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Optimal, constant I/O similarity scaling for full-information and state- feedback control problems. (English) Zbl 0772.49021
The paper discusses the optimal constant scaling problem of full- information \(H_ \infty\) control. The problem arises, because it is impossible to capture all robustness and performance in a single \(H_ \infty\) norm cost function. Alternatively, diagonal similarity scalings of certain closed loop transfer functions can be used. The set of allowable diagonal scalings is problem dependent. The scaling set considered in the paper is a prescribed convex set of positive definite matrices.
The solution to the posed problem is obtained by transforming the original problem into a convex feasibility problem, specifically, a structured, linear matrix inequality. In special cases, solvability of the full-information problem is equivalent to solvability of the state- feedback problem. The paper concludes with a numerical example.
Reviewer: H.Koivo (Tampere)

90C25 Convex programming
93B35 Sensitivity (robustness)
Full Text: DOI
[1] Balas, G.; Packard, A.; Harduval, J., Application of μ synthesis techniques to momentum management and attitude control of the space station, ()
[2] Bland, R.; Goldfarb, D.; Todd, M., The ellipsoid method: A survey, Operations research, 28, 6, 1039-1091, (1980) · Zbl 0474.90056
[3] Boyd, S.; Barratt, C., Linear controller design: limits of performance, (1991), Prentice Hall Englewood Cliffs, NJ · Zbl 0748.93003
[4] Boyd, S.; El Ghaoui, L., Methods of centers for minimizing generalized eigenvalues, (), (special issue on Linear Algebra in Systems and Control) · Zbl 0781.65051
[5] Boyd, S.; Yang, Q., Structured and simultaneous Lyapunov functions for system stability problems, Internat. J. control, 49, 6, 2215-2240, (1989) · Zbl 0683.93057
[6] Davis, C.; Kahan, W.; Weinberger, H., Norm preserving dilations and their applications to optimal error bounds, SIAM J. numerical anal., 19, 3, 445-469, (1982) · Zbl 0491.47003
[7] Doyle, J.C., Analysis of feedback systems with structured uncertainties, (), 242-250, 6
[8] Doyle, J.C., Synthesis of robust controllers and filters, (), 109-114
[9] Doyle, J.C., Matrix interpolation and H∞ performance bounds, (), 129-134
[10] Doyle, J.; Glover, K.; Khargonekar, P.; Francis, B., State space solutions to H2 and H∞ control problems, IEEE trans. automat. control, 34, 8, 831-847, (1989) · Zbl 0698.93031
[11] El Ghaoui, L.; Balakrishnan, V.; Feron, E.; Boyd, S., On maximizing a robustness measure for structured nonlinear perturbations, ()
[12] Glover, K.; Doyle, J.C., State-space formulae for all stabilizing controllers that satisfy an H∞ norm and relations to risk sensitivity, Systems control lett., 11, 167-172, (1988) · Zbl 0671.93029
[13] Mageirou, E.; Ho, Y., Decentralized stabilization via game theoretic methods, Automatica, 13, 393-399, (1977) · Zbl 0358.93030
[14] Khargonekar, P.; Petersen, I.; Zhou, K., Robust stabilization of uncertain linear systems: quadratic stability and H∞ control theory, (), 356-361 · Zbl 0707.93060
[15] Limebeer, D.; Green, M.; Walker, D., Discrete time H∞ control, (), 392-396
[16] Overton, M., Large-scale optimization of eigenvalues, NYU computer science department report no. 505, (May, 1990)
[17] Nesterov, Yu.; Nemirovsky, A., Interior point polynomial methods in convex programming: theory and applications, ()
[18] Packard, A.; Doyle, auJ., Quadratic stability with real and complex perturbations, IEEE trans. automat. control, 35, 2, 198-201, (1990) · Zbl 0705.93060
[19] A. Packard, K. Zhou, P. Pandey and G. Becker, A collection of robust control problems leading to LMIs, Proceedings of the 1991 IEEE CDC.
[20] Power, S.C., Hankel operators on Hilbert space, (1982), Pitman London · Zbl 0489.47011
[21] Safonov, M.G., Stability margins of diagonally perturbed multivariable feedback systems, (), 1472-1478
[22] Safonov, M.G., L∞ sensitivity versus stability margin, (), 115-118
[23] Safonov, M.G., Optimal diagonal scaling for infinity norm optimization, Systems control lett., 7, 257-260, (1986)
[24] Safonov, M.G.; Le, V.X., An alternative solution to the H∞ optimal control problem, Systems control lett., 10, 155-158, (1988) · Zbl 0652.93013
[25] Shamma, J., Robust stability with time-varying structured uncertainty, () · Zbl 0807.93048
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