The Peaking Phenomenon and the Global Stabilization of Nonlinear Systems

IEEE Transactions on Automatic Control

Volumn 36, Issue 4, 1991, Pages 424-440

  Sussmann, H J ^a   Kokotovic, P V ^b  

^a RUTGERS UNIVERSITY   (United States)

^b UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CONTROL SYSTEMS, LINEAR; CONTROL SYSTEMS, NONLINEAR; DYNAMICS; MATHEMATICAL TECHNIQUES - EIGENVALUES AND EIGENFUNCTIONS; SYSTEM STABILITY - LYAPUNOV METHODS;

GLOBAL STABILIZATION; LINEAR PEAKING; NONLINEAR DYNAMICS; NONLINEAR GROWTH; PEAKING PHENOMENON;

CONTROL SYSTEMS, CASCADE;

EID: 0026140962     PISSN: 00189286     EISSN: 15582523     Source Type: Journal    
DOI: 10.1109/9.75101     Document Type: Article

References (44)

1. D. Aeyels, “Stabilization of a class of nonlinear systems by a smooth feedback,” Syst. Contr. Lett., vol. 5, pp. 181-191, 1985.
2. –, “Stabilization by smooth feedback of the angular velocity of a rigid body,” Syst. Contr. Lett., vol. 5, pp. 289-294, 1985.
3. C. I. Byrnes and A. Isidori, “A frequency domain philosophy for nonlinear systems, with application to stabilization and to adaptive control,” in Proc. 23rd IEEE Conf. Decision Contr., 1984, pp. 1569-1573.
4. –, “Global feedback stabilization of nonlinear minimum phase systems,” in Proc. 24th IEEE Conf. Decision Contr., 1985, pp. 1031-1037.
5. –, “Local stabilization of minimum-phase nonlinear systems,” Syst. Contr. Lett., vol. 11, pp. 9-17, 1988.
6. –, “New results and examples in nonlinear feedback stabilization,” Syst. Contr. Lett., vol. 12, 437-442, 1989.
7. –, “Feedback stabilization about attractors and the problem of asymptotic disturbance rejection,” in Proc. 28th IEEE Conf. Decision Contr., 1988, 32-36.
8. –, “Asymptotic stabilization of minimum phase nonlinear systems,” IEEE Trans. Automat. Contr., submitted for publication.
9. J. Carr, Applications of Centre Manifold Theory, New York: Springer-Verlag, 1981.
10. J. H. Chow and P. V. Kokotovic, “A two-stage Lyapunov-Bell-man feedback design of a class of nonlinear systems,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 656-663, 1981.
11. B. A. Francis and K. Glover, “Bounded peaking in the optimal linear regulator with cheap control,” IEEE Trans. Automat. vol.
12. W. Hahn, Stability of Motion. New York: Springer-Verlag, 1967.
13. G. Hewer and C. Kenney, “The sensitivity of the stable Lyapunov equation,” SIAM J. Contr. Optimiz., vol. 26, pp. 321-344, 1988.
14. R. M. Hirschorn, “Invertibility of nonlinear control systems,” SIAM J. Contr. Optimiz., vol. 17, pp. 289-297, 1979.
15. –, “Output tracking in multivariable nonlinear systems,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 593-393, 1981.
16. R. N. Izmailov, “‘Peaking’ effect in stationary linear systems in inputs and outputs,” Automatika i Telemekhanika, vol. 8, pp. 56-62, Aug. 1987.
17. V. Jurdjevic and J. P. Quinn, “Controllability and stability,” J. Diff. Equations, vol. 28, pp. 381-389, 1978.
18. K. Khorasani and P. V. Kokotovic, “A corrective feedback design for nonlinear systems with fast actuators,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 67-69, 1986.
19. –, “Feedback linearization of a flexible manipulator near its rigid body manifold,” Syst. Contr. Lett., vol. 6, pp. 187-192, 1985.
20. H. Khalil and P. V. Kokotovic, Eds., Singular Perturbations in Systems and Control. Piscataway, NJ: IEEE Press, 1986.
21. P. V. Kokotovic, “Recent trends in feedback design: An overview,” Automatica, vol. 21, pp. 225-236, 1985.
22. P. V. Kokotovic, H. Khalil, and J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design. New York: Academic, 1986.
23. P. V. Kokotovic and R. Marino, “On vanishing stability regions in nonlinear systems with high-gain feedback,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 967-970, 1986.
24. P. V. Kokotovic and H. J. Sussmann, “A positive real condition for global stabilization of nonlinear systems,” Syst. Contr. Lett., vol. 12, pp. 125-134, 1989.
25. D. E. Koditschek, “Adaptive techniques for mechanical systems,” in Proc. 5th Yale Workshop on Adaptive Syst., Yale University, New Haven, CT, 1987, 259-265.
26. R. Marino, “High gain feedback in nonlinear control systems,” Int. J. Contr., vol. 42, 1369-1385, 1985.
27. –, “Feedback stabilization of single-input nonlinear systems,” Syst. Contr. Lett., vol. 10, pp. 201-206, 1988.
28. T. Mita, “On zeros and responses of linear regulators and linear observers,” IEEE Trans. Automat. Contr., vol. AC-22, pp. 423-428, 1977.
29. A. Saberi and H. Khalil, “Quadratic-type Lyapunov functions for singularly perturbed systems,” IEEE Trans. Automat. Contr., vol. AC-29, pp. 542-550, 1984.
30. A. Saberi, P. V. Kokotovic, and H. J. Sussmann, “Global stabilization of partially linear composite systems,” SIAM J. Contr. Optimiz., vol. 28, no. 6, pp. 1491-1503, Nov. 1990.
31. V. R. Saksena, J. O’Reilly, and P. V. Kokotovic, “Singular perturbations and two-scale methods in control theory: Survey 1976-1983,” Automatica, vol. 20, pp. 273-293, 1984.
32. S. S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Trans. Automat. Contr., vol. AC-34, pp. 1989.
33. E. D. Sontag, “A Lyapunov-like characterization of asymptotic controllability,” SIAM J. Contr. Optimiz., vol. 21, pp. 462-471, 1983.
34. —, ‘‘Stabilizability, I/O stability, and coprime factorizations,’’ in Proc. IEEE Conf. Decision Contr., Austin, TX, Dec. 1988, pp. 457-458.
35. –, “Smooth stabilization implies coprime factorization,” IEEE Trans. Automat. Contr., vol. 34, pp.435-443, Apr. 1989.
36. —, “Further facts about input to state stabilization,” IEEE Trans. Automat. Contr., to be published.
37. E. D. Sontag and H. J. Sussmann, “Remarks on continuous feedback,” in Proc. 19th IEEE Conf. Decision Contr., 1980, pp. 916-921.
38. M. W. Spong, K. Khorasani, and P. V. Kokotovic, “An integral manifold approach to the feedback control of flexible joint robots,” J. Robot. Automation, vol. RA-3, no. 4, pp. 291-300, 1987.
39. A. Strauss and J. A. Yorke, “Perturbing uniform asymptotically stable nonlinear systems,” J. Differential Equations, vol. 6, pp. 452-483, 1969.
40. H. J. Sussmann, “Limitations on the stabilizability of globally minimum phase systems,” IEEE Trans. Automat. Contr., vol. 35, pp. 117-119, 1990.
41. H. J. Sussmann and P. V. Kokotovic, “The Peaking phenomenon and the global stabilization of nonlinear systems,” Rutgers University Center for Systems and Control, Rep. SYCON 89-05, March 1989.
42. J. Tsinias, “Sufficient Lyapunovlike conditions for stabilization,” Math. Contr. Signals Syst., vol. 2, pp. 343-357, 1989.
43. P. P. Varaiya and R. Liu, “Bounded-input bounded-output stability of nonlinear time-varying systems,” SIAM J. Contr. Optimiz., vol. 4, pp. 698-704, 1966.
44. M. Vidyasagar, “Decomposition techniques for large scale systems with nonadditive interactions: Stability and stabilizability,” IEEE Trans. Automat. Contr., vol. AC-25, pp. 773-779, 1980.