Ch6 root locus method

CHAPTER SIX
CONTROL SYSTEMS ANALYSIS AND DESIGN BYTHE ROOT-
LOCUS METHOD

6-1 INTRODUCTION
• The basic characteristic of the transient response of a closed-loop system is closely
related to the location of the closed-loop poles.
• If the system has a variable loop gain, then the location of the closed-loop poles depends
on the value of the loop gain chosen. It is important, therefore, that the designer know
how the closed-loop poles move in the s plane as the loop gain is varied.
• From the design viewpoint, in some systems simple gain adjustment may move the
closed-loop poles to desired locations. Then the design problem may become the
selection of an appropriate gain value. If the gain adjustment alone does not yield a
desired result, addition of a compensator to the system will become necessary.

6-1 INTRODUCTION
• The closed-loop poles are the roots of the characteristic equation. Finding the roots of the
characteristic equation of degree higher than 3 is laborious and will need computer solution.
However, just finding the roots of the characteristic equation may be of limited value,
because as the gain of the open-loop transfer function varies, the characteristic equation
changes and the computations must be repeated.
• A simple method for finding the roots of the characteristic equation has been developed by
W. R. Evans and used extensively in control engineering. This method, called the root-locus
method, is one in which the roots of the characteristic equation are plotted for all values of
a system parameter.
• The roots corresponding to a particular value of this parameter can then be located on the
resulting graph.

6-1 INTRODUCTION
• Note that the parameter is usually the gain, but any other variable of the open-loop transfer
function may be used. Unless otherwise stated, we shall assume that the gain of the open-loop
transfer function is the parameter to be varied through all values, from zero to infinity.
• By using the root-locus method the designer can predict the effects on the location of the
closed-loop poles of varying the gain value or adding open-loop poles and/or open-loop zeros.
• Therefore, it is desired that the designer have a good understanding of the method for
generating the root loci of the closed-loop system, both by hand and by use of a computer
software program like MATLAB.
• In designing a linear control system, we find that the root-locus method proves to be quite
useful, since it indicates the manner in which the open-loop poles and zeros should be modified
so that the response meets system performance specifications. This method is particularly suited
to obtaining approximate results very quickly.

6-1 INTRODUCTION
• Because generating the root loci by use of MATLAB is very simple, one may think sketching
the root loci by hand is a waste of time and effort. However, experience in sketching the
root loci by hand is invaluable for interpreting computer-generated root loci, as well as for
getting a rough idea of the root loci very quickly.

6-2 ROOT LOCUS PLOTS
• Consider the negative feedback system shown below.The closed-loop transfer function is
• The characteristic equation for this closed-loop system is obtained by setting the
denominator of the right-hand side of Equation equal to zero.That is
or

• Here we assume that G(s)H(s) is a ratio of polynomials in s. Since G(s)H(s) is a complex
quantity, Equation can be split into two equations by equating the angles and magnitudes of
both sides, respectively, to obtain the following:
Angle condition:
Magnitude condition:
• The values of s that fulfill both the angle and magnitude conditions are the roots of the
characteristic equation, or the closed-loop poles.
• A locus of the points in the complex plane satisfying the angle condition alone is the root
locus.

• The roots of the characteristic equation (the closed-loop poles) corresponding to a given
value of the gain can be determined from the magnitude condition. The details of applying
the angle and magnitude conditions to obtain the closed-loop poles are presented later in
this section.
• In many cases, G(s)H(s) involves a gain parameter K, and the characteristic equation may be
written as
• Then the root loci for the system are the loci of the closed-loop poles as the gain K is
varied from zero to infinity.

• Note that to begin sketching the root loci of a system by the root-locus method we must
know the location of the poles and zeros of G(s)H(s). Remember that the angles of the
complex quantities originating from the open-loop poles and open-loop zeros to the test
point s are measured in the counterclockwise direction. For example, if G(s)H(s) is given by

• where –p2 and –p3 are complex-conjugate poles, then the angle of G(s)H(s) is
• where ∅1 , 𝜃1, 𝜃2 , 𝜃3 , and 𝜃4 are measured counterclockwise as shown in previous Figures
(a) and (b).The magnitude of G(s)H(s) for this system is
• where A1 , A2 , A3 , A4 , and B1 are the magnitudes of the complex quantities s+p1 , s+p2 ,
s+p3 , s+p4 , and s+z1 , respectively, as shown in previous Figure (a).

 General Rules for Constructing Root Loci.
• For a complicated system with many open-loop poles and zeros, constructing a root-locus
plot may seem complicated, but actually it is not difficult if the rules for constructing the
root loci are applied.
• By locating particular points and asymptotes and by computing angles of departure from
complex poles and angles of arrival at complex zeros, we can construct the general form of
the root loci without difficulty.
• We shall now summarize the general rules and procedure for constructing the root loci of
the negative feedback control system.

• First, obtain the characteristic equation
• Then rearrange this equation so that the parameter of interest appears as the multiplying
factor in the form
• In the present discussions, we assume that the parameter of interest is the gain K, where
K>0. In the present discussions, we assume that the parameter of interest is the gain K,
where K>0. (If K < 0, which corresponds to the positive-feedback case, the angle condition
must be modified.).

• 1. Locate the poles and zeros of G(s)H(s) on the s plane.
The root-locus branches start from open-loop poles and terminate at zeros (finite zeros or
zeros at infinity). From the factored form of the open-loop transfer function, locate the open-
loop poles and zeros in the s plane. e. [Note that the open-loop zeros are the zeros of
G(s)H(s), while the closed-loop zeros consist of the zeros of G(s) and the poles of H(s).]

• Note that the root loci are symmetrical about the real axis of the s plane, because the
complex poles and complex zeros occur only in conjugate pairs.
• A root-locus plot will have just as many branches as there are roots of the characteristic
equation. Since the number of open-loop poles generally exceeds that of zeros, the number
of branches equals that of poles.
• If the number of closed-loop poles is the same as the number of open-loop poles, then the
number of individual root-locus branches terminating at finite open-loop zeros is equal to
the number m of the open-loop zeros.

• The remaining n-m branches terminate at infinity (n-m implicit zeros at infinity) along
asymptotes.
2. Root loci on the real axis are determined by open-loop poles and zeros lying on it.
The complex-conjugate poles and complex conjugate zeros of the open-loop transfer
function have no effect on the location of the root loci on the real axis because the angle
contribution of a pair of complex-conjugate poles or complex-conjugate zeros is 360° on the
real axis.
• Each portion of the root locus on the real axis extends over a range from a pole or zero to
another pole or zero.

• In constructing the root loci on the real axis, choose a test point on it. If the total number of
real poles and real zeros to the right of this test point is odd, then this point lies on a root
locus. If the open-loop poles and open-loop zeros are simple poles and simple zeros, then
the root locus and its complement form alternate segments along the real axis.
3. Determine the asymptotes of root loci.
If the test point s is located far from the origin, then the angle of each complex quantity may
be considered the same. One open-loop zero and one open-loop pole then cancel the effects
of the other. Therefore, the root loci for very large values of s must be asymptotic to straight
lines whose angles (slopes) are given by

• where n= number of finite poles of G(s)H(s) m= number of finite zeros of G(s)H(s)
• Here, k=0 corresponds to the asymptotes with the smallest angle with the real axis.
Although k assumes an infinite number of values, as k is increased the angle repeats itself,
and the number of distinct asymptotes is n-m.
4. Find the breakaway and break-in points.
Because of the conjugate symmetry of the root loci, the breakaway points and break-in points
either lie on the real axis or occur in complex-conjugate pairs.
If a root locus lies between two adjacent open-loop poles on the real axis, then there exists at
least one breakaway point between the two poles.

• Similarly, if the root locus lies between two adjacent zeros (one zero may be located at –q)
on the real axis, then there always exists at least one break-in point between the two zeros.
• If the root locus lies between an open-loop pole and a zero (finite or infinite) on the real
axis, then there may exist no breakaway or break-in points or there may exist both
breakaway and break-in points.
• Suppose that the characteristic equation is given by
• The breakaway points and break-in points correspond to multiple roots of the characteristic
equation. the breakaway and break-in points can be determined from the roots of

5. Determine the angle of departure (angle of arrival) of the root locus from a complex pole (at a
complex zero).
To sketch the root loci with reasonable accuracy, we must find the directions of the root loci near
the complex poles and zeros.
If a test point is chosen and moved in the very vicinity of a complex pole (or complex zero), the
sum of the angular contributions from all other poles and zeros can be considered to remain the
same.
Therefore, the angle of departure (or angle of arrival) of the root locus from a complex pole (or at
a complex zero) can be found by subtracting from 180° the sum of all the angles of vectors from all
other poles and zeros to the complex pole (or complex zero) in question, with appropriate signs
included.

Angle of departure from a complex pole=180°
– (sum of the angles of vectors to a complex pole in question from other poles)
+ (sum of the angles of vectors to a complex pole in question from zeros)
Angle of arrival at a complex zero=180°
– (sum of the angles of vectors to a complex zero in question from other zeros)
+ (sum of the angles of vectors to a complex zero in question from poles)

6. Find the points where the root loci may cross the imaginary axis.
The points where the root loci intersect the jw axis can be found easily by (a) use of Routh’s
stability criterion or (b) letting s=jw in the characteristic equation, equating both the real part
and the imaginary part to zero, and solving for w and K.
The values of w thus found give the frequencies at which root loci cross the imaginary axis.
The K value corresponding to each crossing frequency gives the gain at the crossing point.

7. Taking a series of test points in the broad neighborhood of the origin of the s plane, sketch
the root loci.
Determine the root loci in the broad neighborhood of the jw axis and the origin. The most
important part of the root loci is on neither the real axis nor the asymptotes but is in the
broad neighborhood of the jw axis and the origin.The shape

• of the root loci in this important region in the s plane must be obtained with reasonable
accuracy.
8. Determine closed-loop poles.
A particular point on each root-locus branch will be a closed-loop pole if the value of K at
that point satisfies the magnitude condition.
Conversely, the magnitude condition enables us to determine the value of the gain K at any
specific root location on the locus.
The value of K corresponding to any point s on a root locus can be obtained using the
magnitude condition, or

6-5 ROOT LOCUS APPROACH TO CONTROL-
SYSTEM DESIGN
 Preliminary Design Consideration.
• In building a control system, we know that proper modification of the plant dynamics may
be a simple way to meet the performance specifications.
• This, however, may not be possible in many practical situations because the plant may be
fixed and not modifiable
• Then we must adjust parameters other than those in the fixed plant. In this book, we
assume that the plant is given and unalterable.
• In practice, the root-locus plot of a system may indicate that the desired performance
cannot be achieved just by the adjustment of gain (or some other adjustable parameter).

SYSTEM DESIGN.(CONT.)
 Preliminary Design Consideration.(Cont.)
• In fact, in some cases, the system may not be stable for all values of gain (or other adjustable
parameter). Then it is necessary to reshape the root loci to meet the performance
specifications.
• The design problems, therefore, become those of improving system performance by
insertion of a compensator. Compensation of a control system is reduced to the design of a
filter whose characteristics tend to compensate for the undesirable and unalterable
characteristics of the plant.

 Design by Root-Locus Method.
• The design by the root-locus method is based on reshaping the root locus of the system by
adding poles and zeros to the system’s open-loop transfer function and forcing the root loci
to pass through desired closed-loop poles in the s plane.
• The characteristic of the root-locus design is its being based on the assumption that the
closed-loop system has a pair of dominant closed-loop poles. This means that the effects of
zeros and additional poles do not affect the response characteristics very much.
• In designing a control system, if other than a gain adjustment (or other parameter
adjustment) is required, we must modify the original root loci by inserting a suitable
compensator.

 Design by Root-Locus Method.(Cont.)
• Once the effects on the root locus of the addition of poles and/or zeros are fully
understood, we can readily determine the locations of the pole(s) and zero(s) of the
compensator that will reshape the root locus as desired.
• In essence, in the design by the root locus method, the root loci of the system are reshaped
through the use of a compensator so that a pair of dominant closed-loop poles can be
placed at the desired location.

 Series Compensation and Parallel (or Feedback) Compensation.
• Figures (a) and (b) show compensation schemes commonly used for feedback control
systems.
• Figure (a) shows the configuration where the compensator Gc(s) is placed in series with the
plant.This scheme is called series compensation.
• An alternative to series compensation is to feed back the signal(s) from some element(s)
and place a compensator in the resulting inner feedback path, as shown in Figure (b). Such
compensation is called parallel compensation or feedback compensation.
• In compensating control systems, we see that the problem usually boils down to a suitable
design of a series or parallel compensator.

 Series Compensation and Parallel (or Feedback)
Compensation.(Cont.)
• The choice between series compensation and parallel compensation depends on the nature
of the signals in the system, the power levels at various points, available components, the
designer’s experience, economic considerations, and so on.
• In general, series compensation may be simpler than parallel compensation; however, series
compensation frequently requires additional amplifiers to increase the gain and/or to
provide isolation. (To avoid power dissipation, the series compensator is inserted at the
lowest energy point in the feedforward path.).

• Note that, in general, the number of components required in parallel compensation will be
less than the number of components in series compensation, provided a suitable signal is
available, because the energy transfer is from a higher power level to a lower level. (This
means that additional amplifiers may not be necessary.).

 Commonly Used Compensators.
• If a compensator is needed to meet the performance specifications, the designer must
realize a physical device that has the prescribed transfer function of the compensator.
• Numerous physical devices have been used for such purposes. In fact, many noble and useful
ideas for physically constructing compensators may be found in the literature.
• If a sinusoidal input is applied to the input of a network, and the steady-state output (which
is also sinusoidal) has a phase lead, then the network is called a lead network.
• If the steady-state output has a phase lag, then the network is called a lag network.
• In a lag–lead network, both phase lag and phase lead occur in the output but in different
frequency regions; phase lag occurs in the low-frequency region and phase lead occurs in
the high-frequency region.

 Commonly Used Compensators.(Cont.)
• A compensator having a characteristic of a lead network, lag network, or lag–lead network
is called a lead compensator, lag compensator, or lag–lead compensator.
• Among the many kinds of compensators, widely employed compensators are the lead
compensators, lag compensators, lag–lead compensators, and velocity-feedback
(tachometer) compensators.

 Commonly Used Compensators.(Cont.)
• It is noted that in designing control systems by the root-locus or frequency-response
methods the final result is not unique, because the best or optimal solution may not be
precisely defined if the time-domain specifications or frequency-domain specifications are
given.

 Effects of the Addition of Poles.
• The addition of a pole to the open-loop transfer function has the effect of pulling the root
locus to the right, tending to lower the system’s relative stability and to slow down the
settling of the response. (Remember that the addition of integral control adds a pole at the
origin, thus making the system less stable.) Figure shows examples of root loci illustrating
the effects of the addition of a pole to a single-pole system and the addition of two poles to
a single-pole system.

 Effects of the Addition of Poles.(Cont.)
• (a) Root-locus plot of a single-pole system; (b) root-locus plot of a two-pole system; (c)
root-locus plot of a three-pole system

 Effects of the Addition of Zeros.
• The addition of a zero to the open-loop transfer function has the effect of pulling the root
locus to the left, tending to make the system more stable and to speed up the settling of the
response. (Physically, the addition of a zero in the feedforward transfer function means the
addition of derivative control to the system. The effect of such control is to introduce a
degree of anticipation into the system and speed up the transient response.) Figure (a)
shows the root loci for a system

 Effects of the Addition of Zeros.(Cont.)
• (a) Root-locus plot of a three-pole system; (b), (c), and (d) root-locus plots showing effects
of addition of a zero to the three-pole system.

 Effects of the Addition of Zeros.(Cont.)
• that is stable for small gain but unstable for large gain. Figures(b), (c), and (d) show root-
locus plots for the system when a zero is added to the open-loop transfer function. Notice
that when a zero is added to the system of Figure(a), it becomes stable for all values of gain.

Eng. Elaf Ahmed Saeed
Email: elafe1888@gmail.com
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Ch6 root locus method

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