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After introducing a quantization block in the simulink model, the output of the controlled system starts to oscillate around the zero fixpoint. I am using a lqr controller to regulate the system, which was designed based on the continuous system. Can I tune the parameters of the lqr design to fix this problem ? Which method is recommended here ?

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The typical method is to choose hardware whose quantization is lower than the system's ambient noise.

You can't damp it out -- it's a nonlinear phenomenon, which, when linearized around a switch in the output, has infinite incremental gain. If for some reason you can't just make the quantization small enough, you can add some dither to it -- either add a sawtooth right before the worst quantization step, or implement a delta-sigma modulator (a 1st-order is enough in my experience).

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