The design using the root-locus method is directly connected to the considerations in section 9. There the specifications of maximum overshoot, rise and settling time for a closed loop having a dominant pair of poles have been converted into the conditions for the damping ratio and for the natural frequency of the related transfer function . With and the poles of the transfer function are directly tightened as shown in Figure 9.2. An open-loop transfer function must now be determined such that the closed loop has a dominant pair of poles at the desired position, which is given by the values of and . Such an approach is also called pole assignment.
The root-locus method is - as generally known - a graphical method, which is used to analyse the position of the closed-loop poles. This method offers the possibility to combine in the complex plane the desired dominant pair of poles with the root locus of the fixed part of the loop and to deform the root locus by adding poles and zeros such that two of the branches traverse through the desired dominant pair of poles at a certain gain . Figure 9.21 shows, how the root locus can be deformed to the right by adding a pole and to the left by adding a zero.
The principal strategy during the controller design by the root-locus method will be shown in the following using two examples.
During the design one will try to increase the damping ratio not unnecessarily high as this step causes an increase in the rise time for a given natural frequency (Figure 9.4). Increasing the natural frequency implies an increase of the speed of the control loop. But this parameter should not be unnecessarily increased, otherwise the dominance of the pair of poles may be lost.
Figure 9.23 shows the root locus of the closed loop using a
It must be mentioned that a complete compensation (cancelling) of the pole cannot be realised exactly, as the plant parameters are not exactly known or may change within some bounds. Thus the root locus will differ in the vicinity of the compensated pole from the ideal case of Figure 9.24, but intersection with the two lines and is almost invariant.
One realises by means of this example that the root-locus method is well suited to provide a fast and first overview about the principal possibilities of corrections. Often the knowledge about the asymptotes is already sufficient. The root-locus method is especially suited for stabilising unstable plants. This will be explained by the second example.
Another possibility is to provide feed back around the unstable plant so that the original unstable pole in the closed loop is shifted into the left-half plane. A simple P controller would yield the root locus shown in Figure 9.26. This configuration is notplane for all gain values. But if a controller is used that has a double zero at and a pole at , the pole at will be substituted by a zero. The root locus will be deformed to the left as shown in Figure 9.27.
This distribution of poles and zeros can be simply realised by a PID controller with the transfer function
As the stability of the closed-loop system is not influenced by the left-half-plane poles, a compensation of these poles is possible. Even if this compensation is not completely possible, the system will be stable. A compensation of the right-half-plane poles - as discussed above - should not be done.