Automatic Control

Higher National Diploma in Mechatronics Technology (Y3)

Dr. Sherif Alhosary

Control Systems

2
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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Closed Loop Designed Using Root Locus

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General Effect of Addition of Poles

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General Effect of Addition of Zeros

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Some Remarks

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 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. 8
Lead and Lag Compensator

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Root Locus design: Basic procedure
1. Translate design specifications into desired positions of dominant poles
2. Sketch RL of uncompensated system to see if desired positions can be achieved
3. 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
4. Evaluate the gain required to put the poles there (magnitude criterion)
5. Evaluate the total system gain so that the 𝑒𝑠𝑠 constants can be determined
6. 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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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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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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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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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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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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Phase-Lead Controller Design
Design Algorithm:
1. Choose a pair of complex conjugate poles in the complex plane that
produces the desired transient response (damping ratio and natural frequency).
2. Find the required phase contribution of a phase-lead controller by using the
corresponding formula.
3. 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
4. Find the controller’s pole by using the corresponding formula.
5. Check that the compensated system has a pair of dominant complex
conjugate closed-loop poles.
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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

17
Phase-Lead Controller Design
Let us locate a zero at -15 , 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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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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Phase-Lead Controller Design
With a zero set at -9 , we have Pc=15.921 The root locus of the
compensated system with a new controller is given in Figure below.

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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.

21
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=41.587 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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Lead Compensation Example

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Lead Comp. Example 2

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Lead Comp. Example 2

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Lead Comp. Example 3
Prop. control, step response

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Lead Comp. Example 2
Lead compensated design

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Lead Comp. Example 2

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Lead Comp. Example 2

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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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Lead Comp. Example 2
Root Locus, new lead comp

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Lead Comp. Example 2
New lead comp.

• Complex conjugate poles still dominate

• Closed-loop zero at -4.47 (which is also an open-loop zero) reduces impact of
closed-loop pole at -5.59
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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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Lead Compensation Example 3

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Lead Compensation Example 3

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Lead Compensation Example 3

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Lead Compensation Example 3

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Lead Compensation Example 3

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Lead Compensator Using Root Locus
Example 4 Consider G p =
10
s ( s + 1)
Design Specifications : P.O  20%; t p  1.0s
To achieve the desired tp, we place the closed - loop poles at s = - 3  j 3.
 = 1/ 2 ; Expect P.O to be 5%; The general formular for the compensator is
K c(s + a)
Gc(s) = ;0 a b
s+b
Gc ( s )G p ( s ) s = −3 j 3 = −180

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Lead Compensator Using Root Locus

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 Lead Compensator Using Root Locus

 −  − 1 −  2 = 180 o ;  −  = 1 +  2 − 180 o = 78.7 o

3
Fix s at - 3;  = 90 - 78.7 = 11.3 ; b = 3 +
o o
= 3 + 15 = 18
o
tan 11.3
K c ( s + 3)  s s + 1 s + 18 
Gc ( s ) = ; Kc =  
 = 7.8
s + 18  10 s + 3  s =3+ j 3
7.8( s + 3)
Gc ( s ) =
s + 18

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Lead Compensator Using Root Locus

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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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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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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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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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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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Lag Compensator Using Root Locus

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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

Basil Hamed 49
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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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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Lag Compensator Using Root Locus

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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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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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Lag Comp. Design via Root Locus
Example 7

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Lag Comp. Design via Root Locus

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Lag Comp. Design via Root Locus

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Lag Comp. Design via Root Locus

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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 -0.104;
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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

Basil Hamed 60
Lag Comp. Design via Root Locus

Basil Hamed 61
Lag Comp. Design via Root Locus
Step response

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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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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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Lead-Lag Compensator Procedures
1. Evaluate the performance of the uncompensated system to determine how
much improvement in transient response is required.
2. Design the lead compensator to meet the transient response specifications.
The design includes the zero location, pole location, and the loop gain.
3. Simulate the system to be sure all requirements have been met.
4. Redesign if the simulation shows 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 the lag compensator to yield the required steady-state error.
7. Simulate the system to be sure all requirements have been met.
8. Redesign if the simulation shows that requirements have not been met.
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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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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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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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Lead-Lag Compensator
So That

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