zbMATH — the first resource for mathematics

Method of passification in adaptive control, estimation, and synchronization. (English. Russian original) Zbl 1195.93117
Autom. Remote Control 67, No. 11, 1699-1731 (2006); translation from Avtom. Telemekh 2006, No. 11, 3-37 (2006).
Summary: The main results of the method of passification which are based on applying the Yakubovich-Kalman frequency theorem to the design of the feedback systems and examples of its use in the problems of adaptive control, state estimation, and synchronization are presented. Various types of the adaptive control algorithms with implicit reference model such as the algorithms of stabilization and tracking with the prescribed dynamics, algorithms with adaptive tuning of the standard control laws, and combined signal-parametric algorithm of adaptive control are described. Brief information about the shunting method in the adaptive control problem is given. The experimental results with the adaptive control on the “Helicopter” benchmark are described. Consideration is given to the problem of adaptive control of the nonlinear plants. Examples of applying the method of passification and adaptive observers to the problems of synchronization of the nonlinear oscillators and message transmission by chaotic signals are presented.

93D15 Stabilization of systems by feedback
93C40 Adaptive control/observation systems
93B40 Computational methods in systems theory (MSC2010)
93D21 Adaptive or robust stabilization
Full Text: DOI
[1] Bogolyubov, N.N. and Mitropol’skii, Yu.A., Asimptoticheskie metody v teorii nelineinykh kolebanii (Asymptotic Methods in the Theory of Nonlinear Oscillations), Moscow: Nauka, 1974. · Zbl 0303.34043
[2] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F., Matematicheskaya teoriya optimal’nykh protsessov (Mathematical Theory of Optimal Processes), Moscow: Nauka, 1974.
[3] Kharitonov, V.L., On Asymptotical Stability of the Equilibrium of the Systems of Linear Differential Equations, Diff. Uravn., 1978, vol. 11, pp. 2086–2088.
[4] Matveev, A.S., Theory of Optimal Control in the Works of V.A. Yakubovich, Avtom. Telemekh., 2006, no. 10, pp. 120–174.
[5] Gusev, S.V. and Likhtarnikov, A.L., Kalman-Popov-Yakubovich Lemma and the S-procedure: A Historical Essay, Avtom. Telemekh., 2006, no. 11, pp. 77–121. · Zbl 1195.93002
[6] Barabanov, N.E., Gelig, A.Kh., Leonov, G.A., et al., Frequency Theorem (Yakubovich-Kalman Lemma) in the Control Theory, Avtom. Telemekh., 1996, no. 10, pp. 3–40. · Zbl 0932.93001
[7] Petrov, B.N., Rutkovskii, V.Yu., Krutova, I.N., and Zemlyakov, S.D., Printsipy postroeniya i proektirovaniya samonastraivayushchikhsya sistem (Principles of Construction and Design of the Self-adjusting Systems), Moscow: Mashinostroenie, 1972.
[8] Yakubovich, V.A., Solution of some Matrix Inequalities in the Automatic Control Theory, Dokl. Akad. Nauk SSSR, 1962, vol. 143, no. 6, pp. 1304–1307. Translated into English as: The Solution of Certain Matrix Inequalities in Automatic Control Theory, Soviet Math. Dokl., 1962, no. 3, pp. 620–623.
[9] Toker, O. and Ozbay, H., On the NP-hardness of Solving Bilinear Matrix Inequalities and Simultaneous Stabilization with Static Output Feedback, in Proc. Am. Control Conf., 1995, pp. 2525, 2526.
[10] Fradkov, A.L., Design of an Adaptive System of Stabilization of a Linear Dynamic Plant, Avtom. Telemekh., 1974, no. 12, pp. 96–103.
[11] Fradkov, A.L., Quadratic Lyapunov Functions in the Problem of Adaptive Stabilization of the Linear Dynamic Plant, Sib. Mat. Zh., 1976, vol. 17, no. 2, pp. 436–445. Translated into English in Siberian Math. J., 1976, vol. 17, no. 2, pp. 341–348. · Zbl 0357.93024 · doi:10.1007/BF00967581
[12] Seron, M.M., Hill, D.J., and Fradkov, A.L., Adaptive Passification of Nonlinear Systems, in Proc. 33rd Conf. Dec. Control, CDC, 1994, pp. 190–195.
[13] Fomin, V.N., Fradkov, A.L., and Yakubovich, V.A., Adaptivnoe upravlenie dinamicheskimi ob”ektami (Adaptive Control of Dynamic Plants), Moscow: Nauka, 1981. · Zbl 0522.93002
[14] Fradkov, A.L., Adaptivnoe upravlenie slozhnymi sistemami (Adaptive Control of Complex Systems), Moscow: Nauka, 1990. · Zbl 0962.93001
[15] Miroshnik, I.V., Nikiforov, V.O., and Fradkov, A.L., Nelininoe i adaptivnoe upravlenie slozhnymi sistemami (Nonlinear and Adaptive Control of Complex Systems), St. Petersburg: Nauka, 2000. Translated into English under the title Nonlinear and Adaptive Control of Complex Systems, Dordrecht: Kluwer, 1999. · Zbl 0962.93001
[16] Andrievskii, B.R., Stotskii, A.A., and Fradkov, A.L., Algorithms of the Speed Gradient in Adaptation and Control, Avtom. Telemekh., 1988, no. 12, pp. 3–39.
[17] Andrievsky, B.R. and Fradkov, A.L., Adaptive Controllers with Implicit Reference Models Based on Feedback Kalman-Yakubovich Lemma, in Proc. 3rd IEEE Conf. Control Appl., Glasgow, 1994, pp. 1171–1174.
[18] Andrievsky, B.R., Churilov, A.N., and Fradkov, A.L., Feedback Kalman-Yakubovich Lemma and Its Applications to Adaptive Control, in Proc. 35th IEEE Conf. Dec. Control, Kobe, 1996.
[19] Andrievskii, B.R. and Fradkov, A.L., Izbrannye glavy teorii avtomaticheskogo upravleniya s primerami na yazyke MATLAB (Selected Chapters of the Theory of Automatic Control with Examples in MATLAB), St. Petersburg: Nauka, 1999. · Zbl 0964.93005
[20] Popov, V.M., On One Problem in the Theory of Absolute Stability of Automatic Systems, Avtom. Telemekh., 1964, vol. 25, no. 9, pp. 1129–1134. · Zbl 0142.36602
[21] Willems, J.C., Dissipative Dynamical Systems, Part I: General Theory, Arch. Rational Mech. Anal., 1972, vol. 45, pp. 321–351. · Zbl 0252.93002 · doi:10.1007/BF00276493
[22] Yakubovich, V.A., Periodic and Almost Periodic Limiting Modes of the Controllable Systems with Several Nonlinearities, Dokl. Akad. Nauk SSSR, 1966, vol. 171, no. 3, pp. 533–536. Translated into Enaglish as: Periodic and Almost-periodic Limit Modes of Controlled Systems with Several, in General Discontinuous, Nonlinearities, Soviet Math. Dokl. 1966, vol. 7, no. 6, pp. 1517–1521.
[23] Fradkov, A.L., Passification of Nonsquare Linear Systems and Yakubovich-Kalman-Popov Lemma, Eur. J. Control, 2003, no. 6, pp. 573–582. · Zbl 1293.93377
[24] Gu, G., Stabilizability Conditions of Multivariable Uncertain Systems via Output Feedback Control, IEEE Trans Automat. Control, 1990, vol. 35, no. 8, pp. 925–927. · Zbl 0719.93068 · doi:10.1109/9.58501
[25] Abdallah, C., Dorato, P., and Karni, S., SPR Design Using Feedback, in Proc. Am. Control Conf., 1990, pp. 1742, 1743.
[26] Weiss, H., Wang, Q., and Speyer, J.L., System Characterization of Positive Real Conditions IEEE Trans. Automat. Control, 1994, vol. 39, no. 3, pp. 540–544. · Zbl 0814.93056 · doi:10.1109/9.280753
[27] Huang, C.H., Ioannou, P.I., Maroulas, J., and Safonov, M.G., Design of Strictly Positive Real Systems Using Constant Output Feedback, IEEE Trans. Automat. Control, 1999, vol. 44, no. 3, pp. 569–573. · Zbl 0965.93082 · doi:10.1109/9.751352
[28] Efimov, D.V. and Fradkov, A.L., Adaptive Tuning to Bifurcation for Time-varying Nonlinear Systems, Automatica, 2006, vol. 42(3), pp. 417–425. · Zbl 1123.93056 · doi:10.1016/j.automatica.2005.09.018
[29] Bobtsov, A.A. and Nikolaev, N.A., Fradkov Theorem-based Design of the Control of Nonlinear Systems with Functional and Parametric Uncertainties, Avtom. Telemekh., 2005, no. 1, pp. 118–129. · Zbl 1130.93347
[30] Saberi, A., Kokotović, P., and Sussmann, H., Global Stabilization of Partially Linear Composite Systems, SIAM J. Control Optim., 1990, vol. 28, pp. 1491–1503. · Zbl 0719.93071 · doi:10.1137/0328079
[31] Bondarko, V.A., Likhtarnikov, A.L., and Fradkov, A.L., Design of Adaptive System of Stabilization of a Distributed-parameter Linear Plant, Avtom. Telemekh., 1979, no. 12, pp. 95–103. · Zbl 0485.93048
[32] Bondarko, V.A. and Fradkov, A.L., Necessary and Sufficient Conditions for Passifiability of the Distributed Linear Systems, Avtom. Telemekh., 2003, no. 4, pp. 3–17. · Zbl 1082.93053
[33] Byrnes, C.I., Isidori, A., and Willems, J.C., Passivity, Feedback Equivalence and the Global Stabilization of Minimum Phase Nonlinear Systems, IEEE Trans. Automat. Control, 1991, vol. AC-36, pp. 1228–1240. · Zbl 0758.93007 · doi:10.1109/9.100932
[34] Fradkov, A.L. and Hill, D.J., Exponential Feedback Passivity and Stabilizability of Nonlinear Systems, Automatica, 1998, no. 6, pp. 697–703. · Zbl 0937.93036
[35] Zemlyakov, S.D. and Rutkovskii, V.Yu., Design of Algorithms to Modify the Rearrangeable Coefficients in the Self-adjusting Systems with Reference Model, Dokl. Akad. Nauk SSSR, 1967, vol. 174, no. 1, pp. 47–49.
[36] Landau, J.D., Adaptive Control Systems. The Model Reference Approach, New York: Dekker, 1979. · Zbl 0475.93002
[37] Petrov, B.N., Rutkovskii, V.Yu., and Zemlyakov, S.D., Adaptivnoe koordinatno-parametricheskoe upravlenie (Adaptive Coordinate-Parametric Control), Moscow: Nauka, 1980. · Zbl 0498.93002
[38] Tsykunov, A.M., Adaptivnoe upravlenie ob”ektami s posledeistviem (Adaptive Control of Delay Plants), Moscow: Nauka, 1984. · Zbl 0574.93034
[39] Popov, A.M. and Fradkov, A.L., Adaptive Control of Singularly Perturbed Plants, in Tr. XI Vses. sov. po probl. upravleniya (Proc. XI All-Union Conf. Control), Erevan, 1983.
[40] Ioannou, P.A. and Kokotović, P.V., Adaptive Systems with Reduced Models, Berlin: Springer, 1983.
[41] Fradkov, A.L., Design of Adaptive Control System for Nonlinear Singularly Perturbed Plants, Avtom. Telemekh., 1987, no. 6, pp. 100–110.
[42] Ilchmann, A., Non-identifier-based Adaptive Control of Dynamical Systems: A Survey, IMA J. Math. Control Info, 1991, no. 8, pp. 321–366. · Zbl 0824.93036
[43] Byrnes, C.I. and Willems, J.C., Adaptive Stabilization of Multivariable Linear Systems, in Proc. 23rd IEEE Conf. Decision and Control, Las Vegas, Nevada, 1984, pp. 1574–1577.
[44] Fradkov, A.L., Scheme of the Speed Gradient and its Application to the Problems of Adaptive Control, Avtom. Telemekh., 1979, vol. 40, no. 9, pp. 1333–1342. · Zbl 0434.93036
[45] Derevitskii, D.P. and Fradkov, A.L., Prikladnaya teoriya diskretnykh adaptivnykh sistem upravleniya (Applied Theory of Discrete Adaptive Control Systems), Moscow: Nauka, 1981. · Zbl 0518.93001
[46] Andrievskii, B.R., Using the Method of Matrix Inequalities to Design the Adaptive Tracking Systems, in Optimal’nye i adaptivnye sistemy (Optimal and Adaptive Systems), Frunze: Frunz. Politekh. Inst., 1979, pp. 20–25.
[47] Utkin, V.I., Skol’zyashchie rezhimy v zadachakh optimizatsii i upravleniya, Moscow: Nauka, 1981. Translated into English under the title Sliding Modes in Control Optimization, Heidelberg: Springer, 1992.
[48] Stotsky, A.A., Combined Adaptive and Variable Structure Control, in Variable Structure and Lyapunov Control, Zinober, A.S.I., Ed., London: Springer, 1994, pp. 313–333. · Zbl 0805.93007
[49] Druzhinina, M.V., Nikiforov, V.O., and Fradkov, A.L., Methods of Adaptive Control of Nonlinear Plants by Output, Avtom. Telemekh., 1996, no. 2, pp. 3–33. · Zbl 0931.93040
[50] Nikiforov, V.O. and Fradkov, A.L., Schemes of Adaptive Control with Extended Error. Review, Avtom. Telemekh., 1994, no. 9, pp. 3–26.
[51] Fradkov, A.L., Adaptive Stabilization of the Minimum Phase Plants with Vector Input without Measuring the Output Derivatives, Dokl. Ross. Akad. Nauk, 1994, vol. 337, no. 5, pp. 592–594. Translated into English in Physics-Doklady, 1994, vol. 39, no. 8, pp. 550–552.
[52] Andrievsky, B.R., Fradkov, A.L., and Stotsky, A.A., Shunt Compensation for Indirect Sliding-mode Adaptive Control, in Proc. 13th IFAC World Congress, San Francisco, 1996, vol. K, pp. 193–198.
[53] Andrievskii, B.R. and Fradkov, A.L., Method of Shunting in the Problem of Adaptive Control of the Unstable and Nonminimum Phase Plants, in Mezhdunar. konf. po probl. upravleniya, posv. 60-letiyu IPU RAN. Tez. dokl. (Int. Conf. Control Problems, Abstracts of Papers), Moscow, 1999, vol. 1, pp. 153, 154.
[54] Andrievsky, B.R. and Fradkov, A.L., Combined Adaptive Autopilot for an UAV Flight Control, in Proc. IEEE Conf. Control Appl., Glasgow, Sept. 2002, pp. 290, 291.
[55] Andrievsky, B.R. and Fradkov, A.L., Combined Adaptive Flight Control System, in Proc. 5th Int. ESA Conf. Spacecraft Guidance, Navigation Control Syst., Frascati, Italy, 2002, ESA-516, Feb. 2003, pp. 299–302.
[56] Andrievsky, B.R. and Fradkov, A.L., UAV Guidance System with Combined Adaptive Autopilot, in Proc. IASTED Int. Conf. ”Intelligent Systems and Control” (ISC 2003), Salzburg: ACTA Press, 2003, pp. 91–93.
[57] Fradkov, A.L. and Andrievskii, B.R., Shunting-based Design of Robust Autopilot, in XI Sankt-Peterb. mezhdunar. konf. po integrirovannym navigatsionnym sistemam (XI St. Petersburg Int. Conf. Integrated Navigation Systems), St. Petersburg: TSNII ”Elektropribor,” 2004, pp. 36–38.
[58] Fradkov, A.L. and Andrievsky, B.R., Shunting Method for Control of Homing Missiles with Uncertain Parameters, Preprint 16th IFAC Symp. on Automatic Control in Aerospace (ACA’2004), St. Petersburg, 2004, vol. 2, pp. 33–38.
[59] Fradkov, A.L. and Andrievsky, B.R., Combined Adaptive Controller for UAV Guidance, Eur. J. Control, 2005, vol. 11, no. 1, pp. 71–79. · Zbl 1293.93445 · doi:10.3166/ejc.11.71-79
[60] Apkarian, J., Internet Control, Circuit Cellar, 1999, vol. 110. The manuscript available on the site http://www.circuitcellar.com .
[61] Quanser Co., URL: http://www.quanser.com/choice.asp .
[62] Laboratoire d’analyse et architechture des systèmes LAAS-CNRS, Toulouse, France, URL: http://www.laas.fr .
[63] Andrievskii, B.R., Arzel’e, D., Fradkov, A.L., and Peaucelle, D., Adaptive Control of Pitch Angle for LAAS ’Helicopter Benchmark,’ Proc. 12th St. Petersburg Int. Conf. ’Integrated Navigation Systems,’ May 23–25, 2005, St. Petersburg: CNRI ”Elektropribor,” 2005.
[64] Andrievsky, B., Fradkov, A., and Peaucelle, D., Adaptive Control Experiments for LAAS ”Helicopter” Benchmark, in Proc. 2nd Int. IEEE Conf. ”Physics and Control,” St. Petersburg, 2005, pp. 760–766. · Zbl 1293.93444
[65] Fradkov, A.L. and Druzhinina, M.V., Output-feedback Nonlinear Adaptive Control with Implicit Reference Model, in Proc. Am. Control Conf., 2001, pp. 3115, 3116.
[66] Blekhman, I.I., Sinkhronizatsiya dinamicheskikh sistem (Synchronization of Dynamic Systems), Moscow: Nauka, 1971. · Zbl 0233.70014
[67] Blekhman, I.I., Vibratsionnaya mekhanika (Vibration Mechanics), Moscow: Nauka, 1994. · Zbl 0867.70002
[68] Pikovskii, A.B., Rozenblyum, M.B., and Kurths, Yu., Sinkhronizatsiya. Fundamental’noe nelineinoe yavlenie (Synchronization. Fundamental Nonlinear Phenomenon), Moscow: Tekhnosfera, 2003.
[69] Leonov, G.A. and Smirnova, V.B., Matematicheskie problemy teorii fazovoi sinkhronizatsii (Mathematical Issues of the Phase Synchronization Theory), St. Petersburg: Nauka, 2000. · Zbl 0961.93001
[70] Lindsey, W., Synchronization Systems in Communications and Control, Englewood Cliffs: Prentice Hall, 1972. Translated under the title Sistemy sinkhronizatsii v svyazi i upravlenii, Moscow: Mir, 1978.
[71] Fradkov, A.L. and Pogromsky, A.Yu., Introduction to Control of Oscillations and Chaos, Singapore: World Scientific, 1998. · Zbl 0945.93003
[72] Pecora, L.M. and Carroll, T.L., Synchronization in Chaotic Systems, Phys. Rev. Lett., 1990, vol. 64, pp. 821–823. · Zbl 0938.37019 · doi:10.1103/PhysRevLett.64.821
[73] Dmitriev, A.S., Panas, A.I., and Starkov, S.O., Dynamic Chasos as the Paradigm of the Modern Communication Systems, Zarub. Radioelektron., 1997, no. 10, pp. 4–26.
[74] IEEE Trans. Circ. Syst., Special Issue on Applications of Chaos in Modern Communication Systems, Kocarev, L., Maggio, G.M., Ogorzalek M., et al., Eds., 2001, vol. 48, no. 12. · Zbl 0991.00029
[75] International Journal Circuit Theory Applications. Special issue: Communications, Information Processing and Control Using Chaos, Hasler, M. and Vandewalle, J., Eds., 1999, vol. 27, no. 6.
[76] Belykh, V.N., Belykh, I.V., and Hasler, M., Hierarchy and Stability of Partially Synchronous Oscillations of Diffusively Coupled Dynamical Systems, Phys. Rev. E, 2000, vol. 62, no. 5, pp. 6332–6345. · doi:10.1103/PhysRevE.62.6332
[77] Rabinovich, M.I., Abarbanel, H.D.I., Huerta, R., et al., Self-regularization of Chaos in Neural Systems: Experimental and Theoretical Results, IEEE Trans. Circ. Syst. I, 1997, vol. 44, pp. 997–1005. · doi:10.1109/81.633889
[78] Fradkov, A.L., Nonlinear Adaptive Control: Regulation, Tracking, Oscillations, in Proc. 1st IFAC Workshop ”New Trends in Design of Control Systems,” Smolenice, 1994, pp. 426–431.
[79] Fradkov, A.L. and Markov, A.Yu., Adaptive Synchronization of Chaotic Systems Based on Speed Gradient Method and Passification, IEEE Trans. Circ. Syst. I, 1997, no. 10, pp. 905–912.
[80] Yuan, J. and Wonham, W., Probing Signals for Model Reference Identification, IEEE Trans. Automat. Control, 1977, vol. AC-22, pp. 530–538. · Zbl 0361.93021 · doi:10.1109/TAC.1977.1101556
[81] Andrievskii, B.R. and Fradkov, A.L., Elementy matematicheskogo modelirovaniya v programmnykh sredakh MATLAB 5 and Scilab (uch. posobie) (Elements of Mathematical Modeling in the MATLAB 5 and Scilab Programming Environments (Tutorial)), St. Petersburg: Nauka, 2001.
[82] Fradkov, A.L., Nikiforov, V.O., and Andrievsky, B.R., Adaptive Observers for Nonlinear Nonpassifiable Systems with Application to Signal Transmission, in Proc. 41th IEEE Conf. Dec. Control, Las Vegas, 2002, pp. 4706–4711.
[83] Van Trees, H.L., Detection, Estimation, and Modulation Theory. Part II. Theory of Nonlinear Modulation, New York: Wiley, 1971. Translated under the title Teoriya obnaruzheniya, otsenok i modulyatsii. T. 2. Teoriya nelineinoi modulyatsii, Moscow: Sovetskoe Radio, 1975.
[84] Fradkov, A.L., Nijmeijer, H., and Markov, A., Adaptive Observer-based Synchronization for Communications, Int. J. Bifurcat. Chaos, 2000, vol. 10(12), pp. 2807–2814. · Zbl 0972.93005 · doi:10.1142/S0218127400001869
[85] Andrievsky, B.R., Adaptive Synchronization Methods for Signal Transmission on Chaotic Carriers, Math. Comput. Simulation, 2002, vol. 58, nos. 4–6, pp. 285–293. · Zbl 0995.65133 · doi:10.1016/S0378-4754(01)00373-1
[86] Peaucelle, D., Fradkov, A., and Andrievsky, B., Robust Passification via Static Output Feedback,–LMI Results, in Preprint 16th IFAC World Congress Automat. Control, Prague, 2005.
[87] Peaucelle, D., Fradkov A., Andrievsky, B., Passification-based Adaptive Control: Robustness Issues, in Preprint 5th IFAC Symp. Robust Control Design (ROCOND’06), Toulouse, France, July 5–7, 2006. · Zbl 1298.93210
[88] Sun, W., Khargonekar, P., and Shim, D., Solution to the Positive Real Control Problem for Linear Time-invariant Systems, IEEE Trans. Automat. Control, 1994, vol. 39, no. 10, pp. 2034–2046. · Zbl 0821.93064 · doi:10.1109/9.280776
[89] Fradkov, A.L., Andrievsky, B., and Evans, R.J., Adaptive Observer-based Synchronisation of Chaotic Systems in Presence of Information Constraints, in Preprint 1st IFAC Conf. Anal. Control Chaotic Syst. ”Chaos 06,” Rheims, France, 2006.
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.