Question

1 a the system stable. For example, in Chapter 2 we derived the transfer function for the inverted pendulum, which, for simple values, might be G(s) for which we have bs)1 and as)-s2-1-(s+)(s 1). Suppose we try Dcl (s) = K Istn . The characteristic equation that results for the system is (4.17) This is the problem that Maxwell faced in his study of governors: Under what conditions on the parameters will all the roots of this equation be in the LHP? The problem was solved by Routh. In our case, a simple solution is to take y and the common (stable) factor cancels. Note that the cancellation is fine in this case, because (s +1) is a stable pole. The resulting second-order equation can be easily solved to place the remaining two poles at any point desired Exercise. If we wish to force the characteristic equation to be s2 +2;wns+ 2 : O, solve for K and δ in terms of ζ and an 4.1.3 Regulation The problem of regulation is to keep the error small when the reference is at most a constant set point and disturbances are present. A quick look at the open-loop block diagram reveals that the controller has no influence at all on the system response to either of the disturbances, w. or v. so this structure is useless for regulation. We turn to the feedback case. From Eq. (4.8). we find e search for a good controller. For example. the term giving the contribution of the plant disturbance to the system error W. To select D to make this term small. we should make Dct as large as possible and intinite if that is feasible. On the other hand. the error term for the sensor noise isV. In this case. unfortunately. if we select Dci to be large, the transfer function tends to unity and the sensor noise is not reduced at al What are we to do? The resolution of the dilemma is to observe that each of these terms is a function of frequency so one of them can be large for some frequencies and small for others. With this in mind. we also note that the frequency content of most plant disturbances occurs at very low frequencies and, in fact, the most common case is a bias. which is all at zero GD e other hand. a good sensor will have no bias and c constructed to have very liule noise over the entire range of low frequencies of interest. Thus. using this information. we design the controller transfer function to be large at the low frequencies. where it will reduce the effect of w, and we make it small at the higher frequencies. where it will reduce the effects of the high frequency sensor noise. The control engineer must determine in each case the best place on the frequency scale to make the crossover from amplification to attenuation. Exercise. Show that if w is a constant bias and if Det has a pole ats - 0 then the error due to this bias will be zero. However, show that if G has a pole at zero, the error due to this bias will not be zero.

Answer 1

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