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Next, consider the system shown in Fig. 4.22. The source N represents the sensor noise (here, N is understood to be the mean square amplitude of the noise). Since from the noise input to the system output, the system can be viewed as a tracking system with unity feedback, the mean square amplitude of noise at the system output is
Fig. 4.22 Sensor noise effect at system's output 
N_{out} = NT/F .
In shaping the loop gain response, the tradeoff is between output noise reduction and disturbance rejection. Larger feedback bandwidth leads to larger output noise, but smaller disturbances. The output mean square error caused by the noise can be found by computer simulation of the output timeresponses. Another way to do this is to find the output noise power by integration (on a computer or even by graphical integration) of the frequencydomain noise responses.
Example 1. Consider the Bode diagram shown in Fig. 4.23. This loop gain response has a rather steep cutoff after f_{b} to reduce the output noise effect, but shallower gain response and smaller feedback at lower frequencies. The phase stability margin is large. The hump on the response of M is small; therefore, the overshoot in the transient response is also small. This response is employed when the plant is already fairly accurate and there is no need for large feedback at lower frequencies, and positive feedback near the crossover frequency should be reduced to reduce the output effect of the sensor noise. In such a system, command feedforward is commonly used to improve the closedloop inputoutput response. When the loop response is steep as shown in Fig. 4.24, the output noise increases because of the positive feedback at the crossover frequency and beyond. This causes a substantial increase in the output noise since the contribution of the noise spectral density to the mean square error is proportional to the noise bandwidth. On the other hand, this response provides better disturbance rejection. The loop response should be therefore shaped in each specific case differently to reduce the total error.


Example 2. Consider the spacecraft attitude control system in Fig. 4.25 which uses a gyro as a sensor. The system is accurate except at the lowest frequencies where the gyro drift causes attitude error. The drift is eliminated by a lowfrequency feedback employing a second sensor, a star tracker. The optimal frequency response for the star tracker loop is that which reduces the total noise from the two sensors, i.e., which reduces the mean square error of the system output variable. The calculations can be performed in the frequency domain or by using the LQG method described in Chapter 8. Since the star tracker noise varies with time, depending on whether bright stars are available in its field of view, the feedback path responses need to be varied to maintain the minimum of the error. Such an adaptive system is illustrated in Chapter 9.  
Fig. 4.25 Spacecraft attitude control system using two sensors 
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