Considering the nonlinear system x_ = and y_ = z z_ = 􀀀az 􀀀 by 􀀀 f(x) (2) which it has an oscillatory behavior is demonstrated in the case that f(􀀀x) = 􀀀f(x), that by replacing the f(x) function f(x + B sin !t), and will -lores of B and ! large enough the system is oscillatory motion large amplitude. In fact all solutions tend to Origin neighborhood so small as you like. To make this demonstration we proceed as follows: Initially disturbed function in terms of x is expressed and B sin(!t), to proceed to calculate the average function. Then test for h(; x) = f(x + B sin ) 􀀀 f0(x;B), There is a continuous function H(; x; 1! ) such that jH(; x; 1 ! )j !(!) where (!) ! 0 when ! ! 1 and performing substitution z = s + 1 !H(t; x; 1 ! ) shows that the perturbed system is equivalent to the following system x_ = and y_ = z +1! H(t; x; 1!) z_ = 􀀀az 􀀀 by 􀀀 F0(x;B) 􀀀 a 􀀀 1! H(t; x; !) 􀀀 1! @H @x and Thus it is proved that for ! large enough, the averaging system is a good approximation of the system disturbed. This is that any solution of the perturbed system is sufficiently close to a solution of averaging system. There is also evidence that B0 such that B > B0, the solution trivial averaging system is asymptotically stable values of ! sufficiently large. Finally it is proved that for B and ! enough large system has disturbed oscillatory motion large amplitude, that is, the disturbance has destroyed the large amplitude oscillations.

background

Effects of Extra Sinusoidal Inputs to Nonlinear Systems

Eliminación de las Grandes Oscilaciones de un Sistema de Ecuaciones Diferenciales

Different methods are put forth for the analysis of stability and performance of nonlinear sampled-data systems, and many methods of stabilization are suggested. The injection of high-frequency oscillations at the input to the non-linear element is suggested as a possible method of stabilizing nonlinear sampled-data systems. Analytical and experimental investigations are carried out, and the advantages of signal stabilization over other methods are established.

http://ieeexplore.ieee.org/iel5/9/24109/01099272.pdf

Signal stabilization of nonlinear sampled-data control systems

Addresses the design of variable structure control to eliminate limit-cycle inherent in nonlinear systems. The design and implementation is rather simple and effective. The given example demonstrates the benefit of this method.

Elimination of limit-cycle via variable structure systems

Digital-to-analog conversion is essential in digital signal processing applications, including closed-loop control schemes. Noise and distortion in digital-to-analog converters result in reduced performance for high-precision mechatronics such as nano-positioning. Glitches are common in practical switched systems such as digital-to-analog converters; observed as an output disturbance. Due to the wide-bandwidth, impulse-like behavior, control law bandwidth is generally too low to provide adequate attenuation; deteriorating open and closed-loop performance. This article demonstrates how large-amplitude high-frequency periodic dither mitigates the effect of glitches in a nano-positioning system under closed-loop control. Simulations are performed using a model that includes significant non-linearities with a response fitted to an off-the-shelf commercial device, as well as using standard linear time-invariant models for other system components fitted to the responses of common, commercially available devices. The results highlight the significance of reconstruction filter design when applying dithering in this setting.

Effects of Extra Sinusoidal Inputs to Nonlinear Systems

Citations

Eliminación de las Grandes Oscilaciones de un Sistema de Ecuaciones Diferenciales

Signal stabilization of nonlinear sampled-data control systems

Elimination of limit-cycle via variable structure systems

Mitigating Non-linear DAC Glitches Using Dither in Closed-loop Nano-positioning Applications

Correspondence on ‘ Applications of high frequency signal injection in non-linear systems’

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