1. Lect. 5 Lead-Lag Control Basil Hamed
Control Systems
Lect. 5 Lead-Lag Control
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2. Control Systems
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3. Compensators
Early in the course we provided some useful guidelines regarding the relationships between the pole positions of a system and certain aspects of its performance
Using root locus techniques, we have seen how the pole positions of a closed loop can be adjusted by varying a parameter
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4. Closed Loop Designed Using Root Locus
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5. General Effect of Addition of Poles
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6. General Effect of Addition of Zeros
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7. Some Remarks
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8. Lead/Lag Compensation
Lead/Lag compensation is very similar to PD/PI, or PID control.
The lead compensator plays the same role as the PD controller, reshaping the root locus to improve the transient response.
Lag and PI compensation are similar and have the same response: to improve the steady state accuracy of the closed-loop system.
Both PID and lead/lag compensation can be used successfully, and can be combined.

9. Lead and Lag Compensator
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10. Root Locus design: Basic procedure
Translate design specifications into desired positions of dominant poles
Sketch RL of uncompensated system to see if desired positions can be achieved
If not, choose the positions of the pole and zero of the compensator so that the desired positions lie on the root locus (phase criterion), if that is possible
Evaluate the gain required to put the poles there (magnitude criterion)
Evaluate the total system gain so that the 𝑒 𝑠𝑠 constants can be determined
If the steady state error constants are not satisfactory, repeat This procedure enables relatively straightforward design of systems with specifications in terms of rise time, settling time, and overshoot; i.e., the transient response.
For systems with steady-state error specifications, Bode (and Nyquist) methods may be more straightforward
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11. Lead Compensator Using Root Locus
A first-order lead compensator can be designed using the root locus. A lead compensator in root locus form is given by
where the magnitude of z is less than the magnitude of p. A phase-lead compensator tends to shift the root locus toward the left half plane. This results in an improvement in the system's stability and an increase in the response speed.
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12. Lead Compensator Using Root Locus
When a lead compensator is added to a system, the value of this intersection will be a larger negative number than it was before.
The net number of zeros and poles will be the same (one zero and one pole are added), but the added pole is a larger negative number than the added zero.
Thus, the result of a lead compensator is that the asymptotes' intersection is moved further into the left half plane, and the entire root locus will be shifted to the left.
This can increase the region of stability as well as the response speed.
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13. Phase-Lead Controller Design
The phase-lead controller works on the same principle as the PD controller. It uses the argument rule of the root locus method, which indicates the phase shift that needs to be introduced by the phase-lead controller such that the desired dominant poles (having the specified transient response characteristics) belong to the root locus. The general form of this controller is given by
By choosing a point for a dominant pole that has the required transient response specifications. First, find the angle contributed by a controller such that the point belongs the root locus, which can be obtained from
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14. Phase-Lead Controller Design
Second, find locations of controller’s pole and zero. This can be done in many ways as demonstrated in Figure below
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15. Phase-Lead Controller Design
All these controllers introduce the same phase shift and have the same impact on the transient response. However, the impact on the steady state errors is different since it depends on the ratio f Since this ratio for a phase lead controller is less than one, we conclude that the corresponding steady state constant is reduced and the steady state error is increased.
Note that if the location of a phase-lead controller zero is chosen, then simple geometry can be used to find the location of the controller’s pole. For example, let be the required zero, then using Figure above the pole
is obtained using:
An algorithm for the phase-lead controller design can be formulated as follows
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16. Phase-Lead Controller Design
Design Algorithm:
Choose a pair of complex conjugate poles in the complex plane that produces the desired transient response (damping ratio and natural frequency).
Find the required phase contribution of a phase-lead controller by using the corresponding formula.
Choose values for the controller’s pole and zero by placing them arbitrarily such that the controller will not damage the response dominance of a pair of complex conjugate poles. Some authors (e.g. Van de Verte, 1994) suggest placing the controller zero at
Find the controller’s pole by using the corresponding formula.
Check that the compensated system has a pair of dominant complex conjugate closed-loop poles.
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17. Phase-Lead Controller Design
Example 1: Consider the following control system represented by its open-loop transfer function
It is desired that the closed-loop system have a settling time of and a maximum percent overshoot of less than we know that the system operating point should be at A controller’s phase contribution is
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18. Phase-Lead Controller Design
Let us locate a zero at , then the compensator’s pole is at The root loci of the original and compensated systems are given in Figure below, and the corresponding step responses in Figure below.
the original (a) compensated (b) systems
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19. Phase-Lead Controller Design
It can be seen that the root locus indeed passes through the point For this operating point the static gain is obtained as ; hence the steady state constants of the original and compensated systems are given by (POSITION ERROR CONSTANT)
and the steady state errors are
Above Figure reveals that for the compensated system both the maximum percent overshoot and settling time are reduced. However, the steady state unit step error is increased, as previously noted analytically.
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20. Phase-Lead Controller Design
With a zero set at -9 , we have Pc= The root locus of the compensated system with a new controller is given in Figure below.
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21. Phase-Lead Controller Design
It can be seen that this controller also reduces both the overshoot and settling time, while the steady state error is slightly increased.
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22. Phase-Lead Controller Design
We can conclude that both controllers produce similar transient characteristics and similar steady state errors, but the second one is preferred since the smaller value for the static gain of the compensated system has to be used. The eigenvalues of the closed-loop system for k= are given by
which indicates that the response of this system is still dominated by a pair
of complex conjugate poles.
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23. Lead Compensation Example
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24. Lead Comp. Example 2
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25. Lead Comp. Example 2
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26. Lead Comp. Example 3
Prop. control, step response
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27. Lead Comp. Example 2
Lead compensated design
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28. Lead Comp. Example 2
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29. Lead Comp. Example 2
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30. Lead Comp. Example 2
We tried hard, but did not achieve the design specs
Let’s go back and re-examine our choices
Zero position of compensator was chosen via rule of thumb
Can we do better? Yes, but two parameter design becomes trickier.
What were other choices that we made?
We chose desired poles to be of magnitude
We could choose them to be further away (faster transient response)
By how much?
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31. Lead Comp. Example 2
Root Locus, new lead comp
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32. Lead Comp. Example 2 New lead comp.
Complex conjugate poles still dominate
Closed-loop zero at (which is also an open-loop zero) reduces impact of closed-loop pole at -5.59
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33. Lead Comp. Example 2
Note faster settling time than prop. controlled loop, However, the CL zero has increased the overshoot a little
Perhaps we should go back and re-design for in order to better control the overshoot
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34. Lead Compensation Example 3
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35. Lead Compensation Example 3
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36. Lead Compensation Example 3
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37. Lead Compensation Example 3
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38. Lead Compensation Example 3
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39. Lead Compensator Using Root Locus
Example 4
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40. Lead Compensator Using Root Locus
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41. Lead Compensator Using Root Locus
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42. Lead Compensator Using Root Locus
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43. Lead Compensator Using Root Locus
RL approach to phase lead design was reasonably successful in terms of putting dominant poles in desired positions; e.g., in terms of ζ and ωn
We did this by positioning the pole and zero of the lead compensator so as to change the shape of the root locus
However, RL approach does not provide independent control over steady-state error constants (details upcoming)
That said, since lead compensators reduce the DC gain (they resemble differentiators), they are not normally used to control steady-state error.
The goal of our lag compensator design will be to increase the steady-state error constants, without moving the other poles too far
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44. Lag Compensator Using Root Locus
A first-order lag compensator can be designed using the root locus. A lag compensator in root locus form is given by where the magnitude of z is greater than the magnitude of p. A phase-lag compensator tends to shift the root locus to the right, which is undesirable. For this reason, the pole and zero of a lag compensator must be placed close together (usually near the origin) so they do not appreciably change the transient response or stability characteristics of the system.
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45. Phase-Lag Controller
The phase-lag controller belongs to the same class as the PI controller.
The phase-lag controller can be regarded as a generalization of the PI controller.
It introduces a negative phase into the feedback loop, which justifies its name.
It has a zero and pole with the pole being closer to the imaginary axis, that is
The phase-lag controller is used to improve steady state errors.
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46. Lag Compensator Using Root Locus
Design Algorithm : 1. Choose a point that has the desired transient specifications on the root locus branch with dominant system poles. Read from the root locus the value for the static gain K at the chosen point, and determine the corresponding steady state errors. 2. Set both the phase-lag controller’s pole and zero near the origin with the ratio obtained such that the desired steady state error requirement is satisfied. 3. In the case of controller, adjust for the static loop gain, i.e. take a new static gain as
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47. Example 5 : Consider the following open-loop transfer function Let the choice of the static gain k=10 produce a pair of dominant poles on the root locus, which guarantees the desired transient specifications. The corresponding position constant and the steady state unit step error are given by
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48. Lag Compensator Using Root Locus
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49. Lag Compensator Using Root Locus
The lag controller’s impact on the steady state errors can be obtained from the expressions for the corresponding steady state constants. Namely, we know that
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50. Lag Compensator Using Root Locus
If we put this controller in series with the system, the corresponding steady state constants of the compensated system will be given by
Now consider a phase-lag controller, that is
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51. Lag Compensator Using Root Locus
The steady state errors of the system considered above can be improved by using a phase-lag controller of the form
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52. Lag Compensator Using Root Locus
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53. Lag Compensator Using Root Locus
Example 6 Consider the following open-loop transfer function Let the choice of the static gain k=20 produce a pair of dominant poles on the root locus that guarantees the desired transient specifications. The system closed-loop poles for k=20 are given by
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54. Lag Compensator Using Root Locus
The absolute value of the real part of the dominant poles (0.5327) is about six times smaller than the absolute value of the real part of the next pole (2.9194), which is in practice sufficient to guarantee poles’ dominance. Since we have a type one feedback control system, the steady state error due to a unit step is zero. The velocity constant and the steady state unit ramp error are obtained as
It can be shown by using MATLAB that the ramp responses of the original and the compensated systems are very close to each other. The same holds for the root loci. Note that even smaller steady state errors can be obtained if we increase the ratio
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55. Lag Comp. Design via Root Locus
Example 7
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56. Lag Comp. Design via Root Locus
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57. Lag Comp. Design via Root Locus
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58. Lag Comp. Design via Root Locus
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59. Lag Comp. Design via Root Locus
Complex conjugate poles still dominate
Closed-loop zero at -0.1 (which is also an open-loop zero) reduces impact of closed-loop pole at ;
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60. Lag Comp. Design via Root Locus
Step response
Note longer settling time of lag controlled loop, and slight increase in overshoot, due to CL zero
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61. Lag Comp. Design via Root Locus
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62. Lag Comp. Design via Root Locus
Step response
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63. Lag Compensator Using Root Locus
It was previously stated that lag controller should only minimally change the transient response because of its negative effect.
If the phase-lag compensator is not supposed to change the transient response noticeably, what is it good for? The answer is that a phase-lag compensator can improve the system's steady-state response.
It works in the following manner. At high frequencies, the lag controller will have unity gain. At low frequencies, the gain will be z0/p0 which is greater than 1.
This factor z/p will multiply the position, velocity, or acceleration constant (Kp, Kv, or Ka), and the steady-state error will thus decrease by the factor z0/p0.
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64. Lead-Lag Compensator
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 add on a lag compensator to improve the steady-state response
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65. Lead-Lag Compensator Procedures
Evaluate the performance of the uncompensated system to determine how much improvement in transient response is required.
Design the lead compensator to meet the transient response specifications. The design includes the zero location, pole location, and the loop gain.
Simulate the system to be sure all requirements have been met.
Redesign if the simulation shows that requirements have not been met.
Evaluate the steady-state error performance for the lead-compensated system to determine how much more improvement in steady-state error is required.
Design the lag compensator to yield the required steady-state error.
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66. Design Phase-Lag-Lead Controller
1. Check the transient response and steady state characteristics of the original system. 2. Design a phase-lead controller to meet the transient response requirements. 3. Design a phase-lag controller to satisfy the steady state error requirements. 4. Check that the compensated system has the desired specifications.
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67. Lead-Lag Compensator
Example 8: In this example we design a phase-lag-lead controller for a control system, that is
such that both the system transient response and steady state errors are improved. We have seen in Example 1 that a phase-lead controller of the form
improves the transient response to the desired one. Now we add in series with
the phase-lead controller another phase-lag controller, which is in fact a dipole
near the origin. For this example we use the following phase-lag controller
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68. Lead-Lag Compensator so that the compensated system becomes
The corresponding root locus of the compensated system and its closed-loop step response are represented in Figures below. We can see that the addition of the phase-lag controller does not change the transient response. However, the phase-lag controller reduces the steady state error from
to since the position constant is increased to
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69. Lead-Lag Compensator
So That
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70. HW 4
P 9.2, P 9.4, P 9.5, P 9.16, P 9.18, P 9.22, P 9.25 Due NEXT Class
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