International Journal of Dynamics and Control https://doi.org/10.1007/s40435-020-00630-1 Some necessary conditions on output redefinition in non-minimum phase systems Fatemeh Jahangiri 1 · Heidar Ali Talebi 2 · Mohammad Bagher Menhaj 2 Received: 25 January 2020 / Revised: 27 March 2020 / Accepted: 31 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020 Abstract This paper derives some necessary conditions for output redefinition. It is shown that for defining a new output it should contain at least one linear term to have a well-defined relative degree. In addition, it is presented that by a static transformation on system output, relative degree and the zero dynamics of the system are not changed. Therefore, output signal is not enough for output redefinition to convert a non-minimum phase system to a minimum phase system and availability of required state variables or at least observability of the states is necessary. Keywords Output redefinition · Relative degree · Zero dynamics · Output signal · Non-minimum phase systems 1 Introduction Minimum phase property has a major role in analysis and control design of dynamical systems [1–3]. Powerful non- linear controllers like feedback linearization and sliding mode methods can not be employed in non-minimum phase systems [4–8]. On the other hand, many systems in appli- cations like flexible structures, aircraft guidance, robots and manufacturing systems are non-minimum phase [9–12]. Therefore, to deal with this problem, several methods have been proposed, like stable inversion [13–16], causal inversion [14], integral manifold [17, 18], approximation methods [19, 20], and output redefinition [21]. In output redefinition, an output is defined such that the zero dynamics become stable since the dynamics depends on the output [22, 23]. Output redefinition is one of the most interesting meth- ods in literature [24–29]. This method has been proposed by Gopalswamy and Hedrick [21]. For output redefinition vari- B Fatemeh Jahangiri fa_jahangiri@sbu.ac.ir Heidar Ali Talebi alit@aut.ac.ir Mohammad Bagher Menhaj menhaj@aut.ac.ir 1 Department of Electrical Engineering, Shahid Beheshti University, Tehran, Iran 2 Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran ous procedures have been suggested. The suggested methods are mainly based on: defining a new output based on physi- cal insight [30], removing right half-plane zeros or shifting right half-plane zeros to the left half-plane in the linearized system [31], adding derivatives of the state variables to the output function [32], augmenting the output with a scaling of the state variables [27], adding the error variables [28], and an approach similar to the pole placement method [33, 34]. In this paper, we derive some necessary general conditions for output redefinition. We show that to have a well-defined realtive degree, the output should have at least one linear term. In addition, availability of some required state vari- ables or at least observability of the states are necessary for output redefinition. In other words, output signal is not enough for converting a non-minimum phase systems to a minimum phase system. More exactly, it is shown that if the new output is defined as a transformation of the system out- put, both relative degree of the system around the equilibrum point and the zero dynamics remain unchanged. Indeed, these results show limitations of constructing new minimum phase outputs. After determinig these limitations, proper methods for deriving the exact minimum phase outputs would be our future work. The rest of the paper is organised as follows. In Sect. 2, we state some preliminaries about zero dynamics and relative degree in nonlinear systems. Section 3 presents the main results of this paper. In Sect. 4, an example is given to show validation of the results and finally, Sect. 5 concludes the paper. 123