Abstract
A number of challenging problems in linear control theory are considered which admit simple formulation and yet lack efficient solution methods. These problems relate to the classical theory of linear systems as well as to the robust theory where the system description contains uncertainty. Various solution methods are discussed and the results of numerical simulations are given.
Similar content being viewed by others
REFERENCES
Spravochnik po teorii avtomaticheskogo upravleniya (Handbook in Automatic Control Theory), Krasovskii, A.A., Ed., Moscow: Nauka, 1987.
Pervozvanskii, A.A., Kurs teorii avtomaticheskogo upravleniya (A Course in Automatic Control Theory), Moscow: Nauka, 1986.
Besekerskii, V.A. and Popov, E.P., Teoriya sistem avtomaticheskogo regulirovaniya (The Theory of Automatic Regulation Systems), Moscow: Nauka, 1966.
Polyak, B.T. and Shcherbakov, P.S., Robastnaya ustoichivost’ i upravlenie (Robust Stability and Control), Moscow: Nauka, 2002.
Bernstein, D.S., Some Open Problems in Matrix Theory Arising in Linear Systems and Control, Linear Algebra Appl., 1992, no. 162–164, pp. 409–432.
Blondel, V. and Tsitsiklis, J.N., NP-hardness of Some Linear Control Design Problems, SIAM J. Control Optimiz., 1997, vol. 35, no.6. pp. 2118–2127.
Blondel, V., Sontag, E., Vidyasagar, M., and Willems, J., Open Problems in Mathematical Systems and Control Theory, London: Springer, 1999.
Nemirovskii, A.A., Several NP-hard Problems Arising in Robust Stability Analysis, Math. Control Sig. Syst., 1994, vol. 6, pp. 99–105.
Aizerman, M.A., On Some Structural Conditions of Stability of Automatic Regulation Systems, Avtom. Telemekh., 1948, no. 11.
Aizerman, M.A. and Gantmakher, F.R., Conditions of Existence of a Stability Domain for One-Loop Automatic Regulation System, Prikl. Mat. Mekh., 1954, vol. XVIII, no.1.
Padmanabhan, P. and Hollot, C.V., Complete Instability of a Box of Polynomials, IEEE Trans. Automat. Control, 1992, vol. 37, no.8, pp. 1230–1233.
Nemirovskii, A.S. and Polyak, B.T., Necessary Conditions for Stability of Polynomials and Their Applications, Avtom. Telemekh., 1994, no. 11, pp. 113–119.
Pujara, L.R., Some Necessary and Sufficient Conditions for Low-Order Interval Polytopes to Contain a Hurwitz Polynomial, Proc. Conf. Decision Control, Phoenix, USA, 1999, pp. 5024–5029.
Moses, R.L. and Liu, D., Determining the Closest Stable Polynomial to an Unstable One, IEEE Trans. Automat. Control, 1991, vol. 39, no.4, pp. 901–907.
Syrmos, V.L., Abdallah, C.T., Dorato, P., and Grigoriadis, K., Static Output Feedback: A Survey, Automatica, 1997, vol. 33, no.2, pp. 125–137.
Garcia, G., Pradin, B., Tarbouriech, S., and Zeng, F., Robust Stabilization and Guaranteed Cost Control for Discrete-Time Linear Systems by Static Output Feedback, Automatica, 2003, vol. 39, pp. 1635–1641.
Kimura, H., Pole Assignment by Gain Output Feedback, IEEE Trans. Automat. Control, 1975, vol. 20, pp. 509–516.
Davison, E.J. and Wang, S.H., On Pole Assignment in Linear Multivariable Systems Using Output Feedback, IEEE Trans. Automat. Control, 1975, vol. 20, pp. 516–518.
Brockett, R. and Byrnes, C., Multivariable Nyquist Criteria, Root Loci and Pole Placement: A Geometric Viewpoint, IEEE Trans. Automat. Control, 1981, vol. 26, pp. 271–284.
Byrnes, C.I. and Anderson, B.D.O., Output Feedback and Generic Stabilizability, SIAM J. Control Optimiz., 1984, vol. 22, pp. 362–380.
Byrnes, C., Pole Assignment by Output Feedback, in Three Decades of Mathematical System Theory, Lect. Notes Control Inf. Sci., 1989, vol. 135, pp. 31–78.
Eremenko, A. and Gabrielov, A., Pole Placement by Static Output Feedback for Generic Linear Systems, SIAM J. Control Optimiz., 2002, vol. 41, no.1, pp. 303–312.
Boyd, S.L., El Ghaoui, L., Feron, E., and Balakrishnan, V., Linear Matrix Inequalities in Systems and Control Theory, Philadelphia: SIAM, 1994.
Kharitonov, V.L., Asymptotic Stability of an Equilibrium Position of a Family of Systems of Linear Differential Equations, Diff. Uravn., 1979, vol. 14, no.11, pp. 1483–1485.
Tsypkin, Ya.Z. and Polyak, B.T., Frequency Domain Criteria for ℓp-robust Stability of Continuous Linear Systems, IEEE Trans. Automat. Control, 1991, vol. 36, no.12, pp. 1464–1469.
Bartlett, A.C., Hollot, C. V., and Lin, H., Root Location of an Entire Polytope of Polynomials: It Suffices to Check the Edges, Mat. Control Sig. Syst., 1988, vol. 1, pp. 61–71.
Barmish, B.R., New Tools for Robustness of Linear Systems, New York: Macmillan, 1994.
Ackermann, J., Robust Control. The Parameter Space Approach, London: Springer, 2002.
Bhattacharyya, S.P., Chapellat, H., and Keel, L., Robust Control: The Parametric Approach, Upper Saddle River: Prentice Hall, 1995.
Hinrichsen, D. and Pritchard A., Stability Radius for Structured Perturbations and the Algebraic Riccati Equation, Syst. Control Lett., 1986, no. 8, pp. 105–113.
Qiu, L., Bernhardsson, B., Rantzer A., et al., A Formula for Computation of the Real Stability Radius, Automatica, 1995, vol. 31, no.6, pp. 879–890.
Zadeh, L.A. and Desoer, C.A., Linear System Theory. The State Space Approach, New York: McGraw-Hill, 1963. Translated under the title Teoriya lineinykh sistem, Moscow: Nauka, 1970.
Kats, A.M., Finding the Parameters of a Regulator from the Desired Characteristic Equation of a Regulation System, Avtom. Telemekh., 1955, no. 3, pp. 269–272.
Volgin, L.N., Elementy teorii upravlyayushchikh mashin (Elemens of the Theory of Governors), Moscow: Sovetskoe Radio, 1962.
Larin, V.M., Naumenko, K.I., and Suntsev, V.N., Spektral’nye metody sinteza lineinykh sistem s obratnoi svyaz’yu (Spectral Methods for Linear Feedback System Design), Kiev: Naukova Dumka, 1971.
Bongiorno, J.J. and Youla, D.C., On the Design of Single-Loop Single-Input-Output Feedback Control Systems in the Complex Frequency Domain, IEEE Trans. Automat. Control, 1977, vol. 22, no.3, pp. 416–423.
Vidyasagar, M., Control System Synthesis: A Factorization Approach, Boston: MIT Press, 1985.
Blondel, V., Simultaneous Stabilization of Linear Systems, London: Springer, 1995.
Osnovy avtomaticheskogo regulirovaniya (Foundations of Automatic Regulation Theory), Solodovnikov, V.V., Ed., Moscow: Mashgiz, 1954.
Dahleh, M. and Pearson, J., l 1-optimal Controllers for MIMO Discrete-Time Systems, IEEE Trans. Automat. Control, 1987, vol. 3, pp. 314–322.
Dahleh, M. and Diaz-Bobillo, I., Control of Uncertain Systems: A Linear Programming Approach, Englewood Cliffs: Prentice Hall, 1995.
Barabanov, A.E., Sintez minimaksnykh regulyatorov (Design of Minimax Regulators), St. Petersburg: S.-Peterburg. Gos. Univ., 1996.
Evans, W.R., Control System Dynamics, New York: McGraw-Hill, 1954.
Uderman, E.G., Metod kornevogo godografa v teorii avtomaticheskogo upravleniya (The Root Locus Method in Automatic Control Theory), Leningrad: Gostekhizdat, 1963.
Maxwell, J.C., Vyshnegradskii, I.A., and Stodola, A., Teoriya avtomaticheskogo regulirovaniya (Automatic Regulation Theory), Moscow: Akad. Nauk SSSR, 1949.
Neimark, Yu.I., Ustoichivost’ linearizovannykh sistem (Stability of Linearized Systems), Leningrad: LKVVIA, 1949.
Siljak, D.D., Nonlinear Systems, New York: Wiley. 1969.
Datta, A., Ho, M.T., and Bhattacharyya, S.P., Structure and Synthesis of PID Controllers, London: Springer-Verlag, 2000.
Nikolaev, Yu.P., The Set of Stable Polynomials of Linear Discrete Systems: Its Geometry, Avtom. Telemekh., 2002, no. 7, pp. 44–53.
Gryazina, E.N., The D-Decomposition Theory, Avtom. Telemekh., 2004, no. 12, pp. 15–28.
Petrov, N.P. and Polyak, B.T., Robust D-Decomposition, Avtom. Telemekh., 1991, no. 11, pp. 41–53.
Frazer, R.A. and Duncan, W.J., On the Criteria for the Stability of Small Motion, Proc. Roy. Soc., Ser. A, 1929, vol. 124, pp. 642–654.
Polyak, B.T. and Tsypkin, Ya.Z., Frequency-Domain Criteria for Robust Stability and Aperiodicity of Linear Systems, Avtom. Telemekh., 1990, no. 9, pp. 45–54.
Barmish, B.R., Corless, M., and Leitmann, G., A New Class of Stabilizing Controllers for Uncertain Dynamical Systems, SIAM J. Control Optimiz., 1983, vol. 21, no.2, pp. 246–255.
Calafiore, G. and Polyak, B.T., Stochastic Algorithms for Exact and Approximate Feasibility of Robust LMIs, IEEE Trans. Automat. Control, 2001, vol. 46, no.11, pp. 1755–1759.
Liberzon, D. and Tempo, R., Common Lyapunov Functions and Gradient Algorithms, IEEE Trans. Automat. Control, 2004, vol. 49, no.6, pp. 990–994.
Meilakhs, A.M., On the Stabilization of Linear Control Systems in the Presence of Uncertainty, Avtom. Telemekh., 1975, no. 2, pp. 182–184.
Petersen, I.R. and Hollot, C.V., A Riccati Equation Approach to the Stabilization of Uncertain Linear Systems, Automatica, 1986, vol. 22, pp. 397–411.
Petersen, I.R. and McFarlane, D.C., Optimal Guaranteed Cost Control and Filtering for Uncertain Linear Systems, IEEE Trans. Automat. Control, 1994, vol. 39, pp. 1971–1977.
Polyak, B.T. and Tempo, R., Probabilistic Robust Design with Linear Quadratic Regulators, Syst. Control Lett., 2001, vol. 43, pp. 343–353.
Polyak, B. and Halpern, M., Optimal Design for Discrete-Time Linear Systems via New Performance Index, Int. J. Adaptive Control Sig. Proc., 2001, vol. 15, no.2, pp. 129–152.
Lozinskii, S.M., Estimating the Errors of the Approximate Solution of Systems of Ordinary Differential Equations, Dokl. Akad. Nauk SSSR, 1953, vol. 92, no.2, pp. 225–228.
Polyak, B.T. and Shcherbakov, P.S., Superstable Linear Control Systems. I. Analysis, Avtom. Telemekh., 2002, no. 8, pp. 37–53.
Rosenbrock, H.H., Computer-Aided Control System Design, London: Academic, 1974.
Voronov, A.A., Osnovy teorii avtomaticheskogo upravleniya. Avtomaticheskoe regulirovanie nepreryvnykh lineinykh sistem (Foundations of Automatic Control Theory. Automatic Regulation of Continuous-Time Linear Systems), Moscow: Energiya, 1980.
Ogata, K., Modern Control Engineering, Englewood Cliffs: Prentice Hall, 1990.
Kaszkurevich, E. and Bhaya, A., Matrix Diagonal Stability in Systems and Computation, Boston: Birkhauser, 2000.
Blanchini, F. and Sznaier, M., A Convex Optimization Approach for Fixed-Order Controller Design for Disturbance Rejection in SISO Systems, IEEE Trans. Automat. Control, 2000, vol. 45, pp. 784–789.
Polyak, B.T. and Shcherbakov, P.S., Superstable Linear Control Systems. II. Design, Avtom. Telemekh., 2002, no. 11, pp. 56–75.
Polyak, B.T., Extended Superstability in Control Theory, Avtom. Telemekh., 2004, no. 4, pp. 70–80.
Polyak, B.T. and Shcherbakov P.S., A New Approach to Robustness and Stabilization of Control Systems via Perturbation Theory, Proc. 14th World Congress of IFAC, Beijing, 1999, vol. C, pp. 13–18.
Polyak, B.T. and Shcherbakov, P.S., Numerical Search of Stable or Unstable Element in Matrix or Polynomial Families: A Unified Approach to Robustness Analysis and Stabilization, in Robustness in Identification and Control, Lect. Notes Control Inf. Sci., 1999, vol. 245, pp. 344–358.
Horn, R. and Johnson, C., Matrix Analysis, New York: Cambridge Univ. Press, 1985. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989.
Anderson, B.D.O., Bose, N.K., and Jury, E.I., Output Feedback Stabilization and Related Problems—Solution via Decision Methods, IEEE Trans. Automat. Control, 1975, vol. 20, no.1, pp. 53–66.
Polak, E. and Wardi, Y., Nondifferentiable Optimization Algorithm for Designing Control Systems Having Singular Value Inequalities, IEEE Trans. Automat. Control, 1982, vol. 18, no.3, pp. 267–283.
Malan, S., Milanese, M., and Taragna, M., Robust Analysis and Design of Control Systems Using Interval Arithmetic, Automatica, 1997, vol. 33, no.7, pp. 1363–1372.
Bobylev, N.A., Emelyanov, S.V., and Korovin, S.K., Estimates of Perturbations of Stable Matrices, Avtom. Telemekh., 1998, no. 4, pp. 15–24.
Kiselev, O.N. and Polyak, B.T., Low-Order Controller Design via the H ∞ Criterion and the Criterion of Maximal Robustness, Avtom. Telemekh., 1999, no. 3, pp. 119–130.
Francis, B., A Coursein H ∞ Control Theory, Berlin: Springer, 1987.
Keel, L.H. and Bhattacharyya, S.P., Robust Stability and Performance with Fixed-Order Controllers, Automatica, 1999, vol. 35, no.10, pp. 1717–1724.
Author information
Authors and Affiliations
Additional information
__________
Translated from Avtomatika i Telemekhanika, No. 5, 2005, pp. 7–46.
Original Russian Text Copyright © 2005 by Polyak, Shcherbakov.
This work was supported by the Russian Foundation for Basic Research, project no. 02-01-00127 and the Presidium of the Russian Academy of Sciences, Complex Programme no. 3.2.1.
Rights and permissions
About this article
Cite this article
Polyak, B.T., Shcherbakov, P.S. Hard Problems in Linear Control Theory: Possible Approaches to Solution. Autom Remote Control 66, 681–718 (2005). https://doi.org/10.1007/s10513-005-0115-0
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10513-005-0115-0