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Given a linear system with initial conditions. If you have conjugate poles in D(s) that lie right on the jw axis, you should, in theory, be able to get sustained oscillations that do not grow unbound nor damp with time. Why would it not be a practical solution? I guess most linear systems would have damping which would imply the oscillations would die out. I tend to think it has something to do with sensitivity of the system's ability to oscillate. For example, if initial conditions are just not perfect when you start the oscillator up, it will not oscillate.

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    \$\begingroup\$ In order to 'not grow unbound nor damp with time' the characteristics of the system must be 100% right. Nothing can do that, so in practice the oscillations will either grow unboundend or damp out. \$\endgroup\$ – Wouter van Ooijen Oct 31 '15 at 16:03
  • \$\begingroup\$ @WoutervanOoijen You should make that an answer. \$\endgroup\$ – Kevin Reid Oct 31 '15 at 16:23
  • \$\begingroup\$ In practice, saturation will limit the amplitude in the case of loop gain >1. \$\endgroup\$ – Chu Oct 31 '15 at 19:15
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(promoted from comment to answer)

In order to 'not grow unbound nor damp with time' the characteristics of the system must be 100% right. Nothing in the real world is ideal, so in practice the oscillations will either grow unboundend or damp out.

In practice a system is used that amplifies a little, and a non-linearity that compensates this by attenuating a high level more than a lower level.

To see this, google 'wien-bridge oscillator', and you will see all sorts of non-linear elements used in the feedback loop: incandescent lamps, diodes, FETs, etc.

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  • \$\begingroup\$ Right. First of all no real physical systems are perfectly linear; linear systems are an abstraction. Even if you are close to a linear system the 'marginally' stable complex poles will always be one side or another of the imaginary axis. So the oscillations either die out or grow. But then nonlinearity comes to the rescue! Only with nonlinearity present can we contrive a stable oscillator. \$\endgroup\$ – docscience Nov 4 '15 at 15:52
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It sounds a bit paradox - however, a good harmonic "linear" oscillator should be as linear as possible and - at the same time - as non-linear as necessary. This is required (a) to start oscillations safely (poles in the righ half of the s-plane) and (b) to bring the poles back to the imag. axis for rising amplitudes. In some (simple) cases this non-linearity is introduced into the circuit by the upper limits due to finite supply voltages (hard limiting), but in most cases we use a nonlinear feedback element like antiparallel diodes, NTC, FET, OTA as resistor,...

"If you have conjugate poles in D(s) that lie right on the jw axis, you should, in theory, be able to get sustained oscillations that do not grow unbound nor damp with time."

Osillators with poles in the RHP exhibit always growing oscillations until the amplitudes are hardly clipped (supply voltage) - unless some means for soft clipping are becoming active.

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