Abstract
Various generalizations and refinements are proposed for a well-known result on robust matrix sign-definiteness, which is extensively exploited in quadratic stability, design of robustly quadratically stabilizing controllers, robust LQR-problem, etc. The main emphasis is put on formulating the results in terms of linear matrix inequalities.
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References
Petersen, I., A Stabilization Algorithm for a Class of Uncertain Systems, Syst. Control Lett., 1987, vol. 8, pp. 351–357.
Xie, L., Output Feedback H ∞ Control of Systems with Parameter Uncertainty, Int. J. Control, 1996, vol. 63, pp. 741–750.
Khargonekar, P.P., Petersen, I.R., and Zhou, K., Robust Stabilization of Uncertain Linear Systems: Quadratic Stabilizability and H ∞ Control Theory, IEEE Trans. Automat. Control, 1990, vol. 35, no. 3, pp. 356–361.
Mao, W.-J. and Chu, J., Quadratic Stability and Stabilization of Dynamic Interval Systems, IEEE Trans. Automat. Control, 2003, vol. 48, no. 6, pp. 1007–1012.
Mao, W.-J. and Chu, J., Correction to “Quadratic Stability and Stabilization of Dynamic Interval Systems,” IEEE Trans. Automat. Control, 2006, vol. 51, no. 8, pp. 1404–1405.
Alamo, T., Tempo, R., Ramirez, D.R., and Camacho, E.F., A New Vertex Result for Robustness Problems with Interval Matrix Uncertainty, in Proc. Eur. Control Conf., Kos, Greece, 2007, paper ThC07.3.
Polyak, B.T., Shcherbakov, P.S., and Topunov, M.V., Invariant Ellipsoids Approach to Robust Rejection of Persistent Disturbances, in Proc. 17th World Congr. IFAC, Seoul, 2008, pp. 3976–3981.
Polyak, B.T., Topunov, M.V., and Shcherbakov, P.S., Robust Rejection of Exogenous Disturbances via Invariant Ellipsoid Technique, in Proc. XIV Int. Workshop on Dynamics and Control, Zvenigorod, 2007, p. 59.
Topunov, M.V. and Shcherbakov, P.S., Ramifications of Petersen’s Lemma on Uncertain Matrices, in Proc. XIV Int. Workshop on Dynamics and Control, Zvenigorod, 2007, p. 66.
Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.
Polyak, B.T. and Shcherbakov, P.S., The D-decomposition Technique for Linear Matrix Inequalities, Avtom. Telemekh., 2006, no. 11, pp. 159–174.
Brickman, L., On the Field of Values of a Matrix, Proc. Am. Math. Soc., 1961, no. 12, pp. 61–66.
Rohn J., Positive Definiteness and Stability of Interval Matrices, SIAM J. Matrix Anal. Appl., 1994, vol. 15, no. 1, pp. 175–184.
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Original Russian Text © M.V. Khlebnikov, P.S. Shcherbakov, 2008, published in Avtomatika i Telemekhanika, 2008, No. 11, pp. 125–139.
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Khlebnikov, M.V., Shcherbakov, P.S. Petersen’s lemma on matrix uncertainty and its generalizations. Autom Remote Control 69, 1932–1945 (2008). https://doi.org/10.1134/S000511790811009X
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DOI: https://doi.org/10.1134/S000511790811009X