Bode�s Integrals
It follows from Equations (5) and(8) that it would be highly desirable to make the sensitivity functions S and T as small as possible. This is unfortunately not possible because it follows from Equation (3) that S+T=1. there are also other constraints on the sensitivities. It was show in [11] that
Where are the right half plane poles of L and are the right half plane zeros of L. These equations imply that the sensitivities can be made small at one frequency only at the expense of increasing the sensitivity at other frequencies. This phenomena is sometimes called the water bed effect. It also follows from the equations that the presences of poles in the right half plane increase the sensitivity and that zeros in the right half plane increase the complementary sensitivity. A fast RHP pole gives higher sensitivity than a slow pole, and a slow RHP zero gives higher sensitivity than a fast zero.
2.4 Bode�s Relations
The amplitude and the phase curves are also related. It is not possible to achieve high phase advance without using high gains and it is not possible to obtain transfer function that decrease rapidly without having large phase lags . these facts are expressed analytically by some relations derived in [10]
Consider a transfer function G(s) with no poles or zeros in the right half plane.
Introduce This means that if the slope of the magnitude curve is constant the phase is /2. this relation appears in
practically all elementary courses in feedback control.
Bode�s relations imposes fundamental limitations on the performance that can be achieved. A simple observation
is that even if it is desirable that the loop gain decreases rapidly at the crossover frequency, it is not possible to have a steeper slope than -2 without violating stability constraints.
An interesting problem is if the limitations imposed by Bode�s relations can be avoided by using nonlinear
systems. The clegg integrator [13]is a nonlinear system where the magnitude curve has the slope -1 and the phase lag is only .
1.4 Bide�s Ideal Loop transfer Function
In his work on design of feedback amplifiers Bode suggested an ideal shape of the loop transfer function. He proposed that the loop transfer function should have the form
 
The Nyquist curve for this loop transfer function is simply a straight line through the origin with
see figure 5. Bode called (15) the ideal cut-off characteristic. In the terminology of automatic control we will call it Bode�s ideal loop transfer function.
One reason why Bode made the particular choice of L (s) given by equation (15) is that it gives a closed-loop system that is insensitive to gain changes. Changes in the process gain will change the crossover frequency but the phase margin is = (1+n/2) for all values of the gain. The amplitude margin is infinite. The slopes n = -1.333,- 1.5and -1.667 correspond controller that are insensitive to gain variations were later generalized by[24] to systems that are insensitive to other variations of the plant, culminating in the QFT method, see [26]
The transfer function given by Equation (15) is an irrational transfer function for non-integer n . it can be
approximated arbitrarily close by rational frequency functions. Bode also suggested that it was sufficient to
approximate L over a frequency range around the desired crossover frequency . assume for example that the gain
of the process varies between and and that it is desired to have a loop transfer function that is close to (15) in the frequency range (). It follows from (15) that
With n=-5/3 and a gain ratio of 100 we get a frequency ratio of about 16 and with n=-4/3 we get a frequency ratio of
32. to avoid having too large a frequency range it is thus useful to have n as small as possible. There is
however, a compromise because the phase margin decreases with decreasing n and the system becomes unstable for n=-2
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