Automatica 49 (2013) 101–110 Contents lists available at SciVerse ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Gain-scheduled control of two-dimensional discrete-time linear parameter-varying systems in the Roesser model ✩ Carlos E. de Souza 1 , Jefferson Osowsky Department of Systems and Control, Laboratório Nacional de Computação Científica (LNCC/MCTI), Av. Getúlio Vargas 333, Petrópolis, RJ 25651-075, Brazil article info Article history: Received 17 October 2011 Received in revised form 17 April 2012 Accepted 3 August 2012 Available online 22 October 2012 Keywords: Two-dimensional systems Gain-scheduled control H ∞ control Guaranteed cost control Linear parameter-varying systems Discrete-time systems Parameter-dependent Lyapunov function abstract This paper is concerned with gain-scheduled control of two-dimensional discrete-time linear parameter- varying systems described by a Roesser state-space model with matrices depending affinely on time- varying scheduling parameters. The parameter admissible values and variations are assumed to belong to given intervals. Linear matrix inequality based methods are devised for designing static state feedback gain-scheduled controllers with either an H ∞ or quadratic regulator-type performance. The control designs build on quadratically parameter-dependent Lyapunov functions and allow for incorporating information on available bounds on the parameters variation. The proposed controller gain can be independent, affine, quadratic, or a matrix fraction of quadratic polynomial matrices in the scheduling parameters. © 2012 Elsevier Ltd. All rights reserved. 1. Introduction Two-dimensional (2-D) processes are quite often encountered in nature and engineering and thus 2-D systems find applications in a wide range of different fields, such as image processing, seis- mographic data processing, gas absorption, water stream heating, thermal processes, etc. (see, for instance, Du & Xie, 2002; Kaczorek, 1985; Lu & Antoniou, 1992, and the references therein). Over the past two decades 2-D discrete-time linear systems have been attracting significant interest within the control com- munity and significant advances have been achieved in the the- ory of control synthesis to solve a variety of problems, such as for instance, dynamic output feedback stabilization (Bisiacco, 1985; Du & Xie, 1999a), H ∞ control (Du, Xie, & Zhang, 2001; Šebek, 1993; Xie, Du, Soh, & Zhang, 2002, and the references therein), stochastic LQ control (Šebek & Kraus, 1995), state feedback ro- bust stabilization (Du & Xie, 1999b; Gao, Lam, Xu, & Wang, 2005), dynamic output feedback robust stabilization (Du et al., 2001; Xie et al., 2002), robust H ∞ control (Du & Xie, 2002), H 2 and mixed ✩ This work was supported by CNPq, Brazil, under grants 30.6270/2011-0/PQ for CEdS and 14.0555/2008-0/GD for JO. The material in this paper was partially presented at the 18th IFAC World Congress, August 28–September 2, 2011, Milano, Italy. This paper was recommended for publication in revised form by Associate Editor Fabrizio Dabbene under the direction of Editor Roberto Tempo. E-mail addresses: csouza@lncc.br (C.E. de Souza), osowsky@lncc.br (J. Osowsky). 1 Tel.: +55 24 22336012; fax: +55 24 22336141. H 2 /H ∞ control (Yang, Xie, & Zhang, 2006), linear quadratic Gaus- sian control (Yang, Zhang, & Xie, 2007), and guaranteed cost control (Dhawan & Kar, 2007a,b, 2010, 2011; Guan, Long, & Duan, 2001). Recently, a linear matrix inequality (LMI) based method for the de- sign of a gain-scheduled state feedback H ∞ control for 2-D linear parameter-varying (LPV) systems in the Roesser model has been developed in Osowsky and de Souza (2011), whereas Wu, Lam, and Wang (2009) has investigated gain-scheduled H ∞ control of LPV systems described by a Fornasini–Marchesini second local state- space model. The latter work deals with dynamic output feedback and has the feature that the control design is based on a parameter- dependent Lyapunov function, however it is given in terms of some parameter-dependent LMIs, and as such the control calculation in- volves solving an infinite number of LMIs, even when the system matrices and Lyapunov function are restricted to be affine in the parameters, and in the case of static state feedback. To overcome this difficulty, the authors apply a gridding technique of the pa- rameter value set and as a consequence, the controller guarantees stability and H ∞ performance only for values of the system param- eters in the chosen set of gridding points. In spite of these devel- opments, to the authors’ knowledge the design of gain-scheduled control for 2-D Roesser models based on parameter-dependent Lyapunov functions is far from being fully resolved, even in the case of systems with affine parameter dependence and for static state feedback, and will be the subject of this paper. This paper deals with gain-scheduled control problems for 2-D discrete-time LPV systems described by a Roesser state-space 0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.09.024