Analysis of quasilinear dynamical systems with periodic coefficients via Liapunov—Floquet transformation

Sinha, Subhash C.; Pandiyan, R.

doi:10.1016/0020-7462(94)90065-5

Cited by 46 publications

(43 citation statements)

References 12 publications

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“…Q(t) is known as the Lyapunov-Floquet (L-F) transformation matrix [11] and the transformation x(t) = Q(t)z(t) produces a real time invariant representation given bẏ…”

Section: Model Order Reduction Techniques In State Spacementioning

confidence: 99%

“…For this set of parameters, the Floquet multipliers are given as (−0.51 ± 0.85ī, −0.88 ± 0.45ī). The L-F transformation matrix Q(t) is computed from the linear part of Equation (30) by the method suggested by Sinha et al [11]. After using the modal transformation, Equation (6) for this case is given by…”

Section: No Resonance Of Any Kindmentioning

confidence: 99%

“…We apply the order reduction approaches discussed above to a special periodic-quasiperiodic system for which L-F transformation is known in a closed form [11]. Consider a coupled periodic-quasiperiodic nonlinear system given bẏ…”

Section: An Applicationmentioning

confidence: 99%

“…This may be accomplished by the use of Lyapunov-Floquet (L-F) transformation that converts the time varying linear system matrix into an equivalent time invariant form. The details of computation and application of L-F transformation can be found in [11,12]. Deshmukh et al [13] recently applied this concept to develop an order reduction technique and control strategy for large-scale time periodic systems in state space form.…”

Section: Introductionmentioning

confidence: 99%

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Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

et al. 2005

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Abstract. The basic problem of order reduction of nonlinear systems with time periodic coefficients is considered in state space and in direct second order (structural) form. In state space order reduction methods, the equations of motion are expressed as a set of first order equations and transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of new set of equations are time invariant. At this stage, four order reduction methodologies, namely linear, nonlinear projection via singular perturbation, post-processing approach and invariant manifold technique, are suggested. The invariant manifold technique yields a unique 'reducibility condition' that provides the conditions under which an accurate nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An alternate approach of deriving reduced order models in direct second order form is also presented. Here the system is converted into an equivalent second order nonlinear system with time invariant linear system matrices and periodically modulated nonlinearities via the L-F and other canonical transformations. Then a master-slave separation of degrees of freedom is used and a nonlinear relation between the slave coordinates and the master coordinates is constructed. This method yields the same 'reducibility conditions' obtained by invariant manifold approach in state space. Some examples are given to show potential applications to real problems using above mentioned methodologies. Order reduction possibilities and results for various cases including 'parametric', 'internal', 'true internal' and 'true combination resonances' are discussed. A generalization of these ideas to periodic-quasiperiodic systems is included and demonstrated by means of an example.

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“…Q(t) is known as the Lyapunov-Floquet (L-F) transformation matrix [11] and the transformation x(t) = Q(t)z(t) produces a real time invariant representation given bẏ…”

Section: Model Order Reduction Techniques In State Spacementioning

confidence: 99%

Section: No Resonance Of Any Kindmentioning

confidence: 99%

Section: An Applicationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

et al. 2005

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“…It has been shown in [31] that the LFT matrix Q(t) can be computed in closed form for a commutative system for which the state transition matrix (STM) is known. In general, the STM of the uncontrolled system in Eq.…”

Section: Y(t) = Q(t)z(t)mentioning

confidence: 99%

Optimal feedback control strategies for periodic delayed systems

Nazari

Butcher

Bobrenkov³

2014

Int. J. Dynam. Control

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In this study, three strategies based on infinitedimensional Floquet theory, Chebyshev spectral collocation, and the Lyapunov-Floquet transformation (LFT) are proposed for optimal feedback control of linear time periodic delay differential equations using periodic control gains. First, a periodic-gain discrete-delayed feedback control is implemented where optimization of the control gains is included to obtain the minimum spectral radius of the closedloop response. Second, a large set of ODEs is obtained using the Chebyshev spectral continuous time approximation, after which optimal (time-varying LQR) control is used to obtain a periodic-gain distributed-delayed feedback control. The third strategy involves the use of both CSCTA and the reduced LFT, along with either pole-placement or time-invariant LQR used on a linear time invariant auxiliary system, to obtain a periodic-gain non-delayed feedback control that asymptotically stabilizes the original system. The delayed Mathieu equation is used as an illustrative example for all three control strategies.

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Analysis of Time-Periodic Nonlinear Dynamical Systems Undergoing Bifurcations

Pandiyan

Sinha

1995

Advances in Nonlinear Dynamics: Methods and Applications

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Analysis of quasilinear dynamical systems with periodic coefficients via Liapunov—Floquet transformation

Cited by 46 publications

References 12 publications

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications

Optimal feedback control strategies for periodic delayed systems

Analysis of Time-Periodic Nonlinear Dynamical Systems Undergoing Bifurcations

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