Digital control system

1. Compensation Techniques
Module 5

2. Compensating Network:
• Compensating Network: A compensating network is one which
makes some adjustments in order to make up for deficiencies in the
system.
• Compensating devices are may be in the form of electrical,
mechanical, hydraulic etc. Most electrical compensator is RC filter.
• The simplest network used for compensator is known as lead, lag
network.
• A compensator is a physical device which may be an electrical
network, mechanical unit, pneumatic, hydraulic or a combination of
various type of devices.

3. • Lead compensator – (to speed up transient response, margin of
stability and improve error constant in a limited way)
• Lag compensator – (to improve error constant or steady-state
behavior – while retaining transient response)
• Lead – Lag compensator – (A combination of the above two i.e. to
improve steady state as well as transient).

4. Compensation techniques
• Root locus approach
• Frequency response approach
• frequency domain responses meet Bode diagram approach.

5. • There are basically two approaches in the frequency-domain design.
• One is the polar plot approach and the other is the Bode diagram
approach.
• When a compensator is added, the polar plot does not retain the
original shape, and, therefore, we need to draw a new polar plot,
which will take time and is thus inconvenient.
• On the other hand, a Bode diagram of the compensator can be simply
added to the original Bode diagram, and thus plotting the complete
Bode diagram is a simple matter.

6. Information Obtainable from Open-Loop
Frequency Response
• The low frequency region (the region far below the gain crossover
frequency) of the locus indicates the steady-state behavior of the
closed-loop system.
• The medium-frequency region (the region near the gain crossover
frequency) of the locus indicates relative stability.
• The high-frequency region (the region far above the gain crossover
frequency) indicates the complexity of the system.

7. Requirements on Open-Loop Frequency
Response.
• We might say that, in many practical cases, compensation is
essentially a compromise between steady-state accuracy and relative
stability.
• To have a high value of the velocity error constant and yet satisfactory
relative stability, we find it necessary to reshape the open-loop
frequency-response curve.
• The gain in the low-frequency region should be large enough, and
near the gain crossover frequency, the slope of the log-magnitude
curve in the Bode diagram should be –20 dB/decade.

9. Basic Characteristics of Lead, Lag, and Lag–
Lead Compensation.
• Lead compensation essentially yields an appreciable improvement in
transient response and a small change in steady-state accuracy.
• It may accentuate high-frequency noise effects.
• Lag compensation, on the other hand, yields an appreciable
improvement in steady-state accuracy at the expense of increasing
the transient-response time.
• Lag compensation will suppress the effects of high-frequency noise
signals.
• Lag–lead compensation combines the characteristics of both lead
compensation and lag compensation.

10. • The use of a lead or lag compensator raises the order of the system
by 1 (unless cancellation occurs between the zero of the compensator
and a pole of the uncompensated open-loop transfer function).
• The use of a lag–lead compensator raises the order of the system by 2
[unless cancellation occurs between zero(s) of the lag–lead
compensator and pole(s) of the uncompensated open-loop transfer
function], which means that the system becomes more complex and
it is more difficult to control the transient-response behavior.

11. LEAD COMPENSATION
Characteristics of Lead Compensators:
• Consider a lead compensator having the following transfer function:
• where α is the attenuation factor of the lead compensator. It has a
zero at s=–1/T
• and a pole at s=–1/(αT). Since 0<α<1, we see that the zero is always
located to the right of the pole in the complex plane.

12. In the lead compensator, The minimum value of α is
usually taken to be about 0.05.
(This means that the maximum phase lead that may be
produced by a lead compensator is about 65°.)

13. the Bode diagram of a lead compensator when Kc=1and α
=0.1.
The corner frequencies for the lead compensator are ω
=1/T and ω =1/(α T)=10/T.
By examining Figure 7–92, we see that ω m is the
geometric mean of the two corner frequencies,

14. Lead Compensation Techniques Based on the
Frequency-Response Approach.

16. • 5. Determine the corner frequencies of the lead compensator as
follows: Zero of lead compensator: ω=1/T
• Pole of lead compensator: ω=1/ αT
• 6. Using the value of K determined in step 1 and that of α determined
in step 4, calculate constant Kc from Kc=K/ αT
• 7. Check the gain margin to be sure it is satisfactory. If not, repeat the
design process by modifying the pole–zero location of the
compensator until a satisfactory result is obtained.

17. Effects of a Lead Compensator:
• 1. Since a Lead compensator adds a dominant zero and a pole, the
damping of a closed loop system is increased.
• 2. The less overshoot, less rise time and less settling time are
obtained due to increase of damping coefficient and hence there is
improvement in the transient response of the closed loop system.
• 3. It improves the phase margin of the closed loop system.
• 4. Bandwidth of the closed loop system is increased and hence the
response is faster.
• 5. The steady state error does not get affected.

18. Limitations:
• 1. Since an additional increase in the gain is required, it results in
larger space, more elements, greater weight and higher cost.
• 2. From a single lead network, the maximum lead angle available is
about 600 . For lead of more than 700 to 900 , a multistage lead
compensator is required.

19. ROOT-LOCUS APPROACH TO CONTROL-
SYSTEMS DESIGN
• In building a control system, we know that proper modification of the
plant dynamics may be a simple way to meet the performance
specifications.
• This, however, may not be possible in many practical situations
because the plant may be fixed and not modifiable.
• In practice, the root-locus plot of a system may indicate that the
desired performance cannot be achieved just by the adjustment of
gain

20. • Then it is necessary to reshape the root loci to meet the performance
specifications.
• The design problems, therefore, become those of improving system
performance by insertion of a compensator.
• Compensation of a control system is reduced to the design of a filter
whose characteristics tend to compensate for the undesirable and
unalterable characteristics of the plant.

21. Design of compensation Circuits by Root-Locus
Method.
• The design by the root-locus method is based on reshaping the root
locus of the system by adding poles and zeros to the system’s open-
loop transfer function and forcing the root loci to pass through
desired closed-loop poles in the s- plane.
• The characteristic of the root-locus design is its being based on the
assumption that the closed-loop system has a pair of dominant
closed-loop poles.
• This means that the effects of zeros and additional poles do not
affect the response characteristics very much.

22. • Once the effects on the root locus of the addition of poles and/or
zeros are fully understood, we can readily determine the locations of
the pole(s) and zero(s) of the compensator that will reshape the root
locus as desired.

23. Series Compensation and Parallel (or
Feedback) Compensation.
• Figures6–33(a) and (b) show compensation schemes commonly used
for feedback control systems.
• Figure 6–33(a) shows the configuration where the compensator Gc(s)
is placed in series with the plant.
• This scheme is called series compensation.

24. • An alternative to series compensation is to feed back the signal(s)
from some element(s) and place a compensator in the resulting inner
feedback path, as shown in Figure 6–33(b).
• Such compensation is called parallel compensation or feedback
compensation.

25. • The choice between series compensation and parallel compensation
depends on the nature of the signals in the system, the power levels at
various points, available components, the designer’s experience,
economic considerations, and so on.
• In general, series compensation may be simpler than parallel
compensation; however, series compensation frequently requires
additional amplifiers to increase the gain and/or to provide isolation.
• (To avoid power dissipation, the series compensator is inserted at the
lowest energy point in the feedforward path.)
• Note that, in general, the number of components required in parallel
compensation will be less than the number of components

26. Commonly Used Compensators.
• If a compensator is needed to meet the performance specifications, the designer
must realize a physical device that has the prescribed transfer function of the
compensator.
• If a sinusoidal input is applied to the input of a network, and the steady-state
output (which is also sinusoidal) has a phase lead, then the network is called a
lead network.
• (The amount of phase lead angle is a function of the input frequency.)
• If the steady-state output has a phase lag, then the network is called a lag
network.
• In a lag–lead network, both phase lag and phase lead occur in the output but in
different frequency regions; phase lag occurs in the low-frequency region and
phase lead occurs in the high-frequency region.
• A compensator having a characteristic of a lead network, lag network, or lag–lead
network is called a lead compensator, lag compensator, or lag–lead compensator.

27. • Frequently used series compensators in control systems are lead, lag,
and lag–lead compensators.
• PID controllers which are frequently used in industrial control
systems.

28. • Effects of the Addition of Poles. The addition of a pole to the open-
loop transfer function has the effect of pulling the root locus to the
right, tending to lower the system’s relative stability and to slow down
the settling of the response.
• Figure 6–34 shows examples of root loci illustrating the effects of the
addition of a pole to a single-pole system and the addition of two
poles to a single-pole system.

29. • Effects of the Addition of Zeros
The addition of a zero to the open-loop transfer function has the effect
of pulling the root locus to the left, tending to make the system more
stable and to speed up the settling of the response.
Figure 6–35 (a) Root-locus plot of a three-pole system; (b), (c), and (d) root-locus plots showing effects of addition of a zero
to the three-pole system.

30. • The root loci for a system that is stable for small gain but unstable for
large gain.
• Figures 6–35(b), (c), and (d) show root-locus plots for the system
when a zero is added to the open-loop transfer function.
• Notice that when a zero is added to the system of Figure 6–35(a), it
becomes stable for all values of gain.

31. LAG COMPENSATION
• Electronic Lag Compensator Using Operational Amplifiers. The
configuration of the electronic lag compensator using operational
amplifiers is the same as that for the lead compensator shown in
Figure 6–36.
• If we choose R2C2>R1C1 in the circuit shown in Figure 6–36, it
becomes a lag compensator.
• Referring to Figure 6–36, the transfer function of the lag
compensator is given by

33. • Design Procedures for Lag Compensation by the Root-Locus Method. The
• procedure for designing lag compensators for the system shown in Figure
6–47 by the root-locus method may be stated as follows (we assume that
the uncompensated system meets the transient-response specifications by
simple gain adjustment; if this is not the case, refer to Section 6–8):
• 1. Draw the root-locus plot for the uncompensated system whose open-
loop transfer function is G(s).
• Based on the transient-response specifications, locate the dominant
closed-loop poles on the root locus.
• 2. Assume the transfer function of the lag compensator to be given by

34. the open-loop transfer function is increased by a factor of b,
where b>1.
3. Evaluate the particular static error constant specified in the problem.
4. Determine the amount of increase in the static error constant necessary to
satisfy the specifications.
5. Determine the pole and zero of the lag compensator that produce the
necessary increase in the particular static error constant without appreciably
altering the original root loci.
6. Draw a new root-locus plot for the compensated system. Locate the desired
dominant closed-loop poles on the root locus. (If the angle contribution of the lag
network is very small—that is, a few degrees—then the original and new root loci are
almost identical.
7. Adjust gain of the compensator from the magnitude condition so that the dominant
closed-loop poles lie at the desired location.
Section 6–7 / Lag Compensation

35. LAG–LEAD COMPENSATION
• Lead compensation basically speeds up the response and increases the
stability of the system.
• Lag compensation improves the steady-state accuracy of the system, but
reduces the speed of the response.
• If improvements in both transient response and steady-state response are
desired, then both a lead compensator and a lag compensator may be used
simultaneously.
• Lag–lead compensation combines the advantages of lag and lead
compensations.
• Since the lag–lead compensator possesses two poles and two zeros, such
a compensation increases the order of the system by 2, unless cancellation
of pole(s) and zero(s) occurs in the compensated system.

36. Electronic Lag–Lead Compensator Using
Operational Amplifiers.