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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. If only solutions near a stable point are of interest, nonlinear systems can often be linearized by approximating them by a linear system obtained by expanding the nonlinear solution in a series , and then linear techniques can be used. 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.

## Nonlinear control systems : analysis and design / Horacio J. Marquez. - Version details - Trove

An example of a nonlinear control system is a thermostat -controlled heating system. 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. Therefore, the furnace is off until the temperature falls below the "turn on" setpoint of the thermostat, when it turns on. Due to the heat added by the furnace, the temperature increases until it reaches the "turn off" setpoint of the thermostat, which turns the furnace off, and the cycle repeats.

This cycling of the temperature about the desired temperature is called a limit cycle , and is characteristic of nonlinear control systems.

## ISBN 13: 9780471427995

Control design techniques for nonlinear 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:. 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:. An early nonlinear feedback system analysis problem was formulated by A. 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.

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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.

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References Publications referenced by this paper. A frequency-domain condition for the stability of feedback systems containing a single time-varying nonlinear element Irwin W. On the Lboundedness of solutions of nonlinear functional equations. On the stability of nonlinear, time-varying feedback systems. George Zames.

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