Stability of Closed Loop Systems
For linear invariant systems such as we have discussed above, the methods available for assessing stability include those used with open loop systems. One such method is to formulate the governing differential equations, find the characteristic equation of the system, and solve for its roots. Another is to find the transfer function from input to output and determine its poles. With the powerful computing methods available, it is feasible to plot loci of the roots (or poles) as one or more of the significant design parameters are varied, as we shall see in examples to follow. For the multivariable highorder systems that commonly occur in aerospace practise this is a very useful technique.
Let us now consider the stability of the loop associated with one particular in – put/output pair in the light of (8.1,6a). Since p = 1 and F and H are scalars, the transfer function is
F(s) Gvr(s) = vr 1 + F(s)H(s) 
(8.2,1) 
The transfer functions F{s) and H(s) are ratios of polynomials in s, NfDx, and H(s) = N2/D2. Equation (8.2,1) then leads to 
that is, F(s) = 
ND2 Gyr{s) ~ DxD2 + NxN2 
(8.2,2) 
The characteristic equation is evidently 

Dx(s)D2(s) + Nt (s)N2(s) = 0 
(8.2,3) 
This should be contrasted with the characteristic equation for the airframe alone, which is D(s) = 0, where D is the denominator of G(s). The block diagram corresponding to (8.2,1) is shown in Fig. 8.2. FH is the open loop transfer function, that is, the ratio of feedback to error, z/ё. Its absolute value FH is the open loop gain.
The stability of the system can be assessed from the frequency response F(ico)H(i(o). It is clear that if there is a frequency and open loop gain for which FH = 1 then un
Figure 8.3 Nyquist diagram. 
der those conditions the denominator of (8.2,1) is zero and Gyr(ico) is infinite. When these conditions hold, the feedback signal that is returned to the junction point is precisely the negative of the error signal that generated it. This means that the system can oscillate at this frequency without any input. This is exactly the situation with the whistling public address system. For then the acoustic signal that returns to the microphone from the loudspeakers, in response to an input pulse, is equal in strength to the originating pulse. Clearly the point (—1,0) of the complex plane has special significance. Nyquist (1932) has shown how the relationship of the frequency response curve (the Nyquist diagram) of FH to this special point indicates stability (see Fig. 8.3). In brief, if the loop gain is <1 when the phase angle is 180°, or if the phase is < 180° when the gain is unity, then the system is stable. The amounts by which the curve misses the critical point define two measures of stability, the gain margin and phase margin, illustrated on the Nichols diagram (see McLean, 1990) of Fig. 8.4. The
examples of Figs. 8.3 and 8.4 are for the open loop transfer function
K(s + 0.125)
.v2(0.15.t2 + 0.8.У + 1)
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