Abstract
This chapter presents the backgrounds on two-dimensional (2-D) systems: first, 2-D representations are discussed, then definitions of 2-D stability are provided and complemented with some previous results that will be useful in the rest of the book; then some techniques used for dealing with actuator saturation are discussed, as used in Chaps. 2 and 4.
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References
E.I. Jury, Stability of multidimensional scalar and matrix polynomials. Proc. IEEE 66, 1018–1047 (1978)
M. Piekarski, Algebraic characterization of matrices whose multivariable characteristic polynomial is Hermitian, in Proceedings of the International Symposium on the Operator Theory of Networks and Systems, Lubbock, Texas, 17–19 August, pp. 121–126 (1977)
J.L. Shanks, S. Treitel, J.H. Justice, Stability and synthesis of two-dimensional recursive filters. IEEE Trans. Audio Electroacoust. 20(2), 115–128 (1979)
J.H. Justice, J.L. Shanks, Stability criterion for N-dimensional digital filters. IEEE Trans. Autom. Control 18(3), 284–286 (1973)
N.K. Bose, Applied Multidimensional Systems Theory (Van Nostrand Reinhold, New York, 1982)
R.N. Bracewell, Two Dimensional Imaging (Prentice Hall Inc., Englewood Cliffs, 1995)
W.S. Lu, A. Antoniou, Two Dimensional Digital Filters, Electrical Engineering and Electronics Series, vol. 80 (Marcel Dekker, New York, 1992)
T. Kaczorek, Two Dimensional Linear Systems (Springer, Berlin, 1985)
J.R. Cui, G.D. Hu, Q. Zhu, Stability and robust stability of 2-D discrete stochastic systems. Discret. Dyn Nat. Soc., Article ID 545361, 11 pp. (2011)
E. Fornasini, G. Marchesini, State-space realization theory of two-dimensional filters. IEEE Trans. Autom. Control 21(4), 484–492 (1976)
E. Fornasini, G. Marchesini, Doubly-indexed dynamical systems: state-space models and structural properties. Math. Syst. Theory 12(1), 59–72 (1978)
R. Eising, Realization and stabilization of 2D systems. IEEE Trans. Autom. Control 23(5), 793–799 (1978)
H. Xu, Y. Zou, \(H_{\infty }\) control for 2-D singular delayed systems. Int. J. Syst. Sci. 42(4), 609–619 (2011)
B.O. Anderson, P. Agathoklis, E.I. Jury, M. Mansour, Stability and the matrix Lyapunov equation for discrete 2-dimensional systems. IEEE Trans. Circuits Syst. CAS-33(3), 261–266 (1986)
K. Galkowski, LMI based stability analysis for 2-D continuous systems, in International Conference on Electronics Circuits and Systems, vol. 3, Dubrovnik, Croatia, 15–18 September, pp. 923–926 (2002)
H.D. Tuan, P. Apkarian, T.Q. Nguyen, T. Narikiys, Robust mixed \(H_{2}/H_{\infty }\) filtering of 2-D systems. IEEE Trans. Signal Process 50(7), 1759–1771 (2002)
T. Ooba, On stability analysis of 2-D systems based on 2-D Lyapunov matrix inequalities. IEEE Trans. Circuits Syst. I 47(8), 1263–1265 (2000)
Y. Zou, S.L. Campbell, The jump behavior and stability analysis for 2-D singular systems. Multidimens. Syst. Signal Process. 11(3), 339–358 (2000)
C. Cai, W. Wang, Y. Zou, A note on the internal stability for 2-D acceptable linear singular discrete systems. Multidimens. Syst. Signal Process. 15(2), 197–204 (2004)
T. Hu, Z. Lin, The equivalence of several set invariance conditions under saturations, in Proceedings of the 41st IEEE Conference on Decision and Control, 10–13 December, Las Vegas, USA (2002)
K. Galkowski, E. Rogers, W. Paszke, D.H. Owens, Linear repetitive process control theory applied to physical example. Int. J. Appl. Math. Comput. Sci. 13(1), 87–99 (2003)
W. Paszke, K. Galkowski, E. Rogers, D.H. Owens, \(H\infty \) control of differential linear repetitive processes. IEEE Trans. Circuits Syst. II: Analog Digit. Signal Process. 53(1), 39–44 (2006)
A. Berman, R.J. Plemmon, Nonnegative matrices in the mathematical sciences. SIAM Class. Appl. Math. 9 (1994)
R. Horn, C. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1991)
R.E. Skelton, T. Iwasaki, K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design (Taylor-Francis, Bristol, 1998)
S. Boyd, L. EI Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Applied Mathematics (SIAM, Philadelphia, 1994)
P. Gahinet, P. Apkarian, A linear matrix inequality approach to \(H_{\infty }\) control. Int. J. Robust Nonlinear Control 4(4), 421–448 (1994)
F. Delmotte, T.M. Guerra, M. Ksantini, Continuous Takagi–Sugeno’s models: reduction of the number of LMI conditions in various fuzzy control design technics. IEEE Trans. Fuzzy Syst. 15(3), 426–438 (2007)
J. Qiu, G. Feng, J. Yang, A new design of delay-dependent robust \(H_{\infty }\) filtering for continuous-time polytopic systems with time-varying delay. Int. J. Robust Nonlinear Control 20(3), 346–365 (2010)
S. Xu, J. Lam, Z. Lin, K. Galkowski, Positive real control for uncertain two-dimensional systems. IEEE Trans. Circuits Syst. I 49(11), 1659–1666 (2002)
K. Gu, An integral inequality in the stability problem of time-delay systems, in The 39th IEEE Conference on Decision Control, Sydney, Australia, 12–15 December, pp. 2805–2810 (2000)
P.A. Bliman, R.C.L.F. Oliveira, V.F. Montagner, P.L.D. Peres, Existence of homogeneous polynomial solutions for parameter-dependent linear matrix inequalities with parameters in the simplex, in Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, CA, USA, pp. 1486–1491, 13–15 December 2006
A. Hmamed, F. Mesquine, M. Benhayoun, A. Benzaouia, F. Tadeo, Stabilization of 2-D saturated systems by state feedback control. Multidimens. Syst. Signal Process. 21(3), 277–292 (2010)
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Benzaouia, A., Hmamed, A., Tadeo, F. (2016). Introduction to Two-Dimensional Systems. In: Two-Dimensional Systems. Studies in Systems, Decision and Control, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-319-20116-0_1
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