Continuous Control Systems: a review

Figure 1.20. Nyquist plot for H₁(s)  1/(1  s) and H ₂ (s)  1/[s(1  s)]

These curves show the gain and phase of the transfer function at different frequencies. The vector joining the origin to a point on the hodograph of the transfer function represents H(j) for a certain frequency .

In this diagram, the point [-1, j0] plays a particularly important role (critical point). We can see in Figure 1.21 that the vector connecting the point [-1, j0] to the plot of the open-loop transfer function H_OL (j) is given by

yp
_S^1 ( j)  1  H_OL
( j )
(1.2.19)

Figure 1.21. Nyquist plot and the critical point

This vector corresponds to the inverse of the output sensitivity function (see Equation 1.2.15). The denominator of the output sensitivity function defines the

poles of the closed-loop transfer function. The stability properties of the system are determined by the denominator of the transfer function. In order that the closed-

loop system be asymptotically stable, all the poles of plane Re s  0 .
S_yp (s) must lie in the half-

If the plot of the open-loop transfer function H_OL(s) passes through the point [-
1, j0], for a value s = j, the denominator of the closed-loop system transfer function will be null at this frequency. The closed-loop system will not be asymptotically stable (more precisely it will have poles on the imaginary axis). It then follows a necessary condition (but not sufficient) for the closed-loop system to be asymptotically stable: the hodograph of H(s) must not pass through the point [- 1, j0]. The Nyquist criterion gives the necessary and sufficient conditions for the asymptotic stability of the closed-loop system.
For systems having open-loop stable poles (Re s  0) the Nyquist stability criterion is expressed as: The plot of the open-loop transfer function H_OL (s)
traversed in the sense of growing frequencies (from  = 0 to  = ) must leave the critical point [-1, j0] on the left.
As a general rule, a controller will be computed for the nominal model of the plant so that the closed-loop system be asymptotically stable, i.e. H_OL (s) will leave the critical point on the left.
It is also obvious that the minimal distance to the critical point will characterize the “stability margin” or “robustness” of the closed-loop system in relation to variations of the system parameters (or uncertainties in parameter values).