Control Systems/Nonlinear Systems

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Nonlinear General Solution[edit]

A nonlinear system, in general, can be defined as follows:

x'(t) = f(t, t_0, x, x_0)
x(t_0) = x_0

Where f is a nonlinear function of the time, the system state, and the initial conditions. If the initial conditions are known, we can simplify this as:

x'(t) = f(t, x)

The general solution of this equation (or the most general form of a solution that we can state without knowing the form of f) is given by:

x(t) = x_0 + \int_{t_0}^t f(\tau, x)d\tau

and we can prove that this is the general solution to the above equation because when we differentiate both sides we get the general solution.

Iteration Method[edit]

The general solution to a nonlinear system can be found through a method of infinite iteration. We will define xn as being an iterative family of indexed variables. We can define them recursively as such:

x_n(t) = x_0 + \int_{t_0}^t f(\tau, x_{n-1}(\tau))d\tau
x_1(t) = x_0

We can show that the following relationship is true:

x(t) = \lim_{n \to \infty}x_n(t)

The xn series of equations will converge on the solution to the equation as n approaches infinity.


Types of Nonlinearities[edit]

Nonlinearities can be of two types:

  1. Intentional non-linearity: The non-linear elements that are added into a system. Eg: Relay
  2. Incidental non-linearity: The non-linear behavior that is already present in the system. Eg: Saturation

Linearization[edit]

Nonlinear systems are difficult to analyze, and for that reason one of the best methods for analyzing those systems is to find a linear approximation to the system. Frequently, such approximations are only good for certain operating ranges, and are not valid beyond certain bounds. The process of finding a suitable linear approximation to a nonlinear system is known as linearization.

Linear Approximation.svg

This image shows a linear approximation (dashed line) to a non-linear system response (solid line). This linear approximation, like most, is accurate within a certain range, but becomes more inaccurate outside that range. Notice how the curve and the linear approximation diverge towards the right of the graph.

← State Machines

Control Systems

Common Nonlinearities →