Algebraic Approaches to Normal Forms and Zero Dynamics

Sammanfattning: The normal form of affine state space systems has been extensively studied during the last decade and it has found several important applications. Two such applications are exact input-output linearization and sliding mode control. A property of affine nonlinear systems that is simple to compute given a normal form is the zero dynamics. The properties of the zero dynamics are important not only in the applications mentioned above but also, for instance, in stabilization. The analysis of the normal form and the zero dynamics as well as the computation is usually done using tools from differential geometry. However, the purpose of this thesis is to study these concepts using some other system descriptions and some other tools.It will be shown how zeros of the sequence of transfer functions that emerge from the Laplace transform of the regular kernels in a Volterra series are connected to the zero dynamics. Given a certain bilinear state space system one part of the numerator zeros is exactly equal to the eigenvalues of a linear part of the zero dynamics. For a class of homogenous systems this relation between zeros of the transfer function and the zero dynamics can be seen without computing the bilinear realization. These results can be viewed as generalizations of the linear case where the zeros of the transfer function are exactly the same as the poles of the zero dynamics.Furthermore, a result is given which shows how differential algebra, in particular the Ritt algorithm, can be used for computing the zero dynamics of affine state space systems. The restriction that differential algebra only can handle differential polynomials is removed by utilizing the fact that many smooth functions can be described by polynomial differential equations.We also consider how a generalized normal form can be defined for a differential algebraic dynamics. This definition is, as in the affine case, based on the notion of a relative degree, i.e., the minimum order of an output derivative that depends explicitly on the input. The tool for computing the relative degree and the normal form is Gröbner bases. For affine systems it is well known that existence of a relative degree is both necessary and sufficient for existence of a static decoupling feedback law. Here we generalize this result to a dynamics which admits a classical state.Finally, as an application we study how these results can be used for the design of sliding mode control laws. In particular, the generalized normal form is shown to be useful in this context.