Matching , linear systems , and the ball and beam Fedor Andreev

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. This basic idea is used in [1-9]. In this paper, we will 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 will present the results from an experiment on a ball and beam system. This work was partially supported by grant CMS 9813182 from the National Science Foundation. 1 Underactuated systems and the matching condition Consider a mechanical system with n-degrees of freedom described by the generalized coordinates, qi. Assume that the kinetic energy is given by T = 1 2 gij(q)q̇ q̇ j , where gij(q) is the mass matrix, and we are using the summation convention. Further assume that there is a damping term depending upon the full state of the system, Ci(q, q̇), a potential energy, V (q), and a control input, ui(q, q̇). The system is called underactuated if there is a non trivial constraint on the allowed input forces. It is convenient to express this constraint as: N (q) uk = 0, (1) where N ik is the orthogonal projection onto the space of unactuated directions. The fact that N ik is an orthogonal projection may be expressed as the requirements: N ik = N and N ik = N N . We assume that Ci(q, q̇) is an odd function of velocity, i.e. Ci(q,−q̇) = −Ci(q, q̇). The equations of motion of the system are given by: gij q̈ j + ∂gij ∂qk q̇ q̇ j − 1 2 ∂gjk ∂qi q̇ j q̇ k + Ci + ∂V ∂qi = ui. (2) It is convenient to denote the inverse of the mass matrix by g, so that gijg jk = δ i . Define a Christoffel symbol of the first kind by: [jk, i] = 1 2 ( ∂gij ∂qk + ∂gki ∂qj − ∂gjk ∂qi ) , (3) and a Christoffel symbol of the second kind by: Γkij = g [ij, l]. (4) Multiplying the equations of motion by the inverse of the mass matrix we get: q̈ k + Γkij q̇ q̇ j + c + g ∂V ∂qi = f, (5) where f i = guj, and c i = gCj.