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1. ., ., . . . – .: , 1989. – 263 .
2. . ., . .
MATLAB. – : , 1999. – 467 .
3. . ., . ., . .
// . – 1996. – № 2. – . 3–33.
4. . ., . ., . .
. – : , 2000. – 549 .
5. . ., . . . //
. – 1994. – № 9. – . 3–22.
6. . . ,
// . . – 1997. – № 2. – . 103–106.
7. . .
// . . – 1997. – № 4. – . 69–73.
8. . . // .
– 1998. – № 9. – . 87–99.
9. . . . – : ,
2003. – 282 .
10. . ., . . . . :
. – Saarbrucken: LAP LAMBERT Academic Publisher, 2012. – 292 .
11. . . // . –
1996. – № 2. – . 117–125.
12.Egart B. Stability of adaptive controllers. – NY: Springer-Verlag, 1979.
13.Feuer A., Morse A.S. Adaptive control of single-input, single-output linear systems // IEEE Trans. Automat. Contr. – 1978. – V. 23. – № 4. – P. 557–569.
14.Kanellakopoulos I., Kokotovich P.V., Morse A.S. Systematic design of adaptive controllers for feedback linearizable systems // IEEE Trans. Automat. Contr. – 1991. – V. 36. – P. 1241–1253.
15.Kreisselmeier G. Adaptive control of a class of slowly time varying plants // Syst. Control Lett. – 1986. – V. 8. – № 2. – P. 97–103.
16.Krstic M., Kanellakopoulos I., Kokotovich P.V. Adaptive nonlinear control without overparametrization // Syst. Control Lett. – 1992. – V. 19. – P. 177–185.
17.Krstic M., Kanellakopoulos I., Kokotovic P. Nonlinear and adaptive control design. – NY: John Weley and Sons Inc., 1995.
18.Krstic M., Kokotovic P. Adaptive nonlinear output-feedback schemes with Marino-Tomei controller // IEEE Trans. Automat. Contr. – 1996. – V. 41. – № 2. – P. 274–280.
19.Marino R., Tomei P. Robust adaptive regulation of linear time-varying systems // IEEE Trans. Automat. Contr. – 2000. – V. 45. – № 7. – P. 1301–1311.
20.Marino R., Tomei P. Adaptive control of linear time-varying systems // Automatica. – 2003. – V. 39. – № 4. – P. 651–659.
22.Morse A.S., Isidori A. and T.J. Tarn. High-order parameter tuner for the adaptive control of linear and non-linear systems // Systems, Models and Feedback: Theory and Applications. – Basel: Birkhauser. – 1992. – P. 339–364.
23.Narendra K.S., Annaswamy A.M. Stable Adaptive Systems. – Englewood Cliffs, NJ: Prentice-Hall, 1989. – 236 p.
24.Narendra K.S., Valavani L.S. Stable adaptive controller design – direct control // IEEE Trans. Automat. Contr. – 1978. – V. 23. – № 4. – P. 570–583.
25.Nikiforov V.O. Adaptive non-linear tracking with complete compensation of unknown disturbances // Euro-pean Journal of Contr. – 1998. – V. 4. – № 2. – P. 132–139.
26.Nikiforov V.O. Robust high-order tuner of simplified structure // Automatica. –1999. – V. 35. – № 8. – P. 1409–1415.
27.Nikiforov V.O., Voronov K.V. Nonlinear adaptive controller with integral action // IEEE Trans. Automat. Contr. – 2001. – V. 46. – № 12. – P. 2035–2037.
28.Qian . and Lin W. Output feedback control of a class of nonlinear systems: a nonseparation principle para-digm // IEEE Trans. Automat. Contr. – 2002. – V. 47. – № 10. – P. 1710–1715.
29.Tsakalis K.S., Ioannou P.A. Adaptive control of linear time-varying plants // Automatica. – 1987. – V. 23. –
№ 4. – P. 459–468.
30.Tsakalis K.S., Ioannou P.A. Adaptive control of linear time-varying plants: A new model reference controller structure // IEEE Trans. Automat. Contr. – 1989. – V. 34. – № 10. – P. 1038–1047.
31.Tsakalis K.S., Ioannou P.A. A new indirect adaptive control scheme for time-varying plants // IEEE Trans. Automat. Contr. – 1990. – V. 35. – № 6. – P. 697–705.
32.Tsakalis K.S., Ioannou P.A. Linear time varying systems: control and adaptation. – Upper Saddle River, NJ: Prentice-Hall, 1993. – 270 p.
33.Zang Z., Bitmead R.R. Transient bounds for adaptive control systems // IEEE Trans. Automat. Contr. – 1994. – V. 39. – № 1. – P. 171–175.
34.Zhang Y., Fidan B., Ioannou P.A. Backstepping control of linear time-varying systems with known and un-known parameters // IEEE Trans. Automat. Contr. – 2003. – V. 48. – № 11. – P. 1908–1925.
35.Arcak M., Kokotovic P. Feasibility conditions for circle criterion design // Systems and Control Letters. – 2001. – V. 42. – № 5. – . 405–412.
36.Arcak M., Larsen M., Kokotovic P. Circle and Popov criteria as tools for nonlinear feedback design // Auto-matica. – 2003. – V. 39. – № 4. – P 643–650.
37.Byrnes C.I., Isidori A. Asymptotic stabilization of minimum phase nonlinear systems // IEEE Trans. Auto-mat. Contr. – 1991. – V. 36. – № 10. – P. 1122–1137.
38.Isidori A. A remark on the problem of semiglobal nonlinear output regulation // IEEE Trans. Automat. Contr. – 1997. – V. 42. – № 12. – P. 1734–1738.
39.Isidori A. A tool for semi-global stabilization of uncertain non-minimum-phase nonlinear systems via output feedback // IEEE Trans. Automat. Contr. – 2000. – V. 45. – № 10. – P. 1817–1827.
40.Isidori A., Byrnes C.I. Output regulation of nonlinear systems // IEEE Trans. Automat. Contr. – 1990. – V. 35. – № 2. – P. 131–140.
41. . ., . ., . .
// . – 1997. – № 6. – . 125–134.
42. . . //
. – 1974. – № 12. – . 96–103.
43. . .
// . . . – 1976. – № 2. – . 436–446.
44.Fradkov A.L. Passification of nonsquare linear systems and Yakubovich-Kalman-Popov Lemma // European Journal of Control. – 2003. – № 6. – P. 573–582.
45. . ., . .
// .
2005. – № 1. – . 118–129.
46. . ., . ., . .
-. – -.: , 1978. – 400 .
47. . . . I.
// . . . – 1996. – № 8–9. – . 50–56; II. // . . . – 1997. – № 4. – . 28–33.
48.Kokotovic P., Murat A. Constructive nonlinear control: a historical perspective // Automatica. – 2001. – V. 37. – № 5. – P 637–662.
49. . .
// . . – 2003. – № 1. – . 35–39.
51. . . . – .: , 1970. – 453 .
52. Bobtsov A.A., Efimov D.V. On robustness property of dynamical systems feedback connection with respect to multiplicative disturbances // IFAC Symposium NOLCOS 2004. – Stuttgart, Germany, 2004. – P. 1457– 1462.
53. . ., . ., . . . //
. – 2000. – № 3. – . 3–37.
54.Fradkov A., Hill D. Exponential feedback passivity and stabilizability of nonlinear systems // Automatica. – 1998. – V. 34. – № 6. – P. 697–703.
55. . . . – .: , 1994. – 344 .
56. . ., . . . – :
( ), 2000. – 245 .
57. . . // « ». – .: ,
1959. – № 3. – . 13–31.
58. . . . – .: , 1965. –
416 .
59. . . . – .: ,
1975. – 308 .
60. . ., . . : . 2. //
. – 2004. – № 4. – . 3–34.
61. . . . – :
, 2011. – 174 .
62. . ., . ., . ., . .
// . . . – 2005. – № 12. – . 8–13.
63.Mascolo S., Grassi G. Controlling chaotic dynamics using backstepping design with application to the Lo-renz system and Chua’s circuit // International Journal of Bifurcation and Chaos. – 1999. – V. 9. – № 7. – P. 1425–1434.
64.Yang T., Chua L.O. Generalized synchronization of chaos via linear transformations // International Journal of Bifurcation and Chaos. – 1999. – V. 9. – № 1. – P. 215–219.
65.Spong M.W., Vidyasagar M. Robot dynamics and control. – New York: Wiley, 1989. – 336 p.
66.Nguang S.K. Robust stabilization of a class of time-delay nonlinear systems // IEEE Trans. Automat. Contr. 2000. – V. 45. – № 4. – P.756–762.
67. . ., . ., . .
// - . – 2003. – . 10. – . 10–18.
68. . ., ., . ., . .
. I: // .
. – 2010. – № 5. – . 125–137; II: // . . – 2010. – № 6. – . 170–181.
69.Azzoni P., Moro D., Ponti F. Engine and load torque estimation with application to electronic throttle con-trol. – SAE technical paper, 1998. – № 980795.
70.HendricksE., Chevalier A., Jensen M. Event based engine control: practical problems and solutions. – SAE paper. – 1995. – № 950008.
71.Hendricks E., Jensen M., Chevalier A., Vesterholm T. Conventional event based engine control. – SAE pa-per. – 1994. – № 940377.
72.Kim Y.-W., Rizzoni G., V. Utkin Automotive engine diagnostics and control via nonlinear estimation // IEEE Control Systems. – 1998. – V. 18. – № 5. – P. 84–99.
73.Stotsky A., Kolmanovsky I. Application of input estimation techniques to charge estimation and control in automotive engines // Control Engineering Practice. – 2002. – № 10. – P. 1371–1383.
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