Abstract
This paper addresses the problem of reachable set bounding for linear discrete-time systems that are subject to state delay and bounded disturbances. Based on the Lyapunov method, a sufficient condition for the existence of ellipsoid-based bounds of reachable sets of a linear uncertain discrete system is derived in terms of matrix inequalities. Here, a new idea is to minimize the projection distances of the ellipsoids on each axis with different exponential convergence rates, instead of minimization of their radius with a single exponential rate. A smaller bound can thus be obtained from the intersection of these ellipsoids. A numerical example is given to illustrate the effectiveness of the proposed approach.
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References
Udwadia, F.E., von Bremen, H., Phohomsiri, P.: Time-delayed control design for active control of structures: principles and applications. Struct. Control Health Monit. 14, 27–61 (2007)
Phat, V.N., Niamsup, P.: Stabilization of linear nonautonomous systems with norm-bounded controls. J. Optim. Theory Appl. 131, 135–149 (2006)
Gao, H., Chen, T.: New results on stability of discrete-time systems with time-varying state delay. IEEE Trans. Autom. Control 52, 328–334 (2007)
Meng, X., Lam, J., Du, B., Gao, H.: A delay-partitioning approach to the stability analysis of discrete-time systems. Automatica 46, 610–614 (2010)
Durieu, C., Walter, E., Polyak, B.: Multi-input multi-output ellipsoidal state bounding. J. Optim. Theory Appl. 111, 273–303 (2001)
Fridman, E., Shaked, U.: On reachable sets for linear systems with delay and bounded peak inputs. Automatica 39, 2005–2010 (2003)
Kim, J.H.: Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances. Automatica 44, 2940–2943 (2008)
Zuo, Z., Ho, D.W.C., Wang, Y.: Reachable set bounding for delayed systems with polytopic uncertainties: the maximal Lyapunov–Krasovskii functional approach. Automatica 46, 949–952 (2010)
Shen, C., Zhong, S.: The ellipsoidal bound of reachable sets for linear neutral systems with disturbances. J. Franklin Inst. 348, 2570–2585 (2011)
Nam, P.T., Pathirana, P.N.: Further result on reachable set bounding for linear uncertain polytopic systems with interval time-varying delays. Automatica 47, 1838–1841 (2011)
Kwon, O.M., Lee, S.M., Park, J.H.: On the reachable set bounding of uncertain dynamic systems with time-varying delays and disturbances. Inf. Sci. 181, 3735–3748 (2011)
Corless, M., Leitmann, G.: Bounded controllers for robust exponential convergence. J. Optim. Theory Appl. 76, 1–12 (1993)
Peng, C., Tian, Y.C.: Delay-dependent robust H ∞ control for uncertain systems with time-varying delay. Inf. Sci. 179, 3187–3197 (2009)
Acknowledgements
The authors would like to thank the reviewers for their very helpful comments and suggestions. This work is supported by the New South Wales Government through its Environmental Trust under grant 2012/RDS/0034 and, in part, by the Vietnam International Education Development (VIED)-UTS Ph.D. Scholarship Program and the National Foundation for Science and Technology Development, Vietnam under grant 101.01.2011.51.
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That, N.D., Nam, P.T. & Ha, Q.P. Reachable Set Bounding for Linear Discrete-Time Systems with Delays and Bounded Disturbances. J Optim Theory Appl 157, 96–107 (2013). https://doi.org/10.1007/s10957-012-0179-2
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DOI: https://doi.org/10.1007/s10957-012-0179-2