Abstract
In this paper we report on some recent results concerning the distance of a stable matrix A from the set of unstable matrices. Related optimization and optimal control problems are discussed in detail and new algorithms are presented for their solution.
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Literature
V.M. Adamjan, D.Z. Arov and M.G. Krein, Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem, Math. USSR Sbornik 15, 31–73, (1971).
R.W. Brockett, Finite Dimensional Linear Systems, J. Wiley, New York (1970)
W.A. Coppel, Matrix quadratic equations, Bull. Austral. Math. Soc. 10, 377–401, (1974)
J.J. Dongarra, J.R. Bunch, C.B. Moler and G.W. Stewart, LINPACK User's Guide, SIAM Publications Philadelphia, PA. (1978).
J.C. Doyle and G. Stein, Concepts for a classical/modern synthesis, IEEE Trans. Aut. Control 26, 4–16, (1981)
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam (1976)
B.A. Francis, The optimal linear quadratic regulator with cheap control, IEEE Trans. Aut. Control AC-24, 616–621, (1979)
E.R. Gantmacher, The Theory of Matrices, vol. 2, New York, (1959)
D. Hinrichsen, A. Ilchmann, A.J. Pritchard, Robustness of stability of time-varying linear systems, Report Nr. 161, Institut für Dynamische Systeme, U. Bremen, (1987)
D. Hinrichsen, A.J. Pritchard, Stability radii of linear systems, Systems & Control Letters 7, 1–10 (1986a)
D. Hinrichsen, A.J. Pritchard, Stability radius for structured perturbations and the algebraic Riccati equation. Systems & Control Letters 8, 105–113 (1986b)
R.E. Kalman, P.L. Falb, M.A. Arbib, Topics in Mathematical System Theory, McGraw-Hill, (1969)
T. Kato, Perturbation theory for linear operators, Grundlehren der mathematischen Wissenschaften 132, Springer, Berlin, Heidelberg, New York, 2nd edition, (1976)
H.W. Knobloch, H. Kwakernaak, Lineare Kontrolltheorie, Springer, Berlin, Heidelberg, New York, Tokyo, (1985)
S.Y. Kung, D.W. Lin, Recent progress in linear system model-reduction via Hankel matrix approximation, in: Circuit Theory and Design (The Hague, 1981) North Holland (1981)
P. Lancaster, L. Rodman, Existence and uniqueness theorems for the algebraic Riccati equation, Int. J. Control 32, 285–309, (1980)
M. Motscha, An algorithm to compute the complex stability radius, Report 168, Institut für Dynamische Systeme, U. Bremen, (1987)
A. Ostrowski, Über Normen von Matrizen, Math. Z. 63, 2–18, (1955)
D.H. Owens, A. Chotai, Robust controller design for linear dynamic systems using approximate models, IEE Proc. 130, 45–56, (1983)
R.V. Patel, M. Toda, Quantitative measures of robustness for multivariable system, Proc. Joint Autom. Control Conf., TD8-A, (1980)
A.J. Pritchard, S. Townley, Robustness of infinite dimensional systems, Control Theory Centre, Report No. 138, U. Warwick, (1986)
L. Qui, E.J. Davison, New perturbation bounds for the robust stability of linear state space models, Proceedings of 25th Conference on Decision and Control, Athens, Greece, 751–755, (1986)
M. Shayman, Geometry of the algebraic Riccati equation, Part I, SIAM J. Control and Optimization 21, 375–394, (1983)
C. Van Loan, How near is a matrix to an unstable matrix, Contemporary Mathematics, 47, 465–478, (1985)
C. Van Loan, A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix, LAA 61, 233–251, (1984)
H.H. Wilkinson, C. Reinsch, Linear algebra, Handbook for automatic computation II, Springer, New York, (1971)
H.K. Wimmer, The algebraic Riccati equation without complete controllability, SIAM J. Alg. Disc. Meth. 3, 1–12, (1982)
W.M. Wonham, Linear Multivariable Control: a Geometric Approach, 2nd edition, Springer-Verlag, Heidelberg (1979)
R.K. Yedavallie, Perturbation bounds for robust stability in linear state space models, Int. J. Control 42, 1507–1517, (1985)
G. Zames, B. Francis, Feedback minimax sensitivity and optimal robustness, IEEE Trans. Aut. Control 28, 585–601, (1983)
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Hinrichsen, D., Motscha, M. (1988). Optimization problems in the robustness analysis of linear state space systems. In: Gómez-Fernandez, J.A., Guerra-Vázquez, F., López-Lagomasino, G., Jiménez-Pozo, M.A. (eds) Approximation and Optimization. Lecture Notes in Mathematics, vol 1354. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0089583
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DOI: https://doi.org/10.1007/BFb0089583
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