβ qγ0 −

1

2∂bgace

cqb

)

+ ∂aDT − Kabeb− Cabe

b, (8)

where gαβ := gαβ

(

qγq′0, qγ)

. This control law achieves asymptotically stable trajectory tracking of the Euler-Lagrange

system with non-quadratic state function.

References

[1] S. S. Chern and Z. Shen, Riemann-Finsler Geometry (World Scientific, London, 2004).

[2] U. Jungnickel, G. Kielau, P. Maißer, A. Müller: A passivity-based control of Euler-Lagrange systems with a non-quadratic Lagrangian,

Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), 88(12), 982-992 (2008).

[3] D. Lovelock and H. Rund, Tensors, Differential Forms and Variational Principles (Dover Publications, New York, 1989).

[4] Ortega, R.; Loria, A.; Nicklasson, P.J.; Sira-Ramirez, H.: Passivity-based Control of Euler-Lagrange-Systems, Mechanical, Electrical

and Electromechanical Applications, Springer-Verlag, London, 1998

[5] H. Rund, Die Hamiltonsche Funktion bei allgemeinen dynamischen Systemen, Arch. Math. III, 207-215 (1952).

[6] van der Schaft, A.J.: L2-Gain and Passivity Technique in Nonlinear Control, Springer-Verlag, London, 2000

c© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.gamm-proceedings.com