Non-linear systems and adaptive filters
Non-linear systems
If the amplitude of the signals encountered in a system lies inside a range small enough for all components of the system to operate in their linear region, then we can say that the linearity assumption is satisfied.
This restriction helps ensure that the principle of superposition is satisfied, hence making the system linear.
Though, when the amplitude does not lie in a close range, we can no more say the system is linear, such as in the case of a transistor amplifier used in a control system. It exhibits input-output characteristics that run into the saturation when the amplitude of the signal applied to the amplifier input is large.
Other sources of nonlinearity in a control system include backlash between coupled gears and friction between the moving parts of the system. When the deviation from linearity is relatively small, the system experiences some form of distortion, this distortion can also be reduced by applying feedback to the system.
Another scenario we need to discuss is if the range of the system is large.
When the range of the system is large, such as in the case of a control application, a linear system is more likely to perform poorly or it may become unstable, as a linear design is not capable of properly compensating for the effects of large deviations from linearity.
In contrast to that, a nonlinear control system can perform better by directly incorporating the nonlinearities into the systems. This thing can be seen clearly by the demonstration of motion control of a robot. In that many nonlinear forces associated with the motion are neglected, leading to the control accuracy decreasing as the speed increases. For example, if the robot had to perform a “pick and place” task, the speed of the motion will be kept relatively slower, so that the robot can accurately perform its task.
And, if the robot needed to perform a task that requires speed instead, its accuracy would be decreased to achieve optimum productivity.
Describing function analysis
When a non-linear system is subjected to a sinusoidal input, the describing function of the system is defined as the complex ratio of the fundamental component of the output to the sinusoidal input. Therefore the main aim of the describing function method is to approximate the nonlinear elements of a nonlinear system and then use the power of frequency-domain techniques to analyze the approximating system.
It is mostly used to predict the occurrence of limit cycles in nonlinear feedback systems. Limit cycles refer to ythe closed trajectory in the phase space onto which other trajectories converge asymptotically, from both outside and inside.
Lyapunov’s method
Another method used for analyzing the stability of a nonlinear system is Lyapunov’s method. It states that the stability properties of the system near the equilibrium point are essentially the same as those obtained through the linearized approximation of the system.
Equilibrium point refers to a point in phase space at which the state vector of the system can reside forever.
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