Frequency response lecture

1. LAB9:FREQUENCY RESPONSE METHOD
LAB 10: LEAD-LAG COMPENSATORS
By :
Eng. Mohammed Abdalnasser Alzaghir
773215130
University of Sana’a
Faculty of Engineering
Mechatronics Department

7. Designing Lead and Lag Compensators
Lead and lag compensators are used quite extensively in control. A lead compensator can increase the
stability or speed of reponse of a system; a lag compensator can reduce (but not eliminate) the steady-state
error. Depending on the effect desired, one or more lead and lag compensators may be used in various
combinations.
Lead, lag, and lead/lag compensators are usually designed for a system in transfer function form.
The conversions page explains how to convert a state-space model into transfer function form.

8. Lead or phase-lead compensator using root locus
A first-order lead compensator C(s) can be designed using
the root locus. A lead compensator in root locus form is
given by
where the magnitude of z0 is less than the magnitude of p0. A phase-lead compensator tends to shift
the root locus toward to the left in the complex s-plane. This results in an improvement in the system's
stability and an increase in its response speed.
In MATLAB a phase-lead compensator in root locus form is implemented using the following
commands (where Kc, z, and pare defined).
s = tf('s’);
C_lead = Kc*(s-z)/(s-p);
We can interconnect this compensator C(s) with a
plant P(s) using the following code.
sys_ol = C_lead*P;

9. Lead or phase-lead compensator using frequency response
A first-order phase-lead compensator can also be
designed using a frequency reponse approach. A lead
compensator in frequency response form is given by the
following.
Note that this is equivalent to the root locus form repeated below
with p =1 / T, z = 1 / aT, and Kc = a. In frequency
response design, the phase-lead compensator adds
positive phase to the system over the frequency range 1
/ aT to 1 / T.

10. Lead or phase-lead compensator using frequency response
A Bode plot of a phase-lead compensator C(s) has the following form.
The two corner frequencies are at 1 / aT and 1 / T; note
the positive phase that is added to the system between
these two frequencies. Depending on the value of a, the
maximum added phase can be up to 90 degrees; if you
need more than 90 degrees of phase, two lead
compensators in series can be employed. The maximum
amount of phase is added at the center frequency, which
is calculated according to the following equation.
The equation which determines the maximum phase is
given below.

11. Lead or phase-lead compensator using frequency response
The equation which determines the maximum phase is
given below.
Additional positive phase increases the phase margin and thus increases the stability of the system. This type of
compensator is designed by determining a from the amount of phase needed to satisfy the phase margin
requirements, and determing T to place the added phase at the new gain-crossover frequency.
Another effect of the lead compensator can be seen in the magnitude plot. The lead compensator increases the
gain of the system at high frequencies (the amount of this gain is equal to a). This can increase the crossover
frequency, which will help to decrease the rise time and settling time of the system (but may amplify high
frequency noise).

12. Lead or phase-lead compensator using frequency response
In MATLAB, a phase-lead compensator C(s) in frequency response form is implemented using the following
code (where a and T are defined).
s = tf('s');
C_lead = (1+a*T*s)/(1+T*s);
We can then interconnect it with a plant P(s) using the
following code.
sys_ol = C_lead*P;

13. Lag or phase-lag compensator using root locus
A first-order lag compensator C(s) can be designed using
the root locus. A lag compensator in root locus form is
given by the following.
This has a similar form to a lead compensator, except now the magnitude of z0 is greater than the
magnitude of p0 (and the additional gain Kc is omitted). A phase-lag compensator tends to shift the
root locus to the right in the complex s-plane, which is undesirable. For this reason, the pole and zero
of a lag compensator are often placed close together (usually near the origin) so that they do not
appreciably change the transient response or stability characteristics of the system.
s = tf('s');
C_lag = (s-z)/(s-p);
We can interconnect this compensator C(s) with a
plant P(s) using the following code.
sys_ol = C_lag*P;
In MATLAB, a phase-lag compensator C(s) in root locus form is
implemented by employing the following code where it is again
assumed that z and p are previously defined.

14. Lag or phase-lag compensator using frequency response
A first-order phase-lag compensator also can be designed
using a frequency response approach. A lag compensator
in frequency response form is given by the following.
The phase-lag compensator looks similar to phase-lead compensator, except that a is now less than 1. The
main difference is that the lag compensator adds negative phase to the system over the specified frequency
range, while a lead compensator adds positive phase over the specified frequency

15. Lag or phase-lag compensator using frequency response
A Bode plot of a phase-lead compensator C(s) has the following form.
The two corner frequencies are at 1 / T and 1 / aT. The main effect
of the lag compensator is shown in the magnitude plot. The lag
compensator adds gain at low frequencies; the magnitude of this
gain is equal to a. The effect of this gain is to cause the steady-state
error of the closed-loop system to be decreased by a factor of a.
Because the gain of the lag compensator is unity at middle and high
frequencies, the transient response and stability are generally not
impacted much.

16. Lag or phase-lag compensator using frequency response
A Bode plot of a phase-lead compensator C(s) has the following form.
The side effect of the lag compensator is the negative phase
that is added to the system between the two corner
frequencies. Depending on the value a, up to -90 degrees of
phase can be added. Care must be taken that the phase
margin of the system with lag compensation is still
satisfactory. This is generally achieved by placing the
frequency of maximum phase lag, wm as calculated below,
well below the new gain crossover frequency.

17. In MATLAB, a phase-lead compensator C(s) in frequency response form is implemented using the following
code (where a and T are defined).
s = tf('s');
C_lag = (a*T*s+1)/(a*(T*s+1));
We can then interconnect it with a plant P(s) using the
following code.
sys_ol = C_lag*P;
Lag or phase-lag compensator using frequency response

18. Lead-lag compensator using either root locus or frequency response
A lead-lag compensator combines the effects of a lead compensator with those of a lag compensator. The
result is a system with improved transient response, stability, and steady-state error. To implement a lead-lag
compensator, first design the lead compensator to achieve the desired transient response and stability, and
then design a lag compensator to improve the steady-state response of the lead-compensated system.

19.  https://ctms.engin.umich.edu/CTMS/index.php?example=Introduction&section=ControlFrequency
 https://ctms.engin.umich.edu/CTMS/index.php?aux=Extras_Leadlag
 https://www.mathworks.com/videos/using-bode-plots-lead-lag-pid-controllers-4-of-5-77061.html
 https://www.youtube.com/watch?v=JiCOwEArmvI
REFERENCES

20. ANY QUESTIONS ?!
Eng. Mohammed Abdalnasser Alzaghir
773215130