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Written by goldeneggs
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Wednesday, 05 November 2008 07:07 |
Non-linear control is a sub-division of control engineering which deals with the control of non-linear systems. The behaviour of a non-linear system cannot be described as a linear function of the state of that system or the input variables to that system. For linear systems, there are many well-established control techniques, for example root-locus, Bode plot, Nyquist criterion, state-feedback, pole-placement etc.
Properties of non-linear systems
Some properties of non-linear dynamic systems are - They do not follow the principle of superposition (linearity and homogeneity).
- They may have multiple isolated equilibrium points (linear systems can have only one).
- They may exhibit properties such as limit-cycle, bifurcation, chaos.
- Finite escape time: The state of an unstable nonlinear system can go to infinity in finite time.
- For a sinusoidal input, the output signal may contain many harmonics and sub-harmonics with various amplitudes and phase differences (a linear system's output will only contain the sinusoid at the output).
Analysis and control of non-linear systems
There are several well-developed techniques for analyzing nonlinear feedback systems: - Describing function method
- Phase plane method
- Lyapunov stability analysis
- Singular perturbation method
- Popov criterion (described in The Lur'e Problem below)
- Center manifold theorem
- Small-gain theorem
- Passivity analysis
Control design techniques for non-linear systems also exist. 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: - Gain scheduling
- Adaptive control
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: - Lyapunov Redesign
- Back-stepping
- Sliding mode control
Nonlinear Feedback Analysis - The Lur'e problem
An early non-linear 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 non-linearity.
The linear part can be characterized by four matrices (A,B,C,D), while the non-linear part is Φ(y) with transfer
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