Control of Dynamic Systems with Time-Periodic Coefficients via the Lyapunov-Floquet Transformation and Backstepping Technique

Abstract

We address the problem of designing controllers that guarantee asymptotic stability of a class of linear as well as nonlinear dynamical systems with time-periodic coefficients. Using a repeated procedure consisting of the Lyapunov-Floquet transformation, the backstepping technique, and Floquet theory, the asymptotic stability of the closed-loop linearized system is guaranteed. Further, a Lyapunov matrix for the closed-loop asymptotically stable linearized system is constructed. This Lyapunov function is then used to design a combination of linear and nonlinear controllers in order to guarantee the asymptotic stability of the nonlinear system. The methodology is illustrated by designing linear and nonlinear control laws for a system consisting of two statically coupled pendula, each subjected to a time-periodic force acting in the axial direction.

Keywords

Nonlinear control Floquet theory Lyapunov-Floquet transformation backstepping

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References

Bittani, S. , Colaneri, P. , and De Nicolao, G. , 1991, The Periodic Riccati Equation, Springer-Verlag, Berlin .

Boghiu, D. , Sinha, S. C. , and Marghitu, D. B. , 1998, “Stability and control of a parametrically excited rotating system. Part II: Controls,” Dynamics and Control 8 (1), 19-35 .

Butcher, E. A. and Sinha, S. C. , 1996, “A hybrid formulation for the analysis of time-periodic linear systems via Chebyshev polynomials,” Journal of Sound and Vibration 95 (3), 518-527 .

Calico, R. A. and Wiesel, W. E. , 1984, “Control of time-periodic systems,” Journal of Guidance, Control and Dynamics 7, 671-676 .

Calise, A. J. , Wasikowski, M. E. , and Schrage, D. P. , 1992, “Optimal output feedback for linear time-periodic systems,” Journal of Guidance, Control and Dynamics 15, 416-423 .

David, A. and Sinha, S. C. , 2003, “Bifurcation control of nonlinear systems with time-periodic coefficients,” Journal of Dynamic Systems, Measurement, and Control, 125, 541-548 .

Deshmukh, V. , Sinha, S. C. , and Joseph, P. , 1999, “Order reduction and control of parametrically excited dynamical systems,” Journal of Vibration and Control 6, 1017-1028 .

Freeman, R. and Kokotovic, P. , 1996, Robust Nonlinear Control Design, Birkhauser, Boston .

Kabamba, P. T. , 1986, “Monodromy eigenvalue assignment in linear periodic systems,” IEEE Transactions on Automatic Control 31, 950-952 .

10.

Krstic, M. , Kanellakopoulos, I. , and Kokotovic, P. , 1995, Nonlinear and Adaptive Control Design, Wiley, NY .

11.

Kwakernaak, H. and Sivan, R. , 1972, Linear Optimal Control Systems, Wiley Interscience, NY .

12.

Lee, Y. J. and Balas, M. J. , 1998, “Controller design of periodic time-varying systems via time invariant methods,” Journal of Guidance, Control and Dynamics 22 (3), 486-488 .

13.

Montagnier, P. , Spiteri, R. J. , and Angeles, J. , 2001, “Pitfalls of a least-squares equivalent controller for linear, time-periodic systems,” International Journal of Control 74 (2), 199-204 .

14.

Pandiyan, R. and Sinha S. C. , 2001, “Time-varying controller synthesis for nonlinear systems subjected to periodic parametric loadings,” Journal of Vibration and Control 7, 73-90 .

15.

Sinha, S. C. and Joseph, P. , 1994, “Control of general dynamic systems with periodically varying parameters via Lyapunov-Floquet transformation,” Journal of Dynamic Systems, Measurement and Control 116, 650-658 .

16.

Sinha, S. C. and Pandiyan, R. , 1994, “Analysis of quasilinear dynamical systems with time-periodic coefficients via Lyapunov-Floquet transformation,” International Journal of Nonlinear Mechanics 29 (5), 687-702 .

17.

Sinha, S. C. , Pandiyan, R. , and Bibb, J. S. , 1996, “Lyapunov-Floquet transformation: computation and applications to periodic systems,” Journal of Vibration and Acoustics 118, 209-219 .

18.

Sinha, S. C. , Marghitu, D. B. , and Boghiu, D. , 1998a, “Stability and control of parametrically excited rotating beam,” Journal of Dynamic Systems, Measurement and Control 120, 462-470 .

19.

Sinha, S. C. , Butcher, E. A. , and David, A. , 1998b, “Computation of dynamically equivalent time-invariant forms for time-periodic systems,” Nonlinear Dynamics 16, 203-221 .

20.

Sinha, S. C. , Henrichs, J. T. , and Ravindra, B. , 2000, “A general approach in the design of active controllers for nonlinear systems exhibiting chaos,” International Journal of Bifurcation and Chaos 10 (1), 165-178 .

21.

Tornambe, A. and Valigi, P. , 1996, “Asymptotic stabilization of a class of continuous-time linear periodic systems,” Systems and Control Letters 28, 189-196 .

22.

Yakubovich, V. A. and Starzhinskii, V. M. , 1975, Linear Differential Equations with Periodic Coefficients, Keter, Jerusalem (translated from Russian).