zbMATH — the first resource for mathematics

Rejection of bounded exogenous disturbances by the method of invariant ellipsoids. (English. Russian original) Zbl 1125.93370
Autom. Remote Control 68, No. 3, 467-486 (2007); translation from Avtom. Telemekh. 68, No. 3, 106-125 (2007).
Summary: Rejection of the bounded exogenous disturbances was first studied by the \(l_{1}\)-optimization theory. A new approach to this problem was proposed in the present paper on the basis of the method of invariant ellipsoids where the technique of linear matrix inequalities was the main tool. Consideration was given to the continuous and discrete variants of the problem. Control of the “double pendulum” was studied by way of example.

93C05 Linear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
93C73 Perturbations in control/observation systems
93B51 Design techniques (robust design, computer-aided design, etc.)
Full Text: DOI
[1] Bulgakov, B.V., On Accumulation of Disturbances in the Linear Systems with Constant Parameters, Dokl. Akad. Nauk SSSR, 1946, vol. 5, no. 5, pp. 339–342.
[2] Ulanov, G.M., Dinamicheskaya tochnost’ i kompensatsiya vozmushchenii v sistemakh avtomaticheskogo upravleniya (Dynamic Precision and Compensation of Disturbances in the Automatic Control Systems), Moscow: Mashinostroenie, 1971.
[3] Yakubovich, E.D., Solution of the Optimal Control Problem for the Linear Discrete Systems, Avtom. Telemekh., 1975, no. 9, pp. 73–79.
[4] Barabanov, A.E., Optimal Control of the Nonminimum-phase Discrete Plant with Arbitrary Bounded Noise, Vestn. Leningr. Gos. Univ., Ser. Mat., 1980, vol. 13, pp. 119–120. · Zbl 0454.93059
[5] Vidyasagar, M., Optimal Rejection of Persistent Bounded Disturbances, IEEE Trans. Automat. Control, 1986, vol. 31, pp. 527–535. · Zbl 0594.93050
[6] Barabanov, A.E. and Granichin, O.N., Optimal Controller for Linear Plants with Bounded Noise, Avtom. Telemekh., 1984, no. 5, pp. 39–46. · Zbl 0561.93027
[7] Dahleh, M.A. and Pearson, J.B., l 1-Optimal Feedback Controllers for MIMO Discrete-time Systems, IEEE Trans. Automat. Control, 1987, vol. 32, pp. 314–322. · Zbl 0622.93041
[8] Glover, D. and Schweppe, F., Control of Linear Dynamic Systems with Set Constrained Disturbances, IEEE Trans. Automat. Control, 1971, vol. 16, pp. 411–423.
[9] Bertsekas, D.P. and Rhodes, I.B., On the Minimax Reachability of Target Sets and Target Tubes, Automatica, 1971, vol. 7, pp. 233–247. · Zbl 0215.21801
[10] Elia, N. and Dahleh, M.A., Minimization of the Worst Case Peak-to-Peak Gain via Dynamic Programming: State Feedback Case, IEEE Trans. Automat. Control, 2000, vol. 45, pp. 687–701. · Zbl 1004.93018
[11] Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.
[12] Polyak, B.T. and Shcherbakov, P.S., Difficult Problems of the Linear Control Theory. Some Approaches, Avtom. Telemekh., 2005, no. 5, pp. 7–46.
[13] Schweppe, F.C., Uncertain Dynamic Systems, New Jersey: Prentice Hall, 1973.
[14] Bertsekas, D.P. and Rhodes, I.B., Recursive State Estimation for a Set-membership Description of Uncertainty, IEEE Trans. Automat. Control, 1971, vol. 16, pp. 117–128.
[15] Kurzhanskii, A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti (Control and Observation under Uncertainty), Moscow: Nauka, 1977.
[16] Chernous’ko, F.L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem (Estimation of the Phase State of the Dynamic Systems), Moscow: Nauka, 1988.
[17] Blanchini, F., Set Invariance in Control–A Survey, Automatica, 1999, vol. 35, pp. 1747–1767. · Zbl 0935.93005
[18] Nazin, A.V., Nazin, S.A., and Polyak, B.T., On Convergence of External Ellipsoidal Approximations of the Reachability Domains of Discrete Dynamic Linear Systems, Avtom. Telemekh., 2004, no. 8, pp. 39–61. · Zbl 1078.93006
[19] Boyd, S., El Ghaoui, L., Ferron, E., and Balakrishnan, V., Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.
[20] Abedor, J., Nagpal, K., and Poola, K., A Linear Matrix Inequality Approach to Peak-to-Peak Gain Minimization, Int. J. Robust Nonlinear Control, 1996, vol. 6, pp. 899–927. · Zbl 0862.93028
[21] Blanchini, F. and Sznaier, M., Persistent Disturbance Rejection via Static State Feedback, IEEE Trans. Automat. Control, 1995, vol. 40, pp. 1127–1131. · Zbl 0832.93024
[22] Polyak, B.T. and Topunov, M.V., Rejection of the Bounded Exogenous Disturbances by the Example of the Double-Pendulum Problem, Abstracts of Papers at the IX E.S. Pyatnitskii Int. Workshop ”Stability and Oscillations of the Nonlinear Control Systems, May 31–June 2, 2006, Moscow: Inst. Probl. Upravlen., 2006, pp. 213–214.
[23] Polyak, B.T., Nazin, A.V., Topunov, M.V., and Nazin, S.A., Rejection of Bounded Disturbances via Invariant Ellipsoids Technique, in Proc. 45th IEEE Conf. Decision Control, San Diego, 2006. · Zbl 1125.93370
[24] Venkatesh, S. and Dahleh, M., Does Star Norm Capture l 1 Norm? in Proc. Am. Control Conf., 1995, pp. 944–945.
[25] Polyak, B.T., Convexity of Quadratic Transformations and Its Use in Control and Optimization, J. Optim. Theory Appl., 1998, vol. 99, pp. 553–583. · Zbl 0961.90074
[26] Gusev, S.V. and Likhtarnikov, A.L., Kalman-Popov-Yakubovich Lemma and the S-procedure: A Historical Essay, Avtom. Telemkh., 2006, no. 11, pp. 77–121. · Zbl 1195.93002
[27] Horn, R. and Johnson, C., Matrix Analysis, New York: Cambridge Univ. Press, 1985. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989. · Zbl 0576.15001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.