A Review note on Compensator Design for Control Education and Engineering

DOI : 10.17577/IJERTV3IS21314

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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, India-783370

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 lag-lead 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, Lag-Lead compensator, root locus approach, frequency response approach.

  1. 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.

  2. 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 sub-systems 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 frequency-based design methods, especially for design based on the Bode plots [9]. Another alternative design in terms of root locus was given in [10]. A non-trial-and-error design technique for lag-lead 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 time-invariant dual- input/single-output systems in continuous time or discrete time is presented to reduce the problem to two single-input/single- output design problems. The development procedure of a fully automated software based One-Touch-and-Go 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 trial-and-error graphical techniques currently used by almost all available textbooks [1- 3] makes the learning and use of lag-lead compensation.

Where (0 < < 1) and ( > 0)

  1. 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 root-locus 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 frequency-response approach. Then finally the open-loop gain of the compensated system from the magnitude condition is determined.

    1. 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.

  2. 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 phase-lead angle to be added to the system is determined. An additional 5° to 12° to the phase-lead 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.

    1. 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 steady-state 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

  3. 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)

  1. Using Root Locus Approach

    The procedure for designing the lag compensator using root locus approach is briefly discussed here [1].

    Firstly by drawing the root-locus plot for the uncompensated system based on the transient-response specifications. The dominant closed-loop 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 root-locus plot for the compensated system is drawn then. And the desired dominant closed-loop 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 closed-loop poles lie at the desired location.

    1. 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

  2. Frequency Response Approach

    The open-loop 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 gain-adjusted 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 open-loop transfer function is equal to (- 180°) plus the required phase margin is measured. The required phase margin is the specified phase margin plus 5° to 12°. This addition of 5° to 12° 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

  3. 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 steady-state responses required improvement, a lag-lead compensator is required. It is a combination of a lag compensator and a lead compensator connected in series. The lag-lead 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 phase-lead portion of the lag-lead compensator is the portion involving the factor 1and the phase-lag portion is the portion having 2 factor.

  1. 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.

    1. 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 LAG-LEAD 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%

  2. Frequency Response Approach

    According to equation (4) the phase-lead portion alters the frequency-response curve by adding phase-lead angle and increasing the phase margin at the gain crossover frequency. The phase-lag portion provides attenuation near and above the gain crossover frequency and thereby allows an increase of gain at the low-frequency range to improve the steady-state performance.

    1. Example

      Uncompensated system is taken as =

      +1 (+2)

      () = (+0.1)(+0.24)

      and the overall transfer function is

      Designed lag-lead 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 lag-lead compensator is too large.

      V. CONCLUSION

      Fig.15. A typical Feed-forward Compensation System

      Fig.13. Bode plot of uncompensated and compensated system with lag-lead compensator

      Fig.14. Output of the compensated and uncompensated system for lag-lead compensator using frequency response analysis

      TABLE VI

      SPECIFICATION PARAMETERS FOR LAG-LEAD COMPENSATOR USING FREQUENCY RESPONSE ANALYSIS

      System

      PM

      GM

      (db)

      Tr (sec)

      Tp (sec)

      %Mp

      Ts (sec)

      Uncompe

      nsated

      -28.1°

      -10.5

      4.25

      6

      9.99

      15

      Compens

      ated

      55.2°

      16.9

      1.655

      2.485

      16.66

      15

  3. 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 steady-state 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

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

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

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

  4. Xu JianXin, New Lead Compensator Designs for Control Education and Engineering, Proceedings of the 27th Chinese Control Conference, Nov., July 16-18, 2008, Kunming,Yunnan, China.

  5. Steven J. Schroeck, William C. Messner, and Robert J. McNab, On Compensator Design for Linear Time-Invariant Dual-Input Single- Output Systems, IEEE/ASME Transactions on Mechatronics, Vol. 6, No. 1, March 2001.

  6. 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 28-31, 2005.

  7. Fei-Yue WANG, A New Non-Trial-and-Error Method for Lag-Lead Compensator Design: A Special Case, International Journal of Intelligent Control and Systems Vol. 11, No. 1, March 2006, 69-76.

  8. Kai Shing Yeung, King W. Wong, and Kan-Lin Chen, A Non-Trial- and-Error Method for Lag-Lead Compensator Design, IEEE Transactions on Education, Vol. 41, No. 1, February 1998.

  9. W. R. Wakeland, Bode compensator design, IEEE Transactions on Automatic Control, Vol. 21, 1974, 771-773.

  10. Teixeira M C M. Direct expressions for Ogata's lead-lag design method using root locus. IEEE Tran. Education. 1994, 37(1):63-64.

  11. 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 (CDC-ECC) Orlando, FL, USA, December 12-15, 2011.

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