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As mentioned in Section 4.2.2, and as will be detailed in
Chapters 10 and 11, the responses shown in Figs. 4.2, 4.12, 4.16,
4.17 are tailored to guarantee stability of a system with a saturation link in
the loop. If, however, the system is furnished with an extra dynamic nonlinear
link of special design (described in Chapters 10, 11, and 13), the
loop phase lag, the slope of the Bode diagram, and the available feedback can be
increased by using the Nyquist-stable system loop response shown in
Fig. 4.18 instead of the phase-stabilizing response shown by the thin line.
Here, *x*_{1} and *x* represent the upper and lower amplitude
stability margins. At frequencies where
*A* > *x*_{1 },
the system is only gain-stabilized. The integral of the phase lag in this system
is larger than in the absolutely stable system, and as a result, the feedback
over the functional bandwidth is larger. This response can be generated by pasting together several elementary responses [9].

(a) | (b) | (c) |

**Fig. 4.18** Comparison of a Nyquist-stable
system with an absolutely stable system (thin lines): (a) Bode diagrams, (b) Nyquist diagrams (not to scale),

(c) Nyquist diagrams on the *L*-plane

Fig. 4.19(a) shows a simplified response which is easier
to implement (although it provides somewhat less feedback). The essential
features of the response are the steep slope of
- 6*n*_{1 }dB*/*oct
before the upper Bode step and the presence of two Bode steps, the width of the
lower step calculated with (4.1), and of the upper step from
*f*_{g} to *f*_{h} with the similarly derived
formula

. | (4.7) |

(a) | (b) |

**Fig. 4.19** Simplified Nyquist-stable loop response:

(a) Bode diagram and (b) Nyquist diagram on the *L*-plane

The larger the integral of phase, the larger is the available
feedback. The phase lag, however, cannot be arbitrarily big. A certain boundary
curve *A*(*B*) exemplified in Fig. 4.20 is specified by the
features of nonlinear links in the loop (nonlinear compensators will be
discussed in Chapters
10 - 13).

**Fig. 4.20** The Nyquist diagram should not penetrate the boundary curve

specified by the properties of the nonlinear dynamic compensator.

In Fig. 4.20, the Nyquist diagram is shown with a loop on it caused by a flexible mode of the plant. At frequencies of this mode, the phase stability margin is excessive. In accordance with the phase integral this reduces the achieved feedback, but the feedback deficit due to the loop is rather small since the mode resonance is narrow and the excess in the integral of phase is small.

Type 1 and Type 2 systems (recall Section 3.7) are Nyquist-stable. The stability in such systems can be achieved with upper and lower Bode steps. In practice, the transition between the steep low-frequency asymptote and the crossover area is most often made gradual to simplify the compensator transfer functions, thus reducing somewhat the available feedback at lower frequencies.

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