An zyxwv 1, Solution for Approximate Feedback Linearization+ zyxw Eric C. zyxwvuts Gwo Electrical Engineering zyxwvutsrq - Systems University of Southern California Los Angeles, CA 90089-2563 Abstract. A numerical approach is studied for the approx- imate feedback linearizationof nonlinear systems which fail to have the conditions for feedback linearization. Requiring derivatives of the system vector fields is a common proce- dure for solving this problem in most of previous work. We formulate this problem zyxwvutsrq as an optimization problem. The evaluations of the system vector fields provide sufficient in- formation for this approach. A tracking control law based on the resulting approximate system can be developed to control the original system. Keywords. feedback linearization, B-spline, Lie deriva- tive, equilibrium manifold, linear controllability. Introduction There arc mamy physical systems [9] which do not satisfy the restrictive conditions for either input-output linearization [4] or feedback linearization [lo]. Thus, we have to adopt the en- gineering approaches. Most of the previous work [l, 11, 13, 151 has focused on local analysis around a particular operating point. Some approaches do take a more global view. The approach given in [7, 91 provides a systematic procedure for constructing approximate systems. The resulting approximate systems, which are feedback linearizable, will stably track a class of desired trajectories close to the equilibrium manifold. Most of the existent techiques require derivatives of the system vector fields. For many physical nonlinear systems, the vector fields may exist as empirical data (lookup tables) or are gener- ated by modeling procedure. It may. not be possible to reliably compute higher order derivatives of these vector fields. Thus, requiring derivatives of the system vector fields becomes serious drawbacks when we apply those techniques. The approach in this paper does not require any derivatives of vector fields. The lookup tables provide sufficient information for our approach to construct approximate systems which are feedback linearizable. The feature of the proposed approach provides more applicably than other approaches from practical point of view. In Section 1, we review some fundamental definitions and properties of B-splines which are useful in the sequel. In Sec- tion 2, we formulate this problem as an zyxwvutsrqpo L, optimization prob- lem. We then convert the formulation into a linear program- ming problem by using sampling data instead of a continuous region. The solution of the linear programming problem can be used as a set of change coordinates for approximately lin- earizing system. In Section 3, we apply our techniques- to a simple example, the ball and beam system, which is neither input-output linearizable nor feedback linearizable. 'Research supported in part by AFOSR under grant AFOSR-91-0255 and by NSF under grant PYI ECS-9157835 and by grants from Rockwell International, Hughes Aircraft Company, and the TRW Foundation. John Hauser Electrical & Computer Engineering University of Colorado Boulder, CO 80309-0425 1 Prelimiaries Let X = zyxwv {zi}h" zyxwvut c R be a partition with zi < zifl for all i and the it'' interval of X is defined as 1. 1 - - [zi,zi+l ), i = 0,1,. .. , k - 1 and Ik = zyx [zk,zk+']. The linear space of polynomials of order T is zyx I P, = p(z) = c,zm-l, Cl,. .. , c,, I E R . (1) { , , Let Djs(z) be the jth derivative of ~(2). The smoothest spline space of order T on the partition X is given by s:(x) = { s : there exist polynomids so, SI,. . . , sk in f', such that s(z) = si(z) for z E I,, i = O,l,. . ., k and Djs;-l(z') = ~js,(zi) for j = 0, I,. . ., T - 2, i = 1,2,. . ., k} . It is well known that S,k(X) is a linear space of dimension T + k [14] and SF(X) is a linear subspace of C'-' on X. Since the dimension of S,k(X) is finite, there exists a basis to span this linear space. Therefore, every function in SF(X) can be expressed by the linear combination of these basis functions. The ith normalized B-spline of order T [14] is denoted by . . E:(z) = - z8)[zZ, , , . , - z);-l for all z E R. where the rth divided difference of a function f, [z', . .. , zi+,]f, is defined recursively. For simplicity, we assume that zi are distinct, then [Xi+* ,.e., z'+r]f - . . . , Z ' + q %itr - %i [z', . . . , Z'+']f = [ . ' I f = f(x') It is clear that &(X) = span{B[}~~~. A space of multidimensional spline functions can be con- structed by taking the tensor-product of single variable spline spaces. For example, we take two variables, 11 and 22, and each B-spline space with dimension 2, then the tensor-product B-spline space is spanned by {JT(zdfT(z2), B;'(zl)BZ~z), B;1(z1)By(z2), B;'(zl)B;2(z2)}. Similarly, we denote tensor-product B-spline space with n vari- ables as 339