## Abstract and Applied Analysis

### Robust Stability Criteria of Roesser-Type Discrete-Time Two-Dimensional Systems with Parameter Uncertainties

#### Abstract

This paper is concerned with robust stability analysis of uncertain Roesser-type discrete-time two-dimensional (2D) systems. In particular, the underlying parameter uncertainties of system parameter matrices are assumed to belong to a convex bounded uncertain domain, which usually is named as the so-called polytopic uncertainty and appears typically in most practical systems. Robust stability criteria are proposed for verifying the robust asymptotical stability of the related uncertain Roesser-type discrete-time 2D systems in terms of linear matrix inequalities. Indeed, a parameter-dependent Lyapunov function is applied in the proof of our main result and thus the obtained robust stability criteria are less conservative than the existing ones. Finally, the effectiveness and applicability of the proposed approach are demonstrated by means of some numerical experiments.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 159745, 6 pages.

Dates
First available in Project Euclid: 6 October 2014

https://projecteuclid.org/euclid.aaa/1412605897

Digital Object Identifier
doi:10.1155/2014/159745

Mathematical Reviews number (MathSciNet)
MR3186949

Zentralblatt MATH identifier
07021837

#### Citation

Zhao, Yan; Zhang, Tieyan; Zhao, Dan; You, Fucai; Li, Miao. Robust Stability Criteria of Roesser-Type Discrete-Time Two-Dimensional Systems with Parameter Uncertainties. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 159745, 6 pages. doi:10.1155/2014/159745. https://projecteuclid.org/euclid.aaa/1412605897

#### References

• D. Henrion, D. Arzelier, D. Peaucelle, and M. Šebek, “An LMI condition for robust stability of polynomial matrix polytopes,” Automatica, vol. 37, no. 3, pp. 461–468, 2001.
• S. Yin, H. Luo, and S. Ding, “Real-time implementation of fault-tolerant control systems with performance optimization,” IEEE Transactions on Industrial Electronics, vol. 64, no. 5, pp. 2402–2411, 2014.
• S. Yin, S. X. Ding, A. H. A. Sari, and H. Hao, “Data-driven monitoring for stochastic systems and its application on batch pro-cess,” International Journal of Systems Science, vol. 44, no. 7, pp. 1366–1376, 2013.
• S. Yin, S. Ding, A. Haghani, H. Hao, and P. Zhang, “A comparison study of basic datadriven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process,” Journal of Process Control, vol. 22, no. 9, pp. 1567–1581, 2012.
• D. Henrion, D. Arzelier, and D. Peaucelle, “Positive polynomial matrices and improved LMI robustness conditions,” Automatica, vol. 39, no. 8, pp. 1479–1485, 2003.
• P. L. D. Peres and J. C. Geromel, “$\mathcal{H}$2 control for discrete-time systems optimality and robustness,” Automatica, vol. 29, no. 1, pp. 225–228, 1993.
• P. L. D. Peres, J. C. Geromel, and J. Bernussou, “Quadratic stabilizability of linear uncertain systems in convex-bounded domains,” Automatica, vol. 29, no. 2, pp. 491–493, 1993.
• P. Gahinet, P. Apkarian, and M. Chilali, “Affine parameter-dependent Lyapunov functions and real parametric uncertainty,” IEEE Transactions on Automatic Control, vol. 41, no. 3, pp. 436–442, 1996.
• E. Fornasini and G. Marchesini, “State–space realization theory of two–dimensional filters,” IEEE Transactions on Automatic Control, vol. 21, no. 4, pp. 484–492, 1976.
• R. P. Roesser, “A discrete state-space model for linear image processing,” IEEE Transactions on Automatic Control, vol. 20, pp.1–10, 1975.
• D. H. Owens, N. Amann, E. Rogers, and M. French, “Analysisof linear iterative learning control schemes: a 2D systems/repetitive processes approach,” Multidimensional Systems and Signal Processing, vol. 11, no. 1-2, pp. 125–177, 2000.
• B. Sulikowski, K. Gałkowski, E. Rogers, and D. H. Owens, “Output feedback control of discrete linear repetitive processes,” Automatica, vol. 40, no. 12, pp. 2167–2173, 2004.
• B. Sulikowski, K. Galkowski, E. Rogers, and D. H. Owens, “PI control of discrete linear repetitive processes,” Automatica, vol. 42, no. 5, pp. 877–880, 2006.
• S. Yin, G. Wang, and H. Karimi, “Data-driven design of robust fault detection system for wind turbines, Mechatronics,” 2013.
• V. Singh, “Elimination of overflow oscillations in 2-D digital filters employing saturation arithmetic: an LMI approach,” IEEE Signal Processing Letters, vol. 12, no. 3, pp. 246–249, 2005.
• V. Singh, “Stability analysis of 2-D discrete systems described by the Fornasini-Marchesini second model with state saturation,” IEEE Transactions on Circuits and Systems II, vol. 55, no. 8, pp. 793–796, 2008.
• A. Dhawan and H. Kar, “Optimal guaranteed cost control of 2-D discrete uncertain systems: an LMI approach,” Signal Processing, vol. 87, no. 12, pp. 3075–3085, 2007.
• V. Singh, “New LMI condition for the nonexistence of overflow oscillations in 2-D state-space digital filters using saturation arithmetic,” Digital Signal Processing, vol. 17, no. 1, pp. 345–352, 2007.
• V. Singh, “Improved LMI-based criterion for global asymptotic stability of 2-D state-space digital filters described by Roesser model using two's complement arithmetic,” Digital Signal Processing, vol. 22, no. 3, pp. 471–475, 2012.
• V. Singh, “On global asymptotic stability of 2-D discrete systems with state saturation,” Physics Letters A, vol. 372, no. 32, pp. 5287–5289, 2008.
• A. Dhawan and H. Kar, “An LMI approach to robust optimal guaranteed cost control of 2-D discrete systems described by the Roesser model,” Signal Processing, vol. 90, no. 9, pp. 2648–2654, 2010.
• A. Dey and H. Kar, “Robust stability of 2-D discrete systemsemploying generalized overflow nonlinearities: an LMI approach,” Digital Signal Processing, vol. 21, no. 2, pp. 262–269, 2011.
• A. Dhawan and H. Kar, “An improved LMI-based criterion for the design of optimal guaranteed cost controller for 2-D discrete uncertain systems,” Signal Processing, vol. 91, no. 4, pp. 1032–1035, 2011.
• V. Singh, “New approach to stability of 2-D discrete systems with state saturation,” Signal Processing, vol. 92, no. 1, pp. 240–247, 2012.
• H. Kar, “A new criterion for the global asymptotic stability of 2-D state-space digital filters with twos complement overflow arithmetic,” Signal Processing, vol. 92, no. 9, pp. 2322–2326, 2012.
• A. Dey and H. Kar, “An LMI based criterion for the global asymptotic stability of 2-D discrete state-delayed systems with saturation nonlinearities,” Digital Signal Processing, vol. 22, no. 4, pp. 633–639, 2012.
• W. Sun, H. Gao Sr., and O. Kaynak, “Finite frequency $\mathcal{H}_{\infty }$ control for vehicle active suspension systems,” IEEE Transactions on Control Systems Technology, vol. 19, no. 2, pp. 416–422, 2011.
• W. Sun, Y. Zhao, J. Li, L. Zhang, and H. Gao, “Active suspension control with frequency band constraints and actuator input delay,” IEEE Transactions on Industrial Electronics, vol. 59, no. 1,pp. 530–537, 2012.
• Y. Suzuki, S. Morioka, and H. Ueda, “Cooking support with information projection over ingredient,” International Journal of Innovative Computing, Information and Control, vol. 9, no. 12, pp. 4753–4763, 2013.
• M. Nakatani and T. Ohno, “An integrated model depicting psychology of active/Non-active internet users: how to motivate people to use internet at home,” International Journal of Innovative Computing, Information and Control, vol. 9, no. 12, pp. 4765–4779, 2013. \endinput