Abstract
The central problem of H ∞-control theory roughly is to optimize (by the choice of compensator in a standard feedback configuration) some worst case (i.e. infinity norm) measure of performance while maintaining stability. For the linear, time-invariant, finitedimensional case, rather complete state space solutions are now available, and work has begun on understanding less restrictive settings. A recent new development has been the establishment of a connection with differential games and the perception of the H ∞ -problem as formally the same as the earlier well established linear quadratic regulator problem, but with an indefinite performance objective. In this article we review the current state of the art for nonlinear systems. The main focus is on the approach through a global theory of nonlinear J-inner-outer factorization and nonlinear fractional transformations being developed by the authors. It turns out that the critical points arising naturally in this theory can also be interpreted as optimal strategies in a game-theoretic interpretation of the control problem.
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
V. Anantharam and C. A. Desoer, On the stabilization of nonlinear systems, IEEE Trans. Aut. Control AC-29 (1984), 569–573.
J. A. Ball and N. Cohen, Sensitivity minimization in H ∞ norm: parameterization of all solutions, Int. J. Control 46 (1987), 785–816.
J. A. Ball, C. Foias, J. W. Helton and A. Tannenbaum, On a local nonlinear commutant lifting theorem, Indiana Univ. Math. J. 36 (1987), 693–709.
J. A. Ball, C. Foias, J. W. Helton and A. Tannenbaum, A Poincare-Dulac approach to a nonlinear Beurling-Lax-Halmos theorem, J. Math. Anal. Appl. 139 (1989), 496–514.
J. A. Ball, I. Gohberg and L. Rodman, Two-sided Nudelman interpolation problem for rational matrix functions, to appear in Marcel-Dekkar volume dedicated to Mischa Cotlar.
J. A. Ball and J. W. Helton, A Beurling-Lax theorem for the Lie group U(m, n)which contains most classical interpolation, J. Operator Theory 9 (1983), 107–142.
J. A. Ball and J. W. Helton, Sensitivity bandwidth optimization for nonlinear feedback systems, in Analysis and Control of Nonlinear Systems (ed. by C. I. Byrnes, C. F. Martin and R. E. Saeks ), North Holland (1988), pp. 123–129.
J. A. Ball and J. W. Helton, Well-posedness of nonlinear causal feedback systems, Proc. 26th IEEE Conf. on Decision and Control, Los Angeles (1987), pp. 152–154.
J. A. Ball and J. W. Helton, Shift invariant manifolds and nonlinear analytic function theory, Integral Equations and Operator Theory 11 (1988), 615–725.
J. A. Ball and J. W. Helton, Factorization of nonlinear systems: toward a theory for nonlinear H ∞ control, Proc. 27th IEEE Conf. on Decision and Control, Austin (1988), 2376–2381.
J. A. Ball and J. W. Helton, Interpolation problems for null and pole structure of nonlinear systems, Proc. 27th IEEE Conf. on Decision and Control, Austin (1988), 14–19.
J. A. Ball and J. W. Helton, Interconnection of nonlinear causal systems, IEEE Trans. Aut. Control, to appear.
J. A. Ball and J. W. Helton, Inner-outer factorization of nonlinear operators, preprint.
J. A. Ball and J. W. Helton, H ∞-control of nonlinear plants: connections with differential games, in preparation.
J. A. Ball, J. W. Helton and C. H. Sung, Nonlinear solutions of Nevanlinna-Pick interpolation problems, Mich. Math. J. 34 (1987), 375–389.
J. A. Ball and A. C. M. Ran, Global inverse spectral problems for rational matrix functions, Linear Alg. Appl. 86 (1987), 237–282.
G. Chen and R. J. P. deFigueiredo, On robust stabilization of nonlinear control systems, Systems & Control Letters 12 (1989), 373–379.
deFC] R. J. P. deFigueiredo and G. Chen, Optimal disturbance rejection for nonlinear control systems, IEEE Trans. Aut. Control, to appear.
J. C. Doyle, Lecture Notes for ONR ( Honeywell Workshop on Advances in Multivariable Control, Minneapolis, MN (1984).
C. A. Desoer and M. G. Kabuli, Right factorization of a class of time varying nonlinear systems, IEEE Trans. Aut. Control 33 (1988), 755–757.
C. Foias and A. Tannenbaum, Weighted optimization theory for nonlinear systems, SIAM J. Control and Opt. 27 (1989), 842–860.
C. Foias and A. Tannenbaum, Iterative commutant lifting for systems with rational symbol, in The Gohberg Anniversary Collection Vol. II: Topics in Analysis and Operator Theory, OT41 Birkhauser, Basel, (1989), pp. 255–277.
B. Francis, A Course in H ∞ Control Theory, Springer-Verlag, Berlin-New York, 1987.
B. Francis and J. Doyle, Linear control theory with an Hoe optimality criterion, SIAM J. Control Opt. 25 (1987), 815–844.
M. Green, K. Glover, D. Limebeer and J. Doyle, A J-spectral factorization approach to H ∞ control, preprint.
K. Glover, All optimal Hankel norm approximations of linear multivariable systems and their Loe error bounds, Int. J. Control 39 (1984), 1115–1193.
K. Glover and J. C. Doyle, State-space formulas for stabilizing controllers, Systems and Control Letters 11 (1988), 167–172.
S. Hara and R. Kondo, Characterization and computation of Hoo-optimal controllers in the state space, Proc. 27th IEEE Conf. on Decision and Control, Austin (1988), 20–25.
J. Hammer, Fractional representations of nonlinear systems: a simplified approach, Int. J. Control 46 (1987), 455–472.
H. Kimura, Conjugation, interpolation and model-matching in H ∞, Int. J. Control (1989), 269–307.
P. R. Khargonekar and K. R. Poola, Uniformly optimal control of linear time-invariant plants: nonlinear time-varying controllers, Systems Control Letters 6 (1986), 303–308.
A. J. Krener, Nonlinear controller design via approximate normal forms, in Proc. IMA Conf. on Signal Processing, Minneapolis (1988), to appear.
D. J. N. Limebeer, B. D. O. Anderson, P. P. Khargonekar and M. Green, A game theoretic approach to H ∞ control for time varying systems, preprint.
E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Aut. Control, 34 (1989), 435–443.
G. Tadmor, Worst case design in the time domain: the maximum principle and the standard H ∞ problem, preprint.
G. Tadmor, The standard H ∞ problem and the maximum principle: the general linear case, preprint.
M. S. Verma, Coprime fractional representations and stability of nonlinear feedback systems, Int. J. Control, to appear.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Birkhäuser Boston
About this chapter
Cite this chapter
Ball, J.A., Helton, J.W. (1990). Nonlinear H ∞ Control Theory: A Literature Survey. In: Kaashoek, M.A., van Schuppen, J.H., Ran, A.C.M. (eds) Robust Control of Linear Systems and Nonlinear Control. Progress in Systems and Control Theory, vol 4. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4484-4_1
Download citation
DOI: https://doi.org/10.1007/978-1-4612-4484-4_1
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8839-8
Online ISBN: 978-1-4612-4484-4
eBook Packages: Springer Book Archive