1 Phase Lead Compensator Design Using Bode Plots
1
Phase Lead Compensator Design
Using Bode Plots
Prof. Guy Beale
Electrical and Computer Engineering Department
George Mason University
Fairfax, Virginia
C ONTENTS
I
INTRODUCTION
II
DESIGN PROCEDURE
II-A
Compensator Structure . . . . . . . . . . . .
II-B
Outline of the Procedure . . . . . . . . . . .
II-C
Compensator Gain . . . . . . . . . . . . . . .
II-D
Making the Bode Plots . . . . . . . . . . . .
II-E
Uncompensated Phase Margin . . . . . . . .
II-F
Determination of ¦Õmax and ¦Á . . . . . . . . .
II-G
Compensated Gain Crossover Frequency . . .
II-H
Determination of zc and pc . . . . . . . . . .
II-I
Evaluating the Design ¨C A Potential Problem
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
2
4
4
5
5
5
8
8
8
DESIGN EXAMPLE
III-A
Plant and Specifications . . . . . . . . .
III-B
Compensator Gain . . . . . . . . . . . .
III-C
The Bode Plots . . . . . . . . . . . . .
III-D
Uncompensated Phase Margin . . . . .
III-E
Determination of ¦Õmax and ¦Á . . . . . .
III-F
Compensated Gain Crossover Frequency
III-G
Compensator Zero and Pole . . . . . . .
III-H
Evaluating the Design . . . . . . . . . .
III-I
Implementation of the Compensator . .
III-J
Summary . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
10
10
11
11
11
12
12
12
13
14
15
III
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
References
17
L IST OF F IGURES
1
2
3
4
5
6
7
8
Magnitude and phase plots for a typical lead compensator. . . .
Bode plots for the system in Example 2. . . . . . . . . . . . .
Polar plot for phase lead compensator with Kc = 1, ¦Á = 0.16.
Bode plots for the lead-compensated system in Example 8. . .
Bode plots for the plant after the steady-state error specification
Bode plots for the compensated system. . . . . . . . . . . . . .
Closed-loop frequency response magnitudes for the example. .
Step responses for the closed-loop systems. . . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
has been satisfied.
. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3
6
7
10
12
14
15
16
These notes are lecture notes prepared by Prof. Guy Beale for presentation in ECE 421, Classical Systems and Control Theory, in the Electrical and
Computer Engineering Department, George Mason University, Fairfax, VA. Additional notes can be found at: .
2
I. INTRODUCTION
As with phase lag compensation, the purpose of phase lead compensator design in the frequency domain generally is to
satisfy specifications on steady-state accuracy and phase margin. There may also be a specification on gain crossover frequency
or closed-loop bandwidth. A phase margin specification can represent a requirement on relative stability due to pure time delay
in the system, or it can represent desired transient response characteristics that have been translated from the time domain into
the frequency domain.
The overall philosophy in the design procedure presented here is for the compensator to adjust the system¡¯s Bode phase
curve to establish the required phase margin at the existing gain-crossover frequency, ideally without disturbing the system¡¯s
magnitude curve at that frequency and without reducing the zero-frequency magnitude value. The unavoidable shift in the gain
crossover frequency is a function of the amount of phase shift that must be added to satisfy the phase margin requirement. In
order for phase lead compensation to work in this context, the following two characteristics are needed:
? the Bode magnitude curve (after the steady-state accuracy specification has been satisfied) must pass through 0 db in
some acceptable frequency range;
? the uncompensated phase shift at the gain crossover frequency must be more negative than the value needed to satisfy
the phase margin specification (otherwise, no compensation is needed).
If the compensation is to be performed by a single-stage compensator, then the amount that the phase curve needs to be
moved up at the gain crossover frequency in order to satisfy the phase margin specification must be less than 90? , and is
generally restricted to a maximum value in the range 55? ¨C65? . Multiple stages of compensation can be used, following the
same procedure as shown below, and are needed when the amount that the Bode phase curve must be moved up exceeds the
available phase shift for a single stage of compensation. More is said about this later.
The gain crossover frequency and bandwidth for the lead-compensated system will be higher than for the plant (even when
the steady-state error specification is satisfied), so the system will respond more rapidly in the time domain. The faster response
may be an advantage in many applications, but a disadvantage of a wider bandwidth is that more noise and other high frequency
signals (often unwanted) will be passed by the system. A smaller bandwidth also provides more stability robustness when the
system has unmodeled high frequency dynamics, such as the bending modes in aircraft and spacecraft. Thus, there is a trade-off
between having the ability to track rapidly varying reference signals and being able to reject high-frequency disturbances.
The design procedure presented here is basically graphical in nature. All of the measurements needed can be obtained from
accurate Bode plots of the uncompensated system. If data arrays representing the magnitudes and phases of the system at
various frequencies are available, then the procedure can be done numerically, and in many cases automated. The examples
and plots presented in this paper are all done in MATLAB, and the various measurements that are presented in the examples
are obtained from the relevant data arrays.
The primary references for the procedures described in this paper are [1]¨C[3]. Other references that contain similar material
include [4]¨C[11].
II. DESIGN PROCEDURE
A. Compensator Structure
The basic phase lead compensator consists of a gain, one pole, and one zero. Based on the usual electronic implementation
of the compensator [3], the specific structure of the compensator is:
?
?
1 (s + zc )
¡¤
¦Á (s + pc )
(s/zc + 1)
(¦Ó s + 1)
= Kc
= Kc
(s/pc + 1)
(¦Á¦Ó s + 1)
Gc_lead (s) = Kc
(1)
with
zc > 0,
pc > 0,
¦Á,
zc
< 1,
pc
¦Ó=
1
1
=
zc
¦Ápc
(2)
Figure 1 shows the Bode plots of magnitude and phase for a typical lead compensator. The values in this example are Kc = 1,
pc = 2.5, and zc = 0.4, so ¦Á = 0.4/2.5 = 0.16. Changing the gain merely moves the magnitude curve by 20 ? log10 |Kc |. The
major characteristic of the lead compensator is the positive phase shift in the intermediate frequencies. The maximum phase
shift occurs at the frequency ¦Ø = ¦Ø max , which is the geometric mean of zc and pc . The shift in the magnitude curve that is
seen at intermediate and high frequencies is undesired but unavoidable. Proper design of the compensator requires placing the
compensator pole and zero appropriately so that the benefits of the positive phase shift are obtained and the magnitude shift
is accounted for. The following paragraphs show how this can be accomplished.
3
Lead Compensator Magnitude: zc = 0.4, pc = 2.5, Kc = 1
16
14
Magnitude (db)
12
10
8
6
4
2
0
?3
10
?2
10
?1
10
0
1
10
Frequency (r/s)
10
10
2
3
10
Lead Compensator Phase: z = 0.4, p = 2.5, K = 1
c
c
c
50
45
40
Phase (deg)
35
30
25
20
15
10
5
0
?3
10
Fig. 1.
?2
10
?1
10
Magnitude and phase plots for a typical lead compensator.
0
10
Frequency (r/s)
1
10
10
2
3
10
4
B. Outline of the Procedure
The following steps outline the procedure that will be used to design the phase lead compensator to satisfy steady-state error
and phase margin specifications. Each step will be described in detail in the subsequent sections.
1) Determine if the System Type N needs to be increased in order to satisfy the steady-state error specification, and if
necessary, augment the plant with the required number of poles at s = 0. Calculate Kc to satisfy the steady-state error.
2) Make the Bode plots of G(s) = Kc Gp (s)/s(Nreq ?Nsys ) .
3) Design the lead portion of the compensator:
a) determine the amount of phase shift in G(j¦Ø) at the gain crossover frequency and calculate the uncompensated
phase margin P Muncompensated ;
b) calculate the values for ¦Õmax and ¦Á that are required to raise the phase curve to the value needed to satisfy the
phase margin specification;
c) determine the value for the final gain crossover frequency;
d) using the value of ¦Á and the final gain crossover frequency, compute the lead compensator¡¯s zero zc and pole pc .
4) If necessary, choose appropriate resistor and capacitor values to implement the compensator design.
C. Compensator Gain
The first step in the design procedure is to determine the value of the gain Kc . In the procedure that I will present, the gain
is used to satisfy the steady-state error specification. Therefore, the gain can be computed from
Kc =
ess_plant
Kx_required
=
ess_specif ied
Kx_plant
(3)
where ess is the steady-state error for a particular type of input, such as step or ramp, and Kx is the corresponding error
constant of the system. Defining the number of open-loop poles of the system G(s) that are located at s = 0 to be the System
Type N , and restricting the reference input signal to having Laplace transforms of the form R(s) = A/sq , the steady-state
error and error constant are (assuming that the closed-loop system is bounded-input, bounded-output stable)
? N +1?q ?
As
ess = lim N
(4)
s¡ú0 s + Kx
where
¡è
?
Kx = lim sN G(s)
s¡ú0
(5)
For N = 0, the steady-state error for a step input (q = 1) is ess = A/ (1 + Kx ). For N = 0 and q > 1, the steady-state
error is infinitely large. For N > 0, the steady-state error is ess = A/Kx for the input type that has q = N + 1. If q < N + 1,
the steady-state error is 0, and if q > N + 1, the steady-state error is infinite.
The calculation of the gain in (3) assumes that the given system Gp (s) is of the correct Type N to satisfy the steady-state
error specification. If it is not, then the compensator must have one or more poles at s = 0 in order to increase the overall
System Type to the correct value. Once this is recognized, the compensator poles at s = 0 can be included with the plant
Gp (s) during the rest of the design of the lead compensator. The values of Kx in (5) and of Kc in (3) would then be computed
based on Gp (s) being augmented with these additional poles at the origin.
Example 1: As an example, consider the situation where a steady-state error of ess_specif ied = 0.05 is specified when the
reference input is a unit ramp function (q = 2). This requires an error constant Kx_required = 1/0.05 = 20. Assume that the
plant is Gp (s) = 200/ [(s + 4) (s + 5)], which is Type 0. Then the compensator must have one pole at s = 0 in order to
satisfy this specification. When Gp (s) is augmented with this compensator pole at the origin, the error constant of Gp (s)/s
is Kx = 200/ (4 ¡¤ 5) = 10, so the steady-state error for a ramp input is ess_plant = 1/10 = 0.1. Therefore, the compensator
requires a gain having a value of Kc = 0.1/0.05 = 20/10 = 2.
¡§
Once the compensator design is completed, the total compensator will have the transfer function
Gc_lead (s) =
Kc
(N
?Nsys )
req
s
¡¤
(s/zc + 1)
(s/pc + 1)
(6)
where Nreq is the total required number of poles at s = 0 to satisfy the steady-state error specification, and Nsys is the number
of poles at s = 0 in Gp (s). In the above example, Nreq = 2 and Nsys = 1.
5
D. Making the Bode Plots
The next step is to plot the magnitude and phase as a function of frequency ¦Ø for the series combination of the compensator
gain (and any compensator poles at s = 0) and the given system Gp (s). This transfer function will be the one used to
determine the values of the compensator¡¯s pole and zero and to determine if more than one stage of compensation is needed.
The magnitude |G (j¦Ø)| is generally plotted in decibels (db) vs. frequency on a log scale, and the phase ¡ÏG (j¦Ø) is plotted
in degrees vs. frequency on a log scale. At this stage of the design, the system whose frequency response is being plotted is
G(s) =
Kc
(N
?Nsys )
req
s
¡¤ Gp (s)
(7)
If the compensator does not have any poles at the origin, the gain Kc just shifts the plant¡¯s magnitude curve by 20?log10 |Kc |
db at all frequencies. If the compensator does have one or more poles at the origin, the slope of the plant¡¯s magnitude curve
also is changed by ?20 db/decade at all frequencies for each compensator pole at s = 0. In either case, satisfying the steadystate error sets requirements on the zero-frequency portion of the magnitude curve, so the rest of the design procedure will
manipulate the phase curve without changing the magnitude curve at zero frequency. The plant¡¯s phase curve is shifted by
?90? (Nreq ? Nsys ) at all frequencies, so if the plant Gp (s) has the correct System Type, then the compensator does not
change the phase curve at all at this point in the design.
The remainder of the design is to determine (s/zc + 1) / (s/pc + 1). The values of zc and pc will be chosen to satisfy
the phase margin specification. Note that at ¦Ø = 0, the magnitude |(j¦Ø/zc + 1) / (j¦Ø/pc + 1)| = 1 ? 0 db and the
phase ¡Ï (j¦Ø/zc + 1) / (j¦Ø/pc + 1) = 0 degrees. Therefore, the low-frequency parts of the curves just plotted will be unchanged, and the steady-state error specification will remain satisfied. The Bode plots of the complete compensated system
Gc_lead (j¦Ø)Gp (j¦Ø) will be the sum, at each frequency, of the plots made in this step of the procedure and the plots of
(j¦Ø/zc + 1) / (j¦Ø/pc + 1).
E. Uncompensated Phase Margin
Since the purpose of the lead compensator is to move the phase curve upwards in order to satisfy the phase margin
specification, we need to determine how much positive phase shift is required. The first step in this determination is to evaluate
the phase margin of the given system in (7). The uncompensated phase margin is the vertical distance between ?180? and the
phase curve of G (j¦Ø) measured at the gain crossover frequency. The gain crossover frequency is defined to be that frequency
¦Ø x where |G (j¦Ø x )| = 1 in absolute value or |G (j¦Ø x )| = 0 in db. This frequency can easily be found on the graphs made in
the previous step. The uncompensated phase margin is
P Muncompensated = 180? + ¡ÏG (j¦Ø x )
?
(8)
?
If the phase curve is above ?180 at ¦Ø x (less negative than ?180 ), then the phase margin is positive, and if the phase curve
is below ?180? at ¦Ø x , the phase margin is negative. A positive value for the uncompensated phase margin means that the
given system is supplying some of the specified phase margin itself. However, if the uncompensated phase margin is negative,
then the lead compensator will need to provide additional phase shift, since it not only has to satisfy all the phase margin
specification, but must also make up for the deficit in phase margin of the system G(s).
Example 2: Consider the transfer function G(s) = 5/ [s (s + 1) (s + 2) (s + 3)]. This represents the system in (7). (Later,
in Example 6, we will assume that Gp (s) = 2/ [(s + 1) (s + 2) (s + 3)] and that the compensator provides 2.5/s in order to
satisfy the steady-state error specification.) The Bode plots for this system are shown in Fig. 2. The gain crossover frequency
is ¦Ø x = 0.65 r/s. At that frequency, the phase shift of G (j¦Ø) is ¡ÏG (j¦Ø) = ?153.2? . Therefore, the uncompensated phase
margin is P Muncompensated = 180? + (?153.2? ) = 26.8? .
¡§
F. Determination of ¦Õmax and ¦Á
Given the value of the uncompensated phase margin from the previous step, we can now determine the amount of positive
phase shift that the lead compensator must provide. The compensator must move the phase curve of G (j¦Ø) at ¦Ø = ¦Ø x upward
from its current value to the value needed to satisfy the phase margin specification. As with the lag compensator, a safety
factor will be added to this required phase shift. Thus, the amount of phase shift that the lead compensator needs to provide
at ¦Ø = ¦Ø x is
¦Õmax = P Mspecif ied + 10? ? P Muncompensated
(9)
The notation ¦Õmax is used to signify that the phase shift provided at ¦Ø = ¦Ø x is the maximum phase shift produced by the lead
compensator at any frequency. A safety factor of 10? is included in (9). In many applications, that will be enough. However,
there are cases where more phase shift is needed from the compensator in order to satisfy the phase margin specification. This
may require the use of multiple stages of compensation. More will be said about this later in this section, in Section II-I, and
in Section III.
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- state the amplitude period phase shift and
- horizontal and vertical shifts of sine and cosine functions
- g sh zen internet
- binary phase shift keying bpsk lecture notes 6 basic
- ferris wheel applications of trigonometric functions
- phase shift full bridge psfb ac dc power supply basic
- design of phase shifted full bridge converter with current
- 1 phase lead compensator design using bode plots
- phase locked loops pll and frequency synthesis
- calculating gain and phase from transfer functions