Local control Lyapunov functions for constrained linear discrete-time systems: The Minkowski algebra approach

IEEE Transactions on Automatic Control

Volumn 54, Issue 11, 2009, Pages 2686-2692

  Raković, Saša V ^a,b   Barić, Miroslav ^c  

^a OTTO VON GUERICKE UNIVERSITY   (Germany)

^b UNIVERSITY OF OXFORD   (United Kingdom)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Constrained control; Lyapunov functions; Stability analysis

Indexed keywords

BASIC SHAPES; BELONG TO; COMPLEXITY ESTIMATES; CONSTRAINED CONTROLS; CONVEX OPTIMIZATION PROBLEMS; CONVEX SET; DESIGN MATRIX; ILLUSTRATIVE EXAMPLES; INVARIANT SET; LINEAR DISCRETE-TIME SYSTEMS; LINEAR PROGRAMS; LOCAL CONTROL; LYAPUNOV; MINKOWSKI; STABILITY ANALYSIS; SYSTEM MATRICES; TECHNICAL NOTES;

CONVEX OPTIMIZATION; DIFFERENTIAL EQUATIONS; DIGITAL CONTROL SYSTEMS; LINEAR TRANSFORMATIONS; LYAPUNOV FUNCTIONS; LYAPUNOV METHODS; SET THEORY;

DISCRETE TIME CONTROL SYSTEMS;

EID: 79952902273     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2009.2031579     Document Type: Article

References (27)

1. A. M. Lyapunov, The General Problem of The Stability of Motion. New York: Taylor & Francis, 1992.
2. W. Hahn, Stability of motions. Berlin, German: Springer, 1967.
3. J. P. LaSalle, The stability of dynamical systems, ser. CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1976.
4. E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, ser. Textbooks in Applied Mathematics, 2nd ed. New York: Springer, 1998.
5. Z. Artstein, "Stabilization with relaxed controls", Nonlinear Analysis, Theory, Methods & Applications, vol. 7, no. 11, pp. 1163-1173, 1983.
6. L. Gurvits, "Stability of discrete linear inclusion", Lin. Alg. and its Applic., vol. 231, pp. 47-85, 1995.
7. E. D. Sontag and H. J. Sussmann, "General classes of control-Lyapunov functions", in Stability Theory, ser. International Series of Numerical Mathematics. Boston, MA: Birkhaüser, 1996, vol. 121.
8. A. Polański, "Further comments on vector norms as Lyapunov functions for linear systems", IEEE Trans. Autom. Control, vol. 43, no. 2, pp. 289-291, Feb. 1998. (Pubitemid 128736481)
9. F. Blanchini and S. Miani, Set-Theoretic Methods in Control, ser. Systems & Control: Foundations & Applications. Boston, MA: Birkhauser, 2008.
10. M. Sznaier, "A set induced norm approach to the robust control of constrained systems", SIAM J. Control and Optimiz., vol. 31, no. 3, pp. 733-746, 1993. (Pubitemid 23710958)
11. A. Zelentsovsky, "Nonquadratic Lyapunov functions for robust stability analysis of linear uncertain systems", IEEE Trans. Autom. Control, vol. 39, no. 1, pp. 135-138, Jan. 1994.
12. F. Blanchini, "Nonquadratic Lyapunov functions for robust control", Automatica, vol. 31, no. 3, pp. 451-461, 1995.
13. J. P. Aubin, Viability theory, ser. Systems & Control: Foundations & Applications. Boston, MA: Birkhaüser, 1991.
14. F. Blanchini, "Set invariance in control-A survey", Automatica, vol. 35, no. 11, pp. 1747-1767, 1999.
15. D. P. Bertsekas, "Infinite-time reachability of state-space regions by using feedback control", IEEE Trans. Autom. Control, vol. AC-17, no. 5, pp. 604-613, Oct. 1972.
16. I. V. Kolmanovsky and E. G. Gilbert, "Theory and computation of disturbance invariant sets for discrete-time linear systems", Mathemat. Problems in Eng., vol. 4, pp. 317-367, 1998. (Pubitemid 128557716)
17. V. M. Kuntsevich and B. N. Pshenichnyi, "Minimal invariant sets of dynamic systems with bounded disturbances", Cybern. and Syst. Analy., vol. 32, pp. 58-64, 1996.
18. S. V. Raković, E. C. Kerrigan, K. I. Kouramas, and D. Q. Mayne, "Invariant approximations of the minimal robustly positively invariant sets", IEEE Trans. Autom. Control, vol. 50, no. 3, pp. 406-410, Mar. 2005. (Pubitemid 40448594)
19. S. V. Raković, "Minkowski algebra and Banach contraction principle in set invariance for linear discrete time systems", in Proc. 46th IEEE Conf. Decision and Control (CDC 2007), New Orleans, LA, Dec. 2007.
20. Z. Artstein and S. V. Raković, "Feedback and invariance under uncertainty via set iterates", Automatica, vol. 44, no. 2, pp. 520-525, 2008.
21. S. V. Raković, "Robust Control of Constrained Discrete Time Systems: Characterization and Implementation", Ph. D. dissertation, Imperial College London, London, U. K., 2005.
22. S. V. Raković, E. C. Kerrigan, D. Q. Mayne, and K. I. Kouramas, "Optimized robust control invariance for linear discrete-time systems: Theoretical foundations", Automatica, vol. 43, pp. 831-841, 2007. (Pubitemid 46507431)
23. R. T. Rockafellar, Convex Analysis. Princeton, NJ: Princeton Univ. Press, 1970.
24. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, ser. Encyclopedia of Mathematics and its Applications. Cambridge, U. K.: Cambridge Univ. Press, 1993, vol. 44.
25. B. Bank, J. Guddat, D. Klatte, B. Kummer, and K. Tammer, Non-linear Parametric Optimization. Boston/Basel/Berlin: Birkhauser, 1983.
26. G. M. Ziegler, Lectures on Polytopes. New York: Springer, 1994.
27. ∈ib: Constrained Matrix-Optimization Problem Library-A Collection of Test Examples for Nonlinear Semidefinite Programs, Control System Design and Related Problems. Trier, Germany: Univ. Trier, 2004 [Online]. Available: www.compleib.de