Control Systems/Nonlinear Systems
Nonlinear General Solution
A nonlinear system, in general, can be defined as follows:
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:
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:
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.
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:
We can show that the following relationship is true:
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:
- Intentional non-linearity: The non-linear elements that are added into a system. Eg: Relay
- Incidental non-linearity: The non-linear behavior that is already present in the system. Eg: Saturation
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.