ROCKWELL T8461 device made of system superposition principle

Linear control theory – This applies to the principle of superposition of systems made from devices that meet the following criteria, roughly meaning that the output is proportional to the input. They are subject to linear differential equations. A major subclass is additional systems with parameters that do not vary with time, called linear time invariant (LTI) systems.

These systems can withstand powerful frequency ranges and very general mathematical techniques, such as the Laplace transform, the Fourier transform, the z transform, Bode diagrams, root locus, and Nyquist stability criteria. This leads to descriptions of systems such as bandwidth, recording amplitude-frequency response, eigen values, increases, resonant frequencies, zeros and poles, providing solutions to system response and design techniques for most systems of interest.

Nonlinear control Theory – This covers a broader category of systems that do not adhere to the superposition principle and applies to more real-world systems because all real-world control systems are nonlinear. These systems are usually composed of nonlinear differential equations. The few mathematical techniques that have been developed to deal with these problems are more difficult and less general, and usually only apply to narrow categories of systems.

These include limit cycle theory, Poincare maps, Lyapunov’s stability theorem, and descriptive functions. Nonlinear systems usually use numerical methods on computers, for example by simulating their operations using analog languages. If you’re only interested in solutions near stable points, then nonlinear systems are often so linearized by approximating their perturbation theory with linear systems, and you can use linear techniques