# Existence Conditions of Asymptotically Stable 2-D Feedback Control Systems on the Basis of Block Matrix Diagonalization

### Giido Izuta

#### Abstract

This work is concerned with the existence of asymptotically stable 2-D (2-dimensional) systems by means of a feedback control model represented by the system of partial difference equations and their Lagrange solutions. Thus, the goal is to establish a controller that provides a feedback control system with state variables depending solely on its Lagrange solution in the sense that the solution to the variable state is not a linear combination of other Lagrange solutions. Roughly speaking, the results showed that, to achieve such a control system, the controller has to diagonalize the block matrices of the matrices composing the system description model. Finally, a numerical example is presented to show how the controller is designed in order to generate a stable feedback control with given Lagrange solutions.

#### References

- Anderson, B. D. O. and Jury, E. I. (1974). Stability of multidimensional recursive filters. In IEEE Trans. Circuits and Syst.Proc. IEEE, number 21, pages 300-304.
- Attasi, S. (1973). Systemes lineaires homogenes a deux indices. In Rapport Laboria, number 31.
- Bose, N. K. (1982). Applied multdimensional systems theory. England: Van Nostradand Reinhold Co.
- Cheng, S. S. (2003). Partial difference equations. London: Taylor & Francis.
- Du, C. and Xie, L. (2002). H-infinity control and filtering of two-dimensional systems. Berlin, Germany: Springer Verlag.
- Elaydi, S. (2005). An introduction to difference equations. USA: Springer Science.
- Fornasini, E. and Marchesini, G. (1978). Doubly indexed dynamical systems: State space models and structural properties. In Math. Syst. Th., number 12, pages 59- 72.
- Fornasini, E. and Marchesini, G. (1980). Stability analysis of 2-d systems. In IEEE Trans. Circ. and Syst., volume CAS, pages 1210-1217.
- Givone, D. D. and Roesser, R. P. (1972). Multidimensional linear iterative circuits - general properties. In IEEE Trans. Comp., volume C, pages 1067-1073.
- Izuta, G. (2007). Stability and disturbance attenuation of 2- d discrete delayed systems via memory state feedback controller. Int. J. Gen. Systems, 36(3):263-280.
- Izuta, G. (2010). Stability analysis of 2-d discrete systems on the basis of lagrange solutions and doubly similarity transformed systems. In Proc. 35th annual conf. IEEE IES, Porto, Portugal.
- Izuta, G. (2012). Networked 2-d linear discrete feedback control systems design on the basis of observer controller. In Proc. 16th Int. Conf. on System Theory, Control and Computing IEEE CS, Sinaia, Romania.
- Izuta, G. (2014a). Controller design for 2-d discrete linear networked systems. In Proc. The Int. Electrical Eng. Congress, Pataya, Thailand.
- Izuta, G. (2014b). On the asymptotic stability analysis of a certain type of discrete-time 3-d linear systems. In Proc. 11th Int. Conf. on Informatics in Control, Automation and Robotics (ICINCO 2014), Vienna, Austria.
- Jerri, A. J. (1996). Linear difference equations with discrete transform methods. Netherlands: Kluwer Acad. Pub.
- Kaczorek, T. (1985). Two-dimensional linear systems. Berlin: Springer Verlag.
- Lim, J. S. (1990). Two-dimensional linear signal and image processing. New Jersey, USA: Prentice Hall.
- Pazke, W., Lam, J., Galkowski, K., and Xu, S. (2004). Robust stability and stabilisation of 2d discrete statedelayed systems. In Systems and Control, volume 51, pages 277-291.
- Piekarski, M. S. (1977). Algebraic characterization of matrices whose multivariable characteristic polynomial is hurwitzian. In Proc. Int. Symp. Operator Theory, pages 121-126.
- Rogers, E., Krzysztof, G., and Owens, D. H. (2007). Control Systems Theory and Applications for Linear Repetitive Processes. Springer, London.
- Tzafestas, S. G. (1986). Multidimensional systems - Techniques and Applications. New York: Marcel Dekker.
- Zerz, E. (2000). Topics in Multidimensional Linear Systems Theory. Springer.

#### Paper Citation

#### in Harvard Style

Izuta G. (2016). **Existence Conditions of Asymptotically Stable 2-D Feedback Control Systems on the Basis of Block Matrix Diagonalization** . In *Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,* ISBN 978-989-758-198-4, pages 463-470. DOI: 10.5220/0005975604630470

#### in Bibtex Style

@conference{icinco16,

author={Giido Izuta},

title={Existence Conditions of Asymptotically Stable 2-D Feedback Control Systems on the Basis of Block Matrix Diagonalization},

booktitle={Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,},

year={2016},

pages={463-470},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005975604630470},

isbn={978-989-758-198-4},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics - Volume 1: ICINCO,

TI - Existence Conditions of Asymptotically Stable 2-D Feedback Control Systems on the Basis of Block Matrix Diagonalization

SN - 978-989-758-198-4

AU - Izuta G.

PY - 2016

SP - 463

EP - 470

DO - 10.5220/0005975604630470