Automatica 41 (2005) 173 – 176 www.elsevier.com/locate/automatica Technical Communique The Schur stability via the Hurwitz stability analysis using a biquadratic transformation  Mahdi Jalili-Kharaajoo, Babak N. Araabi ∗ Control and Intelligent Processing Center of Excellence, Electrical and Computer Engineering Department, University of Tehran, Tehran 14395/515, Iran Received 24 July 2003; accepted 7 September 2004 Available online 30 October 2004 Abstract In this note, it is shown that instead of a bilinear transformation, a biquadratic transformation can be used to determine the Schur stability of a given discrete-time polynomial by determining the Hurwitz stability of the corresponding continuous-time polynomial. However, using a biquadratic transformation, the analysis is not limited by any restriction on the root positions. Several numerical examples are provided to illustrate the procedure.  2004 Elsevier Ltd. All rights reserved. Keywords: Schur stability; Hurwitz stability; Discrete-time systems; Bilinear transformation; Parametric controller design 1. Introduction The stability analysis is one of the most important chal- lenges in the design of control systems. In the design of discrete-time control systems, the Schur stability of a discrete-time polynomial should be insured. There are some methods to check the Schur stability of a given discrete- time polynomial (see Kuo, 1992; Ogata, 1995). However, in some cases, instead of direct analysis on the discrete-time system, it is more convenient to perform stability analysis on an equivalent continuous-time system. This indirect ap- proach is particularly effective when dealing with paramet- ric design of control systems. Indeed, in many applications, it is easier to use the Hurwitz stability analysis to determine the Schur stability (see Xu, Datta, & Bhattacharyya, 2001, 2002; Clapperton, Crusca, & Aldeen, 1996; Lind, 1993). Using a bilinear transformation (Ogata, 1995; Shiomi, Otsuka, Inaba, & Ishii, 1999), the determination of the Schur  The paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor S. Lall under the direction of Editor P. Van den Hof. ∗ Corresponding author. Tel.: +98 21 8630024; fax: +98 21 8633029. E-mail addresses: mahdijalili@ece.ut.ac.ir (M. Jalili-Kharaajoo), araabi@ut.ac.ir (B.N. Araabi). 0005-1098/$ - see front matter  2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2004.09.004 stability of a given discrete-time polynomial can be con- verted to the determination of the Hurwitz stability of an equivalent continuous-time polynomial. Through this ap- proach, the powerful continuous-time design techniques can be applied to the discrete-time domain. This method has been utilized to calculate all the stabilizing PID gains for a discrete-time system (Xu et al., 2001, 2002), for model reduction in generalized singular perturbation (Clapperton et al., 1996), and for digital filter design (Lind, 1993). Some earlier applications can be found in Groutage, Volfson, and Schneider (1984) and Power (1970). Although, bilinear transformation provides a tool to de- termine the Schur stability via the Hurwitz stability; how- ever, its application is restricted by a pathological case. This method is not applicable to discrete-time polynomials with one or more roots at z = 1. Indeed, the Horowitz stabil- ity analysis in this case, does not provide conclusive results about the Schur stability of the original discrete-time poly- nomial. In this note, instead of the bilinear transformation, a biquadratic transformation without any pathological case is introduced. It is shown that this transformation maps the unit circle in the discrete-time domain to the imaginary axis in the continuous-time domain and vice versa. Inside and outside of the unit circle are mapped to the open left-half