A computation method of formal linearization for time-variant nonlinear systems via Chebyshev interpolation

2004 ◽  
Author(s):  
K. Komatsu ◽  
H. Takata
Keyword(s):  
Nonlinear Systems ◽  
Computation Method ◽  
Chebyshev Interpolation ◽  
Formal Linearization

A New Approach for Computing the Normal Forms of High Dimensional Nonlinear Systems

2010 ◽  
Author(s):  
Shuping Chen ◽  
Wei Zhang ◽  
Minghui Yao
Keyword(s):  
Nonlinear Systems ◽  
Normal Form ◽  
Cantilever Beam ◽  
Normal Forms ◽  
Nonlinear Differential Equations ◽  
Real Life ◽  
High Dimensional ◽  
Computation Method ◽  
Normal Form Theory ◽  
Averaged Equations

Normal form theory is very useful for direct bifurcation and stability analysis of nonlinear differential equations modeled in real life. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. The method developed here uses the lower order nonlinear terms in the normal form for the simplifications of higher order terms. In the theoretical model for the nonplanar nonlinear oscillation of a cantilever beam, the computation method is applied to compute the coefficients of the normal forms for the case of two non-semisimple double zero eigenvalues. The normal forms of the averaged equations and their coefficients for non-planar non-linear oscillations of the cantilever beam under combined parametric and forcing excitations are calculated.

Further Reduction of Normal Forms for High Dimensional Nonlinear Systems and Application to a Composite Laminated Piezoelectric Plate

2013 ◽  
Vol 291-294 ◽  
pp. 2662-2665
Author(s):  
Shu Ping Chen ◽  
Wei Zhang
Keyword(s):  
Nonlinear Systems ◽  
Normal Forms ◽  
Nonlinear Oscillations ◽  
Piezoelectric Plate ◽  
Nonlinear Differential Equations ◽  
High Dimensional ◽  
Computation Method ◽  
Normal Form Theory ◽  
Averaged Equations ◽  
Form Theory

Normal form theory is robust and useful for direct bifurcation and stability analysis of nonlinear differential equations in real engineering problems. This paper develops a new computation method for obtaining a significant refinement of the normal forms for high dimensional nonlinear systems. In the theoretical model for the nonlinear oscillation of a composite laminated piezoelectric plate, the computation method is applied to compute the coefficients of the normal forms for the case of one double zero and a pair of pure imaginary eigenvalues. The algorithm is implemented in Maple V and the normal forms of the averaged equations and their coefficients for nonlinear oscillations of the composite laminated piezoelectric plate under combined parametric and transverse excitations are calculated.

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.

scholarly journals An Augmented Automatic Choosing Control of a Formal Linearization Filter Type for Nonlinear Systems

2014 ◽  
Vol 18 (1) ◽  
pp. 63-70
Author(s):  
Tomohiro Hachino ◽  
Hitoshi Takata ◽  
Soichiro Osako ◽  
Kazutomo Yunokuchi ◽  
Hiromi Miyajima ◽  
...  
Keyword(s):  
Nonlinear Systems ◽  
Filter Type ◽  
Formal Linearization
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