Control with Positivity Constraint for 2D Continuous-Time Systems in Roesser Model
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Control with Positivity Constraint for 2D Continuous-Time Systems in Roesser Model M. Alfidi1 · A. Hmamed2 · F. Tadeo4 · A. Benzaouia3 Received: 28 June 2020 / Revised: 5 September 2020 / Accepted: 8 October 2020 © Brazilian Society for Automatics–SBA 2020
Abstract This paper deals with the stability synthesis for a class of 2D continuous-time systems described by the Roesser model. Conditions for stability and stabilization of positive continuous-time Roesser systems are derived, for which the states take non-negative values whenever the initial conditions are non-negative. As well, the synthesis of state-feedback controllers, including the requirement of positiveness of the controllers, and its extension to uncertain plants are solved in terms of linear programming. In addition, the synthesis problem with non-symmetrical bounds and stabilization are also treated. Numerical examples are included to illustrate the proposed approach. Keywords 2D continuous-time systems · Stabilization · Bounded controls · Positive systems · Positive control · Roesser model · Linear programming
1 Introduction Two-dimensional (2D) dynamical systems exist in many practical applications, such as in digital data filtering, image processing Roesser (1975), thermal processes, gas absorption, modeling of partial differential equations Marszalek (1984), fractional order systems Dami et al. (2017) and stochastic systems Duan et al. (2014).
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M. Alfidi [email protected] A. Hmamed [email protected] F. Tadeo [email protected] A. Benzaouia [email protected]
1
Engineering, Systems, and Applications Laboratory, National School of Applied Sciences, Sidi Mohamed Ben Abdellah University, Fez, Morocco
2
LESSI, Department of Physics Faculty of Sciences, Dhar El Mehraz, B.P 1796, 30000 Fes-Atlas, Morocco
3
LAEPT-URAC 28, Faculty of Sciences Semlalia, University Cadi Ayyad, B.P. 2390, Marrakech, Morocco
4
Department of Ingenieria de Sistemas y Automatica, Universidad de Valladolid, 47005 Valladolid, Spain
In the study of distributed parameter system, partial differential equations arise: these equations actually correspond to 2D or nD continuous-time systems Mastorakis et al. (2000), Mastorakis and Swamy (2002). Therefore, the study of 2D continuous systems is of both practical and theoretical importance. Up to date, some results for 2D continuous-time systems described by the Roesser 2D continuous systems have been given in, for instance, Badie et al. (2018a), Badie et al. (2020), Hmamed et al. (2016), Lam et al. (2004), Xu et al. (2005a, b), Duan et al. (2014). In connection with Roesser (1975) and Fornasini and Marchesini (1978) models, some important problems such as realization, controllability, minimum energy control Kaczorek (1997), finite frequency H∞ filtering Li and Gao (2012), dissipativity control and filtering Ki et al. (2015) have been extensively investigated. However, the stabilization problem is still not completely solved. It is also worth pointing out that all of the above-mentioned results were obtained i
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