Convergence within a polyhedron: controller design for time-delay systems with bounded disturbances
- Author(s): Phan Thanh Nam 1 ; Pubudu Nishantha Pathirana 2 ; Hieu Trinh 2
-
-
View affiliations
-
Affiliations:
1:
Department of Mathematics, Quynhon University, Binhdinh, Vietnam;
2: School of Engineering, Deakin University, Geelong, VIC 3217, Australia
-
Affiliations:
1:
Department of Mathematics, Quynhon University, Binhdinh, Vietnam;
- Source:
Volume 9, Issue 6,
13 April 2015,
p.
905 – 914
DOI: 10.1049/iet-cta.2014.0628 , Print ISSN 1751-8644, Online ISSN 1751-8652
This study considers linear systems with state/input time-varying delays and bounded disturbances. The authors study a new problem of designing a static output feedback controller which guarantees that the state vector of the closed-loop system converges within a pre-specified polyhedron. Based on the Lyapunov–Krasovskii method combining with the free-weighting matrix technique, a new sufficient condition for the existence of a static output feedback controller is derived. The author's condition is expressed in terms of linear matrix inequalities with two parameters need to be tuned and therefore can be efficiently solved by using a two-dimensional search method combining with convex optimisation algorithms. To be able to obtain directly an output feedback control matrix from the derived condition, they propose an appropriate combination between a state transformation with a choice of a special form of the free-weighting matrices. The feasibility and effectiveness of the derived results are illustrated through five numerical examples.
Inspec keywords: delay systems; Lyapunov methods; time-varying systems; convex programming; vectors; search problems; feedback; convergence; linear matrix inequalities; linear systems; control system synthesis
Other keywords: free-weighting matrix technique; linear systems; state transformation; bounded disturbances; two-dimensional search method; free-weighting matrices; output feedback control matrix; Lyapunov-Krasovskii method; state/input time-varying delays; convergence; time-delay systems; linear matrix inequalities; prespecified polyhedron; static output feedback controller designing; closed-loop system; state vector; convex optimisation algorithms
Subjects: Stability in control theory; Algebra; Control system analysis and synthesis methods; Optimisation techniques; Distributed parameter control systems; Time-varying control systems
References
-
-
1)
-
22. Zuo, Z., Wang, Z., Chen, Y., Wang, Y.: ‘A non-ellipsoidal reachable set estimation for uncertain neural networks with time-varying delay’, Commun. Nonlinear Sci. Numer. Simul., 2014, 19, (4), pp. 1097–1106 (doi: 10.1016/j.cnsns.2013.08.015).
-
-
2)
- E. Fridman , M. Dambrine , N. Yeganefar . On input-to-state stability of systems with time delay: a matrix inequality approach. Automatica , 9 , 2364 - 2369
-
3)
- O.M. Kwon , S.M. Lee , J.H. Park . On the reachable set bounding of uncertain dynamic systems with time-varying delays and disturbances. Inf. Sci. , 3735 - 3748
-
4)
- E. Fridman , U. Shaked . On reachable sets for linear systems with delay and bounded peak inputs. Automatica , 2005 - 2010
-
5)
- J.P. Richard . Time-delay systems: an overview of some recent advances and open problems. Automatica , 1667 - 1694
-
6)
-
7. Du, B., Lam, J., Shu, Z.: ‘Stabilization for state/input delay systems via static and integral output feedback’, Automatica, 2010, 46, (12), pp. 2000–2007 (doi: 10.1016/j.automatica.2010.08.005).
-
-
7)
- E. Fridman , U. Shaked , K. Liu . New conditions for delay-derivative-dependent stability. Automatica , 3 , 2723 - 2727
-
8)
-
28. Zuo, Z., Fu, Y., Wang, Y.: ‘Results on reachable set estimation for linear systems with both discrete and distributed delays’, IET Control Theory Appl., 2012, 6, (14), pp. 2346–2350 (doi: 10.1049/iet-cta.2012.0491).
-
-
9)
-
8. Han, X., Fridman, E., Spurgeon, S.K.: ‘Sliding mode control in the presence of input delay: a singular perturbation approach’, Automatica, 2012, 48, (8), pp. 1904–1912 (doi: 10.1016/j.automatica.2012.06.016).
-
-
10)
-
26. Lam, J., Zhang, B., Chen, Y., Xu, S.: ‘Reachable set estimation for discrete-time linear systems with time delays’, Int. J. Robust Nonlinear Control, 2014, DOI: 10.1002/rnc.3086.
-
-
11)
-
29. Zuo, Z., Chen, Y., Wang, Y., Ho, D.W.C., Chen, M.Z.Q., Li, H.: ‘A note on reachable set bounding for delayed systems with polytopic uncertainties’, J. Franklin Inst., 2013, 350, (7), pp. 1827–1835 (doi: 10.1016/j.jfranklin.2013.04.025).
-
-
12)
-
32. Nam, P.T., Pathirana, P.N., Trinh, H.: ‘Exponential convergence of time-delay systems in the presence of bounded disturbances’, J. Optim. Theory Appl., 2013, 157, (3), pp. 843–852 (doi: 10.1007/s10957-012-0240-1).
-
-
13)
-
4. Nam, P.T., Phat, V.N.: ‘Robust stabilization of linear systems with delayed state and control’, J. Optim. Theory Appl., 2009, 140, (2), pp. 287–299 (doi: 10.1007/s10957-008-9453-8).
-
-
14)
-
38. Du, B., Lam, J., Shu, Z.: ‘Strong stabilisation by output feedback controller for linear systems with delayed input’, IET Control Theory Appl., 2012, 6, (10), pp. 1329–1340 (doi: 10.1049/iet-cta.2011.0312).
-
-
15)
-
30. Zuo, Z., Fu, Y., Chen, Y., Wang, Y.: ‘A new method of reachable set estimation for time delay systems with polytopic uncertainties’, Appl. Math. Comput., 2013, 221, (15), pp. 639–646 (doi: 10.1016/j.amc.2013.06.099).
-
-
16)
- C. Shen , S. Zhong . The ellipsoidal bound of reachable sets for linear neural systems with disturbances. J. Franklin Inst. , 2570 - 2585
-
17)
-
20. Seuret, A., Gouaisbaut, F.: ‘Wirtinger-based integral inequality: application to time-delay systems’, Automatica, 2013, 49, (9), pp. 2860–2866 (doi: 10.1016/j.automatica.2013.05.030).
-
-
18)
-
3. Zhang, X.M., Li, M., Wu, M., She, J.H.: ‘Further results on stability and stabilisation of linear systems with state and input delays’, Int. J. Syst. Sci., 2009, 40, (1), pp. 1–10 (doi: 10.1080/00207720802088223).
-
-
19)
-
27. Zhang, B., Lam, J., Xu, S.: ‘Reachable set estimation and controller design for distributed delay systems with bounded disturbances’, J. Franklin Inst., 2014, 351, (6), pp. 3068–3088 (doi: 10.1016/j.jfranklin.2014.02.007).
-
-
20)
- H. Gao , J. Lam , C. Wang , Y. Wang . Delay-dependent output-feedback stabilization of discrete-time systems with time-varying state delay. IEE Proc. Control Theory Appl. , 6 , 691 - 698
-
21)
- A. Haidar , E. Boukas , S. Xu , J. Lam . Exponential stability and static output feedback stabilization of singular time-delay systems with saturating actuators. IET Control Theory Appl. , 9 , 1293 - 1305
-
22)
-
31. Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnam, V.: ‘Linear matrix inequalities in systems and control theory’ (SIAM, Philadelphia, PA, 1994).
-
-
23)
- Z. Zuo , D.W.C. Ho , Y. Wang . Reachable set bounding for delayed systems with polytopic uncertainties: the maximal Lyapunov-Krasovskii functional approach. Automatica , 949 - 952
-
24)
- J.H. Kim . Improved ellipsoidal bound of reachable sets for time-delayed linear systems with disturbances. Automatica , 2940 - 2943
-
25)
- P. Park , J.W. Ko , C. Jeong . Reciprocally convex approach to stability of systems with time-varying delays. Automatica , 1 , 235 - 238
-
26)
- Y. He , Q.G. Wang , L. Xie , C. Lin . Further improvement of free-weighting matrices technique for systems with time-varying delay. IEEE Trans. Autom. Control , 2 , 293 - 299
-
27)
-
33. Seuret, A., Gouaisbaut, F.: ‘Integral inequality for time-varying delay systems’. Proc. European Control Conf., 2013, pp. 3360–3365.
-
-
28)
- P.T. Nam , P.N. Pathirana . Further result on reachable set bounding for linear uncertain polytopic systems with interval time-varying delays. Automatica , 1838 - 1841
-
29)
-
39. 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., 2013, 157, (1), pp. 96–107 (doi: 10.1007/s10957-012-0179-2).
-
-
30)
- X.M. Zhanga , M. Wua , J.H. She , Y. He . Delay-dependent stabilization of linear systems with time-varying state and input delays. Automatica , 8 , 1405 - 1412
-
31)
-
23. Hien, L.V., An, N.T., Trinh, H.: ‘New results on state bounding for discrete-time systems with interval time-varying delay and bounded disturbance inputs’, IET Control Theory Appl., 2014, 8, (14), pp. 1405–1414 (doi: 10.1049/iet-cta.2013.0980).
-
-
32)
-
28. Boyd, S., Vandenberghe, L.: ‘Convex optimization’ (Cambridge University Press, 2004).
-
-
33)
-
25. Nam, P.T., Pathirana, P., Trinh, H.: ‘Linear functional state bounding for perturbed time-delay systems and its application’, IMA J. Math. Control Inf., 2013, DOI: 10.1093/imamci/dnt039.
-
-
34)
-
24. Hien, L.V., Trinh, H.: ‘A new approach to state bounding for linear time-varying systems with delay and bounded disturbances’, Automatica, 2014, 50, (6), pp. 1735–1738 (doi: 10.1016/j.automatica.2014.04.025).
-
-
35)
-
6. Phat, V.N.: ‘Switched controller design for stabilization of nonlinear hybrid systems with time-varying delays in state and control’, J. Franklin Inst., 2010, 347, (1), pp. 195–207 (doi: 10.1016/j.jfranklin.2009.09.006).
-
-
1)