1. Root Loci Analysis (3): Root Locus Approach to Control System Design
Chapter 6
Root Loci Analysis (3):
Root Locus Approach to Control System Design

2. 6-5 Root Locus Approach to Control System Design
The primary objective of this section is to present procedures for the design and compensation of LTI control systems based on root-locus method.
Performance Specifications
Control systems are deigned to perform specific tasks, whose requirements on the systems are usually given by ts, Mp and ess in step response (dominant poles can therefore be determined).

3. System Compensation 
In many practical cases, the adjustment of the gain alone may not provide sufficient alteration of the system behavior to meet the given specifications. In that case, a compensator is necessary.
Think of the region in which the desired closed loop poles are located.

4. 1) Series compensation
Gc(s)
The block diagram shows the configuration where the compensator Gc(s) is placed in series with the plant. This scheme is called series compensation.

5. 2) Feedback compensation
Gc(s)
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 the block diagram. Such a compensation is called feedback compensation.

6. Design based on Root-Locus Method.
The design 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 pass through desired closed-loop poles, usually, dominant poles. The addition of poles and zeros forms a compensator. For instance, 
2+j23
2j23 

7. 4. Commonly used compensators
Among the many kinds of compensators, widely employed compensators are
Lead compensator;
Lag compensator;
Lag-lead compensator;
Velocity-feedback (tachometer) compensator.
The names of lead, lag or lag-lead compensators will become clear in the frequency domain analysis.

8. 5. 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 time of the response.
Conclusion: Adding open loop poles tend to push the root loci to the right of s plane.

9. 6. 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 time of the response.  
Conclusion: Zeros attract root loci, and thus suitably placed zeros can move root loci to the desired region.

10. 6-6 Lead Compensation
1. Mathematical model: Re j 

11. 2. Design Procedure: is introduced via the following example.
from which we obtain that ts=3.5s, Mp=16.3%.

12. Step 1: Performance specifications: to determine the desired locations for the dominant closed-loop poles.
It is required to modify the closed-loop poles so that ts=1.75s, Mp=16.3%, which can be converted into
=0.5 (=600), n=4 rad/s
Hence, the desired closed-loop poles are
s=2j23
Step 2: By drawing the root-locus of the original system to ascertain whether or not the K* alone can yield the desired closed-loop poles.

13. 2+j23
Step 3: Obviously, a lead compensator is needed for reshaping the root loci as desired locations.  
=600 
2j23
Step 4: Determine the location of the zero and pole of the lead compensator:

14. the lead compensator must contribute =300 at s=2j23.
Determine the angle deficiency  so that the total sum of angles is 1800(2k+1). In this example, since 
2+j23
2j23 
900
1200
the lead compensator must
contribute =300 at
s=2j23.
Determine the zero and pole locations of the compensator.

15. 
2+j23
2j23 
Note that there are infinitely many choices as long as the angle between the two vectors is (=300). 
2+j23    
=(=300)

16. o Draw horizontal line PA and the line PO; P (2+j23)
Bisect the angle APO
to obtain line PB; o
P (2+j23) A 
/2
/2  B C D
Draw two lines PC and PD that make angles /2. The intersections of PC and PD with x-axis give the locations of the pole and zero of the compensator.

17. Determine the values of  and T.
In this example, by measuring the intersection points yields
zero at s=2.9;
pole at s=5.4  T=1/2.9=0.345, T=1/5.4=0.185  =0.573.
Step 5: Determine Kc of the compensator.
where K*=4Kc, which can be evaluated by

18. This completes our lead compensator design.
Hence, 
2+j23
2j23 
This completes our lead compensator design.

19. 6-7 Lag Compensation Motivation for introducing lag compensator:j 
6-7 Lag Compensation
Motivation for introducing lag compensator:
If a system exhibits satisfactory transient response characteristics but unsatisfactory steady-state characteristics, a lag compensator can be applied.
Adding a pair of a pole and a zero with a pole on the right of the zero (Lag controllers):} We shall now compare Plot (1) with Plot (8) and Plot (9). In contrast to the lead\vadjust{\vfill} controller case, Plot (8) and Plot (9) are obtained from Plot (1) by adding a pair of a pole and a zero with the pole on the right of the zero. It is clear that the total effect is somewhat similar to adding a pole. Thus, part of the root-locus plot is pushed to the right to a certain degree. It is also important to note that the root-locus plot will not change much (except the local
part near the pair of a pole and a zero) if the pair of a pole and a zero are much closer than other poles and
zeros. This is indeed how a lag or PI controller works. By choosing the pair of a pole and a zero much closer to one another (usually also close to origin), this controller will essentially not change the dominant pole locations. Thus, it will not change the transient response much if it is suitably designed, but will improve steady-state tracking by increasing the system gain with the ratio of zero over the pole. (This is why the pole and zero are sometimes chosen to be close to the origin so that a large ratio can be obtained
For example, the system with two dominant poles has satisfactory transient response but its static velocity error does not meet the requirement.

20. Basic requirements
Do not appreciably change the dominant closed-loop poles of the original system;
The open-loop gain should be increased as much as needed.
This can be accomplished if a lag compensator is put in cascade with the given feedforward transfer function.

21. 1. Mathematical model : Re j 

22. 2. Design Procedure: is introduced via the following example.
Performance specifications:
=0.5 (=600);
2) Steady-state performance specification: Static velocity error ess=0.2 (Kv=5s1).

23. The root loci for K*: 0+ is:
Step 1: Based on the transient-response specifications, locate the dominant closed-loop poles on the root locus by drawing the root-loci of the original system. -1 -2 j 
The root loci for K*: 0+ is:
The closed-loop transfer function is:

24. From =0. 5, draw the line having 600 with the negative real axis
From =0.5, draw the line having 600 with the negative real axis. The dominant closed-loop poles for this example are:
s1,2=0.33j0.59,
and
s3=2.34,
Kv=0.53s1,
K*=1.06. -1 -2 j 

25. Therefore, the original system
satisfies =0.5 (=600) but does not satisfy ess=0.2 (Kv=5s1).
Step 2: Obviously, from Step 1, a lag compensator is needed with the form:

26. Step 3: Evaluate the particular static error constant specified in the problem.
In this example, by requirement, Kv=5s1 (about ten times of the original Kv =0.53s1).
Step 4: Determine the amount of increase in the static error constant necessary to satisfy the specifications.
In this example, to meet Kv=5, we must increase the static velocity error constant by a factor of 10, that is, , which will become clear later.

27. Step 5: Determine the pole and zero of the compensator as shown in the Figure below.80 E o E 
Draw a line from the dominant pole with an angle that intersects with x-axis at E. Then choose a point to the right of E to locate the zero of the compensator: zero=0.05; hence, pole=0.005.

28. Step 6: Draw the new root locus plot for the compensated system and locate the desired dominant closed-loop poles. -1 -2 j
Compensated  

29. In this example, we assume the damping ratio of the new dominant closed-loop poles is kept the same (=0.5 (=600), then the poles obtained from the new root-locus plot are
s1,2=0.31j0.55
Step 7: Adjust gain of the compensator from the magnitude condition so that the dominant closed-loop poles lie at the desired locations.
First, K* can be determined by magnitude condition:

30. That is
Since
we have,
The transfer function of the lag compensator is thus obtained as

31. The compensated system has the following open-loop transfer function:
The static velocity error constant Kv is
which is almost ten times of the original one (Kv=0.53). This completes the design.

32. 6-8 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, usually we use a single lag-lead compensator.
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. Rather than introducing both a lead compensator and a lag compensator as separate elements, however, it is economical to use a single lag-lead compensator. Lag lead compensator design goes beyond our syllabus (teaching program).

33. Summary
Purpose of root locus method: To investigate the closed-loop stability and the system compensator design through the open-loop transfer function with variation of a certain system parameter, commonly, but not limited to, the open-loop gain.

34. 2. Root loci construction rules
The number of root locus branches is equal to the order of the characteristic equation.
The loci are symmetrical about the real axis.
Negative feedback
Positive feedback
Rule 1. The root locus branches start from open-loop poles and end at open-loop zeros or zeros at infinity.
Rule 1. The same.
The table summarizes the differences between negative feedback and positive feedback root loci construction rules.

35. Rule 2. If the total number of real poles and real zeros to the right of a test point on the real axis is odd, then the test point lies on a root locus.
Rule 2. If the total number of real poles and real zeros to the right of a test point on the real axis is even, then the test point lies on a root locus.
Rule 3.

36. Rule 4. Breakaway point:
Rule 4. The same.
Rule 5.

37. Rule 6. Intersection of the root loci with the imaginary axis.
Rule 7. If nm+2, the sum of poles remains unchanged as K* varies form zero to infinity
Rule 7. The same.

38. 3. Compensator Design Base on Root Locus Method
1. Lead Compensation: Re j 
Zero is closer to the imaginary axis implies that the compensator dominates by PD control action

39. which is able to make compensated system more stable and improve system transient performance.
2+j23
2j23 

40. 2. Lag Compensation: Re j 
Pole is closer to the imaginary axis implies that the compensator dominates by PI control action

41. which is able to improve system’s steady state performance while keeping a satisfactory transient response. -1 -2 j
These are the main points of this chapter.