Control of time-periodic systems via symbolic computation with application to chaos control

Sinha, Subhash C.; Gourdon, Emmanuel; Zhang, Y.

doi:10.1016/j.cnsns.2004.06.001

Cited by 20 publications

(4 citation statements)

References 14 publications

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“…Tor 2 periodic T − L-F transformation matrix [12]. For the details on computation of the L-F transformation and its applications, we refer the reader to references [13]- [15].…”

Section: ( )mentioning

confidence: 99%

Reducibility of Periodic Quasi-Periodic Systems

Ezekiel¹,

Redkar²

2014

IJMNTA

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In this work, the reducibility of quasi-periodic systems with strong parametric excitation is studied. We first applied a special case of Lyapunov-Perron (L-P) transformation for time periodic system known as the Lyapunov-Floquet (L-F) transformation to generate a dynamically equivalent system. Then, we used the quasi-periodicnear-identity transformation to reduce this dynamically equivalent system to a constant coefficient system by solving homological equations via harmonic balance. In this process, we obtained the reducibility/resonance conditions that needed to be satisfied to convert a quasi-periodic system in to a constant one. Assuming the reducibility is possible, we obtain the L-P transformation that can transform original quasi-periodic system into a system with constant coefficients. Two examples are presented that show the application of this approach.

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“…Tor 2 periodic T − L-F transformation matrix [12]. For the details on computation of the L-F transformation and its applications, we refer the reader to references [13]- [15].…”

Section: ( )mentioning

confidence: 99%

Reducibility of Periodic Quasi-Periodic Systems

Ezekiel¹,

Redkar²

2014

IJMNTA

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“…In this section, we briefly outline the Floquet theory and a symbolic controller design technique for linear timeperiodic systems as introduced by Sinha et al (2005). Consider an n-dimensional linear periodic system in the state-space form…”

Section: Controller Design For Linear Time-periodic Systemsmentioning

confidence: 99%

Construction of reduced order controllers for nonlinear systems with periodic coefficients

Gabale

Sinha

2010

Journal of Vibration and Control

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This study provides a methodology for reduced order controller design for nonlinear dynamic systems with time-periodic coefficients. The proposed methodology is quite general in a sense that it can be easily modified for nonlinear systems with constant coefficients. The equations of motion are represented by quasi-linear differential equations in state space, containing a time-periodic linear part and nonlinear monomials of states with periodic coefficients. The Lyapunov—Floquet (L-F) transformation is used to transform the time-varying linear part of the system into a time-invariant form. Eigenvalue decomposition of the time-invariant linear part can then be used to identify the dominant/nondominant dynamics of the system. The nondominant states of the system are expressed as a nonlinear, time-periodic, manifold relationship in terms of the dominant states. As a result, the original large system can be expressed as a lower order system represented only by the dominant states. A reducibility condition is derived to provide conditions under which a nonlinear order reduction is possible. Then a proper coordinate transformation and state feedback can be found under which the reduced order system is transformed into a linear, time-periodic, closed-loop system. This permits the design of a time-varying feedback controller in linear space to guarantee the stability of the system. The case of systems with constant coefficients is treated as a subset of its periodic counterpart. In this case one does not have to use the L-F transformation; moreover, the manifold and all other transformations are purely spatial. The proposed methodology is illustrated by designing a two dimensional reduced order controller for a 4-degrees of freedom, inverted pendulum subjected to a periodic force for which the equations of motion are time-periodic. An example involving a 2-dof spring-mass-damper system is also presented which yields the equations of motion with constant coefficients.

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“…Sinha & Butcher 1997), we obtain that if k 1 R5.18, it guarantees asymptotic stability. More details of controller design for such systems by symbolic computation may be found in the paper by Sinha et al (2005). The uncontrolled as well as controlled trajectories are shown in figure 1.…”

Section: Linear Full-state Feedback Control Using Symbolic Computationmentioning

confidence: 99%

Control of chaos in nonlinear systems with time-periodic coefficients

Sinha

Dávid

2006

Phil. Trans. R. Soc. A.

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In this study, some techniques for the control of chaotic nonlinear systems with periodic coefficients are presented. First, chaos is eliminated from a given range of the system parameters by driving the system to a desired periodic orbit or to a fixed point using a full-state feedback. One has to deal with the same mathematical problem in the event when an autonomous system exhibiting chaos is desired to be driven to a periodic orbit. This is achieved by employing either a linear or a nonlinear control technique. In the linear method, a linear full-state feedback controller is designed by symbolic computation. The nonlinear technique is based on the idea of feedback linearization. A set of coordinate transformation is introduced, which leads to an equivalent linear system that can be controlled by known methods. Our second idea is to delay the onset of chaos beyond a given parameter range by a purely nonlinear control strategy that employs local bifurcation analysis of time-periodic systems. In this method, nonlinear properties of post-bifurcation dynamics, such as stability or rate of growth of a limit set, are modified by a nonlinear state feedback control. The control strategies are illustrated through examples. All methods are general in the sense that they can be applied to systems with no restrictions on the size of the periodic terms.

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Control of time-periodic systems via symbolic computation with application to chaos control

Cited by 20 publications

References 14 publications

Reducibility of Periodic Quasi-Periodic Systems

Reducibility of Periodic Quasi-Periodic Systems

Construction of reduced order controllers for nonlinear systems with periodic coefficients

Control of chaos in nonlinear systems with time-periodic coefficients

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