7.3.  MINOR-LOOP FEEDBACK-COMPENSATION TECHNIQUES

Let us consider the general system illustrated in Figure 7.2. The compensating element in this case is the transfer function B(s). In order to have a basis of comparison, we will follow an analysis for minor-loop feedback compensation similar to that performed for the case of phase lead-network cascade compensation.

The minor-loop feedback element B(s) usually represents rate feedback or acceleration feedback. In general, phase-lag, -lead, and/or lag–lead networks may also be cascaded with B(s).

The stabilizing effect of minor-loop feedback compensation can easily be demonstrated for a simple second-order system. We assume that the system illustrated in Figure 7.2 contains simple rate feedback. The specific transfer functions for the system are

The system is redrawn with these transfer functions and shown in Figure 7.14a.

Without any rate feedback, the configuration represents a simple second-order system whose damping ratio is ζ and undamped natural frequency is ω_n. The resulting system transfer function with rate-feedback compensation is given by

Comparing the denominator of Eq. (7.26) with that of Eq. (4.3), we observe that it is still of second order and ω_n remains the same, but ζ is greater due to the increase in the coefficient of s in the denominator. The equivalent damping ratio with rate feedback added can be obtained by setting the coefficients of the s terms equal to each other, as follows:

where ζ_eq = an equivalent damping ratio with rate feedback added. Solving for ζ_eq we obtain

Therefore, we can conclude that the addition of the minor loop using rate feedback has increased the damping ratio from ζ to ζ_eq by an amount equal to ω_nb/2. This assumes that b is positive (negative feedback).

It is important at this point to compare Eqs. (7.15) and (7.28). Note that they are very similar, and they imply that

The fact that rate feedback behaves as the approximated phase lead network, as defined by Eq. (7.12) (porportional plus derivative controller), can be easily demonstrated from Figure 7.14a and b. Let us assume that there is zero input to both systems, because we are concerned only with the system poles. Clearly, in both cases, there are two negative-feedback paths in parallel around G₀(s). In the cascade-compensation case, the total feedback around G₀(s) is 1 + αTs; in the rate-feedback-compensation case, the total feedback around G₀(s) is 1 + bs. Therefore, the stabilizing effects of αT and b are equivalent. We discuss proportional plus derivative (PD) controllers fully in the next section, 7.4, on proportional-plus-integral-plus-derivative (PID) compensators.

Let us next determine the steady-state error resulting from the use of minor-loop rate-feedback compensation. We assume that the input to this system is a unit ramp in order to have a finite steady-state response error and a basis for comparison. From our discussion of cascade compensation in Section 7.2 we know from Eq. (7.18) that the resulting steady-state error of this system without any compensation (b = 0) is 2ζ/ω_n. For the case of minor-loop rate-feedback compensation, the resulting expression for E(s) is given by

Applying the final-value theorem, the steady-state error is found to be

Therefore, the steady-state response error of the system with minor-loop rate-feedback compensation has increased by a factor of b. This unfavorable result can easily be remedied by placing a high-pass filter in cascade with the rate device. Such a filter would block the steady-state value of the rate output. This technique is illustrated in Figure 7.15.

As in the preceding section, it is important to emphasize that the relationships derived apply only to the simple system considered. Problems 7.5 and 7.6 illustrate how these relationships change if a zero factor is added to the basic system transfer function considered.

Figure 7.15   Illustration of minor-loop feedback compensation using a rate device in cascade with a high-pass filter