Feedback Control With Not Enough Sensor Sample Rate


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I am doing feedback control of a mass-spring-damper system. As we all know, the input for this system is force, output is displacement or position. The displacement value is obtained using a sensor. The problem is I want to control the output displacement as 1000Hz sine wave fluctuation, in order to obtain accurate displacement value the sensor's sampling rate need to at least 20kHz, but unfortunately, this is beyond the capability of the sensor I use. A paper talking about the same problem with me, then author said "An alternative way to control the plant that acts as a stiffness is by only using an integral action. The crossover frequency, the frequency at which the open-loop transfer function crosses 0 dB, of the integral controller is chosen several factors below the first eigenfrequency of the system, which means the required sample rate is not extremely high in this case." I want to use this method, but I don't understand why does this method work? I really don't see there is any relationship between low pass integrator and low sampling rate. Could anyone have a clear explanation of this?