This document contains lecture notes on control system design using root locus analysis. It discusses: 1) Transient response characteristics like percent overshoot, peak time, and settling time for second-order systems. 2) Design of transient response via gain adjustment by sketching the root locus and finding the gain value corresponding to the desired damping ratio. 3) Using a proportional-derivative controller to improve transient response speed by adding a zero to the compensator transfer function to place system poles in a desired location not reachable by gain adjustment alone. Two design examples are provided to illustrate the process.

1. Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
Note 11
Design via Root Locus
Part I
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 1

2. Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
1. Transient Response
For a second order system
ωn
2
G ( s) = 2
s + 2ζω n s + ω n
2
The poles are the roots of the characteristic equation of s 2 + 2 ζω n s + ω n2 = 0 , i.e.,
s1, 2 = −ζω n ± jω n 1 − ζ 2
Representing these two poles in the complex plane
Im
s1 = −ζω n + jω n 1 − ζ 2 × + jω n 1 − ζ 2
180o-θ
θ
Re
− ζω n
s2 = −ζω n − jω n 1 − ζ 2 × − jω n 1 − ζ 2
Recall that the time response of a second order system to a unit step input is :
c(t ) = 1 −
1−ζ
1
2
(
e −ζω nt sin ω n 1 − ζ 2 t + θ )
 1−ζ 2 
where : θ = tan  −1 
 ζ 
 
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 2

3. Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
The typical step response of a second order system is given as follow.
Percent overshoot, %OS. The percent overshoot is defined as the amount that the
waveform at the peak time overshoots the steady-state value, which is expressed as a
percentage of the steady-state value.
1−ζ 2 )
%OS = e −(ζπ / × 100
For given %OS, the damping ratio can be solved from the above equation
− ln(%OS / 100)
ζ =
π 2 + ln 2 (%OS / 100)
Peak time, TP. The peak time is the time required for the response to reach the first
peak.
π π
TP = =
ωn 1 − ζ 2 Im(s )
Setting time, Ts. The settling time is the time required for the amplitude of the sinusoid
to decay to 2% of the steady-state value.
4 4
Ts = =
ζω n Re( s )
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 3

4. Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
It should be noted that the formulae to characterize the transient response, given in the
proceeding page, were derived for only a second order system, or a system with two
poles. If a system has more than two poles, generally speaking, we cannot use the
formulae to characterize the system transient response.
However, under certain conditions, a system with more than two poles can be
approximated as a second-order system. Consider a three-pole system with two complex
poles s1, 2 = −ζω n ± jω n 1 − ζ 2 and a real pole s3 = −a , as shown in the following figure. If
the pole s3 is five times farther to the left than the complex poles (or a > 5ξω n ), then the
system can be approximated as a second-order system with the poles of s1 and s2. In this
case, the poles of s1 and s2 are called dominate poles.
Im
s1
s3 ×
× Re
×
s2
In this course, we limit our discussion on higher order systems that can be approximated
by using second-order systems. Therefore, we can use the formulae for a second order
system to characterize the transient response of the higher order system.
2. Transient Response Design via Gain Adjustment
In order that a system has the desired transient response, the design procedure via gain
adjustment usually consists of the following three steps:
(1) Sketch the root locus for the system.
(2) Find the damping ratio, which results in the desired transient response; and then
draw a radial line to represent the damping ratio.
(3) Find the dominate pole, which is the intersection of the root locus and the
damping ratio line, as well as the corresponding value of gain, K.
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 4

5. Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
Design Problem 1
Consider the system shown in the following figure. Determine the value of gain, K, to
yield 4.60% overshoot. After that, estimate the settling time of the transient response.
R(s) C(s)
+ 1
K
_ ( s + 1)( s + 3)( s + 10)
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 5

6. Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
3. Improving Transient Response – Using PD
Controller
Motivation: Consider the following closed-loop system
Compensator Plant
R(s)
+ C(s)
K G (S )
_
Assuming that the desired transient response, defined by percent overshoot and settling
time, is represented by point B in the following figure (a). Figure (b) shows us the desired
transient response, as well as the transient response defined by point A. It is noted that
point A is on the root locus, as shown in figure (a). These two responses have the same
percent overshoot (due to the same ζ), but the response defined by the point B is faster
than that defined by the point A (due to the larger real part of B than A).
Problem: the point B is not on the root locus, so we cannot design the system by simply
adjusting the gain, K.
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 6

7. Lecture Notes of Control Systems I - ME 431/Analysis and Synthesis of Linear Control System - ME862
Solution: Add a zero to the transfer function of the controller such that the system has a
root locus that goes through the desired pole location (i.e., the point B). The new control
system is shown in the following.
PD Controller Plant
R(s) + E(s) M(s) C(s)
K(s+a) G (S )
_
The new controller is called an ideal derivative compensator. It has the transfer function
Gc ( s ) = K ( s + a) = Ka + Ks
The value of a is to be determined by using the angle criterion, i.e., the angle of the new
open-loop transfer function Gc ( s )G ( s ) is an odd multiple of 180o. And the value of K is
to be determined by using the magnitude criterion, i.e., the magnitude of Gc ( s )G ( s ) is 1.
Systems that feed the error forward to the plant are called proportional (P) control
systems. Systems that feed the integral of error forward to the plant are called integral
(I) control systems. Systems that feed the derivative of error forward to the plant are
called derivative (D) control systems.
Thus, the above ideal derivative compensator is also called a proportional-plus-
derivative (PD) controller.
Design Problem 2
Consider the system shown in the following figure. Design a PD controller to yield
4.60% overshoot, with a settling time of 1.0 sec.
R(s) C(s)
+ 1
K
_ ( s + 1)( s + 3)( s + 10)
Department of Mechanical Engineering, University Of Saskatchewan, 57 Campus Drive, Saskatoon, SK S7N 5A9, Canada 7