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In MIMO and even in SISO control systems, several paths are often connected in parallel, especially when several actuators or sensors are employed. As was demonstrated in Chapter 3, if two stable m.p. links W_{1} and W_{2} are connected in parallel as shown in Fig. 4.43(a), then the total transfer function W_{1} + W_{2} can become n.p. The frequency responses of the parallel channels should be shaped properly for the combined channel to be m.p.
(a) | (b) | (c) |
Fig. 4.43 (a) Parallel channels, (b) Bode diagrams for W_{1}, W_{2}, and W_{2} + W_{1}, and
(c) s-plane root loci for W_{1} + W_{2}
Example 1. The low-pass links W_{1} and W_{2 }are connected in parallel as shown in Fig. 4.43(a). The steep roll-off W_{1} and the three versions of shallower roll-off W_{2} are shown in Fig. 4.43(b). At frequency f_{1} where the gains are equal, the phase difference between the two channels is, respectively, less than , equal to , and more than . The thin lines show the logarithmic responses of |W_{1} + W_{2}| obtained by vector addition of the links' output signals. When the phase difference between the channels at f_{1} is , the outputs of the links cancel each other and therefore, the composite link transfer function has a pair of purely imaginary zeros ±j2f_{1}. If the slope of W_{2} is gradually changed, the root loci for the zeros of the transfer function cross the j-axis as shown in Fig. 4.43(c) and the total transfer function becomes n.p. |
As has been proven, the sum W_{1} + W_{2} is m.p. if and only if the Nyquist diagram for W_{1}/W_{2} does not encompass the point -1. Since the ratio W_{1}/W_{2} is also stable and m.p., one can determine whether the Nyquist diagram encloses the critical point by examining the Bode diagram for W_{1}/W_{2}.
When the tolerances of the parallel channel transfer functions are not negligible, they can produce large variations in W_{1} + W_{2 }. The sensitivities of the sum to the components,
and
,
become unlimited as the ratio W_{1}/W_{2} approaches -1. To constrain the sensitivities, the hodograph of W_{1}/W_{2} should be required not to penetrate the safety margin around the point -1.
Analogous to the stability margins, the phase safety margin is defined as y, and the amplitude safety margins, as x and x_{1}.
A common practical reason for using two parallel links is that one of the links (actuators, or sensors) works better at lower frequencies, and the second link works better at higher frequencies. Combining them with frequency selection filters generates a link (actuator, or sensor) that is good over a wide frequency range. The composed link transfer function must be m.p. so that it can be included in the feedback loop. However, excessive selectivity of the filters can make the composite link n.p.
Fig. 4.44(a) shows the responses of the low-pass link W_{1} and the high-pass link W_{2 }, with different selectivities. Fig. 4.44(b) shows the Bode diagrams for the ratio W_{1}/W_{2}. It is seen that when the difference in the slope between W_{1} and W_{2} responses increases, the Bode diagram for the ratio steepens, the related phase lag increases, and the critical point becomes enclosed by the related Nyquist diagram.
(a) | (b) |
Fig. 4.44 Bode diagrams for (a) W_{2 }, W_{1} and (b) W_{1}/W_{2}
In order to preserve sufficient safety margins while keeping the slope of W_{1}/W_{2} steep, the Bode diagram for W_{1}/W_{2} can be shaped as in a Nyquist-stable system, Fig. 4.45(b). Then, W_{1} and W_{2} can be as illustrated as in Fig. 4.45(a). Responses of this kind are particularly useful for systems with the main-vernier actuator arrangement described in Section 2.7.
An alternative to shaping the responses and then approximating them with rational functions is direct calculation of the channel transfer functions. Given the transfer function of the first link, the transfer function of the second link can be found directly as W_{2} = 1 - W_{1}. This method works well if the links are precise (as when sensors' readings are combined). If the links are imprecise (like actuators and different signal paths through the plant), and the selectivity is high, then the link parameter variations should be accounted for and sufficient safety margins introduced.
(a) | (b) |
Fig. 4.45 (a) Frequency-selective responses for W_{1} and W_{2} and
(b) Nyquist-stable shape of Bode diagram for
W_{1}/W_{2}
which preserve m.p. character of W_{1} + W_{2}
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