Approximate Input-Output Linearization of Nonlinear Systems

Abstract

In this paper, we will proceed to solve an order ρ input-output linearization problem. We will define an order ρ input-output (I/O) linear system whose Volterra series expansion (V.S.E) around the equilibrium point x₀ is
where W(k) is the k-th order term of the V.S.E. This system represents approximately a linear input-output response when the terms of order larger than ρ are negligible. We will define an order ρ input-output linearization problem as finding nonlinear feedback which make the system order ρ I/O linear. We will propose an order ρ structure algorithm to determine the feedback and identify the class of systems which can be transformed into order ρ I/O linear systems. We will also show that, under suitable conditions, an order ρ I/O linear system can be represented in appropriate coordinates as
ξ=Fξ+Gv+O(ξ, η, v)^ρ+1
η=f(ξ, η)+g(ξ, η)v
y=Hξ
where F, G and H are matrices of real numbers. This implies that the state ξ which dominates the input-output response can be represented in the form of an order ρ state space linear system.