Robust Stability and Stabilization of 2D Positive System Employing Saturation
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Robust Stability and Stabilization of 2D Positive System Employing Saturation Jinling Wang1
· Yuxiao Hou1 · Lihui Jiang2 · Linzhong Zhang1
Received: 4 February 2020 / Revised: 11 August 2020 / Accepted: 14 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This work is concerned with the issue of stability analysis and controller synthesis for a class of Fornasini–Marchesini second-type systems with polytopic uncertainty, state delays and saturation nonlinearity. Firstly, by utilizing the co-positive Lyapunov function method, sufficient conditions in the linear programming setting are presented assuring the positivity and robust asymptotic stability of the proposed systems. Then, both the state feedback controller and the observer-based state feedback controller are considered based on the obtained results. In addition, the design schemes of the related controller gain matrices are also explicitly provided. Finally, two illustrative examples are given to demonstrate the effectiveness of the method outlined in this paper. Keywords Fornasini–Marchesini second model · Positive systems · Saturation nonlinearity · Uncertainty · Asymptotic stability
1 Introduction Two-dimensional (2D) systems are totally different from the traditional onedimensional (1D) systems in that the state of the former has two independent variables, whereas the latter has only one variable [14,40]. In recent years, the study of 2D systems has attracted a great deal of interest due to their capability of modeling a wide
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Jinling Wang [email protected] Yuxiao Hou [email protected] Lihui Jiang [email protected] Linzhong Zhang [email protected]
1
School of Science, Anhui Agricultural University, Hefei 230036, PR China
2
College of Artificial Intelligence and Big Data, Hefei University, Hefei 230601, PR China
Circuits, Systems, and Signal Processing
range of practical systems, such as those in water stream heating, thermal processes, image data processing and so on [9,36,39]. In general, there are three kinds of famous 2D state-space models, i.e., the Roesser model [26], the Fornasini–Marchesini first (FM-I) model [12] and the Fornasini–Marchesini second (FM-II) model [13]. It is worth noting that the former two models can be converted into the form of the last one by means of the variable transformation [16]. That is to say, the FM-II model is the most general model among them. Stability of 2D discrete-time systems represented by the FM-II model has been considered in [30,32,38]. In particular, the finite-region stability and finite-region boundedness for this kind of 2D systems have been tackled in [42]. On the other hand, it is necessary to introduce the time delays since they exist universally, have a major impact on the performance of systems and even lead to instability [3,41]. The H∞ control problem for 2D systems with constant delays and time-varying delays has been discussed, respectively, in [8] and [15]. And the issue of leakage delay-dependent stability for a class of com
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