Effects of Addition of Poles and Zeros

Effects of Addition of Poles and Zeros

Figure: Effect of Adding Poles

Figure: Effect of Adding Zero

1. Addition of poles pulls the root locus to the right
2. Additional zero pulls the root-locus to the left

Controller Design

A compensator or controller placed in the forward path of a control system will modify the shape of the loci if it contains additional poles and zeros.

In compensator design, hand calculation is cumbersome, and a suitable computer pakage is generally used.

Compensator / Characteristics
PD / One additional zero
PI / One additional zero and a pole at origin
PID / Two additional zeros and a additional pole at origin

Example

A control system has the open-loop transfer function,

A PD compensator of the form is to be introduced in the forward path to achieve the performance specification:

Overshoot less than 5%, settling time less than 2seconds

Determine the values of and to meet the specification.

Original Controller

For

For

corresponds to the controller gain of .

Settling time condition is not met.

PD compensator design

When PD compensator is used, we actually add a zero in the open-loop transfer function. Potential locations of zero include:

1. At , 2. At , 3. At

sec,

Here, the closed-loop pole in the root locus branch between 0 and -1 dominate the time response with oscillatory 2nd-order response superimposed.

sec,

sec,

Summary

Of the three compensators considered, only option 2 meets the performance specifications. The recommended compensator is therefore, .

The time-domain responses for the four conditions are shown in Figure below.

Realization of Compensator using Passive Components

It is not possible to design isolated zero or pole at origin using passive components. In that case a pair of pole and zero is produced. Compensators may be of four types: Lead compensator, Lag compensator, Lag-lead compensator, and feedback compensator. [Cascade compensator]*

A. Lead Compensator

where and .

;

The pole-zero configuration is shown in figure above on the right side. The zero frequency gain is cancelled by an amplifier of gain .

B. Lag Compensator

where, and

The pole-zero configuration is shown in figure (a) above.

C. Lag-lead Compensator

where,

Comparing left and right side we get,

Therefore, .

And, .

The pole-zero configuration is shown in figure below.

Cascade Compensation in Time Domain

Here the design specifications are converted to and of a complex conjugate pairof closed-loop poles based on the assumption that the system will be dominated by these two complex poles and therefore its dynamic behavior can be approximated by that of a second-order system. A compensator is designed so that closed-loop poles other than the dominant poles are located very close to the open-loop zeros or far away from the -axis so that they make negligible contribution to the system dynamics.

Lead Compensation

Let us consider a unity feedback system with a forward path transfer function . Let, be the location of dominant complex closed-loop pole. If the angle condition at that point is not met for uncompensated system, a compensator has to be designed so that the compensated root locus passes through . Let the compensator has transfer function . Applying angle condition we get

For a given there is no unique location for the pole-zero pair.

Procedure for designing lead compensator is as follows:

1. From specification determine .
2. Draw the root-locus plot of uncompensated system and see whether only gain adjustment can yield the desired closed-loop poles. If not, calculate the angle deficiency, . This angle will be contributed by the lead compensator.
3. ;
4. Locate and so that the lead compensator will contribute necessary .
5. Determine of the compensated system from magnitude condition.

If large error constant is required, cascade a lag network.

Example 01 Let, Compensate the system so as to meet the transient response specifications: settling time second. Peak overshoot for step input .

This specifications imply that, and .

The desired dominant roots lie at

The angle contribution required for the lead compensator is,

.

As is large, a double lead network is appropriate. Each section of double lead network will then contribute an angle of at .

Let us now locate compensator zero at Join the compensator zero to and locate the compensator pole by making an angle of . The pole is found to be at -19.8.

The open-loop transfer function of the compensated system thus becomes,

.

Dominance of the closed-loop poles (-1 ± j2) is preserved.

Example 02 .

The desired dominant closed-loop poles are located at, . The angle condition required from the lead compensator pole-zero pair is,

Place a compensator zero close to the pole -1 at s = -1.2. Join the zero to sd and make an angle of 60 to the left of the line. The compensator pole will be found at -4.95. The open-loop transfer function of the transfer function becomes,

.

The gain can be evaluated using magnitude condition at .

=

or,

We can select to any suitable value.

Lag Compensation

A lag compensator is used to improve the steady-state behavior of a system while preserving satisfactory transient response. This compensation scheme is useful where there is satisfactory transient response but unsatisfactory steady-state response. Let us consider a unity feedback system with a forward path transfer function of

.

The desired closed-loop pole location is indicated in figure below. It is required to improve the system error constant to a specified value without impairing its transient response. To accomplish this a lag compensator with pole-zero pair close to each other is required such that it contribute a negligible angle at. Apart from being closed to each other, the pole-zero pair is also located close to origin.

The gain of the uncompensated system at is given by

.

For the compensated system, the system gain at is

.

As , . The error constant of the compensated system is given by

Following the above equation, . Thus,  of the lag compensator is nearly equal to the ratio of the specified error constant to the error constant of the uncompensated system.

Procedure for designing lag compensator is as follows:

1. Draw the root locus plot of uncompensated system.
2. Translate the transient response specifications into a pair of complex dominant roots. Locate these roots in the uncompensated root locus plot.
3. Calculate the gain of the uncompensated system at the dominant root, and also evaluate the error constant.
4. Determine the factor by which the error constant of the uncompensated system should be increased to meet the specified value. Select a larger value.
5. Select zero of the compensator sufficiently close to the origin. As a guide we may construct a line making an angle less than 10 with the -line from .
6. The compensated pole may be located at . The pole-zero pair should contribute an angle  less than 5 at.

Example Consider the system with The system is to be compensated to meet the following specifications: Damping ratio  = 0.5, Settling time ts = 10 sec.,

Velocity error constant .

Using the above data, . Thus, the desired dominant closed-loop poles are required to be located at, .

Now, ; and .

Now, . We take as 10.

Locate on the root locus taking an angle of 6 from sd. From plot, .

Thus,

Now, for the compensated system locate the point that lies on the -line and the compensated root locus using angle criterion. For the compensated system at

Thus, the open-loop transfer function of the compensated system is,

Feedback Compensation

Though cascade compensation is quite satisfactory and economical in most cases, feedback compensation may be warranted due to the following factors:

1. In nonelectrical systems, suitable cascade devices may not be available.
2. Feedback compensation often provides greater stiffness against load disturbances.

The net effect of applying feedback is to apply a zero to the open-loop transfer function, which is the principal of lead compensation.

Example Consider the system with open-loop transfer function,

.

Reducing the minor feedback loop we get, .

The characteristic equation of the system is

It may be rewritten as,

From this equation we have,

The previous equation shows that the net effect of rate feedback is to add a zero at s = -1/.

The root locus plot of the uncompensated system is shown in figure below.

The desired dominant roots from the given specifications are calculated as, -1 ± j1.34. (sd)

The angle contribution of the open-loop poles at this point is -2 x 128 - 8 = -264. Therefore for the point sd on the root locus, the compensating zero should make an angle of The line cuts the real axis at s = -1.1. Thus the open-loop transfer function of the compensated system becomes,

The value of at sdis found as, 17.4. The velocity error constant is given by,

If this is acceptable, then the design is complete; otherwise an amplifier is to be introduced in the forward path outside the minor feedback loop.