Is there any scientific comparison between linear and nonlinear systems?

I often hear that

Nonlinear control is more sluggish than linear control.

which makes sense. But is there any research or any claim based on practical experience which supports this claim?

  • $\begingroup$ Ahem. Linear systems can be solved in one pass. Nonlinear systems, generally can not. $\endgroup$ – joojaa May 3 '18 at 12:37

"Linear" imposes a set of restrictions. "Non-linear" simply means there are no restrictions.

Many non-linear control schemes can be faster than linear ones. Linear control schemes are restricted to "smoothly" transitioning. Non-linear control can be implemented by suddenly slamming a digital value, for example.

A good example of a fast-responding non-linear control scheme is a typical thermostat. When the room is below the set point, the heater is turned on fully. When above the set point, the heater is turned off fully.

This is obviously the fastest way to get to the desired temperature. Such a system may be less "good" than a linear controller in other metrics, like overshoot, but that's not what you asked about.

The reason linear control schemes are sometimes chosen is not for their speed or other control property, but because they can be mathematically analyzed more easily. There has been much theory and techniques developed around linear systems that don't apply to non-linear systems. Consider Laplace transforms and S-parameter analysis, for example.

  • $\begingroup$ One could argue that settling into the desired value is the target. But Your right since nonlinear is everything else it also includes such things that are for example astable and dont need much control at all. $\endgroup$ – joojaa May 3 '18 at 13:35
  • $\begingroup$ Could you give an example of a "linear control system" what is that? In my field, linear systems are pretty much non existent. $\endgroup$ – Salomon Turgman May 3 '18 at 23:40
  • 2
    $\begingroup$ @Sal: Any PID controller is linear. You might implement a cruise control in a car that way, for example. $\endgroup$ – Olin Lathrop May 4 '18 at 11:35
  • $\begingroup$ The answer treats sluggishness in the control signal as opposed to the intended meaning of the OP which is slugishness in the output. From this viewpoint, the response to a stable (or stabilized) linear system is exponential (indicating being fast) while the response to the nonlinear system is likely not exponential (indicating being slow) due to the nonlinearity including that of the dynamics, due to state and linear constraints. $\endgroup$ – kbakshi314 Mar 18 at 14:02

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