Nonlinear control

Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dynamical systems with inputs, and how to modify the output by changes in the input using feedback, feedforward, or signal filtering.

Linear control theory applies to systems made of devices which obey the superposition principle.

Nonlinear control theory covers a wider class of systems that do not obey the superposition principle.

The mathematical techniques which have been developed to handle them are more rigorous and much less general, often applying only to narrow categories of systems.

Even if the plant is linear, a nonlinear controller can often have attractive features such as simpler implementation, faster speed, more accuracy, or reduced control energy, which justify the more difficult design procedure.

A building heating system such as a furnace has a nonlinear response to changes in temperature; it is either "on" or "off", it does not have the fine control in response to temperature differences that a proportional (linear) device would have.

These can be subdivided into techniques which attempt to treat the system as a linear system in a limited range of operation and use (well-known) linear design techniques for each region: Those that attempt to introduce auxiliary nonlinear feedback in such a way that the system can be treated as linear for purposes of control design: And Lyapunov based methods: An early nonlinear feedback system analysis problem was formulated by A. I. Lur'e.

Control systems described by the Lur'e problem have a forward path that is linear and time-invariant, and a feedback path that contains a memory-less, possibly time-varying, static nonlinearity.

Consider: The Lur'e problem (also known as the absolute stability problem) is to derive conditions involving only the transfer matrix H(s) and {a,b} such that x = 0 is a globally uniformly asymptotically stable equilibrium of the system.

There are two well-known wrong conjectures on the absolute stability problem: Graphically, these conjectures can be interpreted in terms of graphical restrictions on the graph of Φ(y) x y or also on the graph of dΦ/dy x Φ/y.

[2] There are counterexamples to Aizerman's and Kalman's conjectures such that nonlinearity belongs to the sector of linear stability and unique stable equilibrium coexists with a stable periodic solution—hidden oscillation.

There are two main theorems concerning the Lur'e problem which give sufficient conditions for absolute stability: The Frobenius theorem is a deep result in differential geometry.

When applied to nonlinear control, it says the following: Given a system of the form where

A feedback control system . It is desired to control a system (often called the plant ) so its output follows a desired reference signal. A sensor monitors the output and a controller subtracts the actual output from the desired reference output, and applies this error signal to the system to bring the output closer to the reference. In a nonlinear control system at least one of the blocks, system, sensor, or controller, is nonlinear.
Lur'e problem block diagram