Relative degree of interconnected SISO nonlinear control systems

The concept of relative degree plays an important role in nonlinear control theory. It provides, for example, a necessary and sufficient condition for the existence of a feedback linearizing control law for a single-input, single-output (SISO) input-affine nonlinear state space system. It also gives a sufficient condition under which a left inverse exists. In applications it is common for systems to be composed of smaller interconnected subsystems. It is known that various feedback structures preserve relative degree, but it is largely unknown how to compute the relative degree of interconnected systems in general. So the goal of this paper is to determine the relative degree of two nonlinear control systems having relative degree and interconnected in a variety of different ways. A collection of illustrative examples is given.

I. Introduction

In linear system theory, the notion of relative degree plays a fairly modest role. It is usually defined in terms of a system's transfer function written as either a rational function or in terms of a power series about the origin [14]. In the context of nonlinear control systems, the concept plays a more significant role. For example, it provides a necessary and sufficient condition for the existence of a feedback linearizing control law for a single-input, single-output (SISO) input-affine nonlinear state space system [13]. It also gives a sufficient condition under which a left inverse exists [12], which is useful for solving output tracking control problems. In this context, relative degree is defined using a state space model, but, as in the linear systems case, relative degree can also be described in a purely input-output setting using Chen-Fliess functional series (also called Fliess operators) [7], [8]. This definition is consistent with the state space notion of relative degree, but not every Fliess operator is realizable. So the series definition is in fact more general.