Control System Engineering Lecture 17

1. ME 176
Control Systems Engineering
Department of
Mechanical Engineering
Design Via Root Locus

2. Design: Root Locus - PID Controller
Proportional - Plus - Integral - Plus - Derivative (PID) controller : compensation with
two zeros plus a pole at the origin. One zero can be first designed as the derivative
compensator, then other zero and one pole at the origin can be designed as ideal
integrator.
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3. Design: Root Locus - PID Controller
Design Steps:
1. Evaluate performance of uncompensated system to determine how much improvement in
transient response is required.
2. Design PD controller, include zero location and loop gain.
3. Simulate the system shows the requirements have been met.
4. Redesign if the simulations show that requirements have not been met.
5. Design PI controller to yield the required stead state error.
6. Determine the gains K1, K2, and K3.
7. Simulate the system shows the requirements have been met.
8. Redesign if the simulations show that requirements have not been met.
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4. Design: Root Locus - PID Controller
Example:
Design a PID controller so that the
system can operate with a peak time
that is two thirds of uncompensated
system at 20% overshoot with zero steady
state input.
Step 1. Evaluate performance of uncompensated
system to determine how much improvement in
transient response is required:
● Find equivalent damping ration line.
● Find the dominant pole location:
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Mechanical Engineering

5. Design: Root Locus - PID Controller
Example:
Design a PID controller so that the
system can operate with a peak time
that is two thirds of uncompensated
system at 20% overshoot with zero steady
state input.
Step 1. Evaluate performance of uncompensated
system to determine how much improvement in
transient response is required:
● Find uncompensated peak time.
Department of
Mechanical Engineering

6. Design: Root Locus - PID Controller
Example:
Design a PID controller so that the
system can operate with a peak time
that is two thirds of uncompensated
system at 20% overshoot with zero steady
state input.
Step 2. Design PD controller, include zero
location and loop gain:
● Find compensated peak time.
● Find compensated dominant pole:
● Get PD zero location - angle:
Department of
Mechanical Engineering

7. Design: Root Locus - PID Controller
Example:
Design a PID controller so that the
system can operate with a peak time
that is two thirds of uncompensated
system at 20% overshoot with zero steady
state input.
Step 2. Design PD controller, include zero
location and loop gain:
● Get PD zero location - real axis:
● Get resulting Gain:
Department of
Mechanical Engineering

8. Design: Root Locus - PID Controller
Example:
Design a PID controller so that the
system can operate with a peak time
that is two thirds of uncompensated
system at 20% overshoot with zero steady
state input.
Step 3 and 4 validation:
Step 5 : Design PI controller to yield the
required stead state error. Any ideal
integral compensator will work as long
as the zero is placed close to the origin.
- Get dominant pole location of damping
ration line, and get the gain.
Department of
Mechanical Engineering

9. Design: Root Locus - PID Controller
Example:
Design a PID controller so that the
system can operate with a peak time
that is two thirds of uncompensated
system at 20% overshoot with zero steady
state input.
Step 6: Determine the gains K1, K2,
and K3.
Department of
Mechanical Engineering

10. Design: Root Locus - Lag Lead Compensator
Design Steps:
1. Evaluate performance of uncompensated system to determine how much improvement in
transient response is required.
2. Design lead compensator to meet the transient response specifications. The design includes
the zero location, pole location, and loop gain.
3. Simulate the system shows the requirements have been met.
4. Redesign if the simulations show that requirements have not been met.
5. Evaluate the steady state error performance for the lead-compensated system to determine
how much more improvement in steady-state error is required.
6. Design lag compensator to yield the required stead state error.
7. Simulate the system shows the requirements have been met.
8. Redesign if the simulations show that requirements have not been met.
Department of
Mechanical Engineering

11. Design: Root Locus - Lag Lead Compensator
Example:
Design a lag lead compensator so that the system
can operate with a 20% overshoot and a twofold
reduction in settling time. Compensated system
should also have a tenfold improvement in steady-state error for ramp input.
Step 1: Evaluate performance of uncompensated system to determine how much
improvement in transient response is required.
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Mechanical Engineering

12. Design: Root Locus - Lag Lead Compensator
Example:
Design a lag lead compensator so that the system
can operate with a 20% overshoot and a twofold
reduction in settling time. Compensated system
should also have a tenfold improvement in steady-state error for ramp input.
Step 2: Design lead compensator to meet the transient response specifications.
The design includes the zero location, pole location, and loop gain.
● Get location of compensated "dominant pole" - note that settling time is inversely
proportional to pole real part.
imaginary part given by:
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Mechanical Engineering

13. Department of
Mechanical Engineering
Design: Root Locus - Lag Lead Compensator
Example:
Design a lag lead compensator so that the system
can operate with a 20% overshoot and a twofold
reduction in settling time. Compensated system
should also have a tenfold improvement in steady-state error for ramp input.
Step 2: Design lead compensator to meet the transient response specifications.
The design includes the zero location, pole location, and loop gain.
● Design lead compensator: select location of zero to cancel out a pole at -6,
leaving the system with two original poles, simplifying the search for the other
lead pole: Summing angles of all known poles : -164.65 degrees
Summing with multiple of (2k+1) 180 : 180
Angle of remaining pole: -15.35
Pc
= -29.1

14. Design: Root Locus - Feedback Compensation
Characteristics:
- Design is more complicated, but could yield faster response.
- May not require additional amplification, since signal comes from high output.
Procedures:
- Normally involves finding K, K1, and Kf.
- Two appoaches:
- Similar to cascade forward, using closed-loop poles and zeros.
- Sub-system analysis, controlling smaller section of overall system.
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15. Design: Root Locus - Feedback Compensation
Approach 1:
● Basically involves system reduction to that analyzed for cascade configuration.
● Solving for the loop gain, using this to analyze the root locus.
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16. Design: Root Locus - Feedback Compensation
Approach 2:
Instead of modifying the root locus with additional poles and zeros, the loop gain of
a minor loop is used to modify the system poles.
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17. Definition: Physical Realization of Compensation
Active-Circuit Realization
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18. Definition: Physical Realization of Compensation
Passive Circuit Realization
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Mechanical Engineering
The type of realization for a specific tranfer
function is know by looking at the form of the
transfer function.
Transfer function with very small poles with a
greater, but likewise small zero uses the lag
compensation network:
Transfer function with negative zero, and a
larger negative pole uses the lead
compensation:
Transfer functions with 2nd order numerators
and denominators are Lag-lead compensators.

19. Definition: Physical Realization of Compensation
Active-Circuit Realization of Unrestricted :
Lag-Lead Compensator
Passive-Circuit Realization of Unrestricted :
Lag-Lead Compensator
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20. Design: Root Locus - Summary
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21. Design: Root Locus - Summary
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22. Problem Set 1:
Design PI controller to derive the step response error to zero for the unity feedback
system with a damping ration of 0.5 where:
Compare: %OS, Settling Time, and Kp.
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23. Problem Set 2:
Design PD controller to reduce the settling time be a factor of 4 while continuing to
operate the system with 20% overshoot.
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Mechanical Engineering

24. Problem Set 3:
Design a PID controller that will yield a peak time of 1.047s and a damping ratio of
0.8, with zero error for step input.
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25. Problem Set 4:
Design a minor-loop rate feedback compensation to yield a damping ration of 0.7 for
the minor loop dominant poles and a damping ratio of 0.5 for the closed-loop
system's dominant pole.
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26. Problem Set 5:
Identify and realize the following controller with operational amplifier:
(PI Controller)
Identify and realize the following controller with passive networks:
(Lag Compensator)
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Mechanical Engineering