Automatica 44 (2008) 2214–2220 www.elsevier.com/locate/automatica Technical communique Design of static H ∞ loop shaping controller in four-block framework using LMI approach ✩ S. Patra, S. Sen, G. Ray ∗ Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, 721302, India Received 28 June 2006; received in revised form 14 November 2007; accepted 1 January 2008 Available online 3 March 2008 Abstract This paper attempts to develop an alternate but simple technique for designing a static output feedback H ∞ loop shaping controller from four-block H ∞ framework. Sufficient conditions are established for the existence of a static H ∞ loop shaping controller. These conditions are in LMI form, and are different from those obtained in [Prempain, E. (2004). Static H ∞ loop shaping control. Control 2004. UK: University of Bath; Prempain, E., & Postlethwaite, I. (2004). Static H ∞ loop shaping control of fly-by-wire helicopter. In Conference on decision and control (pp. 1188–1195); Prempain, E., & Postlethwaite, I. (2005). Static H ∞ loop shaping control of fly-by-wire helicopter. Automatica, 41, 1517–1528] using normalized coprime factorization approach. Numerical examples are considered to illustrate the results of this paper. c  2008 Elsevier Ltd. All rights reserved. Keywords: Static H ∞ loop shaping controller; Linear matrix inequality 1. Introduction For single-input-single-output systems, it is well known that the effective design of a compensator plays an important issue to reshape the frequency response of the open-loop transfer function that will meet the feedback system stability, good performance and certain robustness. However, a direct consequence of this technique to multi-variable systems is not straightforward due to cross coupling and lack of phase information of multi-input-multi-output (MIMO) systems. For MIMO systems, a popular H ∞ loop shaping design procedure has the advantage of offering levels of robust performance with the proper selection of weighting function matrices. However, various factors influence the choice of these weighting function matrices, such as bandwidth, roll-off rates, and low frequency gain magnitude. The choice of weighting function matrices depends on closed-loop performance requirements and for different choices, different stability margins will be obtained, and hence, the controller parameters also will be changed. For ✩ This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Mayuresh Kothare under the direction of Editor Andr´ e Tits. ∗ Corresponding author. Tel.: +91 03222 283078. E-mail address: gray@ee.iitkgp.ernet.in (G. Ray). an uncertain shaped plant, the robust controller is synthesized either in the normalized coprime factor form or in its equivalent four-block H ∞ framework (McFarlane & Glover, 1990, 1992), as shown in Fig. 1. Both of these frameworks are more general for considering unstructured uncertainty, however, former approach may not be always straightforward for controller design considering input saturation. In the presence of input saturation, the normalized coprime factors of the shaped plant are difficult to obtain as it appears in between the pre- compensator and the nominal plant, and also, it is hard to realize the uncertainty structure as shown in Fig. 1(a). On the other hand, the equivalent four-block structure, shown in Fig. 1(b), can provide an easier framework. The controller, K is designed by minimizing the ∞-norm of the transfer function matrix from  w T 2 w T 1  T to  z T 2 z T 1  T , and it effectively ensures closed- loop stability of the system against unstructured uncertainty as described in Fig. 1(a) (McFarlane & Glover, 1992). In four- block structure, the saturation nonlinearity can be realized in between G and W ; and some well-known techniques, like, transforming the system into a Lur’e type system (Hindi & Boyd, 1998), polytopic approach (Henrion, Tarbouriech, & Garcia, 1999) etc. can easily be applied to design H ∞ loop shaping controller with input saturation constraint. Moreover, the latter approach provides a sequence of design steps if the 0005-1098/$ - see front matter c  2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.01.008