L2-Gain Analysis of Nonlinear Systems and Nonlinear State Feedback H∞ Control

IEEE Transactions on Automatic Control

Volumn 37, Issue 6, 1992, Pages 770-784

  der Schaft, A J ^a  

^a UNIVERSITY OF TWENTE   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

CONTROL SYSTEMS, OPTIMAL; MATHEMATICAL TECHNIQUES - EIGENVALUES AND EIGENFUNCTIONS; MATHEMATICAL TECHNIQUES - STATE SPACE METHODS; SYSTEM STABILITY;

H-INFINITY CONTROL; HAMILTON-JACOBI EQUATIONS; HAMILTONIAN VECTOR FIELD; INVAERIANT MANIFOLDS; L-2-GAIN ANALYSIS; SMOOTH NONLINEAR SYSTEMS;

CONTROL SYSTEMS, NONLINEAR;

EID: 0026883666     PISSN: 00189286     EISSN: 15582523     Source Type: Journal    
DOI: 10.1109/9.256331     Document Type: Article

References (40)

1. B.D.O. Anderson, “An algebraic solution to the spectral factorization problem,” IEEE Trans. Automat. Contr., vol. AC-12, pp. 410–414, 1967.
2. V. I. Arnold, Mathematical Methods of Classical Mechanics. Berlin: Springer-Verlag, 1978 (translation of the 1974 Russian edition).
3. R. A. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed. Reading, MA: Benjamin/Cummings, 1978.
4. R. W. Brockett, Finite Dimensional Linear Systems. New York: Wiley, 1970.
5. P. Brunovsky, “On the optimal stabilization of nonlinear systems,” Czech. Math. J., vol. 18, pp. 278–293, 1968.
6. J. A. Ball and J. W. Helton, “Factorization of nonlinear systems: Towards a theory for nonlinear H ∞ control,” in Proc. 27th CDC, Austin, TX, 1988, pp. 2376–2381.
7. J. A. Ball and J. W. Helton, “H ∞ control for nonlinear plants: Connections with differential games,” in Proc. 28th CDC, Tampa, FL, 1989, pp. 956–962.
8. T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory. New York: Academic, 1982.
9. C. I. Byrnes, A. Isidori, and J. C. Willems, “Passivity, feedback equivalence and the global stabilization of minimum phase nonlinear systems,” IEEE Trans. Automat. Contr., vol. 36, pp. 1228–1240, 1991.
10. C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Out-put Properties. New York: Academic, 1975.
11. J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, “State space solutions to standard H 2 and H ∞ control problems,” IEEE Trans. Automat. Contr., vol. 34, 831–846, 1990.
12. B. A. Francis, A Course in H ∞ Control Theory (Lect. Notes Contr. Inf. Sci., Vol. 88). Berlin: Springer-Verlag, 1987.
13. S. T. Glad, “Robustness of nonlinear state feedback—A survey,” Automatica, vol. 23, pp. 425–435, 1987.
14. K. Glover and J. C. Doyle, “A state space approach to H ∞ optimal control,” in Three Decades of Mathematical System Theory, H. Nijmeijer and J. M. Schumacher, Eds. (Lect. Notes Contr. Inf. Sci., Vol. 135). Berlin: Springer-Verlag, 1989, pp. 179–218.
15. J. W. Grizzle and P. P. Khargonekar, personal communication, 1990.
16. D. Hill and P. Moylan, “The stability of nonlinear dissipative systems,” IEEE Trans. Automat. Contr., vol. AC-21, pp. 708–711, 1976.
17. D. J. Hill and P. J. Moylan, “Connections between finite gain and asymptotic stability,” IEEE Trans. Automat. Contr., vol. AC-25, pp. 931–936, 1980.
18. A. Isidori, Nonlinear Control Systems, 2nd ed. Berlin: Springer-Verlag, 1989.
19. H. Kwakernaak, “A polynomial approach to minimax frequency domain optimization of multivariable feedback systems,” Int. J. Contr., vol. 41, pp. 117–156, 1986.
20. P. P. Khargonekar and K. M. Nagpal, “Filtering and smoothing in an H ∞ setting,” in Proc. 28th CDC, Tampa, FL, 1989, pp. 415–420.
21. P. P. Khargonekar, I. R. Petersen, and M. A. Rotea, “H ∞ optimal control with state feedback,” IEEE Trans. Automat. Contr., vol. 33, pp. 786–788, 1988.
22. D. L. Lukes, “Optimal regulation of nonlinear dynamical systems,” SIAM J. Contr., vol. 7, 75–100, 1969.
23. P. J. Moylan, “Implications of passivity in a class of nonlinear systems,” IEEE Trans. Automat. Contr., vol. AC-19, pp. 373–381, 1974.
24. P. J. Moylan and B. D. O. Anderson, “Nonlinear regulator theory and an inverse optimal control problem,” IEEE Trans. Automat. Contr., vol. AC-18, pp. 460–464, 1973.
25. P. J. Moylan and D. J. Hill, “Stability criteria for large-scale systems, „ IEEE Trans. Automat. Contr., vol. AC-23, pp. 143–149, 1978.
26. R. Marino, W. Respondek, and A. J. van der Schaft, “Almost disturbance decoupling for single-input single-output nonlinear systems,” IEEE Trans. Automat. Contr., vol. 34, pp. 1013–1017, 1989.
27. H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems. New York: Springer-Verlag, 1990.
28. C. Scherer, “H ∞ control by state feedback: An iterative algorithm and characterization of high-gain occurrence,” Syst. Contr. Lett., vol. 12, pp. 383–391, 1989.
29. M. Vidyasagar, “L 2 stability of interconnected systems using a reformulation of the passivity theorem,” IEEE Trans. Circuit. Syst., vol. 24, pp. 637–645, 1977.
30. A. J. van der Schaft, “On a state space approach to nonlinear H ∞ control,” Syst. Contr. Lett., vol. 16, pp. 1–8, 1991.
31. A. J. van der Schaft, “On the Hamilton-Jacobi equation of nonlinear H ∞ optimal control,” in Proc. 1st EEC, Grenoble, July 1991, Hermes, Paris, pp. 649–654.
32. A. J. van der Schaft, “Relations between (H ∞ -) optimal control of a nonlinear system and its linearization,” in Proc. 30th CDC, Brighton, UK, 1991, pp. 1807–1808.
33. J. C. Willems, The Analysis of Feedback Systems. Cambridge, MA: M.I.T. Press, 1971.
34. J. C. Willems, “The generation of Lyapunov functions for input-output stable systems,” SIAM J. Contr., vol. 9, pp. 105–133, 1971.
35. J. C. Willems, “Least squares stationary optimal control and the algebraic Riccati equation,” IEEE Trans. Automat. Contr., vol. AC-16, pp. 621–634, 1971.
36. J. C. Willems, “Dissipative dynamical systems, Part I: General theory,” Arch. Rat. Mech. Anal., vol. 45, pp. 321–351, 1972.
37. J. C. Willems, “Dissipative dynamical systems, Part II: Linear systems withquadratic supply rates.” Arch. Rat. Mech. Anal., vol. 45, pp. 352-393. 1972
38. S. Weiland, “Theory of approximation and disturbance attenuation for linear systems,” Ph.D. thesis, Rijksuniversiteit Groningen, 1991
39. G. Zames, “Feedback and optimal sensitivity: Model reference transformations, multiplicative seminorms, and approximate inverses,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 301-320, 1981
40. ∞ optimization,” Syst. Contr. Lett., vol. II, pp. 85-91, 1988.