Control Systems/Nonlinear Systems

Nonlinear General Solution
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 origin equation.

Iteration Method
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
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
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.



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.