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On the robustness of linear stabilizing feedback control for linear uncertain systems: Multi-input case

Abstract

The robust stabilization of linear systems with constant uncertainties against structured perturbations using Lyapunov's theory is investigated. The only information needed on the uncertainties is the knowledge of their boundaries. The matching conditions of the uncertain systems are not required to be satisfied. It is first shown that, under some assumptions, the system can be transformed into a certain canonical controllable companion form. Then, under some additional assumptions, the existence of a linear controller which stabilizes the system based on Lyapunov's theory is shown.

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References

  1. 1.

    Barmish, B. R., Petersen, I. R., andFeuer, A.,Linear Ultimate Boundedness Control of Uncertain Dynamical Systems, Automatica, Vol. 19, No. 5, pp. 523–532, 1983.

  2. 2.

    Gutman, S., andLeitmann, G.,Stabilizing Feedback Control for Dynamical Systems with Bounded Uncertainty, Proceedings of the IEEE Conference on Decision and Control, Clearwater, Florida, 1976.

  3. 3.

    Leitmann, G.,Guaranteed Ultimate Boundedness for a Class of Uncertain Linear Dynamical Systems, IEEE Transactions on Automatic Control, Vol. AC-23, No. 6, pp. 1109–1110, 1978.

  4. 4.

    Gutman, S.,Uncertain Dynamical Systems—A Lyapunov Min-Max Approach, IEEE Transactions on Automatic Control, Vol. AC-24, No. 3, pp. 437–443, 1979.

  5. 5.

    Leitmann, G.,Guaranteed Asymptotic Stability for Some Linear Systems with Bounded Uncertainties, Journal of Dynamical Systems, Measurement, and Control, Vol. 101, No. 3, pp. 212–216, 1979.

  6. 6.

    Leitmann, G.,On the Efficacy of Nonlinear Control on Uncertain Linear Systems, Journal of Dynamical Systems, Measurement, and Control, Vol. 102, No. 2, pp. 95–102, 1981.

  7. 7.

    Barmish, B. R., Corless, M., andLeitmann, G.,A New Class of Stabilizing Controllers for Uncertain Dynamical Systems, SIAM Journal on Control and Optimization, Vol. 21, No. 2, pp. 246–255, 1983.

  8. 8.

    Leitmann, G.,Deterministic Control of Uncertain Systems, Paper Presented at the 4th International Conference on Mathematical Modeling in Science and Technology, Zurich, Switzerland, 1983.

  9. 9.

    Petersen, I. R.,Structural Stabilization of Uncertain Systems: Necessity of the Matching Condition, SIAM Journal on Control and Optimization, Vol. 23, No. 2, pp. 286–296, 1985.

  10. 10.

    Stalford, H. L.,Necessary and Sufficient Conditions for Matching Conditions in Uncertain Systems: Scalar Input, Proceedings of the 1987 American Control Conference, Minneapolis, Minnesota, pp. 897–903, 1987.

  11. 11.

    Barmish, B. R., andLeitmann, G.,On Ultimate Boundedness Control of Uncertain Systems in the Absence of Matching Assumptions, IEEE Transactions on Automatic Control, Vol. AC-27, No. 1, pp. 153–158, 1982.

  12. 12.

    Chen, Y. H.,On the Deterministic Performance of Uncertain Dynamical Systems, International Journal of Control, Vol. 43, No. 5, pp. 1557–1579, 1986.

  13. 13.

    Stalford, H. L., andGarrett, F. E., Jr.,Robust Nonlinear Control for High Angle-of Attack Flight, Paper No. AIAA-87-0346, AIAA 25th Aerospace Sciences Meeting, Reno, Nevada, 1987.

  14. 14.

    Stalford, H. L.,Target Tracking at High α Using Certain Robust Nonlinear Controllers, Paper No. AIAA-87-2407, AIAA Guidance, Navigation, and Control Conference, Monterey, California, 1987.

  15. 15.

    Stalford, H. L.,On Robust Control of Wing Rock Using Nonlinear Control, Proceedings of the 1987 American Control Conference, Minneapolis, Minnesota, pp. 1890–1899, 1987.

  16. 16.

    Stalford, H. L.,Robust Control of Uncertain Systems in the Absence of Matching Conditions: Scalar Input, Proceedings of the 1987 Conference on Decision and Control, Los Angeles, California, pp. 1298–1307, 1987.

  17. 17.

    Stalford, H. L., andChao, C. H.,On the Robustness of Linear Stabilizing Feedback Control for Linear Uncertain Systems, Journal of Optimization Theory and Applications, Vol. 63, No. 2, pp. 239–246, 1989.

  18. 18.

    Stalford, H. L., andChao, C. H.,A Necessary and Sufficient Condition in Lyapunov Robust Control, Journal of Optimization Theory and Applications, Vol. 63, No. 2, pp. 225–238, 1989.

  19. 19.

    Stalford, H. L., andChao, C.-H.,Necessary and Sufficient Condition in Lyapunov Robust Control: Multi-Input Case, Journal of Optimization Theory and Applications, Vol. 66, No. 1, 1990.

  20. 20.

    Stalford, H. L.,Stability Conditions for Nonlinear Control Processes Using Lyapunov Functions with Discontinuous Derivatives, Journal of Mathematical Analysis and Applications, Vol. 84, No. 2, pp. 365–371, 1981.

  21. 21.

    Kailath, T.,Linear Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1980.

  22. 22.

    Popov, V. M.,Invariant Description of Linear, Time-Invariant Controllable Systems, SIAM Journal on Control, Vol. 10, No. 2, pp. 252–264, 1972.

  23. 23.

    Chao, C. H.,Robust Stabilization of Linear Time-Invariant Uncertain Systems via Lyapunov Theory, PhD Dissertation, Virginia Polytechnic Institute and State University, 1988.

  24. 24.

    Stalford, H. L.,On Robust Control of Uncertain Linear Systems in the Absence of Matching Conditions, Modeling and Control of Systems in Engineering, Quantum Mechanics, Economics, and Biosciences, Edited by A. Blaquière, Springer-Verlag, Berlin, Germany, pp. 15–36, 1989.

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Communicated by G. Leitmann

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Chao, C.H., Stalford, H.L. On the robustness of linear stabilizing feedback control for linear uncertain systems: Multi-input case. J Optim Theory Appl 64, 229–244 (1990). https://doi.org/10.1007/BF00939447

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Key Words

  • Robust control
  • uncertain systems
  • Lyapunov functions
  • linear control