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Dynamics and Control of Time-Periodic Mechanical Systems via Floquet-Lyapunov Theory

Abstract:

Many practical problems in engineering can be modelled as linear dynamical systems with periodically varying coefficients. This thesis proposes a new design method for the control of these linear time-periodic systems.

First, Floquet-Lyapunov theory is used to derive the Floquet factors of the state-transition matrix of a given system. We introduce a novel approach to obtain every real representation. It is demonstrated that the periodicity of the periodic factor can be determined a priori using a constant matrix, which we call the Yakubovich matrix, based upon the signs of the eigenvalues of the monodromy matrix. We then introduce a novel method for the numerical computation of the Floquet factors, relying upon a boundary-value problem formulation and the Yakubovich matrix.

In the second part, we use the invertibility of the controllability Gramian and a specific form for the feedback gain matrix to build a novel control law for the closed-loop system. The new controller can be full-state or observer-based and allows the control engineer to assign all the invariants of the system, i.e. the full monodromy matrix. Deriving the feedback matrix requires first solving a matrix integral equation for the periodic Floquet factor of the new state-transition matrix of the closed-loop system. This is achieved via a spectral method, which can then be further refined by a boundary-value problem formulation. Computational efficiency of the scheme may be further improved by performing the controller synthesis on the transformed system obtained from the reducibility theorem.

Finally, the effectiveness of the method is illustrated with an application to a quick-return mechanism using a software toolbox developed for MATLAB(TM).

Author: Pierre Montagnier-Michau

Advisors: Jorge Angeles, Raymond J. Spiteri

Download: pmontagnier_phd_thesis