Nita H. Shah; Mahesh A. Yeolekar

The motion of a biped robot can be explained by a set of nonlinear ordinary differential equations. In this paper, we investigate the linearised form of a system of nonlinear ordinary differential equations with impulse effect which modelled a simple planer biped robot without knee. It demonstrated the periodic walking of biped robot in a sagittal plane in absence of external forces except gravity. This paper explains the bifurcation study for the system of biped robot with respect to the bifurcation parameters, mass and length. The results exhibit that the stable symmetric gait leads to chaotic gait by the continuous change in the values of parameters. We observed that the symmetric gaits of robot are more responsive for the values of length of legs than the values of masses of robot.]]>

Bhawna Tandon; Shiv Narayan; Jagdish Kumar

This study proposes a feedback linearisation based on the back-stepping method to design a nonlinear controller with a goal of improving both steady state and transient stability of a magnetic levitation system. The feedback linearisation based on back-stepping technique is the combination of the two techniques mentioned i.e. feedback linearisation and backstepping. It uses backstepping design process, designs a sequence of 'virtual' systems of relative degree one, reduces the relative degree by one by choosing a 'virtual' input, achieves passivity with respect to a 'virtual output', and the last 'virtual output' is used to close feedback loop. Unlike the direct feedback linearisation method, the proposed method does not require a linear controller. Moreover, there is no need to know the exact nonlinear model of the system. Back stepping process of the controller guarantees its robustness against disturbances and uncertainties.]]>

Shwet Nisha; Pradip K. Parida

In this paper, we have developed an improved regula falsi method of order four for finding simple roots of nonlinear equations f(x) = 0, where f : [a; b] ⊂ R → R is a given continuously differentiable function. This is done by combining a Newton-like method of order four to solve f(x) = 0 and the usual regula-falsi method. Convergence analysis for the method has been given in this paper. Finally some numerical examples are presented and comparison has been made with existing results.]]>

Abdelhamid Rabhi; Mohammed Belkheiri; JÃ©rÃ´me Bosche; Ahmed El Hajjaji

A new control approach is proposed to address the tracking problem of a class of uncertain nonlinear systems. In this approach, one relies first on a partially known model of the system to be controlled using a backstepping control strategy. The obtained controller is then augmented by an online artificial neural network (ANN) that serves as an approximator for the neglected dynamics and modelling errors. Thus, the developed method combines backstepping approach and ANN to address the tracking problem for uncertain systems. The proposed approach is systematic, and exploits the known nonlinear dynamics to derive the stepwise virtual stabilising control laws. At the final step, an augmented Lyapunov function is introduced to derive the adaptation laws of the network weights. The suggested control algorithm is tested experimentally on a Laboratory ABS system showing satisfactory results although the system is highly nonlinear and with unknown physical parameters. ]]>

Najeeb Alam Khan; Chein-Shan Liu; Fatima Riaz

In this paper, a novel multiple-scale polynomial-Fourier-series method (PFSM) is developed to be used in the data interpolation, in which the multiple-scale Rk can be determined exactly and optimally in terms of the data nodes. For solving the nonlinear Duffing equation, an optimally scaled harmonic balance method (OSHB) is derived, which is better than the classic harmonic balance method (HB). In terms of the OSHB, the periodic solutions of the Duffing oscillator, and reconstruction of the frequency response curves, which exhibit a hysteresis within which the multiple solutions can happen in an interval of frequency near to the resonant frequency can be precisely solved. The PFSM is further adapted to solve the initial value problem of the Duffing equation, and the periodic solution can be obtained more accurately than the HB. The proposed method has also been tested to solve the boundary value problem and the initial value problem of some nonlinear ordinary differential equations (ODEs). The conclusion can be drawn that the present OSHB and PFSM are effective to solve nonlinear ODEs, including the nonlinear Duffing equation as a demonstrative example.]]>

HÃ¼seyin Budak; Mehmet Zeki Sarikaya

In this paper, a companion of trapezoid inequality for functions of two independent variables with bounded variation is established and some applications for general cubature formula are given.]]>