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Controller design using the root-locus method

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.

Figure 9.21: Deforming of the root locus (a) to the right by an additional pole, (b) to the left by an additional zero in the open loop

The principal strategy during the controller design by the root-locus method will be shown in the following using two examples.

Example 9.4.1   Given is a plant described by the transfer function

(9.52)

For this plant a controller must be designed, such that the step response of the closed loop shows the following properties:

    and    

First, these specifications will be transformed. The conditions for and are from Figures 9.3 and 9.4

    and    

In order to have a geometrical interpretation one should consider Figure 9.22, where a pair of complex poles

   

is shown. The distance of both poles from the origin is

(9.53)

The angle is

(9.54)

or

(9.55)

where for the current case of the condition is met. The damping ratio describes the angle , the frequency the distance of the dominant pair of poles from the origin.
Figure 9.22: Pair of conjugate complex poles in the  plane

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

Figure 9.23: Root locus of (plant with P controller) and potential positions of the dominant pairs of poles (blue thick lines)
P controller. Potential positions for the dominant pair of poles are drawn by the two thick blue lines and . It is obvious that the design using a pure P controller (changes in the gain ) does not lead to the goal, as the root locus does not traverse the two lines and . Equally it is clear which steps have to be taken, such that the two branches of the root locus under consideration traverse the two lines and . If the two poles and are shifted further left, the centre of gravity of the poles will move left and with it the total curve without changing the structure of the system. One possibility is to perform this shift by a simple lead element which compensates the pole by a zero and a pole . The resulting controller transfer function is

(9.56)

and the transfer function of the modified open loop

(9.57)

The root locus of the closed loop is shown in Figure 9.24.
Figure 9.24: Root locus of with modified controller according to Eq. (9.56)
It traverses the line at . The associated gain at this point can be determined via the distances to the three poles from Eq. (6.22)

   

as

   

The step response of the closed loop in Figure 9.25 shows that the specifications are achieved.
Figure 9.25: Closed-loop step response with modified controller according to Eq. (9.56)

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.

Example 9.4.2   Given is an unstable plant with the transfer function

(9.58)

At first, it is close to compensate the pole by a corresponding zero. This would be possible, e.g. using a first-order all-pass element with the transfer function

   

But because of the reason given above, the compensation of the unstable pole will be practically never complete. So it must be performed without this procedure for stability reasons.

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 not

Figure 9.26: Root locus of the closed loop consisting of an unstable plant and a P controller
stabilisable, as the two branches on the right-hand side remain in the right-half  plane 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.
Figure 9.27: Root locus of the closed loop consisting of an unstable plant and a PID controller

This distribution of poles and zeros can be simply realised by a PID controller with the transfer function

   
     

From Figure 9.27 it can be seen for higher gain values than the critical gain the closed loop is stable, as then all poles remain in the left-half  plane.

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.

Demonstration Example 9.1   A virtual experiment stabilising a pendulum



Next: Compensator design methods Up: Design of controllers using Previous: Application of the design   Contents
Christian Schmid 2005-05-09