 Open Access
 Total Downloads : 307
 Authors : Awadhesh Kumar Singh, Mrigen Das, Dipak Basumatary, Ganesh Roy
 Paper ID : IJERTV3IS21314
 Volume & Issue : Volume 03, Issue 02 (February 2014)
 Published (First Online): 08032014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Review note on Compensator Design for Control Education and Engineering
Awadhesh Kumar Singh, Mrigen Das, Dipak Basumatary, Ganesh Roy Instrumentation Engineering Department, Central Institute of Technology, Kokrajhar Kokrajhar, BTAD, Assam, India783370
Abstract The compensators are one of the most important aspects in any undergraduate control engineering courses owing to their widespread industrial applications. The degree of convergence of the output waveform for any compensator depends upon proper selection and tuning of the compensator. In this paper the compensator parameters (, , ) of the lead, lag and laglead compensators are tuned to their optimum values using the conventional root locus as well as frequency response approach. The compensator algorithm is studied using MATLAB and usefulness of these compensators for controlling process variables are demonstrated using proper tuning. The comparative studies showing promising results are discussed with suitable examples.
Keywords Lead compensator, Lag compensator, LagLead compensator, root locus approach, frequency response approach.

INTRODUCTION
Feedback control system with compensator is an important
In this paper, the specifications based on time domain analysis such as settling time, rise time, peak time, maximum overshoot etc. are measured from conventional root locus plots of the uncompensated as well as compensated system. In another way the frequency domain specifications like phase margin, gain margin, bandwidth are measured from bode plot of uncompensated as well as compensated system. Finally all the specifications results obtained from root locus approach and frequency response approach are compared and which technique is the best is discussed. And hopefully this technique will facilitate the control system engineer to select the precise way to design the compensators.

LEAD COMPENSATOR
The lead compensator is designed to satisfy the transient response of any system. The compensator having a transfer function of the following form is known as lead compensator
technique that is widely used in the process industries. Their main advantage is that corrective action occurs as soon as the
[1], [2].= +1
=
+1
(1)
controlled variable deviates from the set point, regardless of the
+1
1
+
source and type of disturbance.
Compensators are corrective subsystems introduced into the system to compensate for the deficiency in the performance of the plant. So given a plant and a set of specifications, suitable compensators are to be designed so that the overall system will meet the given specification. Due to its simplicity and effectiveness, phase lag and lead compensation is essential for various frequencybased design methods, especially for design based on the Bode plots [9]. Another alternative design in terms of root locus was given in [10]. A nontrialanderror design technique for laglead compensator is developed in [7] and [8]. In [4] a new lead compensator design algorithm is presented by using computation technology to facilitate the course teaching and implementation. In [5] a new method for the design of compensators for linear timeinvariant dual input/singleoutput systems in continuous time or discrete time is presented to reduce the problem to two singleinput/single output design problems. The development procedure of a fully automated software based OneTouchandGo PID controller and Lead/Lag compensator design tool is presented in [6] to help industrial process control engineers. Also A graphical procedure for the compensator design on the Nyquist plane is presented in [11] with some numerical examples.
However, the nature of traditional trialanderror graphical techniques currently used by almost all available textbooks [1 3] makes the learning and use of laglead compensation.
Where (0 < < 1) and ( > 0)

Using Root Locus Approach
At first from the performance specifications, the desired locations for the dominant closed loop poles are determined. And secondly by drawing rootlocus plot of the uncompensated system ascertain whether or not the gain adjustment alone can yield the desired closed loop poles. If not, the angel of deficiency is calculated. This angel must be contributed by the lead compensator if the new root locus is to pass through the desired locations for the dominant closed loop poles. If static error constants are not specified, the location of pole and zero of the lead compensator is determined so that the lead compensator will contribute the necessary angle . If not other requirements are imposed on the system to make the value of as large as possible. A large value of generally results in larger value of Kv,(velocity constant) which is desirable . If a particular static error constant is specified, it is generally simpler to use the frequencyresponse approach. Then finally the openloop gain of the compensated system from the magnitude condition is determined.

Example
The uncompensated system is taken as = .
(+1)
The root locus plot is given in Fig.1. After considering above stated procedure the designed compensator is
TABLE I
System
Settling Time, Ts(sec)
Rise Time, Tr(sec)
Pick Time, Tp(sec)
Maximum Overshoot,
%Mp
Uncompensated
8
2.42
3.63
16.30
Compensated
1.57
0.5
0.805
17.16
SPECIFICATION PARAMETERS FOR LEAD COMPENSATOR USING ROOT LOCUS ANALYSIS
() = 31.56(+2)
(+9)
and the overall transfer function is
= 31.56(+2) .
(+1)(+9)
The root locus plot of this transfer function is given in Fig.2 All the Specifications regarding uncompensated and compensated are provide in TABLE I. The unit step response is also shown in Fig.3.


Frequency Response Approach
For designing the lead compensator firstly assumed =
and therefore = +1
+1
according [1], [2]. The open
loop transfer function of the compensated system is now
()() = +1 () = +1 1() (2)
+1 +1
Where 1 = ()
Fig.1. Root locus plot of uncompensated system [G(s)= K/s(s+1)]
Fig.2. Root locus plot of the product of compensated system [G(s)= K/s(s+1)] and [Gc(s)=31.56(s+2)/(s+9)]
Gain K is determined to satisfy the requirement on the given static error constant. Using the gain K, a Bode diagram of 1() is drawn. The phase margin is evaluated from the plot. Then the necessary phaselead angle to be added to the system is determined. An additional 5Â° to 12Â° to the phaselead angle is required, because the addition of the lead compensator shifts the gain crossover frequency to the right and decreases the phase margin.
The attenuation factor is determined by use of equation
sin = 1. The frequency where the magnitude of the
1+
uncompensated system 1() is equal to 20log (1/) is determined and selected as the new gain crossover frequency. This frequency corresponds to = 1/ and the maximum phase shift occurs at this frequency.
The corner frequencies of the lead compensator are determined as follows:
p>Zero of lead compensator: = 1
Pole of lead compensator: = 1
c
Using the value of K and ( = ) K is determined. The
gain margin is checked to be sure for satisfactory. If not, repeat
the design process by modifying the polezero location of the compensator until a satisfactory result is obtained.

Example
Uncompensated system is taken as =
(+1)
Designed lead compensator following the above procedure
is = 0.377+1. Hence the overall transfer
0.125 +1
functions = 12(0.377+1) . All the specifications
(+1)(0.125 +1)
regarding the systems are listed in TABLE II. The bode plots of the compensated and uncompensated system are depicted in Fig.4. Also the unit step response is given in the Fig.5.
Fig.3.Output of the compensated and uncompensated system for lead compensator using root locus analysis
III. LAG COMPENSATOR
The lag compensator is used to satisfy steadystate characteristics. The transfer function of lag compensator is in the following form [1], [2].
=
+1
+1
+1
+
1
=
(3)
Fig.4. Bode plot of uncompensated [G(s)= K/s(s+1)] and compensated system
Fig.5. Output of the compensated and uncompensated system for lead compensator using frequency response analysis
TABLE II
SPECIFICATION PARAMETERS FOR LEAD COMPENSATOR USING FREQUENCY RESPONSE ANALYSIS
System
Phase Margin, PM
(deg)
Gain Margin, GM
(db)
Ts (sec)
Tr (sec)
Tp (sec)
%Mp
Uncompe nsated
16.4
8
2.42
3.63
16.3
Compens
ated
42.9
1.67
0.38
0.645
30.39


Comparative Study on the Specifications
Depending upon the unit step response given in Fig.3 the specifications TABLE I is prepared. Similarly on the basis of step response in Fig.5 the TABLE II is listed. After comparing both the table it is observed that in the lead compensator using the root locus technique the maximum peak overshoot is quiet less than that of using Bode plot technique. It means that root locus technique is much more accurate for this design. While the changes of the rise time, the settling time and the peak time are very small amount comparing in both the techniques. Hence they are not so important aspects of consideration.
Where ( > 1) and ( > 0)

Using Root Locus Approach
The procedure for designing the lag compensator using root locus approach is briefly discussed here [1].
Firstly by drawing the rootlocus plot for the uncompensated system based on the transientresponse specifications. The dominant closedloop poles on the root locus plot are located. The particular static error constant specified in the problem is evaluated and to satisfy the specifications the static error constant is increased and the amount is determined. The pole and zero of the lag compensator that produce the necessary increment in the particular static error constant without appreciably altering the original root loci is determined. A new rootlocus plot for the compensated system is drawn then. And the desired dominant closedloop poles on the root locus must located on that. If the angle contribution of the lag network is very small (a few degrees) then the original and new root loci are almost identical. Otherwise, there will be a slight discrepancy between them. The gain of the compensator from the magnitude condition is adjusted so that the dominant closedloop poles lie at the desired location.

Example
The uncompensated system is taken as = 1.06 .
+1 (+2)
The root locus plot is given in Fig.6. After considering above stated procedure the designed compensator is
() = 0.966(+0.05) and the overall transfer function is
(+0.005 )
= 1.02396 (+0.05)
(+2)(+1)( +0.005 )
The root locus for overall transfer function is given in Fig.7. Also the step responses are depicted in Fig.8. All the specifications are listed in TABLE III.
Fig.6. Root locus plot of uncompensated system [G(s)= 1.06/s(s+1)(s+2)]
lag compensator. This frequency is chosen as the new gain crossover frequency. To prevent detrimental effects of phase lag due to the lag compensator, the pole and zero of the lag compensator must be located substantially lower than the new gain crossover frequency. Therefore, the corner frequency
= 1
which is corresponding to the zero of the lag
Fig.7. Root locus plot of the product of uncompensated system [G(s)=
1.06/s(s+1)(s+2)] and compensator [Gc(s)=0.966(s+0.05)/(s+0.005)]
compensator is selected 1 octave to 1 decade below the new gain crossover frequency. The attenuation necessary to bring the magnitude curve down to 0 dB at the new gain crossover frequency is determined. Determining the value of from this as the attenuation is 20log the other corner frequency (corresponding to the pole of the lag compensator) is
determined from = 1 .
B.1.Example
Uncompensated system is taken as = 0.53
+1 (0.5+1)
Designed lag compensator following the above procedure
is = 0.943(0.377+0.1). Hence the overall transfer functions
(+0.01)
= 5(10+1) .
Fig.8. Output of the compensated and uncompensated system for lag compensator using root locus analysis
TABLE III
SPECIFICATION PARAMETERS FOR LAG COMPENSATOR USING ROOT LOCUS ANALYSIS
(100 +1)(+1)(0.5+1)
All the specifications regarding the systems are listed in TABLE IV. The bode plots of the compensated and uncompensated system are depicted in Fig.9. Also the step response is given in the Fig.10.
System
Kv
(1/sec)
Tr
(sec)
Tp
(sec)
Ts
(sec)
%Mp
Uncompensated
0.53
4.04
6
14
13.13
Compensated
5
3.88
6.5
25.8
20.06
Fig.9. Bode plot of uncompensated [G(s)= 0.53/s(s+1)(0.5s+1)] and compensated system


Frequency Response Approach
The openloop transfer function of the compensated system is
() = +1 () = +1 1() (4)
+1 +1
Where 1 = () and =
The required gain K to satisfy static velocity error constant is determined. If the gainadjusted but uncompensated system
1() = () does not satisfy the specifications on the phase and gain margins, then the frequency point where the phase angle of the openloop transfer function is equal to ( 180Â°) plus the required phase margin is measured. The required phase margin is the specified phase margin plus 5Â° to 12Â°. This addition of 5Â° to 12Â° compensates the phase lag of the
Fig.10. Output of the compensated and uncompensated system for lag compensator using frequency response analysis
TABLE IV
SPECIFICATION PARAMETERS FOR LAG COMPENSATOR USING FREQUENCY RESPONSE ANALYSIS
System
PM
(deg)
GM
(db)
Ts (sec)
Tr (sec)
Tp (sec)
%Mp
Uncomp ensated
13
4.44
14
4.04
6.1
13.1
Compen sated
41.6
14.3
13
4.17
6.9
11.92

Comparative Study on the Specifications
Considering the TABLE III and TABLE IV and the step responses of the lag compensator there is considerable changes in settling time occurred. In the frequency response technique the system settles down in a much lesser time. Hence the frequency response technique is the better choice than root locus technique with respect to Ts. Although the step response depending upon root locus approach is more smooth than other since no undershoot is present here. So with respect to that point root locus approach may also be adopted.
IV. LAG LEAD COMPENSATOR
When both the transient and steadystate responses required improvement, a laglead compensator is required. It is a combination of a lag compensator and a lead compensator connected in series. The laglead compensator has a transfer function of the form
Fig.11. Root locus plot of uncompensated [G(s) = K/s(s+1)(s+2)] and compensated system
Fig.12. Output of the compensated and uncompensated system for lag lead compensator using root locus analysis
=
(1+1)(2 +1) ; > 1 (4)
(1+1)( 2 +1)
Here the phaselead portion of the laglead compensator is the portion involving the factor 1and the phaselag portion is the portion having 2 factor.

Using Root Locus Approach
In equation (4) there are two parts are available. In the root locus technique at first the design of lead section is done to meet the specifications on transient response. From this a suitable value of is found. The error constant is determined for the lead compensator system. If the error constant is satisfactory then the design is complete. If the specified error constant is much higher than that obtained by lead compensator then the design of lag section is proceeded such that the overall system meet the specification on steady state performance. Since is already determined the error constant is increased by a factor by the lag section.

Example
The uncompensated system is taken as = .
+1 (+2)
The root locus plot is given in Fig.11. After considering above stated procedure the designed compensator is
TABLE V
SPECIFICATION PARAMETERS FOR LAGLEAD COMPENSATOR USING ROOT LOCUS ANALYSIS
System
Kv (1/sec)
Tr (sec)
Tp (sec)
Ts (sec)
%Mp
Uncomp
ensated
0.5
4.25
6
14
9.99%
Compen
sated
7.36
6.04
2.5
30
6.39%


Frequency Response Approach
According to equation (4) the phaselead portion alters the frequencyresponse curve by adding phaselead angle and increasing the phase margin at the gain crossover frequency. The phaselag portion provides attenuation near and above the gain crossover frequency and thereby allows an increase of gain at the lowfrequency range to improve the steadystate performance.

Example
Uncompensated system is taken as =
+1 (+2)
() = (+0.1)(+0.24)
and the overall transfer function is
Designed laglead compensator following the above
(+4.3)(+0.005 )
13.2(+0.1)(+0.24)
procedure is = +0.7 +0.15 . Hence the overall transfer
=
+7 +0.015
(+1)(+2)(+4.3)(+0.005 )
functions = 20 +0.7 +0.15 .
The root locus for overall transfer function is given in Fig.11. Also the step responses are depicted in Fig.12. All the specifications are listed in TABLE V.
+1 +2 +7 +0.015
All the specifications regarding the systems are listed in TABLE V. The bode plots of the compensated and
uncompensated system are depicted in Fig.13. Also the step response is given in the Fig.14.
lead compensator bode plot technique is more suitable for higher order system. Because in higher order system the angle contributed by lead portion of laglead compensator is too large.
V. CONCLUSION
Fig.15. A typical Feedforward Compensation System
Fig.13. Bode plot of uncompensated and compensated system with laglead compensator
Fig.14. Output of the compensated and uncompensated system for laglead compensator using frequency response analysis
TABLE VI
SPECIFICATION PARAMETERS FOR LAGLEAD COMPENSATOR USING FREQUENCY RESPONSE ANALYSIS
System
PM
GM
(db)
Tr (sec)
Tp (sec)
%Mp
Ts (sec)
Uncompe
nsated
28.1Â°
10.5
4.25
6
9.99
15
Compens
ated
55.2Â°
16.9
1.655
2.485
16.66
15


Comparative Study on the Specifications
By comparing TABLE V and TABLE VI it can be noted that in root locus approach the transient response specifications like maximum overshoot, rise time etc. are not so satisfactory. Although the steady state response is quite normal. But in the other hand in the frequency response technique both the transient as well as steadystate behaviour are quite satisfactory. Therefore the selection of frequency response technique will be more technically appropriate. Also in lag
Every process control system has a general block diagram like a feed back control system shown on the Fig.15. It contains a controller and a plant which have to control with the help of the controller. Such controller is known as compensator. The physical parameters of a control system which are generally controlled in the industries and laboratories are temperature, pressure, flow, level etc. The proper selection of a compensator is a very crucial issue. After selecting the proper compensator the control of these plant variables will become quite effortless.
REFERENCES

Katsuhiko Ogata, Modern Control Engineering, 5th ed., Prentice Hall, Delhi, 2010.

I. J. Nagrath, and M. Gopal, Control System Engineering, 5th ed., New Age Techno Press, New Delhi, 2011.

Norman S. Nise, Control System Engineering, 4th ed., Wiley, USA, 2004.

Xu JianXin, New Lead Compensator Designs for Control Education and Engineering, Proceedings of the 27th Chinese Control Conference, Nov., July 1618, 2008, Kunming,Yunnan, China.

Steven J. Schroeck, William C. Messner, and Robert J. McNab, On Compensator Design for Linear TimeInvariant DualInput Single Output Systems, IEEE/ASME Transactions on Mechatronics, Vol. 6, No. 1, March 2001.

P. D. Rohitha S. Senadheera and Jeff K. Pieper, Fully Automated PID and Lead/Lag Compensator Design tool for Industrial Use, in Proceedings of the 2005 IEEE Conference on Control Applications Toronto, Canada, August 2831, 2005.

FeiYue WANG, A New NonTrialandError Method for LagLead Compensator Design: A Special Case, International Journal of Intelligent Control and Systems Vol. 11, No. 1, March 2006, 6976.

Kai Shing Yeung, King W. Wong, and KanLin Chen, A NonTrial andError Method for LagLead Compensator Design, IEEE Transactions on Education, Vol. 41, No. 1, February 1998.

W. R. Wakeland, Bode compensator design, IEEE Transactions on Automatic Control, Vol. 21, 1974, 771773.

Teixeira M C M. Direct expressions for Ogata's leadlag design method using root locus. IEEE Tran. Education. 1994, 37(1):6364.

Roberto Zanasi, Stefania Cuoghi and Lorenzo Ntogramatzidis, Lead Lag compensators: analytical and graphical design on the Nyquist plane, 50th IEEE Conference on Decision and Control and European Control Conference (CDCECC) Orlando, FL, USA, December 1215, 2011.