Barra-Esfera

Automatica 38 (2002) 2147 – 2152

www.elsevier.com/locate/automatica

Brief Paper

Matching, linear systems, and the ball and beam
F. Andreeva ; 1 , D. Aucklyb;1 , S. Gosavic;1 , L. Kapitanskib ; ∗ ;1; 2 , A. Kelkard;1 , W. Whitec;1
a Department

of Mathematics, Western Illinois University, Macomb, IL 61455, USA
of Mathematics, Kansas State University, Manhattan, KS 66506, USA
cDepartment of Mechanical and Nuclear Engineering, Kansas State University, Manhattan, KS 66506, USA
d Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
b Department

Received 25 July 2000; received in revised form 22 May 2002; accepted 28 June 2002

Abstract
A recent approach to the control of underactuated systems is to look for control laws which will induce somespeciÿed structure on the
closed loop system. In this paper, we describe one matching condition and an approach for ÿnding all control laws that ÿt the condition.
After an analysis of the resulting control laws for linear systems, we present the results from an experiment on a nonlinear ball and beam
system.
? 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Underactuated systems;Control design; Lambda-method

1. Underactuated systems and the matching condition
Over the past 5 years several researchers have proposed
nonlinear control laws for which the closed-loop system
assumes some special form, see the controlled Lagrangian
method of Bloch, Leonard, and Marsden (1997, 2000) and
Bloch, Chang, Leonard, and Marsden (2001), the generalized matching conditions ofHamberg (1999, 2000a,b), the
interconnection and damping assignment passivity based
control of Blankenstein, Ortega, and van der Schaft (2001),
the -method of Auckly, Kapitanski, and White (2000),
and Auckly and Kapitanski (2002), and the references
therein. In this paper we describe the implementation of the
-method of Auckly et al. (2000) on a ball and beam system. For the readers conveniencewe start with the statement
of the main theorem on -method matching control laws
(Theorem 1). We also present an indicial derivation of the
main equations. We then prove a new theorem showing that
the family of matching control laws of any linear time invariant system contains all linear state feedback control laws
This paper was not presented at any IFAC meeting. This paper was
recommendedfor publication in revised form by Associate Editor Andrew
Teel under the direction of Editor Hassan Khalil.
∗ Corresponding author.
E-mail address: levkapit@math.ksu.edu (L. Kapitanski).
1 Supported in part by NSF grant CMS 9813182.
2 Supported in part by NSF grant DMS 9970638.

(Theorem 2). We next present the general solution of the
matching equations for the Quanser ball and beamsystem.
(Note, that this system is di erent from the system analyzed
by Hamberg, 1999.) As always, the general solution contains
several free functional parameters that may be used as tuning
parameters. We chose these arbitrary functions in order to
have a fair comparison with the manufacturer’s linear control
law. Our laboratory tests conÿrm the predicted stabilization.
This was our ÿrstexperimental test of the -method. We later
tested this method on an inverted pendulum cart (Andreev,
Auckly, Kapitanski, Kelkar, & White, 2000c).
Consider a system of the form
grj xj + [ jk; r ] xj xk + Cr +
˙˙

@V
= ur ;
@xr

(1)

r = 1; : : : ; n, where gij denotes the mass-matrix, Cr the dissipation, V the potential energy, [ij; k ] the Christo el symbol
of the ÿrst kind
[ jk; i] =

12

@gij
@gjk
@gki
+
−
k
j
@q
@qi
@q

(2)

and ur is the applied actuation. To encode the fact that some
degrees of freedom are unactuated, the applied forces and/or
i
i
torques are restricted to satisfy Pj g jk uk = 0, where Pj is
a g-orthogonal projection. The matching conditions come
from this restriction together with the requirement that the

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