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)+??(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.