Abstract
This paper considers the robust stability of a linear time-invariant state space model subject to real parameter perturbations. The problem is to find the distance of a given stable matrix from the set of unstable matrices. A new method, based on the properties of the Kronecker sum and two other composite matrices, is developed to study this problem; this new method makes it possible to distinguish real perturbations from complex ones. Although a procedure to find the exact value of the distance is still not available, some explicit lower bounds on the distance are obtained. The bounds are applicable only for the case of real plant perturbations, and are easy to compute numerically; if the matrix is large in size, an iterative procedure is given to compute the bounds. Various examples including a 46th-order spacecraft system are given to illustrate the results obtained. The examples show that the new bounds obtained can have an arbitrary degree of improvement over previously reported ones.
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References
N. Becker and M. Grimm, Comments on “Reduced conservatism in stability robustness bounds by state transformation,”IEEE Trans. Automat. Control,24 (1988), 223–224.
R. Bellman,Introduction to Matrix Analysis, McGraw-Hill, New York, 1960.
J. W. Brewer, Kronecker products and matrix calculus in system theory,IEEE Trans. Circuits and Systems,25 (1978), 772–781.
R. Byers, A bisection method for measuring the distance of a stable matrix to the unstable matrices,SIAM J. Sci. Statist. Comput.,9 (1988), 875–881.
J. K. Cullum and R. A. Willoughby,Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. 1, Birkhauser, Basel, 1985.
A. T. Fuller, Conditions for a matrix to have only characteristic roots with negative real parts,J. Math. Anal. Appl.,23 (1968), 71–98.
I. C. Gohberg and M. G. Krein,Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, Providence, RI, 1969.
G. Golub, F. T. Luk, and M. L. Overton, A block Lanczos method for computing the singular values and corresponding singular vectors of a matrix,ACM Trans. Math. Software,7 (1981), 149–169.
A. Graham,Kronecker Products and Matrix Calculus with Applications, Ellis Horwood, Chichestes, 1981.
D. Hinrichsen and M. Motscha, Optimization Problems in the Robustness Analysis of Linear State Space Systems, Report No. 169, Institut fur Dynamische Systeme, University of Bremen, 1987.
D. Hinrichsen and A. J. Pritchard, Stability radii of linear systems,Systems Control Lett. 7 (1986), 1–10.
R. A. Horn and C. A. Johnson,Matrix Analysis, Cambridge University Press, Cambridge, 1985.
A. Jennings,Matrix Computation for Engineers and Scientists, Wiley, London, 1977.
E. I. Jury,Inners and Stability of Dynamic Systems, Wiley, New York, 1974.
P. Lancaster and H. K. Farahat, Norms on direct sums and tensor products,Math. Comp. 26 (1972), 401–414.
W. H. Lee, Robustness Analysis for State Space Models, Report, TP-151, Alphatech Inc., 1982.
C. C. MacDuffee,The Theory of Matrices, Chelsea, New York, 1950.
M. Marcus,Finite Dimensional Multilinear Algebra, Part, Marcel Dekker, New York, 1973.
J. M. Martin, State-space measure for stability robustness,IEEE Trans. Automat. Control,32 (1987), 509–512.
C. C. Paige, Bidiagonalization of matrices and solution of linear equations,SIAM J. Numer. Anal.,11 (1984), 197–209.
B. N. Parlett,The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980.
R. V. Patel and M. Toda, Quantitative measures of robustness of multivariable system,Proceeding of the Joint Automatic Control Conference, San Francisco, CA, 1980, Paper TD8-A.
L. Qiu and E. J. Davison, New perturbation bounds for the robust stability of linear state space models,Proceedings of the 25th IEEE Conference on Decision and Control, Athens, 1986, pp. 751–755.
L. Qiu and E. J. Davison, A new method for the stability robustness of state space models with real perturbations, Systems Control Group Report No. 8801, Department of Electrical Engineering, University of Toronto, 1988.
L. Qiu and E. J. Davison, Computation of the stability robustness of large state space models with real perturbations,Proceedings of the 27th IEEE Conference on Decision and Control, Austin, TX, 1988, pp. 1380–1385.
L. Qiu and E. J. Davison, Stability robustness of generalized eigenvalues,Proceedings of the 28th IEEE Conference on Decision and Control, Tampa, FL, 1989, pp. 1902–1907,IEEE Trans. Automat. Control, to appear.
L. Saydy, A. L. Tits, and E. H. Abed, Guardian maps and the generalized stability of parametrized families of matrices and polynomials,Math. Control Signals Systems,3 (1990), 345–371.
G. B. Sincarsin, Laboratory Development of Control Techniques for 3rd Generation Spacecraft: Detailed Design, Dynacon Report, Daisy 9, DOC-CR-SP-84-010, 1984.
C. Stéphanos, Sur une extension du calcul des substitutions linéaires.J. Math. Pures Appl.,6, (1900), 73–128.
C. Van Loan, How near is a stable matrix to an unstable matrix,Contemporary Math.,47 (1985), 465–477.
R. K. Yedavalli, Perturbation bounds for robust stability in linear state space models,Internat. J. Control,42 (1985), 1507–1517.
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This work has been supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A4396.
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Qiu, L., Davison, E.J. The stability robustness determination of state space models with real unstructured perturbations. Math. Control Signal Systems 4, 247–267 (1991). https://doi.org/10.1007/BF02551280
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DOI: https://doi.org/10.1007/BF02551280