Abstract
Existing design methodologies based on infinite-dimensional linear programming generally require an iterative process often involving progressive increase of truncation length, in order to achieve a desired accuracy. In this chapter we consider the fundamental problem of determining a priori estimates of the truncation length sufficient for attainment of a given accuracy in the optimal objective value of certain infinite-dimensional linear programs arising in optimal feedback control. The treatment here also allows us to consider objective functions lacking interiority of domain, a problem which often arises in practice.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series (Pitman, London, 1984).
H. Attouch and R. J.–B. Wets, (1991), Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc., 328 (1991), 695–729.
H. Attouch and R. J.–B. Wets, (1993), Quantitative stability of variational systems: II. A framework for nonlinear conditioning, SIAM J. Optim., 3, No. 2 (1993), 359–381.
H. Attouch and R. J.–B. Wets, Quantitative stability of variational systems: III. ∈–approximate solutions, Math. Programming, 61 (1993), 197–214.
D. Azé and J.–P. Penot, Operations on convergent families of sets and functions, Optimization, 21, No. 4 (1990), 521–534.
S. P. Boyd and C. H. Barratt, Linear Controller Design: Limits of Performance (Prentice–Hall, Englewood Cliffs, NJ 1991).
G. Beer, Topologies on Closed and Closed Convex Sets, Mathematics and its Applications, 268 (Kluwer Academic Publishers, Dordrecht, 1993).
J. M. Borwein and A. S. Lewis, Partially–finite convex programming, Part I: Quasi relative interiors and duality theory, Math. Programming, 57 (1992), 15–48.
M. A. Dahleh and J. B. Pearson, l 1-Optimal feedback controllers for MIMO discrete–time systems, IEEE Trans. Automat. Control, AC-32 (1987), 314–322.
M. A. Dahleh and I. J. Diaz-Bobillo, Control of Uncertain Systems: A Linear Programming Approach (Prentice–Hall, Englewood Cliffs, NJ 1995).
G. Deodhare and M. Vidyasagar, Control system design by infinite linear programming, Internat. J. Control, 55, No. 6 (1992), 1351–1380.
C. A. Desoer and M. Vidyasagar, Feedback Systems: Input–Output Properties (Academic Press, New York, 1975).
A. C. Eberhard and R. B. Wenczel, Epi–distance convergence of parametrised sums of convex functions in non–reflexive spaces, J. Convex Anal., 7, No. 1 (2000), 47–71.
N. Elia and M. A. Dahleh, Controller design with multiple objectives, IEEE Trans. Automat. Control, AC-42, No. 5 (1997), 596–613.
R. D. Hill, A. C. Eberhard, R. B. Wenczel and M. E. Halpern, Fundamental limitations on the time–domain shaping of response to a fixed input, IEEE Trans. Automat. Control, AC-47, No. 7 (2002), 1078–1090.
R. B. Holmes, Geometric Functional Analysis and its Applications, Graduate Texts in Mathematics, 24 (Springer-Verlag, New York, 1975).
V. Jeyakumar, Duality and infinite–dimensional optimization, Nonlinear Analysis: Theory, Methods and Applications, 15, (1990), 1111–1122.
V. Jeyakumar and H. Wolkowicz, Generalizations of Slater’s constraint qualification for infinite convex programs, Math. Programming, 57 (1992), 85–102.
J. S. McDonald and J. B. Pearson, l 1–Optimal control of multivariable systems with output norm constraints, Automatica J. IFAC, 27, No. 2 (1991), 317–329.
R. T. Rockafellar, Conjugate Duality and Optimization (SIAM, Philadelphia, PA, 1974).
R. T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, NJ, 1970).
R. T. Rockafellar and R. J.–B. Wets, Variational systems, an introduction, Multifunctions and Integrands, G. Salinetti (Ed.), Springer-Verlag Lecture Notes in Mathematics No. 1091 (1984), 1–54.
H. Rotstein, Convergence of optimal control problems with an H ∞–norm constraint, Automatica J. IFAC, 33, No. 3 (1997), 355–367.
M. Vidyasagar, Control System Synthesis (MIT Press, Cambridge MA, 1985).
M. Vidyasagar, Optimal rejection of persistent bounded disturbances, IEEE Trans. Automat. Control, AC-31 (1986), 527–534.
R. B. Wenczel, PhD Thesis, Department of Mathematics, RMIT, 1999.
R. B. Wenczel, A. C. Eberhard and R. D. Hill, Comments on “Controller design with multiple objectives,” IEEE Trans. Automat. Control, 45, No. 11 (2000), 2197–2198.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag New York
About this chapter
Cite this chapter
Wenczel, R., Eberhard, A., Hill, R. (2009). Convergence of truncates in l 1 optimal feedback control 61. In: Pearce, C., Hunt, E. (eds) Optimization. Springer Optimization and Its Applications, vol 32. Springer, New York, NY. https://doi.org/10.1007/978-0-387-98096-6_4
Download citation
DOI: https://doi.org/10.1007/978-0-387-98096-6_4
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98095-9
Online ISBN: 978-0-387-98096-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)