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4.2.7   Nyquist-stable systems

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, x1 and x represent the upper and lower amplitude stability margins. At frequencies where A > x1 , 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 - 6ndB/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 fg to fh 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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