Abstract
This paper presents a brief summary of some old ideas concerning the control of uncertain systems based on dynamic programming. We show some recent developments and mention some open problems in the area.
This is a preview of subscription content, log in via an institution.
Preview
Unable to display preview. Download preview PDF.
References
J. Abedor, K. Nagpal and K. Poolla, “A linear matrix inequality approach to peak-to-peak gain minimization. Linear matrix inequalities in control theory and applications”, Internat. J. Robust Nonlinear Control, 6, no. 9–10, pp. 899–927, 1996.
N.E. Barabanov, “Lyapunov indicator of discrete inclusion”, Autom. and Rem. Contr. parts I,II,III, Vol. 49, no. 2, pp. 152–157; no. 4, pp. 283–287; no. 5, pp. 558–566, 1988.
T. Basar, “Dynamic noncooperative games”, Acad. Press, New York, 1992.
B.R. Barmish, I. Petersen and A. Feuer, “Linear Ultimate Boundedness Control of Uncertain Systems”, Automatica, Vol. 19, No. 5, pp. 523–532, 1983.
B.R.Barmish, M.Corless and G.Leitmann, “A new class of stabilizing controllers for uncertain dynamically systems”, SIAM J. on Contr. and Optim. Vol. 21, No.2, March 1983.
B.R. Barmish and J. Sankaran, “The propagation of parametric uncertainty via polytopes”, IEEE Trans. Automat. Control, Vol. 24, no. 2, pp. 346–349, 1979.
A. Bemporad, A. Casavola and E. Mosca, “Nonlinear Control of constrained linear systems with predictive reference management”, IEEE Trans. on Autom. Contr., Vol. 42, no. 3 March, pp. 340–349, 1997.
A. Bemporad and E. Mosca, “Fulfilling hard constraints in uncertain linear systems by reference managing”, Automatica, Vol. 34, no. 4, 451–461, 1998.
J. Bernussou, P.L.D. Peres and J. Geromel, “On a convex parameter space method for linear control design of uncertain systems”, SIAM J. on Contr. and Opt., 29, pp. 381–402, 1991.
D.P. Bertsekas and I.B Rhodes “On the minmax reachability of target set and target tubes”, Automatica, Vol. 7, pp. 233–247, 1971.
D.P. Bertsekas and I.B. Rhodes, “Recursive state estimation for a setmembership description of uncertainty”, IEEE Trans. Automatic Control, Vol. 16, pp. 117–128, 1971.
D.P. Bertsekas, “Infinite-time reachability of state-space regions by using feedback control”. IEEE Trans. Automatic Control, Vol. 17, no. 5, pp. 604–613, 1972.
D.P. Bertsekas, “Dynamic programming and optimal control”, Athena Scientific, Belmont, Ma, 1995.
A. Bhaya and F.C. Mota “Equivalence of stability concepts for discrete time-varying systems” International J. on Robust and Nonlinear Control, Vol. 4, no. 6, p. 725–740, 1994.
F. Blanchini, “Control Synthesis of Discrete Time Systems with Control and State Bounds in the Presence of Disturbance” Journal of Optimization Theory and Applications, vol. 65, no. 1, p. 29–40, April 1990.
F. Blanchini “Ultimate boundedness control for discrete-time uncertain system via set-induced Lyapunov functions”, IEEE Trans. on Autom. Contr., Vol. 39, no. 2, 428–433, 1994.
F. Blanchini “Nonquadratic Lyapunov function for robust control”, Automatica, Vol. 31, no. 3, p. 451–461, 1995.
F.Blanchini, “Set invariance in control — a survey, Automatica, to appear, 1999.
F.Blanchini and S.Miani, (1995a) “On the transient estimate for linear systems with time-varying uncertain parameters”, IEEE Trans. on Circ. and Syst., Part I, to appear.
F.Blanchini and S.Miani, “Piecewise-linear functions in robust control”, in Robust control via variable structure and Lyapunov methods, Edited by F. Garofalo and L. Glielmo, Springer Verlag, pp. 213–240, 1996.
F.Blanchini and S.Minai, “A universal class of smooth functions for robust control”, IEEE Transactions on Automatic Control, March 1999, to appear.
F. Blanchini and M. Sznaier, “Persistent disturbance rejection via static state feedback”, IEEE Trans. on Autom. Contr., Vol. 40, no. 6, pp. 1127–1131, 1995.
F.Blanchini, S.Miani and M.Sznaier, “Worst case l ∞ to l ∞ gain minimization: dynamic versus static state feedback”, Proceedings of Proceedings of the 35th Conference on Decision and Control, 1996, Special session on Set Theoretic Methods in Control and Estimations, pp. 2395–2400
F.Blanchini and A.Megretski, “Robust state feedback control of lti systems: nonlinear is better than linear”, IEEE Transactions on Automatic Control, 1999, accepted for publication.
F. Blanchini and W. Ukovich, “A Linear Programming Approach to the Control of Discrete-Time Periodic System with State and Control Bounds in the Presence of Disturbance', Journal of Optimization Theory and Applications, Vol. 73, no. 3, p. 523–539, September 1993.
F.Blaquiere, F.Gerard and G.Leitmann, “Quantitative and qualitative games”, Academic Press, 1969.
S.Boyd, L.El Ghaoui, E.Feron and V.Balakrishnan, “Linear matrix inequality in system and control theory”, SIAM Studies in Applied Mathematics, Philadelphia, 1994.
R.K. Brayton C.H. Tong, “Constructive stability and asymptotic stability of dynamical systems”, IEEE Trans. on Circ. and Syst., Vol. CAS-27, no. 11, pp. 1121–1130, 1980.
T.S. Chang and D.E. Seborg, “A Linear Programming Approach for Multivariable Feedback Control with Inequality Constraints”, International Journal of Control, Vol. 37, No. 3, pp. 583–597, 1983.
L. Chisci, A. Garulli and G. Zappa, “Recursive state bounding by parallelotopes”, Automatica, vol. 32, pp. 1049–1056, 1996.
M.Corless, “Robust analysis and controller design via quadratic Lyapunov functions”. In A. Zinober, editor, Variable structure and Lyapunov Control, Vol 193, Lecture notes in in control and information science, Chapter 9, pp. 181–203, Springer Verlag, 1994.
M.A. Dahleh and J.B. Pearson, “l 1-Optimal feedback controllers for MIMO discrete-time systems,” IEEE Trans. Automat. Contr., Vol. AC-32, no.4, pp. 314–322, 1987.
J.H. De Vleger, H.B Verbuggen and P.M Bruijn, “A time-optimal control algorithm for digital computer control”, Automatica, Vol. 18, no. 2, pp. 219–244, 1982.
P. Dorato, R. Tempo and G. Muscato, “Bibliography on robust control”, Automatica, Vol. 29, no. 1, p. 201–214, 1993.
J.D. Glover and F.C. Schweppe, “Control of linear dynamic systems with set constrained disturbances”, IEEE Trans. on Autom. Contr., Vol AC-16, no. pp. 411–423, 1971.
B. Grumbaum, “Convex polytopes”, John Wiley & Sons, New York, 1969.
K. Gu, Y.H. Chen, M.A. Zohdy and N.K. Loh, “Quadratic stabilizability of uncertain systems: a two level optimization setup”, Automatica, Vol. 27, no. 1, pp. 161–165, 1991.
S. Gutman, “Uncertain dynamic systems — a Lyapunov min-max approach”, IEEE Trans. on Autom. Contr., Vol. 24, no. 3, pp. 437–443, 1979.
P.O. Gutman and M. Cwikel, “Admissible sets and feedback control for discrete-time linear systems with bounded control and states”, IEEE Trans. on Autom. Contr., Vol. AC-31, No. 4, pp. 373–376, 1986.
P.O. Gutman and M. Cwikel, “An algorithm to find maximal state constraint sets for discrete-time linear dynamical systems with bounded control and states”, IEEE Trans. on Autom. Contr., Vol. 32, No. 3, pp. 251–254, 1987.
S.Gutman and G.Leitmann, “On a class of linear differential games”, Int. Jour. of Opt. Th. and Appl., Vol. 17, no. 5/6, 1975.
P. Kargonekar, I.R. Petersen and K. Zhou, “Robust stabilization of uncertain systems and H ∞ optimal control”, IEEE Trans. on Autom. Contr., Vol. AC-35, pp. 356–361, 1990.
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., Vol. 57, no. 2, pp. 265–293, 1988.
S.S. Keerthi and E.G. Gilbert, “Computation of minimum-time feedback control laws for discrete-time systems with state-control constraints”, IEEE Trans. on Autom. Contr., Vol. 32, no. 5, pp. 432–435, 1987.
Krasovskii and Krasovskii “Control under lack of information”, Birkauser, 1995.
G. Leitmann, “Guaranteed asymptotic stability for some linear systems with bounded uncertainties”, Trans. of The A.S.M.E. Vol. 101, p. 212–216, 1979.
Y. Lin, E.D. Sontag and Y. Wang, “A smooth converse Lyapunov theorem for robust stability”, SIAM J. on Contr. and Opt., Vol. 34, no. 1, pp. 124–160, 1996.
D.Q. Mayne and H. Michalska, “Receding horizon control of nonlinear systems”. IEEE Trans. Automat. Control, 35, no. 7, pp. 814–824, 1990.
A.M. Meilakhs, “Design of stable systems subject to parametric perturbations”, Automation and Remote Contr., Vol. 39, no. 10, pp. 1409–1418, 1979.
A.N. Michael, B.H.Nam and V.Vittal, “Computer generated Lyapunov Functions for Interconnected Systems: Improved Results with applications to power systems”, IEEE Trans. on Circ. and Syst. Vol. 31, no. 2, 1984.
M. Milanese and G. Belforte, “Estimation Theory and Uncertain Intervals evaluation in the presence of Unknown bub Bounded Errors: Linear Families of Models end Estimators”, IEEE Trans. on Autom. Contr., Vol. AC-27, pp. 408–414, 1982.
M. Milanese and A. Vicino “Optimal Estimation Theory for Dynamic Systems with Set Membership Uncertainty: an Overview”, Automatica, Vol. 27, pp. 997–1009, 1991.
A.P. Molchanov and E.S. Pyatnitskii, “Lyapunov functions specifying necessary and sufficient conditions of absolute stability of nonlinear nonstationary control system”, Autom. and Rem. Contr., parts I,II,III, Vol. 47, no. 3, pp. 344–354; no. 4, pp.443–451; no. 5, pp. 620–630, 1986.
R. Morris and R.F. Brown, Extension of Validity of the GRG Method in Optimal Control Calculation, IEEE Transactions on Automatic Control, Vol. 21, pp. 420–422, 1976.
Y.Ohta, H.Imanishi, L.Gong and H.Haneda, “Computer generated Lyapunov functions for a class of nonlinear systems”, IEEE Trans. on Circ. and Syst. Vol. 40, no. 5, 1993.
I.R. Petersen, “Quadratic stabilizability of uncertain systems: existence of a nonlinear stabilizing control does not imply the existence of a linear stabilizing control”, IEEE Trans. on Autom. Contr. Vol. 30, no. 3, pp. 292–293, 1995.
I.R. Petersen and C. Hollot, “A Riccati equation approach to the stabilization of uncertain systems”, Automatica, Vol. 22, p. 397–411, 1986.
A.I. Propoi, “Use of linear programming methods for synthesizing sampled data automatic systems”, Autom. and Remote Control, vol. 24, pp. 837–844, 1963.
A. Rantzer and M. Johansson, “Computation of piecewise quadratic functions for hybrid systems”, IEEE Trans. on Autom. Contr. Vol. 43, no. 4, pp. 555–559, 1998.
M.A. Rotea and P.P. Kargonekar, “Stabilization of uncertain systems with norm-bounded uncertainty: a Lyapunov function approach”, SIAM J. on Contr. and Opt., Vol. 27, p.1462–1476, 1989.
D.D. Siljak “Parameter space methods for robust control design: a guided tour”, IEEE Trans. on Autom. Contr., Vol 34, no. 7, p.674–688, 1989.
J.S. Shamma “Nonlinear state feedback for l 1 optimal control”, Syst. and Contr. Lett., Vol. 21, pp. 40–41, 1994.
J.S. Shamma, “Optimization of the l ∞-induced norm under full state feed-back”, IEEE Trans. Automat. Control 41, no. 4, pp. 533–544, 1996.
J.S.Shamma and D.Xiong, “Control of Rate Constrained Linear Parameter Varying Systems”, Proc. of the 34th CDC, New Orleans, La, Dec 1995, pp. 2515–2520.
J.S.Shamma and K.Y.Tu, “Set-valued observer and optimal disturbance rejection”, IEEE Trans, to appear, 1999.
A.A. Stoorvogel, “Nonlinear L 1 optimal controllers for linear systems”, IEEE Trans. Automat. Control, Vol. 40, no. 4, pp. 698–690, 1996.
M. Sznaier and M. Damborg, “Control of Constrained Discrete Time Linear Systems Using Quantized Controls”, Automatica, Vol. 25, no. 4, pp. 623–628, 1989.
R. Tempo, “Robust Estimation and Filtering in the Presence of Bounded Noise”, IEEE Trans. On Autom. Contr., Vol. 21, pp. 420–422, 1988.
P.B. Usoro, F.C. Schweppe, L.A. Gould and Wormley D.N., “A Lagrange Approach to Set Theoretic Control Synthesis”, IEEE Trans. on Autom. Control, Vol. 27, No. 2, pp. 393–399, 1982.
R.K. Yedavalli and Z. Liang, “Reduced conservatism is stability robustness bounds by state transformations”, IEEE Trans. on Autom. Contr. Vol. 31, pp. 863–866, 1986.
A.L. Zelentsowsky, “Nonquadratic Lyapunov functions for robust stability analysis of linear uncertain systems”, IEEE Trans. on Autom. Control, vol. 39, no. 1, pp. 135–138, 1994.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1999 Springer-Verlag London Limited
About this paper
Cite this paper
Blanchini, F., Miani, S. (1999). Dynamic programming for robust control: Old ideas and recent developments. In: Garulli, A., Tesi, A. (eds) Robustness in identification and control. Lecture Notes in Control and Information Sciences, vol 245. Springer, London. https://doi.org/10.1007/BFb0109882
Download citation
DOI: https://doi.org/10.1007/BFb0109882
Published:
Publisher Name: Springer, London
Print ISBN: 978-1-85233-179-5
Online ISBN: 978-1-84628-538-7
eBook Packages: Springer Book Archive