Multi-output maps with associated asymptotically stable zero dynamics

Abstract

Conceptions of relative degree and minimum phase are connected to many problems. To apply these conceptions one needs to know an output map which renders an affine system to be minimum phase. Necessary and sufficient conditions of the existence of such outputs are presented in the multy-output case. Obtained conditions result in new setting of the stabilization problem and in a new set of static and dynamic stabilizing feedbacks. Lyapunov functions are designed for close-loop systems. An example of a hovercraft stabilization is considered. I. INTRODUCTION One of stabilizing design methods for a nonlinear system is based on changes of variables in the state space and in the space. The goal of these changes of variables usually consists in obtaining an equivalent system for which we know how to solve the problem. For example, one can often try to transform the original nonlinear system into a linear one or a partially linear one (2). It is well-known that the stabilization problem can be solved by the zero value output stabilization if the system under investigation is a minimum phase one (1), (2), (8). It is worth to note that even in case of the complete knowledge of the state vector the use of properties of minimum phase systems often allows essentially enlarge a set of stabilizing feedbacks. In this paper any smooth output map of the system will be called a output similarly to the use of term virtual control in the backstepping method (7). This output may not be a real output of the system. An asymptotical stability of zero dynamics depends on choise of a output. So one needs to know where an affine system can be provided by outputs with asymptotically stable zero dynamics and how to find such outputs if any exists. For autonomous affine systems with scalar one can find solutions of these problems in (3), (4), (5). The main contribution of this paper consists in finding a multy-output for an affine system with multiple whose associated zero dynamics is asymptotically stable.