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4.2.2  Bode step

In physical plants, the uncertainty of the plant parameters increases and the gain drops at higher frequencies. In electrical circuits, this happens due to stray capacitances and inductances, in thermal systems, due to thermal resistances and thermal capacitances, and in mechanical systems, the same happens due to the plant flexibility. Because of this and because of the sensor noise which will be studied in Sections 4.3.2 and 4.3.3, the loop gain must sharply decrease at higher frequencies. For the purpose of analysis, the loop gain Bode diagram can be sufficiently well approximated at these frequencies by a line with a constant asymptotic slope of -6n dB/oct. The slope is rather steep in physical feedback systems with n  2. In other words, |T(j)| decreases at higher frequencies at least as -2, and the integral of feedback (3.7) is therefore zero.

The crossover region studied in this section is the region of transition between the functional frequency band and the higher-frequency band where the feedback becomes negligible. As was stated in corollaries in Section 3.9.2, shaping the Bode diagram in the crossover region (step A1 from Section 4.2.1) is critical in achieving a maximum area of positive feedback near the crossover and, therefore, achieving a maximum area of negative feedback in the functional feedback band.

Physical systems include actuators with saturation. If a system contains no other nonlinear links, absolute stability is required and the stability margin boundary must be as shown in Fig. 4.1(a). From the Bode integral of phase it follows that to maximize the feedback, the phase lag must be the maximum allowable, i.e., the Nyquist diagram should follow the boundary curve as closely as possible. Such a diagram is shown by the thin line. The gain monotonically decreases with increasing frequency and eventually degenerates into the high-frequency asymptote with the associated phase shift -n90°.

It will be shown below that this Nyquist diagram corresponds to the Bode diagram shown in Fig. 4.1(b). The loop gain response is piece-linear with corner frequencies fd and fc. The related loop phase lag is less than (1 - y)180° until the loop gain becomes smaller than -x dB.

The system is phase-stabilized with the margin not less than y180° up to the frequency fd where the loop gain drops to -x. Because of the Bode phase-gain relation, the slope of the Bode diagram at these frequencies is approximately -12(1 - y) dB/oct.

The high-frequency asymptotic loop response is considered known. It is defined by:
(a) the asymptotic slope - 6n dB/oct,
(b) the point on this asymptote with coordinates (fc , -x) as shown in Fig. 4.2(b), and
(c) the nonminimum phase lag at this frequency Bn(fc).

 
(a) (b)

Fig. 4.2  Bode step: (a) absolute stability boundary on the L-plane, (b) piece-linear gain response with related phase lag response that produce the Nyquist diagram shown in (a) which approximates the boundary

The transition between the slope -12(1 - y)dB/oct and the high-frequency asymptotic slope must be as short as possible to increase the loop selectivity: to maximize the loop gain in the functional frequency range while reducing the loop gain at higher frequencies. The transition is provided by the Bode step made at the gain level of -xdB as shown in Fig. 4.2(b).

Without the step, the phase lag in the crossover area would be too large due to the steep high-frequency asymptote and the non-minimum phase lag. The step reduces the phase lag at the crossover frequency - but also reduces the loop selectivity (i.e., given the high-frequency asymptote, reduces the feedback bandwidth). Therefore, the length of the step must not be excessive.

The nonminimum phase lag Bn(fc) is assumed to be less than 1 radian which is true in well-designed systems. With the linear approximation (3.17), the nonminimum phase lag at frequencies lower than fc is

.

The phase lag related to the asymptotic slope ray which starts at fc can be expressed with (3.14) as, approximately,

.

Consider next the "discarded" dashed-line ray which is the extension of the main slope line beyond the frequency of the beginning of the step fd. The phase lag related to this ray is expressed with (3.14) as, approximately,

.

To make the loop phase lag at frequencies f < fd equal to (1 - y)180°, the sum of the phase contribution of the asymptotic slope and the nonminimum phase lag should equal the phase contribution of the "discarded" dashed-line ray. This consideration is expressed as

.

From this equation, the Bode step frequency ratio is

. (4.1)

For the typical phase stability margin of 30°, i.e., y = 1/6,

fc / fd 0.6n + Bn(fc) . (4.2)
 
   

Example 1. When the specified stability margins are 30° and 10 dB, and the crossover frequency fb = 6.4 kHz, then the slope is -10 dB/oct and fd = 2fb = 12.8 kHz. Further, when n = 3 and Bn(fc) = 0.5 rad, then from (4.2) fc/fd = 2.3 so that fc  30 kHz.

The Nyquist diagram for the Bode step response closely follows the stability boundary in Fig. 4.2(a). In practice, this response is approximated by a rational transfer function, and the corners of the Nyquist diagram become rounded. Examples of loop responses with Bode steps will be given in Sections 5.6, 5.7, and 5.11, in Chapter 13, and in Appendix 13.

The loop response with a Bode step should be employed when the dominant requirement is maximizing the disturbance rejection, i.e., maximizing the feedback in the functional frequency range. This case is common but not ubiquitous. Noise reduction requirements and certain implementation issues may require Bode diagrams to be differently shaped in the crossover area.

In any case, the Bode diagram must be made shallow over some range in the crossover frequency region to ensure the desired phase stability margin. There are several options for where to do this: to the right of fd by the Bode step, to the left of the crossover frequency as in systems where the sensor noise is critical, and over a frequency range nearly symmetrically situated about the crossover as in the so-called PID controller which will be discussed in Chapter 6. Loop responses without any Bode step typically provide 4 to 20 dB less feedback in the functional frequency range.

The output transient response to a step-disturbance in a homing system with a Bode step and 30° to 40° phase stability margin has substantial overshoot. If this overshoot exceeds the specifications, the loop gain response in the neighborhood of fb should be made shallower. However, this will reduce the available feedback and the disturbance rejection.

For a system which has an explicit command input, a prefilter or one of its equivalents can be introduced to ensure good step-responses without reducing the available feedback.

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