On the relative degree and normal forms of linear systems by output transformation with applications to tracking

Abstract

The normal form, which is also known as Byrnes–Isidori normal form, of a linear system is very important in control theory, as it is a fundamental tool for solving a couple of important problems. It is known that the normal form exists only if the system has a relative degree and a nonsingular coupling matrix, which is not always satisfied. Though both the relative degree and the non-singularity of the coupling matrix are invariant under state and input transformations, they can be changed by output transformation. This paper establishes novel necessary and sufficient conditions for the existence of an output transformation such that the transformed system has a well defined relative degree and nonsingular coupling matrix, namely, the existence of a normal form for linear multivariable systems. The conditions are expressed by two rank identities in terms of the coefficient matrices of the linear system. When these two rank conditions are satisfied, an algorithm for constructing the output transformation matrix is also provided. The output tracking problem is revisited with the help of the obtained results. Examples are given to illustrate the obtained results.