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4.2.4  Reshaping the feedback response

We proceed next with step A2 from Section 4.2.1, i.e., with shaping the loop response over the functional feedback bandwidth.

In many practical cases the disturbances' amplitude decreases with frequency, and the loop gain should also decrease with frequency. If, further, the system is phase-stabilized with a constant stability margin, the loop gain response is similar to that shown in Fig. 4.2 and to response (3) in Fig. 4.12(a).

To match a specified disturbance rejection response, the constant-slope gain response must be modified. Fig. 4.12(a) gives several examples of feasibly reshaped loop gain responses. The responses redrawn in Fig. 4.12(b) on the arcsin f scale have the same area under them over the frequency interval [0, 1]. Therefore, the phase response at frequencies f > 1 and the stability margins can be the same for all shown gain responses (recall (3.12) from Section 3.9.5).

 
(a) (b)

Fig. 4.12  Reshaped loop gain responses on (a) the logarithmic frequency scale and
(b) the arcsin f scale

Any frequency can be chosen to be the normalized frequency f = 1 (or  = 1). For control system design it is commonly convenient to use as the normalized frequency the frequency at which the constant-slope response gain is approximately 10 dB.

   
Fig. 4.13  Reshaping the loop gain responses

Example 1. Propellant sloshing in Cassini spacecraft's tanks causes structural modes with gain magnitude up to 8 dB. To provide gain stabilization with 8 dB upper stability margin over the range where the modes can be, the nominal loop gain needs to be at least 16 dB. Fig. 4.13 shows the loop gain response to suit the problem, obtained by reshaping the constant slope response. Numerically, the loss in feedback at lower frequencies can be estimated by application of the rule (3.12) (preservation of the area of the loop gain) when the plot is redrawn on arcsin f scale, with the normalized frequency  = 1 at the upper end of the slosh mode range.

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