1 Compensator Design to Improve Transient Performance Using Root Locus Prof. Guy Beale Electrical and Computer Engineering Department George Mason University Fairfax, Virginia Correspondence concerning this paper should be sent to Prof. Guy Beale, MSN 1G5, Electrical and Computer Engineering Department, George Mason University, 4400 University Drive, Fairfax, VA 22030-4444, USA. Fax: 703-993-1601. Email: [email protected]
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1
Compensator Design to Improve Transient
Performance Using Root Locus
Prof. Guy Beale
Electrical and Computer Engineering Department
George Mason University
Fairfax, Virginia
Correspondence concerning this paper should be sent to Prof. Guy Beale, MSN 1G5, Electrical and Computer Engineering
Department, George Mason University, 4400 University Drive, Fairfax, VA 22030-4444, USA. Fax: 703-993-1601. Email:
2
Contents
I INTRODUCTION 4
II DESIGN PROCEDURE 4
A. Compensator Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
B. Outline of the Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
C. System Type N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
D. Selecting a Dominant Closed-Loop Pole Location . . . . . . . . . . . . . . . . . . . . . . 8
E. Determining the Compensator’s Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 11
1. Plant Phase Shift at s1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2. Compensator Phase Shift at s1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3. Placing the Compensator Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4. Placing the Compensator Pole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5. Determining the Compensator Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6. Compensator Phase Shift Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
7. Simultaneous Placement of Compensator Pole and Zero . . . . . . . . . . . . . . . 21
8. Multi-Stage Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
III Design Example 25
A. Phase Lead Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1. Given System and Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2. Selection of the Dominant Closed-Loop Pole . . . . . . . . . . . . . . . . . . . . . . 26
3. Designing the Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
B. Phase Lag Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1. Given System and Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2. Selection of the Dominant Closed-Loop Pole . . . . . . . . . . . . . . . . . . . . . . 30
3. Designing the Compensator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
References 35
List of Figures
1 Allowable region for s1 in order to satisfy specifications on overshoot, settling time, and
frequency of oscillation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Calculating the phase shift of G(s) at the chosen point s = s1. . . . . . . . . . . . . . . . 13
3
3 One possible solution for locating the compensator zero and pole. . . . . . . . . . . . . . . 15
4 Comparison of closed-loop step responses for 5 compensator designs with s1 = −4 + j5.4575. 19
5 Root locus plot for four compensator designs. . . . . . . . . . . . . . . . . . . . . . . . . . 20
6 Comparison of lag and lead compensation for a particular system. . . . . . . . . . . . . . 22
7 Selecting the compensator zero and pole to maximize α = zc/pc. . . . . . . . . . . . . . . 24
8 Step response of the uncompensated system for the phase lead design example. . . . . . . 26
9 Root locus of the uncompensated system and the desired closed-loop pole s1 = −0.25+j0.488. 2710 Lead compensated root locus and step response for the design example. . . . . . . . . . . 29
11 Comparison of root locus plots with two lag compensator designs. . . . . . . . . . . . . . 33
12 Comparison of closed-loop step responses for two lag compensators. . . . . . . . . . . . . . 34
4
I. INTRODUCTION
The purpose of compensator design using root locus methods generally is to establish a specified point
in the s-plane, s = s1, as a closed-loop pole. The assumption is that time-domain transient specifications,
such as settling time and overshoot, will be satisfied if s1 is a dominant closed-loop pole. In the simplest
case, s1 is already on the root locus of the uncompensated system. The compensator is then just a gain
Kc that is chosen to satisfy the magnitude criterion at the point s1.
More often, the point s1 is not on the uncompensated root locus, so the compensator must add enough
phase shift at the point s1 to satisfy the phase angle criterion so that the compensated root locus does
pass through s1. This is done by choice of the compensator’s poles and zeros. The compensator gain is
then chosen to satisfy the magnitude criterion at the point s1.
In many cases, the speed of response and/or the damping of the uncompensated system must be
increased in order to satisfy the specifications. This requires moving the dominant branches of the root
locus to the left. A phase lead compensator (providing positive phase shift at s1) is used for this purpose.
If the branches need to be moved to the right, a phase lag compensator (providing negative phase shift
at s1) is used. The design techniques are identical for the two types of compensator; the roles of the
compensator’s poles and zeros are just reversed. Because of this similarity in the design methods, phase
lead compensation will be discussed in detail here. An example of each type of compensation will be given
after the general design procedure is described.
Conceptually, the design procedure presented here is graphical in nature. The process of locating the
compensator’s poles and zeros to satisfy the phase requirements can be visualized from the trigonometric
relationships that must be satisfied at the desired dominant closed-loop pole. The computations can be
easily done by calculator. If data arrays representing the numerator and denominator polynomials of
the open-loop system are available, then the procedure can be done using a software package such as
MATLAB, and in many cases it can be automated. The examples and plots presented here are all done
in MATLAB, and the various measurements that are presented in the examples are obtained from the
arrays storing the appropriate variables.
The primary references for the procedures described in this paper are [1] — [3]. Other references that
contain similar material are [4] — [11].
II. DESIGN PROCEDURE
A. Compensator Structure
The basic phase lead or phase lag compensator consists of a gain, one real pole, and one real zero. Based
on the usual electronic implementation of the compensator [3], the circuit for a lead or lag compensator is
the series combination of two inverting operational amplifiers. The first amplifier has an input impedance
that is the parallel combination of resistor R1 and capacitor C1 and a feedback impedance that is the
5
parallel combination of resistor R2 and capacitor C2. The second amplifier has input and feedback resistors
R3 and R4, respectively.
Assuming that the op amps are ideal, the transfer function for this circuit is
Gc(s) =Vout(s)
Vin(s)=
Kc (s− zc)
(s− pc)=
Kx
µs
−zc + 1¶
µs
−pc + 1¶ =
Kx (τs+ 1)
(ατs+ 1)(1)
=R4R2R3R1
· (sR1C1 + 1)(sR2C2 + 1)
=R4R2R3R1
· R1C1R2C2
· (s+ 1/R1C1)(s+ 1/R2C2)
=R4C1R3C2
· (s+ 1/R1C1)(s+ 1/R2C2)
The zero and pole of the compensator are located at s = zc at s = pc, respectively. Therefore, zc and
pc are negative if they are located in the left-half of the s-plane and positive if they are in the right-half
plane. The following relationships1 can be obtained by inspecting Eq. (1).
Kx =R4R2R3R1
, τ = R1C1, ατ = R2C2, α =zcpc=
R2C2R1C1
(2)
zc = −1/R1C1, pc = −1/R2C2, Kc =Kx
α=
R4C1R3C2
The zero zc is to the right of the pole pc for a phase lead compensator, and it is to the left of the pole for
a phase lag compensator. This is true whether the pole and zero are in the left-half plane (the usual case)
or in the right-half plane. Assuming that zc < 0 and pc < 0 (both in the left-half plane), then α < 1 for
a lead compensator, and α > 1 for a lag compensator.
At any point s = s1, the compensator provides a magnitude
|Gc (s1)| = |Kc| · |s1 − zc||s1 − pc| =
|Kc| ·q[Re (s1)− zc]
2 + Im2 (s1)q[Re (s1)− pc]
2 + Im2 (s1)(3)
and a phase angle
∠Gc (s1) = ∠Kc +∠ (s1 − zc)− ∠ (s1 − pc) = ∠Kc + tan−1·
Im (s1)
Re (s1)− zc
¸− tan−1
·Im (s1)
Re (s1)− pc
¸(4)
Only positive values of compensator gain will be discussed in this paper, so with Kc > 0, the magnitude
and phase of the gain are |Kc| = Kc and ∠Kc = 0◦. In applications whereKc < 0, the we have |Kc| = −Kc
and ∠Kc = 180◦.
1In my descriptions of the design of compensators using Bode plots (Phase Lag Compensator Design Using Bode Plots,
Phase Lead Compensator Design Using Bode Plots), a slightly different definition for the compensator transfer function is
used, although it refers to the same electronic circuit shown in [3].
6
B. Outline of the Procedure
The following steps outline the procedure that will be used to design either a lead or a lag compensator
using root locus methods in order to satisfy transient performance specifications, such as settling time
and percent overshoot. Compensator design to satisfy steady-state error requirements is discussed in a
separate paper, Compensator Design to Improve Steady-State Error Using Root Locus.
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. This
should be done at this point in the design, even though numerically satisfying the steady-state
error specification is done later. If the System Type is not taken care of now, the remainder of the
design procedure may be ineffective when the System Type is changed at the end of the design.
2. Choose a point in the s-plane to be the location for a dominant closed-loop pole. The selection of
this point is based on the transient performance specifications and should produce a closed-loop
system that will satisfy those specifications.
3. Design the compensator:
(a) Compute the phase shift of the plant (including any additional poles at s = 0 needed to
satisfy the steady-state error specification) at the chosen point s = s1. If the phase shift is an odd
integer multiple of 180◦ (180◦mod360◦), then the selected point is on the uncompensated system’s
root locus for positive gain. If the phase shift is an even integer multiple of 180◦ (0◦mod360◦),
then the selected point is on the uncompensated system’s root locus for negative gain. In either
case, the only compensation needed for the transient performance specifications is the proper
gain, and the procedure can jump to step 3(e).
(b) Assuming that the selected point s1 is not already on the root locus (for either positive or
negative gain), compute the amount of phase shift that the compensator must provide at s = s1
in order to make that point lie on the root locus. The compensator’s phase shift (usually) will
be the shortest distance from the plant’s phase shift at s1 to an odd integer multiple of 180◦ for
positive gain (the most usual case) or to an even integer multiple of 180◦ for negative gain. It
should be noted that having the compensator provide this amount of phase shift only guarantees
that the point s1 will lie on the compensated root locus; it does not guarantee that the closed-loop
system will be stable when s1 is a closed-loop pole. Additional factors must be taken into account
for that.
(c) Select the location for either the compensator zero zc or pole pc. If the compensator phase
shift at s1 computed in the previous step is positive, phase lead compensation is required. This
places the compensator zero to the right of the pole. If the compensator’s phase shift is negative,
then phase lag compensation is needed, and the zero is to the left of the pole. One of these factors
(zero or pole) can be placed with some freedom; once it is placed, the location of the other factor
7
is fixed.
(d) Compute the horizontal distance from the point s = s1 to the location of the compensator
factor not already placed (pole or zero), and place that factor at that location. This pole/zero
pair in series with the plant makes the root locus pass through the point s1.
(e) Compute the compensator gain Kc. The value of the gain is chosen to satisfy the magnitude
criterion at s1 (|Gc (s1)Gp (s1)| = 1) in order for s1 to be a closed-loop pole.4. If necessary, choose appropriate resistor and capacitor values to implement the compensator
design.
To illustrate the design procedure, the following system model and specifications will be used:
Gp(s) =8
s+ 4(5)
• steady-state error specification for a unit ramp input is ess_specified = 0.1;
• step response settling time specification is Ts−specified = 1 second;
• step response overshoot specification POspecified ≈ 10%.
C. System Type N
The first step in the design of the compensator will be to determine if the plant Gp(s) has the correct
System Type to satisfy the steady-state error specification. Defining the number of open-loop poles of
a system 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)
ess = lims→0
·AsN+1−q
sN +Kx
¸(6)
where
Kx = lims→0
£sNG(s)
¤(7)
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. Therefore, for a system to have a non-zero, finite steady-state error for a specified reference
input, the System Type must satisfy N = Nreq = q − 1. This is the total number of open-loop poles atthe origin needed for the compensated system to satisfy the steady-state error specification. If the System
Type of the plant Gp(s) is Nsys, then the compensator must have (Nreq −Nsys) poles at the origin. These
poles would be included with the plant model during the design of the rest of the compensator, and then
they would be implemented as part of the compensator after the design is complete. The system that will
be evaluated during the rest of the design process will be
G(s) =1
s(Nreq−Nsys)·Gp(s) (8)
8
Example 1:
The plant transfer function in (5) has no poles at the origin, so it has System Type Nsys = 0. The
steady-state error specification is for a unit ramp input which has a Laplace Transform R(s) = 1/s2, so
for this input q = 2. Therefore, in order for the steady-state error specification to be satisfied, the total
number of poles at the origin must be Nreq = q − 1 = 2 − 1 = 1. Thus, the compensator must provideNreq −Nsys = 1− 0 = 1 pole at the origin. The system G(s) corresponding to Eq. (8) for this example
is G(s) = 8/ [s (s+ 4)]. ¨
Note that satisfying the numerical value of the steady-state error is not done at this point in the design.
That will be done as a separate task.
D. Selecting a Dominant Closed-Loop Pole Location
The real engineering design takes place in this step. This is the mapping of the performance specifica-
tions that must be satisfied into the design parameter that influences the rest of the steps in the design.
For other than very simple systems, sound engineering judgement is needed in order for this mapping to
yield a compensator design that actually allows the specifications to be satisfied.
In many cases, the dominant closed-loop poles are chosen to be a complex conjugate pair, so the design
point s = s1 is a complex number with negative real part and positive imaginary part. The starting
point in choosing s1 generally is to make use of the relationships between time-domain characteristics and
closed-loop pole locations that exist for the standard second-order system, shown in (9).
G(s) =ω2n
s (s+ 2ζωn), TCL(s) =
ω2ns2 + 2ζωns+ ω2n
(9)
where ζ is the dimensionless damping ratio, and ωn is the undamped natural frequency (rad/ sec). For
0 < ζ < 1, the closed-loop system is underdamped, the closed-loop poles are complex conjugates, and the
step response exhibits overshoot and damped sinusoidal transient behavior. The closed-loop poles are at
p1, p2 = −ζωn ± jωn
q1− ζ2 (10)
In this case, specifications on the amount of overshoot and the settling time for the step response are
natural specifications. These types of specifications will be used in the design procedure presented in this
paper. A steady-state error specification on the ramp response is also appropriate; that will be discussed
in a separate paper.
Percent overshoot (PO) in the step response of the standard second-order system is only a function of
the damping ratio. The value of overshoot (%) and the value of ζ are related by
PO =he−πζ/
√1−ζ2
i· 100%, ζ =
¯̄ln¡PO100
¢¯̄qπ2 + ln2
¡PO100
¢ (11)
9
Points in the s-plane that correspond to a constant value of ζ lie on a radial line from the origin making
an angle θ with respect to the negative real axis, with θ = cos−1 (ζ). Therefore, a specification of percent
overshoot establishes a fixed relationship between the imaginary part of the dominant closed-loop pole
and the real part of the pole. This relationship is given by tan(θ), which is the slope of the radial line
corresponding to the value of ζ, so that Im [s1] = −Re [s1] · tan£cos−1 (ζ)
¤.
Defining settling time Ts as the time required for the step response to get within and stay within a fixed
percentage of the final value, the settling time for the standard second-order system is a function of the
product of damping ratio and undamped natural frequency. In this paper, the band about the final value
used to define Ts is ±2%, and the settling time is
Ts =4
ζωn(12)
Comparing (12) and (10) shows that settling time is related to the real part of the closed-loop poles, so
a settling time specification corresponds to the dominant closed-loop poles lying on a vertical line in the
s-plane located at
Re [s1] =−4Ts
(13)
If the settling time specification is an upper bound, then the closed-loop poles must lie on or to the left
of the vertical line given in (13).
The damped natural frequency is defined to be ωd = ωnp1− ζ2, which is seen in (10) to be the
imaginary part of the closed-loop poles. This parameter is also the frequency of oscillation in the transient
part of the step response. Therefore, a specification on the frequency of oscillation imposes a restriction
on the imaginary part of the dominant closed-loop poles, which corresponds to a horizontal line in the
s-plane.
If two or more specifications are imposed on the system, then there are multiple constraints on the
location of the dominant closed-loop pole s1. An acceptable location for that pole must satisfy each of the
constraints. Therefore, s1 must lie in the intersection of the regions defined by the various specifications.
Example 2:
The following specifications are to be imposed on a system’s closed-loop step response: (1) 5% ≤ PO ≤25%, (2) 0.5 s ≤ Ts ≤ 2 s, (3) ωd ≤ 12.6 r/s. Assuming that the equations for the standard second-ordersystem will hold for the actual system, then the following constraints are imposed on the location of s1.
From (11), the lower bound of 5% overshoot corresponds to a damping ratio ζ = 0.6901 and an angle
θ = 46.4◦. The point s = s1 must lie on or above this radial line. The upper bound of 25% overshoot
corresponds to a damping ratio ζ = 0.4037 and an angle θ = 66.2◦. The point s = s1 must lie on or below
this radial line. From (13), the lower bound of Ts = 0.5 s corresponds to real part of Re [s1] = −8. Thepoint s1 must lie on or to the right of this vertical line. The upper bound of Ts = 2 s corresponds to real
10
-15 -10 -5 0 5
0
5
10
15
20
Real Axis
Imag
Axi
s
Locating Closed-Loop Poles in the s-plane to Satisfy Specifications
POmin = 5% zetamax = 0.6901
Thetamin = 46.4 deg
POmax = 25%
zetamin = 0.4037
Thetamax = 66.2 deg
Ts-min = 0.5 s
zeta*wn = 8
Ts-max = 2 s
zeta*wn = 2
wd = 12.6 r/s
Fig. 1. Allowable region for s1 in order to satisfy specifications on overshoot, settling time, and frequency of oscillation.
part of Re [s1] = −2. The point s1 must lie on or to the left of this line. The upper bound of ωd = 12.6r/s corresponds to a horizontal line at j12.6. The closed-loop pole must lie on or below this line. The
dominant closed-loop pole s1 must lie in the intersection of these 5 constraint curves. The dotted region
in Figure 1 is the set of acceptable locations for s1 for this example. ¨
Example 3:
In Example 1 the original Gp(s) augmented with one pole at the origin to satisfy the steady-state
error requirement is G(s) = 8/[s(s + 4)]. The transient specifications are Ts−specified = 1 second and
PO ≈ 10%. The damping ratio associated with 10% overshoot is ζ = 0.5912. The corresponding angle
and slope of the radial line from the origin through s1 are θ = 53.76◦ and 1.3644, respectively. From (13),
the real part of s1 is s = −4. Combining these two requirements places the dominant closed-loop pole at
11
s1 = −4 + j5.4575. ¨
The relationships between dominant pole locations and step response characteristics presented above
are only for the standard second-order system of (9). These relationships may provide good starting points
for the selection of s1 to satisfy specifications, but they can only be used as general guidelines for more
complex system models. Even if the given system Gp(s) is modeled by (9), any compensation in series
with Gp(s), other than a pure gain, will result in a more complex model. The overshoot equation (11)
generally is not a good prediction of what the actual overshoot will be in a higher-order system or in a
system with zeros. The actual overshoot may be less than predicted by (11), but it will more likely be
larger. In choosing a value for s1 to satisfy a percent overshoot specification, the recommended approach
is to be very conservative — that is, use a significantly smaller value of overshoot than the specified value
when computing the effective damping ratio from (11).
The prediction of the real part of s1 computed from (13) to satisfy a settling time specification is
generally fairly accurate. This is due to the fact that the decay of the transient response is controlled
by the real part of the dominant closed-loop poles. This is how the expression in (13) was derived and
accounts for its accuracy in predicting the settling time of higher-order systems.
In many actual design problems with specifications on overshoot, settling time, frequency of oscillation,
etc., the proper choice of location for the dominant closed-loop pole (or complex conjugate pair) will be
achieved only through iteration. Engineering design is an iterative process, and the selection of s1 is a
major step in the design of the compensator that will allow the specifications to be satisfied.
E. Determining the Compensator’s Parameters
Once the point s = s1 is chosen to be a dominant closed-loop pole, the design of the compensator is
an exercise in trigonometry. The pole(s) and zero(s) of the compensator are chosen to satisfy the phase
angle criterion at s1 so that the total phase shift of the series combination of the plant and compensator
at s1 is an odd integer multiple of 180◦ if K > 0 or an even integer multiple of 180◦ if K < 0. This
makes the root locus pass through the point s1. Once the pole(s) and zero(s) are placed, the gain of the
compensator is selected to satisfy the magnitude criterion. This makes the point s = s1 be a closed-loop
pole. However, as previously mentioned, making s = s1 be a closed-loop pole does not guarantee that
the closed-loop system is stable. There may still be closed-loop poles in the right-half plane. Thus, there
is more to choosing the locations of the compensator’s pole(s) and zero(s) than just satisfying the phase
angle criterion.
12
E.1 Plant Phase Shift at s1
The starting point in the compensator design is to determine the phase shift of the augmented plant of
Eq. (8) at the point s1. The phase shift of G(s) at s1 is
∠G (s1) = ∠K +∠N (s1)− ∠D (s1) (14)
= ∠K +mXi=1
∠ (s1 − zi)−nXi=1
∠ (s1 − pi)
= ∠K +mXi=1
tan−1·Im (s1)− Im (zi)Re (s1)−Re (zi)
¸−
nXi=1
tan−1·Im (s1)− Im (pi)Re (s1)−Re (pi)
¸where the zi and pi are the open-loop zeros and poles, respectively. We will assume that K > 0, so
∠K = 0.
Care must be used when evaluating the tan−1 function when using the normal atan function in MATLAB
or on calculators. If the denominator of the tan−1 function is negative, the atan function will return the
incorrect angle. To obtain the correct angle in this case, 180◦ (π rad) must be added to the angle returned
by the atan function. The function atan2 returns the correct value since it takes two input arguments;
its syntax is θ = atan2 (Im,Re).
Example 4:
From Example 1, the augmented plant model is G(s) = 8/ [s (s+ 4)], so there are no zeros zi and
the open-loop poles are p1 = 0 and p2 = −4. We will assume that the desired dominant closed-looppole is at s = s1 = −4 + j5.4575, based on Example 3. The phase angle of G(s) at s1 is: ∠G (s1) =∠8 − ∠s1 − ∠ (s1 + 4) = 0 − tan−1 [(5.4575− 0) / (−4− 0)] − tan−1 [(5.4575− 0) / (−4 + 4)]. ∠G (s1) =− tan−1 [5.4575/ (−4)]− tan−1 (5.4575/0) = − (−53.76◦ + 180◦)−90◦ = −216.24◦ = 143.76◦. Both formsfor the phase shift, ∠G (s1) = −216.24◦ or ∠G (s1) = 143.76◦, are acceptable ways to represent the angle.Figure 2 shows the relationships between the open-loop poles and the point s1. The angles that are
computed are the angles of the vectors drawn from the poles to the point s1 measured counter-clockwise
from the positive real axis. ¨
Note the 180◦ added to the phase shift returned by the atan function in the above example for the pole
p1. This will be the case when the point s1 is to the left of the pole or zero under consideration.
E.2 Compensator Phase Shift at s1
If the point s1 is on the root locus of the uncompensated system G(s) (which is Gp(s) augmented with
any poles at the origin that were needed to produce the correct System Type), then ∠G (s1) is an oddinteger multiple of 180◦ (for K > 0). In this case ∠Gc (s1) = 0
◦, and the only (additional) compensation
that is needed in order to make s = s1 a closed-loop pole is a gain Kc. That would be chosen to satisfy
the magnitude criterion |KcG (s1)| = 1.
13
-10 -8 -6 -4 -2 0 2-6
-4
-2
0
2
4
6
Real Axis
Imag
Axi
s
G(s) = 8/[s(s+4)], s 1 = -4 + j5.4575
s1
p1p2
Fig. 2. Calculating the phase shift of G(s) at the chosen point s = s1.
Most often, s1 will not be on the original root locus, so the compensator must also provide a phase shift
at s1 so that the total phase shift of the plant—compensator combination is an odd integer multiple of
180◦ at that point. This is done by choice of the compensator’s pole and zero. Satisfying the phase angle
criterion must be done first. After that, the value of Kc is computed to satisfy the magnitude criterion.
Given the value ∠G (s1) from the previous step, the required compensator angle ∠Gc (s1) can be easily
computed. Generally, the compensator phase shift at s1 is chosen to be the angle with the smallest
absolute value such that ∠G (s1) + ∠Gc (s1) is an odd integer multiple of 180◦. Thus, if ∠G (s1) = 150◦
or ∠G (s1) = −210◦, then ∠Gc (s1) = +30◦ will place s1 on the root locus, and the compensator is phase
lead. Likewise if ∠G (s1) = −150◦ or ∠G (s1) = 210◦, then ∠Gc (s1) = −30◦ will place s1 on the rootlocus, and the compensator is phase lag. However, in some cases it might be necessary (or desirable) to
14
use a particular type of compensator, such as phase lead, in order to satisfy other objectives. In a situation
like this, if ∠G (s1) = −60◦ for example, then the compensator phase shift at s1 would be chosen to be180◦ − (−60◦) = 240◦ (phase lead) rather than −180◦ − (−60◦) = −120◦ (phase lag). A more detailed
example illustrating this situation is given in Section II-E.6.
The necessary compensator phase shift to make s1 lie on the root locus is given by
∠Gc (s1) = 180◦ (2l + 1)− ∠G (s1) (15)
= ∠Kc +∠Nc (s1)− ∠Dc (s1)
= ∠Kc +
mcXi=1
∠ (s1 − zci)−ncXi=1
∠ (s1 − pci)
Assuming that Kc > 0, then ∠Gc (s1) = ∠Nc (s1)− ∠Dc (s1).
Example 5:
From Example 4, the phase shift of G(s) at s1 = −4 + j5.4575 is ∠G (s1) = −216.24◦ = 143.76◦. Thephase angle of the compensator needed to place s = s1 on the root locus (using the smallest absolute value
for the compensator phase angle) is ∠Gc (s1) = −180◦− (−216.24◦) = 180◦−143.76◦ = 36.24◦. Since thisangle is positive, the compensator is phase lead. The pole and zero of the compensator will be placed on
the real axis such that ∠ (s1 − zc)− ∠ (s1 − pc) = 36.24◦. Methods of choosing appropriate locations are
presented in the next section. One possible solution is to place the compensator zero at s = −5, whichprovides 79.62◦ of phase shift at s1, and to place the compensator pole at s = −9.78, which provides43.38◦ at s1. The difference between those angles is the required value of 36.24◦. Figure 3 illustrates this
solution. ¨
E.3 Placing the Compensator Zero
There is some flexibility in choosing the location of the compensator zero, but not complete freedom.
Assuming that a single stage of compensation is desired, the left-most location of the zero is governed
by the compensator’s phase shift requirement at s1. Because ∠Gc (s1) = ∠ (s1 − zc) − ∠ (s1 − pc), the
compensator zero must provide more phase shift at s1 than the total compensator. Therefore, the left-most
allowed position of the compensator zero is constrained by ∠Gc (s1).
The right-most location of the zero is generally constrained by transient performance specifications,
and ultimately by the requirement for closed-loop stability. Moving the compensator zero to the right
also moves a closed-loop pole to the right. For some choices for the location of the zero, there may be a
closed-loop pole closer to the jω axis than the selected point s1. When this happens, s1 may no longer
be the dominant pole in terms of settling time. If the compensator zero is in the right-half plane, then
the closed-loop system may be unstable.
15
-10 -8 -6 -4 -2 0 2-6
-4
-2
0
2
4
6
Real Axis
Imag
Axi
s
Providing 36.24 degrees at s 1 = -4 + j5.4575
s1
zcpc
Fig. 3. One possible solution for locating the compensator zero and pole.
Unfortunately, there is not a rule for placing the compensator zero. The amount of overshoot can vary
widely depending on its location. For systems with one open-loop pole at the origin and with one or more
additional poles on the negative real axis, a rule of thumb is to place the compensator zero at or to the
left of the second real-axis open-loop pole. This rule of thumb is intended to ensure that s = s1 is the
dominant closed-loop pole, but it does not guarantee that overshoot specifications will be satisfied.
Example 6:
For the system G(s) defined in the previous examples and the point s1 = −4 + j5.4575, the required
phase shift of the compensator was found to be ∠Gc (s1) = 36.24◦. Thus, the compensator zero must
provide more than 36.24◦ at s1. In order to use a single stage of compensation, the left-most point for
the zero is s = Re [s1] − [Im [s1] / tan (36.4◦)] = −11.45. The right-most point for the compensator zero
16
is just to the left of the origin. The zero cannot be placed at s = 0 since that would cancel the open-loop
pole there, and internal stability would be lost. Placing the compensator zero in the right-half plane
would also result in an unstable closed-loop system. Some possibilities for the location of the zero are
zc = −9 [∠ (s1 − zc) = 47.51◦], zc = −7 [∠ (s1 − zc) = 61.20
◦], zc = −5 [∠ (s1 − zc) = 79.62◦], zc = −3
[∠ (s1 − zc) = 100.4◦], zc = −1 [∠ (s1 − zc) = 118.8
◦]. The effects of these different choices for zc will be
shown in the following examples. ¨
The first three locations for the zero in the above example (zc = −9, zc = −7, zc = −5) satisfy therule of thumb mentioned earlier. For these choices, the closed-loop poles at s = s1 and its complex
conjugate will be closer to the jω axis than the third pole. Therefore, the settling time of the step
response will be governed by the choice of s1, as desired. The remaining two choices for the compensator
zero (zc = −3, zc = −1) do not satisfy the rule of thumb. The poles at s1 will no longer be closest tothe jω axis, and so will not be controlling the settling time. This does not mean that the settling time
specification will be violated if the rule of thumb is not satisfied, but as the zero is moved farther and
farther to the right, at some point, the settling time specification will be violated.
E.4 Placing the Compensator Pole
Once the location for the compensator zero has been selected, there is no more freedom in the design;
the compensator pole and gain are now fixed. The location of the pole is constrained by the phase angle
requirement on the compensator at the point s1, and once the pole location is determined, the gain is
constrained by the magnitude criterion at s1.
Since the total phase shift of the compensator at s = s1 is the angle produced by the zero minus the
angle produced by the pole, the phase shift of the compensator pole at s1 is
∠ (s1 − pc) = ∠ (s1 − zc)− ∠Gc (s1) (16)
The value of this angle can be computed once the location of the zero is chosen. There is only one location
for the compensator pole that will produce the angle ∠ (s1 − pc) at s = s1. The distance dpc from the
real-axis projection of s1 to the compensator pole can be computed easily from
dpc =Im [s1]
tan [∠ (s1 − pc)](17)
The compensator pole is located to the left of s1 if dpc > 0 and to the right of s1 if dpc < 0. The
location of the compensator pole at s = pc is given by
pc = Re [s1]− dpc (18)
17
Example 7:
From Example 5, the compensator must provide 36.24◦ of phase shift at s1. In Example 6, five possible
choices for the compensator zero were presented, namely, zc = {−9,−7,−5,−3,−1}. For each of thesechoices, the compensator pole can be located by using Eqs. (16) — (18). The angles, distances, and
locations of the compensator pole are shown in the following table for the various choices of compensator
zero.
zc ∠ (s1 − pc) dpc pc
−9 11.27◦ 27.4 −31.4−7 24.96◦ 11.7 −15.7−5 43.38◦ 5.78 −9.78−3 64.14◦ 2.64 −6.64−1 82.56◦ 0.713 −4.713
The table shows that as the compensator zero moves to the right, the required phase shift of the pole
increases, and the pole also moves to the right. ¨
E.5 Determining the Compensator Gain
Now that the pole and zero of the compensator have been selected, the root locus plot will pass through
the point s = s1. Therefore, s1 is a potential closed-loop pole. In order for that point to actually be a
closed-loop pole, the magnitude criterion must be satisfied at s1 by the series combination of plant and
compensator. The compensator gain is used for this purpose.
The magnitude criterion states that
|Kc| · |s1 − zc||s1 − pc| · |G (s1)| = 1 (19)
if s = s1 is a closed-loop pole, with G(s) defined in (8). The compensator gain is the only parameter that
is undetermined at this point. Therefore, to make s = s1 be a closed-loop pole, the compensator gain
must be
|Kc| = |s1 − pc||s1 − zc| · |G (s1)| (20)
Note that (20) only provides the magnitude (absolute value) of the compensator gain. If the system
requires that the gain be negative, then Kc = − |Kc|.
Example 8:
For each of the combinations of compensator zero and pole from Examples 6 and 7, there is a unique
value for the gain Kc. The value of gain that places one of the closed-loop poles at s1 is computed from
(20). The magnitude of the augmented plant at s1 is |G (s1)| = 8/ (|s1| · |s1 + 4|) = 0.2166. Using the
18
values for zc and pc from the table in Example 7, the corresponding gain values are given in the following
table.
zc |s1 − pc| |s1 − zc| Kc
−9 27.94 7.402 17.42
−7 12.93 6.228 9.585
−5 7.946 5.548 6.611
−3 6.065 5.548 5.045
−1 5.504 6.228 4.080
For each of these combinations, the compensator transfer function, including the pole at s = 0 needed
for the steady-state error specification, is Gc(s) = Kc (s− zc) / [s (s− pc)]. With the values used here,
the closed-loop system is stable, and two of the closed-loop poles are at the specified location of s1 =
−4+ j5.4575 and its complex conjugate. The location of the third closed-loop pole varies with the choice
of compensator parameters. For the values used in these examples, the third pole is at s = {−27.4,−11.7, −5.78, −2.64, −0.713}. Note that for the particular structure of Gc(s)Gp(s) in these examples,
the absolute value of the third closed-loop pole is equal to the distance between the real-axis projection
of s1 and the compensator’s open-loop pole. The closed-loop step responses for these five compensator
designs are shown in Fig. 4. The settling times for the first four compensator designs are all approximately
the same, slightly less than 1 second. The settling time for the fifth compensator design is significantly
longer and does not satisfy the specifications. Of the compensator designs that do satisfy the settling time
requirement, only the fourth design (zc = −3) also satisfies the overshoot specification. The dashed linesin the figure show the ±2% interval for the settling time specification and the 10% overshoot specification.Figure 5 shows the root locus plots for four of the compensator designs. The triangles shown in the plots
represent the closed-loop poles. The point s1 is a closed-loop pole in each case. It is easily seen in the
figure that the third closed-loop pole moves to the right as zc does, and when zc = −1 the third pole isfar to the right of s1. This causes the much longer settling time evident in Fig. 4. ¨
These examples have illustrated the procedure for mapping a set of transient response specifications
into a desired location s = s1 for the dominant closed-loop pole and for designing a compensator to
make that desired location actually be a closed-loop pole. The examples have shown that not every
compensator design that places a closed-loop pole at s1 will satisfy all of the specifications. Closed-loop
stability is not even guaranteed if the compensated system is third-order or higher. Therefore, the design
of the compensator will generally be an iterative process. Although there are general guidelines that
can be followed, a design needs to be validated through simulation to determine whether or not it is an
acceptable design.
19
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
1.2
1.4
0.98
1.02
1.10
Time (s)
Ampl
itude
Closed-Loop Step Responses
1 2
3
4
5Curve 1: z c = -9Curve 2: z c = -7Curve 3: z c = -5
Curve 4: z c = -3
Curve 5: z c = -1
Fig. 4. Comparison of closed-loop step responses for 5 compensator designs with s1 = −4 + j5.4575.
E.6 Compensator Phase Shift Revisited
It was mentioned in Section II-E.2 that the compensator phase shift at s1 is generally taken to be
the smallest absolute value that will yield an odd integer multiple of 180◦ for the plant—compensator
combination. It was also mentioned that there are instances when such a choice would be inappropriate.
The following example illustrates when closed-loop stability requirements force a change to this general
procedure.
Example 9:
Consider the system G(s) = 5(s+2)/[s2(s+0.4)(s+6)] and the desired closed-loop pole s1 = −1+j0.8.
The phase shift of G(s) at s1 is ∠G (s1) = −19.98◦. A compensator phase ∠Gc1 (s1) = −160.02◦ wouldplace s1 on the root locus for Kc > 0. This would require a phase lag compensator. On the other hand,
a compensator phase ∠Gc2 (s1) = 199.98◦ would also place s1 on the root locus for Kc > 0. This would
20
-20 -10 0 10-20
-15
-10
-5
0
5
10
15
20
Real Axis
Imag
Axi
szc = -7
-10 -5 0 5-15
-10
-5
0
5
10
15
Real Axis
Imag
Axi
s
zc = -5
-8 -6 -4 -2 0 2-10
-8
-6
-4
-2
0
2
4
6
8
10
Real Axis
Imag
Axi
s
zc = -3
-5 -4 -3 -2 -1 0 1-8
-6
-4
-2
0
2
4
6
8
Real Axis
Imag
Axi
s
zc = -1
Fig. 5. Root locus plot for four compensator designs.
21
require a phase lead compensator. Which one should be used? In answering this question, it should be
remembered that open-loop poles tend to repel branches of the root locus, and open-loop zeros tend to
attract them. The system G(s) has two open-loop poles at the origin, the boundary of stability. Using a
phase lag compensator, with the pole to the right of the zero (closer to the jω axis assuming they are in the
left-half plane) would tend to move the branches of the root locus that begin at s = 0 into the right-half
plane, resulting in an unstable closed-loop system. A lead compensator, on the other hand, would tend
to draw those branches into the left-half plane. Figure 6 illustrates this. The top two plots are the root
loci for the system with phase lag and phase lead compensation. The compensator transfer functions are
Gc−lag(s) = 0.39 (s+ 5.54)2/ (s+ 1)
2 and Gc−lead(s) = 71.9 (s+ 0.75)2/ (s+ 7.19)
2. It is clearly seen
that the lag compensator produces an unstable system for all K > 0. The lead compensator is able to
stabilize the system. It should be noted that s = s1 is a closed-loop pole in both designs. The bottom two
plots are the closed-loop step responses. The instability of the design with the lag compensator is clearly
indicated. ¨
E.7 Simultaneous Placement of Compensator Pole and Zero
As we discussed in the previous sections, there is some freedom in placing the compensator zero and
pole, as long as the total phase angle of the compensator has the correct value. That freedom can be
utilized to vary the amount of overshoot or the value of the settling time. Unfortunately, there is generally
not a clear decision procedure for selecting the location of the pole or zero.
There is a procedure that places both the pole and zero at the same time. There is no freedom of
choice with this procedure, but it does have the advantage of maximizing the value of α = zc/pc for a
lead compensator or minimizing α for a lag compensator. This is advantageous because it minimizes the
range of resistor and capacitor values used to implement the compensator.
We will define the phase shift of the desired closed-loop pole at s1 to be the angle of the radial line
drawn from the origin to s1, measured in the counter-clockwise direction from the positive real axis, and
denote it by ∠s1. With that definition, the phase shifts of the compensator zero and pole at s1 are
∠ (s1 − zc) =∠s1 +∠Gc (s1)
2, ∠ (s1 − pc) =
∠s1 − ∠Gc (s1)
2(21)
Therefore, the distances from the real-axis projection of s1 to the zero and pole are
dzc =Im [s1]
tan [∠ (s1 − zc)], dpc =
Im [s1]
tan [∠ (s1 − pc)](22)
and the compensator zero and pole are located at
zc = Re [s1]− dzc , pc = Re [s1]− dpc (23)
22
-8 -6 -4 -2 0 2-4
-3
-2
-1
0
1
2
3
4
Real Axis
Imag
Axi
sLag Compensated
s1
-10 -5 0-10
-8
-6
-4
-2
0
2
4
6
8
10
Real Axis
Imag
Axi
s
Lead Compensated
s1
0 2 4 6 8 10-60
-50
-40
-30
-20
-10
0
10
Time (s)
Ampl
itude
Lag Compensated
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Ampl
itude
Lead Compensated
Fig. 6. Comparison of lag and lead compensation for a particular system.
23
Examination of (21) shows that ∠ (s1 − zc) − ∠ (s1 − pc) = ∠Gc (s1) as it must. Since ∠Gc (s1) is
computed to provide the proper phase shift at s1 to make the root locus pass through that point, the
compensator parameters in (23) provide a valid solution to the design problem. This solution is not
necessarily any better than any other solution other than the fact that it optimizes the ratio zc/pc.
Example 10:
From Example 5, the compensator must provide 36.24◦ at the point s1 = −4+j5.4575. The phase shift
of s1 itself is ∠s1 = 126.24◦. From (21), the angles of the compensator zero and pole are 81.24◦ and 45◦,
respectively. The horizontal distances from s1 to the zero and pole are dzc = 0.8411 and dpc = 5.4575.
This places the zero zc at s = −4.8411 and the pole pc at s = −9.4575. The value of α for this pole—zerocombination is α = zc/pc = 0.5119. Figure 7 shows the geometry of this solution. ¨
E.8 Multi-Stage Compensation
In the examples and discussion thus far, it has been assumed the compensator had one pole and one
zero. In some cases it may be desirable or necessary to use multiple stages of compensation. If the required
compensator phase shift ∠Gc (s1) is very large, then the left-most allowed location for the compensator
zero may be close to the jω axis or even in the right-half plane. Moving the zero to the left of this
point means that one zero will not provide enough positive phase shift, so that two or more zeros will be
required.
Even with moderate values of ∠Gc (s1) it may be desirable to use multiple stages of compensation. This
would allow the zero and pole to be moved farther to the left, away from the dominant pole location. In
this way, the equations that apply to second-order systems might be more applicable to the higher-order
system.
The easiest way to design a multi-stage compensator is to assume that each stage provides the same
amount of phase shift at s1. Since the phase shift of a product of complex numbers is the sum of the
individual phase shifts, using this approach means that the phase shift of each stage of the compensator is
the total compensator phase shift divided by the number of stages. The most usual case would probably
be two stages; therefore, each stage would provide one-half of the total required phase shift.
Once the total phase shift is divided by the number of stages, the design of each stage follows the
procedure discussed in the previous sections. The location of the compensator zero is selected to provide
a phase shift larger than the phase shift required by an individual stage. The compensator pole is
computed to provide the correct phase shift for an individual stage. The total compensator numerator
then has the formQnum_stages
i−1 (s− zci), and similarly for the compensator denominator. The gain Kc is
then computed to satisfy the magnitude criterion. In implementation, the gain can be assigned equally
to each stage by letting Kc−stage = num_ stages√Kc.
24
-10 -8 -6 -4 -2 0 2-4
-2
0
2
4
6
8
s1
zcpc
Real Axis
Imag
Axi
s
Alternative Placement of Compensator Pole and Zero
Fig. 7. Selecting the compensator zero and pole to maximize α = zc/pc.
Example 11:
The compensator phase shift computed in the previous examples is ∠Gc (s1) = 36.24◦. Although this
compensator can be designed with a single stage of compensation, it was shown in Example 8 that there
was a restricted range of locations for the compensator zero that allowed both the overshoot and settling
time specifications to be satisfied. To use a single stage of compensation, the zero must provide more than
36.24◦ at s1.
In order to move the additional closed-loop poles far to the left of s1, a two-stage compensator will be
designed. This means that each stage of compensation must provide 36.24◦/2 = 18.12◦ at s1. The phase
angle for the zero of each stage will be chosen as ∠ (s1 − zc) = 19◦. Therefore, the angle of each pole must
be ∠ (s1 − pc) = 0.88◦. The horizontal distances from s1 to the compensator zero and pole are 15.85 and
25
355.1, respectively. Therefore, the zero and pole are zc = −19.85 and pc = −359.1. Using two poles andzeros at these locations, the gain is Kc = 2071.9, so the compensator is
Gc =2071.9 (s+ 19.85)2
(s+ 359.1)2 =
·45.52 (s+ 19.85)
(s+ 359.1)
¸2(24)
Using this compensator with the system G(s) = 8/ [s (s+ 4)] produces an overshoot of 11.6%, which is
less than three of the single-stage compensators. The settling time is approximately the same as the first
four designs presented in Example 8. With the two-stage compensator, there are now four closed-loop
poles, rather than three as before. Two of them are at s1 and its complex conjugate. The remaining two
closed-loop poles are at s = −357.1± j123. Although the overshoot is still a bit higher than specified, it
is smaller than the other designs that placed the compensator zero to the left of s1, and the additional
closed-loop poles are much farther to the left of s1 than in any of the single-stage designs. ¨
III. Design Example
Both phase lead and phase lag examples will be presented in this section. The lead example will be
presented first since that is the compensator type most often used because it increases the system damping.
After that, the lag compensator example will be given. It will be seen that the design procedures for the
two types of compensators are identical; just the role of the pole and zero are reversed.
A. Phase Lead Example
A.1 Given System and Specifications
The open-loop transfer function for the system to be controlled is
Gp(s) =0.375(s+ 0.8)
s(s+ 0.2)(s+ 1)(s+ 1.5)(25)
The system is Type 1, so it will have zero steady-state error for a step input and a non-zero, finite steady-
state error for a ramp input. If unity feedback is placed around Gp(s), the closed-loop poles are located at
s = {−0.054± j0.4405, −0.9004, −1.6916}. Therefore, the original system is closed-loop stable with unityfeedback and Gc(s) = 1. However, the damping ratio of the complex conjugate pair of closed-loop poles is
ζ = 0.122, which corresponds to 68% overshoot for the standard second-order system. The settling time
is dominated by those complex conjugate poles since they are the closest to the jω axis, and the predicted
settling time is 74 seconds. The uncompensated (plant with unity feedback) step response is shown in
Fig. 8, and it is clear that the predictions on overshoot and settling time are accurate for this system.
The specifications that are imposed on the system are:
• overshoot in the response to a step input: PO ≈ 20%;
• settling time for the response to a step input: Ts ≈ 16 seconds.
26
0 10 20 30 40 50 60 70 800
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Time (s)
Am
plitu
deUncompensated Step Response
Fig. 8. Step response of the uncompensated system for the phase lead design example.
Thus, the compensator needs to reduce the overshoot and the settling time significantly. This requires
that the effective damping of the system be increased, which requires a phase lead compensator.
A.2 Selection of the Dominant Closed-Loop Pole
Using the equations for the standard second-order system, the damping ratio that corresponds to an
overshoot of 20% can be computed from (11), and is ζ = 0.4559. The angle and slope of the radial line
associated with this value of ζ are θ = 62.87◦ and 1.952, respectively. The settling time specification
requires the dominant closed-loop poles to have a real part computed from (13) of Re [s1] = −4/16 =−0.25. Combining these requirements places the desired closed-loop pole at s = s1 = −0.25 + j0.488.
The uncompensated root locus is shown in Fig. 9, along with the point s1. It is clear from the plot
that the dominant branches of the root locus need to be moved to the left in order to pass through s1,
again indicating the need for a lead compensator.
27
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Real Axis
Imag
Axi
s
s1
Uncompensated Root Locus
Fig. 9. Root locus of the uncompensated system and the desired closed-loop pole s1 = −0.25 + j0.488.
A.3 Designing the Compensator
The first step in designing the compensator is to determine the phase shift of Gp(s) at s = s1. This
angle is ∠Gp (s1) = tan−1 [0.488/0.55]−tan−1 [0.488/− 0.25]−tan−1 [0.488/− 0.05]−tan−1 [0.488/0.75]−
tan−1 [0.488/1.25], so ∠Gp (s1) = 41.58◦ − (−62.87◦ + 180◦) − (−84.15◦ + 180◦) − 33.05◦ − 21.33◦ =−225.77◦ = 134.23◦. Therefore, in order for s1 to be on the root locus, the compensator must provide aphase shift of ∠Gc (s1) = 45.77
◦ at s = s1. This can be done with a single pole—zero pair.
In order to use a single stage of compensation, the compensator zero must provide more than ∠Gc (s1) =
45.77◦ at s1, so the left-most location for the zero is s = −0.725. Following the rule of thumb that the zeroshould be at or to the left of the second real-axis open-loop pole, the compensator zero will be placed at
the second pole, namely at s = −0.2. This location produces an angle of 95.85◦ at s1, so the compensatorpole must have an angle at s1 of ∠ (s1 − pc) = 95.85
◦ − 45.77◦ = 50.08◦.The distance from the real-axis projection of s1 to the compensator pole needed to provide this angle is
28
equal to dpc = Im [s1] / tan [50.08◦] = 0.4083. Therefore, the compensator pole is located at s = −0.6583.
At this stage in the design, the compensator is Gc(s) = Kc (s+ 0.2) / (s+ 0.6583).
The compensator gain is determined from the magnitude criterion |Gc (s1)Gp (s1)| = 1. The gain
calculation is
Kc =|s1| · |s1 + 1| · |s1 + 1.5| · |s1 + 0.6583|
0.375 · |s1 + 0.8| = 1.5192 (26)
Note that the plant pole and compensator zero at s = −0.2 have been omitted from the calculation since
they would cancel out exactly. The final lead compensator for this example is
Gc(s) =1.5192 (s+ 0.2)
(s+ 0.6583)(27)
The compensated root locus and step response are shown in Fig. 10. The overshoot is 19.4%, and
the settling time is 16.3 seconds, so the specifications have been satisfied. For the particular Gp(s) in
this example, the dominant closed-loop pole location s1 is sufficiently farther to the right than the other
closed-loop poles to make the second-order equation for overshoot hold for this higher-order system.
The following table summarizes the performances of the original system and the final compensated sys-
tem. It is seen that the compensator defined in (27) allows both of the transient performance specifications
to be satisfied.
Uncompensated Compensated Specification
PO 67.6% 19.4% ≈ 20%
Ts 74.7 sec 16.3 sec ≈ 16 sec
Tr 2.63 sec 2.98 sec None
ess−ramp 1 2.17 None
closed-loop poles −0.054± j0.4405 −0.25± j0.488 s1 = −0.25 + j0.488
−0.9004 −0.2 None
−1.6916 −0.8285 None
−1.8298 None
Note that one of the compensated closed-loop poles is at s = −0.2, the location of one of the plant’sopen-loop poles. The reason for this is the fact that the compensator zero was placed at that same location.
A compensator zero and a plant pole (or vice versa) at the same location always results in a closed-loop
pole at that location also. That pole does not affect the settling time since there is a closed-loop zero at
that point also. Two of the closed-loop pole for the compensated system are at s = s1 and its complex
conjugate as desired. The lead-compensated system has larger steady-state error for a ramp input than
the uncompensated system, but since that characteristic was not specified, this does not cause a problem.
If there was a specification on steady-state error, it would be considered at this point in the design process,
using the technique described in my paper Compensator Design for Steady-State Error Using Root Locus.
29
-2 -1 0 1-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Real Axis
Imag
Axi
sLead Compensated Root Locus
s1
0 5 10 15 20 250
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (s)
Am
plitu
de
Lead Compensated Step Reponse
PO = 19.4%
Ts = 16.3 s
Fig. 10. Lead compensated root locus and step response for the design example.
B. Phase Lag Example
B.1 Given System and Specifications
The system to be controlled has the open-loop transfer function
Gp(s) =2(s+ 1)
(s+ 2)(28)
The system is Type 0 so it will have a non-zero, finite steady-state error for a step input. Specifically,
ess = 0.5 from Eqs. (6) and (7). If unity feedback is placed around Gp(s), the closed-loop pole is at
s = −1.33. Therefore, with Gc(s) = 1 the closed-loop system is stable.
The specifications that are imposed on the system are:
• overshoot in the response to a step input: PO < 5%;
• settling time for the response to a step input: Ts ≤ 2 seconds;
30
• steady-state error for a step input: ess = 0.
The steady-state error specification requires a Type 1 system (or higher). Therefore, the compensator will
have to have a pole at the origin as its minimum configuration. For purposes of designing the complete
compensator, this pole may be included with the plant model to form the following augmented model
corresponding to Eq. (8).
G(s) =2(s+ 1)
s(s+ 2)(29)
The transfer function in (29) is the one that will be evaluated relative to the transient performance
specifications for the design of the compensator.
B.2 Selection of the Dominant Closed-Loop Pole
Once again, the equations for second-order systems may be used to choose the location of the dominant
closed-loop pole. The overshoot specification imposes the constraint on the damping ratio of ζ > 0.6901,
and the settling time specification imposes the constraint Re [s1] ≤ −2. For convenience, the dampingratio will be ζ = 0.707 since that corresponds to an angle of 45◦ and an overshoot of 4.3%. The dominant
closed-loop pole will be placed at s1 = −2 + j2. For the standard second-order system, this choice for s1
would allow both of the transient performance specifications to be satisfied. Although the system in (29)
is not the model of (9), and will be further modified by any additional compensation other than a gain,
we will use this choice as at least a starting point.
Since the root locus for G(s) lies entirely on the real axis (for both positive and negative K), the point
s1 is not on the root locus. Therefore, the compensator will need to provide phase shift at s1 in order to
make a branch of the root locus pass through that point.
B.3 Designing the Compensator
The first step will to compute the phase shift of the augmented system at s1. This value is
∠G (s1) = ∠ (s1 + 1)− ∠s1 − ∠ (s1 + 2) (30)
= 116.57◦ − 135◦ − 90◦
= −108.43◦ = 251.57◦
In order to place s = s1 on the root locus, the compensator must provide a phase shift at s1 of
∠Gc (s1) = −180◦ − (−108.43◦) (31)
= 180◦ − 251.57◦
= −71.43◦
31
Since ∠Gc (s1) < 0, phase lag compensation is needed, so the compensator pole will be to the right
of the compensator zero. The method described in Section II-E.7 will be used to choosing the pole and
zero so that the value of α = zc/pc will be as small as possible. The phase angle of the point s = s1 is
∠s1 = 135◦, so the necessary angles of the compensator pole and zero are
∠ (s1 − zc) =135◦ + (−71.43◦)
2= 31.72◦ (32)
∠ (s1 − pc) =135◦ − (−71.43◦)
2= 103.29◦
The distances from the real-axis projection of s1 to the zero and pole and the locations of the compensator
zero and pole are
dzc =2
tan [31.72◦]= 3.2361 (33)
dpc =2
tan [103.29◦]= −0.4721
and
zc = −5.2361, pc = −1.5279 (34)
so the total compensator at this point in the design (including the pole needed for the steady-state error
specification) is
Gc1 =Kc1 (s+ 5.2361)
s (s+ 1.5279)(35)
The gain Kc1 is computed from the magnitude criterion for the series combination of the original plant
model in (28) and the compensator in (35). The value of the gain is Kc1 = 0.6833, so the final compensator
is
Gc1 =0.6833 (s+ 5.2361)
s (s+ 1.5279)(36)
A second compensator will be designed before the results of this first compensator are evaluated. Then
a comparison of the two designs will be made. Since ∠Gc (s1) < 0, and the total phase shift of the
compensator is the angle of the numerator minus the angle of denominator, this compensator can be
implemented without a zero, using only a pole (in addition to the one at s = 0) and a gain. If this is
done, the phase angle of the numerator is 0, so the phase angle of denominator is the total phase shift of
the compensator.
With this approach, ∠ (s1 − pc2) = 0 − ∠Gc (s1) = 71.43◦. The distance from the real-axis projection
of s1 to the compensator pole is dpc = 0.6667, so the pole is located at s = −2.6667. From the magnitude
criterion, the gain is computed to be Kc2 = 2.6667, so the final form of this compensator is
32
Gc2 =2.6667
s (s+ 2.6667)(37)
Both compensators produce third-order systems when placed in series with Gp(s), and s = s1 is a
closed-loop pole with each of the designs. Figure 11 shows the compensated root locus plots for each of
these designs. The small triangles on the plots are the actual closed-loop poles. It is clear that the design
parameter of having s1 be a closed-loop pole is satisfied. Two points should be noted when comparing
these root locus plots. The first point is the differences in the numbers of asymptotes in the plots. The
closed-loop system with Gc1(s) has only 1more pole than zero (relative degree = 1), so the only asymptote
is 180◦. The system with Gc2(s) has 2 more poles than zeros, so the asymptotes have angles of ±90◦.This may be an important difference if something causes the plant or compensator gain to increase in
value. This would reduce the effective damping ratio, leading to more overshoot. Although the closed-loop
system would remain stable with this Gp(s) for all K > 0, this would not be true for other systems. If the
second compensator design increased the relative degree of a system from 2 to 3 (rather than from 1 to 2
as in this example), increases in gain would lead to an unstable closed-loop system if those increases were
sufficiently large. Therefore, the presence of the zero in Gc1(s) provides additional stability robustness for
the closed-loop system.
The second point to note is the location of the third-closed loop pole for the two designs. With Gc1(s),
that pole is very close to the open-loop zero at s = −1. Therefore, its effect on the transient response willbe small, even though the pole is closer to the jω axis than s1. By contrast, the third closed-loop pole
with Gc2(s) is farther away from the open-loop zero and also even closer to the jω axis. The effect of this
pole on the transient response might be significant.
Figure 12 shows the step responses for these two compensator designs, as well as for the closed-loop
response of the augmented system in (29). The only system that satisfies the settling time specification
is the one with compensator Gc1(s). That system also has a much shorter rise time than the other two
designs.
The following table summarizes the results from this example.
Gp(s) G(s) Gc1(s)Gp(s) Gc2(s)Gp(s) Specification
PO 16.67% 0% 0.65% 0% < 5%
Ts 2.12 sec 5.38 sec 1.075 sec 4.66 sec ≤ 2 secTr undefined 2.67 sec 0.749 sec 1.88 sec None
ess−step 0.5 0 0 0 0
closed-loop poles −1.33 −0.5858 −2± j2 −2± j2 s1 = −2 + j2
−3.4142 −0.8944 −0.6667 None
The overshoot and settling time for the uncompensated system are based on the final value for the output
of 0.5. Since the uncompensated system has the same number of poles and zeros, there is an impulse in the
33
-10 -5 0-4
-3
-2
-1
0
1
2
3
4
Real Axis
Imag
Axi
sUsing Lag Compensator G c1(s)
s1
-3 -2 -1 0 1 2-5
-4
-3
-2
-1
0
1
2
3
4
5
Real Axis
Imag
Axi
s
Using Lag Compesator G c2(s)
s1
Fig. 11. Comparison of root locus plots with two lag compensator designs.
step response, and the maximum value (0.6667) occurs at t = 0. This value was used in the computation
of overshoot. Since the output never takes on a value equal to 10% of its final value, the rise time is
undefined.
Augmenting the system with the pole at the origin satisfies the steady-state error and overshoot specifi-
cations, but not the settling time specification. As previously mentioned, the system using Gc1(s) satisfies
all of the specifications. The second compensator design fails on the settling time specification due to the
closed-loop pole at s = −0.6667.For the augmented system of (29) and the transient response specifications in this example, gain com-
pensation can be used. Both the specifications are satisfied for all Kc ≥ 4.55. With this approach, thetotal compensator would be Gc(s) = Kc/s. The point s1 would not be a closed-loop pole; both poles
would be real. Very short rise and settling times can be achieved by using large gain values. However, a
34
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
1.2
1.4
Gc1(s)
Gc2(s)
Gc(s) = 1/s
Time (s)
Am
plitu
deClosed-Loop Step Responses
Fig. 12. Comparison of closed-loop step responses for two lag compensators.
potential drawback with this approach is large acceleration in the system output. The control signal would
change values rapidly near the beginning of the response causing the output to change rapidly also, with
the output signal approaching a discontinuity for large gains. If the output signal represented mechanical
position of some object, a large acceleration might be objectionable. The rapid change in the value of the
control signal might also exceed physical limitations of the actuator providing motion to the system. The
control signals generated by Gc1(s) and Gc2(s) change more smoothly and limit the acceleration.
This example illustrates the design technique of satisfying transient performance specifications using
root locus. Even though the compensated system was third-order and contained one zero, the selection of
the point s1 yielded a compensator design Gc1(s) that more than satisfied the specifications. If there was
a specification on steady-state error for a ramp input, it would be considered at this point in the design
process, using the “special lag” design described in my paper Compensator Design for Steady-State Error
Using Root Locus.
35
References
[1] J. D’Azzo and C. Houpis, Linear Control System Analysis and Design. New York: McGraw-Hill, 4th ed., 1995.
[2] R. C. Dorf and R. H. Bishop, Modern Control Systems. Reading, MA: Addison-Wesley, 7th ed., 1995.
[3] K. Ogata, Modern Control Engineering. Upper Saddle River, NJ: Prentice Hall, 3rd ed., 1997.
[4] G. Franklin, J. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems. Reading, MA: Addison-Wesley,
3rd ed., 1994.
[5] G. Thaler, Automatic Control Systems. St. Paul, MN: West, 1989.
[6] W. A. Wolovich, Automatic Control Systems. Fort Worth, TX: Holt, Rinehart, and Winston, 3rd ed., 1994.
[7] J. V. de Vegte, Feedback Control Systems. Englewood Cliffs, NJ: Prentice Hall, 3rd ed., 1994.
[8] B. C. Kuo, Automatic Controls Systems. Englewood Cliffs, NJ: Prentice Hall, 7th ed., 1995.
[9] N. S. Nise, Control Systems Engineering. New York: John Wiley & Sons, 3rd ed., 2000.
[10] C. Phillips and R. Harbor, Feedback Control Systems. Upper Saddle River, NJ: Prentice Hall, 4th ed., 2000.
[11] G. C. Goodwin, S. F. Graebe, and M. E. Salgado, Control System Design. Upper Saddle River, NJ: Prentice Hall,
2001.
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