Approximate input-output linearization of nonlinear systems

We will define an approximate input-output linearization problem -- an order ¿ input-output linearization problem. To solve this, we must find a feedback u=¿(x)+ß(x)v for the system x=f(x)+g(x)u, y =h(x) such that the input-output response will be order ¿ input-output linear, i.e. its Volterra series expansion (V.S.E.) will be y(t) = W0(t) + ¿i=1 m ¿0 t{¿p=0 ¿K(i,p)(t-¿)p/p!}vi(¿)d¿ +¿j=1 ¿W(¿+j) where W(k) is k-th order term of V.S.E. This system will be approximately linear if the kernels of order larger than ¿ are negligible. We will identify, using a modified structure algorithm, the class of nonlinear systems which can be transformed into order ¿ input-output linear systems. We will also show that, under suitable conditions, an order ¿ input-output linear system can be expressed in an appropriate state as ¿ = F¿ + Gv +od(¿, ¿, v)¿+1 ¿ = f¿(¿, ¿) + ¿(¿, ¿)v y = H¿ where F, G and H are matrices of real numbers.