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On the improvement of linear discrete system stability: the maximal set of the \(F\)-admissible initial states. (English) Zbl 1066.93046

A discrete linear control system is considered under the assumption that its output is stabilizable by a given state-feedback control law. The author characterizes the set of all initial states \(x_0\) for which the output function \(y(i)\) of the closed-loop system satisfies the constraints \(\| y(i)\| \leq \alpha_i\), for all positive integer numbers \(i\). Here, the positive numbers \(\alpha_i\) are appropriately chosen and can be interpreted as “a desired degree” of stability. An algorithm for constructing this set of initial states is proposed and some numerical simulations are presented. The case of discrete-time delayed control systems is also studied.

MSC:

93D15 Stabilization of systems by feedback
93C55 Discrete-time control/observation systems
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[1] L. Baghdadi, Stabilisation des systèmes linéaires dans les espaces de Hilbert , Thèse de Magister, Département de Mathématiques, Université d’Oran, 1985.
[2] A.V. Balakrishnan, Applied functional analysis , Springer-Verlag, Berlin, 1976. · Zbl 0333.93051
[3] S. P. Banks, State-space and frequency-domain methods in the control of distributed parameter systems , Peter Peregrinus, London, 1983. · Zbl 0575.93037
[4] A. Benzaouia, The regulator problem for a class of linear systems with constrained control , Systems Control Lett. 10 (1988), 357-363.
[5] G. Bitsoris, On the positive invariance of polyhedral sets for discrete-time systems , Systems Control Lett. 11 (1988), 243-248. · Zbl 0661.93045 · doi:10.1016/0167-6911(88)90065-5
[6] R.F. Curtain and A.J. Pritchard, Infinite dimensional linear systems theory , Springer-Verlag, Berlin, 1978. · Zbl 0389.93001
[7] E.G. Gilbert and Tin Tan, Linear systems with state and control constraints : The theory and application of maximal output admissible sets , IEEE Trans. Automat. Contr. 36 (1991), 1008-1019. · Zbl 0754.93030 · doi:10.1109/9.83532
[8] P.O. Gutman and M. Cwikel, An algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded controls and states , IEEE Trans. Automat. Contr. AC -30, 251-254. · Zbl 0618.49017 · doi:10.1109/TAC.1987.1104567
[9] P.O. Gutman and P. Hagander, A new design of constrained controllers for linear systems , IEEE Trans. Automat. Contr. AC -30 (1985), 22-33. · Zbl 0553.93052 · doi:10.1109/TAC.1985.1103785
[10] C.D. Johnson and W.M. Wonham, On a problem of Letov in optimal control , Trans. ASME J. Basic Engrg. Ser. D 87 (1965), 81-89.
[11] S.S. Keerthi, Optimal feedback control of discrete-time systems with state-control constrains and general cost functions , Ph.D. Dissertation, Computer Informat. Control Engrg., University of Michigan, Ann Arbor, Michigan, 1986. · Zbl 0592.93048
[12] S.S. Keerthi and E.G. Gilbert, Optimal infinite horizon feedback laws for a general class of constrained discrete-time systems : Stability and moving-horizon approximations , J. Optim. Theory Appl. 57 (1988), 265-293. · Zbl 0622.93044 · doi:10.1007/BF00938540
[13] A.J. Pritchard and J. Zabczyk, Stability and stabilizability of infinite dimensional systems , SIAM Rev. 23 (1981), 25-52. JSTOR: · Zbl 0452.93029 · doi:10.1137/1023003
[14] R. Rabah and D. Ionescu, Stabilization problem in Hilbert spaces , Internat. J. Control 46 (1987), 2035-2042. · Zbl 0634.93063 · doi:10.1080/00207178708934032
[15] M. Rachik, A. Abdelhak and J. Karrakchou, Discrete systems with delays in state, control and observation : The maximal output sets with state and control constraints , Optimization 42 (1997), 169-183. · Zbl 0892.93045 · doi:10.1080/02331939708844357
[16] M. Rachik, E. Labriji, A. Abkari and J. Bouyaghroumni, Infected discrete linear systems : On the admissible sources , Optimization 48 (2000), 271-289. · Zbl 0978.93044 · doi:10.1080/02331930008844506
[17] M. Rachik, M. Lhous, A. Tridane and A. Abdelhak, Discrete nonlinear systems : On the admissible nonlinear disturbances , J. Franklin Inst. 338 (2001), 631-650. · Zbl 1012.93042 · doi:10.1016/S0016-0032(01)00018-7
[18] M. Rachik, A. Tridane and M. Lhous, Discrete infected controlled nonlinear systems : On the admissible perturbation , SAMS 41 (2001), 305-323. · Zbl 1017.93066
[19] J.M. Schumacher, A direct approach to compensator design for distributed parameter systems , SIAM J. Control Optim. 21 (1983), 823-836. · Zbl 0524.93054 · doi:10.1137/0321050
[20] R. Triggiani, On the stabilizability problem in Banach spaces , J. Math. Anal. Appl. 52 (1975), 383-403. · Zbl 0326.93023 · doi:10.1016/0022-247X(75)90067-0
[21] M. Vassilaki, J.C. Hennet and G. Bistoris, Feedback control of linear discrete time systems under state and control constrains , Internat. J. Control 47 (1988), 1727-1735. · Zbl 0644.93046 · doi:10.1080/00207178808906132
[22] V.I. Vorotnokov, Partial stability and control , Birkhauser, Boston, 1998.
[23] W.M. Wonham, Linear multivariable control , A geometric approach , Springer-Verlag, New York, 1985. · Zbl 0609.93001
[24] K. Yosida, Functional analysis , Springer-Verlag, New York, 1980. · Zbl 0435.46002
[25] K. Yoshida, Y. Nishimura and Y. Yonezawa, Variable gain feedback control for linear sampled-data systems with bounded control , Control Theory Adv. Tech. 2 -2 (1986), 313-323.
[26] K. Yoshida, H. Kawabe, Y. Nishimura and Y. Yonezawa, A design of saturating control systems with state and input constrains , Proc. 12th IFAC World Congress (Sydney), Vol. 1, 1993, pp. 81-86.
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