Bifurcation control of nonlinear systems with time-periodic coefficients

1999 ◽  
Keyword(s):  
Nonlinear Systems ◽  
Bifurcation Control ◽  
Periodic Coefficients ◽  
Time Periodic

Control of chaos in nonlinear systems with time-periodic coefficients

2006 ◽  
Vol 364 (1846) ◽  
pp. 2417-2432 ◽  
Author(s):  
S.C Sinha ◽  
Alexandra Dávid
Keyword(s):  
Nonlinear Systems ◽  
Periodic Orbit ◽  
Nonlinear Control ◽  
State Feedback ◽  
State Feedback Control ◽  
Control Technique ◽  
Periodic Coefficients ◽  
Full State ◽  
Full State Feedback ◽  
Time Periodic

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.

Development of a Feedback Linearization Technique for Parametrically Excited Nonlinear Systems via Normal Forms

2006 ◽  
Vol 2 (2) ◽  
pp. 124-131 ◽  
Author(s):  
Yandong Zhang ◽  
S. C. Sinha
Keyword(s):  
Nonlinear Systems ◽  
Feedback Linearization ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Original System ◽  
Computation Method ◽  
Normal Form Theory ◽  
Periodic Coefficients ◽  
Quasi Linear System ◽  
Time Periodic

The problem of designing controllers for nonlinear time periodic systems via feedback linearization is addressed. The idea is to find proper coordinate transformations and state feedback under which the original system can be (exactly or approximately) transformed into a linear time periodic control system. Then a controller can be designed to guarantee the stability of the system. Our approach is designed to achieve local control of nonlinear systems with periodic coefficients desired to be driven either to a periodic orbit or to a fixed point. The system equations are represented by a quasi-linear system containing nonlinear monomials with periodic coefficients. Using near identity transformations and normal form theory, the original close loop problem is approximately transformed into a linear time periodic system with unknown gains. Then by using a symbolic computation method, the Floquet multipliers are placed in the desired locations in order to determine the control gains. We also give the sufficient conditions under which the system is feedback linearizable up to the rth order.

Bifurcation Control of Nonlinear Systems With Time-Periodic Coefficients

2003 ◽  
Vol 125 (4) ◽  
pp. 541-548 ◽  
Author(s):  
Alexandra Da´vid ◽  
S. C. Sinha
Keyword(s):  
Bifurcation Control ◽  
Nonlinear Feedback ◽  
Periodic Load ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Periodic Coefficients ◽  
Codimension One ◽  
Form Theory ◽  
Time Invariant ◽  
Time Periodic

In this study, a method for the nonlinear bifurcation control of systems with periodic coefficients is presented. The aim of bifurcation control is to stabilize post bifurcation limit sets or modify other nonlinear characteristics such as stability, amplitude or rate of growth by employing purely nonlinear feedback controllers. The method is based on an application of the Lyapunov-Floquet transformation that converts periodic systems into equivalent forms with time-invariant linear parts. Then, through applications of time-periodic center manifold reduction and time-dependent normal form theory completely time-invariant nonlinear equations are obtained for codimension one bifurcations. The appropriate control gains are chosen in the time-invariant domain and transformed back to the original variables. The control strategy is illustrated through the examples of a parametrically excited simple pendulum undergoing symmetry-breaking bifurcation and a double inverted pendulum subjected to a periodic load in the case of a secondary Hopf bifurcation.

Galerkin Approximations for Stability of Delay Differential Equations With Time Periodic Coefficients

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Anwar Sadath ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximate Model ◽  
Periodic Coefficients ◽  
Galerkin Approximations ◽  
Approximate System ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Least Squares Approach

A numerical method to determine the stability of delay differential equations (DDEs) with time periodic coefficients is proposed. The DDE is converted into an equivalent partial differential equation (PDE) with a time periodic boundary condition (BC). The PDE, along with its BC, is then converted into a system of ordinary differential equations (ODEs) with time periodic coefficients using the Galerkin least squares approach. In the Galerkin approach, shifted Legendre polynomials are used as basis functions, allowing us to obtain explicit expressions for the approximate system of ODEs. We analyze the stability of the discretized ODEs, which represent an approximate model of the DDEs, using Floquet theory. We use numerical examples to show that the stability charts obtained with our method are in excellent agreement with those existing in the literature and those obtained from direct numerical simulation.

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