Matching, linear systems, and the ball and beam

system takes the form grj x ¨j + [j k, r]x ˙jx ˙ k + Cr + ∂V = 0, r = 1, . . . , n, ∂xr
j Second, λi j Pk and gij satisfy s r Pk Pt gℓs
∂λℓ r i + [ℓj, s]λℓ r − [rj, i]λs ∂q j = 0, (6)
1
Underactuated systems and the matching condition
Over the past five years several researchers have proposed nonlinear control laws for which the closed loop system assumes some special form, see the controlled Lagrangian method of [8, 9, 10] the generalized matching conditions of [11, 12, 13], the interconnection and damping assignment passivity based control of [7], the λ-method of [6, 5], and the references therein. In this paper we describe the implementation of the λ-method of [6] on a ball and beam system. For the readers convenience we start with the statement of the main theorem on λ-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
− gℓi
r ℓ (∂Pt λr ) ∂qj
and (9)
λℓ r gℓi = gri
in (8) to obtain (7). To derive (6), first, differentiate (9) with respect to q j to get λℓ r ∂λℓ ∂gri ∂ gℓi r = − gℓi j . j j ∂q ∂q ∂q (10)
arXiv:math/0006121v2 [math.OC] 8 Nov 2002
Matching, linear systems, and the ball and beam
F. Andreev1,3 , D. Auckly1,4 , vi1,5 , W. White1,5 , A. Kelkar1,6
Abstract
A recent approach to the control of underactuated systems is to look for control laws which will induce some specified structure on the closed loop system. In this paper, we describe one matching condition and an approach for finding all control laws that fit 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.
(7)
Ptr λℓ r
∂ gij ∂ gℓi ∂ gjℓ − Ptr λℓ − Ptr λℓ r r ∂q ℓ ∂q j ∂q i ∂gij ∂gri ∂gjr = Ptr r − Ptr j − Ptr i . ∂q ∂q ∂q
r ℓ ∂ (Pt λr gℓi ) ∂qj
(8)
∂ gℓi Use Ptr λℓ r ∂qj =
in part by NSF grant CMS 9813182 in part by NSF grant DMS 9970638
and ur is the applied actuation. To encode the fact that some degrees of freedom are unactuated, the applied forces and/or torques are restricted to satisfy Pji g jk uk = 0, where Pji is a g -orthogonal projection. The matching conditions come from this restriction together with the requirement that the closed loop 1
1 Supported 2 Supported
r = 1, . . . , n, where gij denotes the mass-matrix, Cr the dissipation, V the potential energy, [i j, k ] the Christoffel symbol of the first kind, [jk, i] = 1 2 ∂gij ∂gki ∂gjk + − ∂q k ∂q j ∂q i , (2)
r g ki Ci − g ki Ci = 0, Pk r Pk
∂V g ki i ∂q
−
∂V g ki i ∂q
(3) = 0,
r r ∂ gnm ∂ (λℓ ∂ (λℓ r Pt ) r Pt ) + gℓn + gℓm ℓ m n ∂q ∂q ∂q ℓ ∂Pt ∂Ptℓ partialgnm + +gℓn m gℓm n . = Ptℓ ℓ ∂q ∂q ∂q
for some choice of g , C , and V . The matching conditions read
ki ki r k r Γk Pk ij − Γij = 0, Pk g Ci − g Ci = 0, r Pk
+gir
r λℓ r Pt
∂λi s ℓ + [ij, r]λi s − [sj, ℓ]λr ∂q j
of matching control laws of any linear time invariant system contains all linear state feedback control laws (Theorem 2). We next present the general solution of the matching equations for the Quanser ball and beam system. (Note, that this system is different from the system analyzed by Hamberg, [11].) 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 manifacturer’s linear control law. Our laboratory tests confirm the predicted stabilization. This was our first experimental test of the λ-method. We later tested this method on an inverted pendulum cart, [3]. Consider a system of the form grj x ¨j + [j k, r] x ˙j x ˙ k + Cr + ∂V = ur , ∂xr (1)
6
Department of Mathematics, Western Illinois University, Macomb, IL 61455, USA Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA 5 Department of Mechanical and Nuclear Engineering, Kansas State University, Manhattan, KS 66506, USA Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
Although the proof of this proposition may be found in [6], [4], and [5], for convenience, we include an k where Γij is the Christoffel symbol of the second kind, indicial notation derivation of equations (6) and (7). k Substitute equations (2), (4) for both Γk ij and Γij into k kℓ Γij = g [ij, ℓ]. (4) the first of equations (3) and multiply the result by the scalar 2 to obtain: If the matching conditions (3) hold, the control law r kℓ ∂ gℓi r kℓ ∂ gjℓ r kℓ ∂ gij − Pk g − Pk g Pk g will be given by ∂q ℓ ∂q j ∂q i r kℓ ∂gij r kℓ ∂gℓi r kℓ ∂gjℓ k = Pk g − Pk g − Pk g . ur = grk (Γk ˙ iq ˙ j + Cr − Cr ℓ j ij − Γij )q ∂q ∂q ∂q i (5) Multiply by g and use that P is self-adjoint i.e., ∂V ∂V rt + grk g ki i − g ki i . Pik gkj = gik Pjk , to get ∂q ∂q The motivation for this method is that H = 1 ˙ iq ˙ j + V is a natural candidate for a Lyapunov 2 gij q d H = −gij ci q ˙ j . Following [6], infunction because dt k troduce new variables λi = gij g jk . We have Theorem 1 The functions gij , V , and C satisfy (3) in a neighborhood of x0 if and only if