11

COMPENSATORS

11.1 Introduction

The ultimate goal of examining the time domain analysis, frequency domain analysis and stability of a system in the previous chapters is to design a control system. The specific tasks for a system or requirements of a system can be achieved by a proper design of control systems. The performance specifications of a system generally relate to accuracy, stability and response time and these also vary based on the domain by which the system is to be designed. When the performance specifications of a system are given in time domain, the design can be carried out using any time-domain techniques and also if the performance specifications of a system are given in frequency domain, then the design can be carried out using any frequency domain techniques.

The general step followed in designing a control system is adjusting the gain of the system to meet its requirements. In practical situations, adjustment of gain may lead a system to unstable region or it may not be sufficient to meet the desired specifications. In such situations, additional devices or components can be added to the system to meet the specifications or to make the system stable. The additional devices or components that are added to a system are called compensators. The procedure for designing the compensators of a single-input single-output linear time invariant control system is discussed in this chapter. The steps to be followed in designing a control system are:

1. Determine how the system functions and what the system has to do.
2. Determine the controller or compensator that can be more suitable for the given ­system.
3. Determine the parameters or constants present in the controller or compensator transfer function that has been chosen in the previous step.

11.2 Compensators

The compensator circuits basically introduce a pole and/or zero to the existing system to meet the desired specifications. The compensators may be designed using electrical, mechanical, pneumatic or any other components. The basic idea of determining the ­transfer function of the compensator is to suitably place the dominant closed-loop poles of a ­system. Compensated systems can be classified into six categories based on the location of ­compensators in the system as follows:

1. Series or cascade compensation
2. Feedback or parallel compensation
3. Load or series-parallel compensation
4. State feedback compensation
5. Forward compensation with series compensation
6. Feed-forward compensation

11.2.1 Series or Cascade Compensation

In this type of compensated system, the compensator is included in the feed-forward path of the system as shown in Fig. 11.1(a).

Fig. 11.1(a) ∣ Series or cascade compensation

The addition of compensator in the feed-forward path adjusts the gain of a system which reduces the response time and peak overshoot of the system. In addition, the stability of the system gets reduced.

11.2.2 Feedback or Parallel Compensation

In this type of compensated system, the compensator is included in the feedback path of a system as shown in Fig. 11.1(b).

Fig. 11.1(b) ∣ Parallel or feedback compensation

The addition of compensator in the feedback path increases the response time of the ­system that makes it accurate and more stable.

11.2.3 Load or Series-Parallel Compensation

The combination of both series and parallel compensation shown in Fig. 11.1(c) is known as load or series - parallel compensation.

Fig. 11.1(c) ∣ Series-parallel or load compensation

11.2.4 State Feedback Compensation

In this type of compensation, the control signal in the form of state variable is fed back as a control signal through the constant real gain as shown in Fig. 11.1(d).

Fig. 11.1(d) ∣ State feedback compensation

The implementation of the state feedback compensation is costly and impractical for higher order systems.

11.2.5 Forward Compensation with Series Compensation

When a simple closed-loop system is in series with the feed-forward controller, , the resultant compensation is the forward compensation with series compensation that is shown in Fig. 11.1(e).

Fig. 11.1(e) ∣ Forward compensation with series compensation

11.2.6 Feed-forward Compensation

When the feed-forward controller is placed in parallel with the forward path of a simple closed-loop system, the resultant compensation is the feed-forward compensation that is shown in Fig. 11.1(f).

Fig. 11.1(f) ∣ Feed-forward compensation

The compensated systems shown in Figs. 11.1(a), 11.1(b) and 11.1(d) have one degree of freedom which intimates that the system has single controller. The disadvantage of one degree of freedom controller is that the performance criteria realized using these ­compensation techniques are limited.

In simple words, the compensators introduce additional poles/zeros to an ­existing system so that the desired specification is achieved.

11.2.7 Effects of Addition of Poles

The following are the effects of addition of poles to an existing system:

1. The root locus of a compensated system will be shifted towards the right-hand side of the s-plane.
2. Stability of a system gets lowered.
3. Settling time of a system increases.
4. Accuracy of a system is improved by the reduction of steady-state error.

11.2.8 Effects of Addition of Zeros

The following are the effects of addition of zeros to an existing system:

1. The root locus of a compensated system will be shifted towards the left-hand side of the s-plane.
2. Stability of a system gets increased.
3. Settling time of a system decreases.
4. Accuracy of a system is lowered as steady-state error of the system increases.

In this chapter, three types of compensators used in the electrical systems are discussed. They are:

1. Lag compensators
2. Lead compensators
3. Lag–lead compensators

11.2.9 Choice of Compensators

The choice of compensators from the different categories discussed in the previous sections is based on the following factors:

1. Nature of signal to the system
2. Available components
3. Experience of the designer
4. Cost
5. Power levels at different points and so on

11.3 Lag Compensator

The lag compensator is one that has a simple pole and a simple zero in the left half of the s-plane with the pole nearer to the origin. The term lag in the lag compensator means that the output voltage lags the input voltage and the phase angle of the denominator of the transfer function is greater than that of numerator. The general transfer function of the lag compensator is given by

, (11.1)

where are the constants and .

The pole-zero configuration of the lag compensator is shown in Fig. 11.2.

Fig. 11.2 ∣ Pole-zero configuration of G_la(s)

The corner frequencies present in the lag compensator, whose transfer function is given by Eqn. (11.1) are at and . The Bode plot and polar plot of the lag compensator are shown in Figs. 11.3(a) and 11.3(b) respectively.

Fig. 11.3 ∣ Plots of lag compensator

The Bode plot shown in Fig. 11.3(a) is plotted with and . It is inferred that (i) magnitude of lag compensator is high at low frequencies and (ii) magnitude of lag compensator is zero at high frequencies. Hence, from the above conclusions, it is clear that the lag compensator behaves like a low-pass filter. The value of is chosen between 3 and 10. The magnitude plot and phase plot of the compensator with different values of are shown in Figs. 11.4(a) and 11.4(b) respectively.

Fig. 11.4 ∣ Bode plot for different values of (a) magnitude plot and (b) phase plot

11.3.1 Determination of Maximum Phase Angle Φ_m

The modified transfer function of the lag compensator is

(11.2)

The magnitude and phase angle of the lag compensator are

(11.3)

and (11.4)

The maximum phase angle occurs at

Differentiating Eqn. (11.4) with respect to ω, we obtain

Solving the above equation, we obtain

(11.5)

Substituting the above equation in Eqn. (11.4), we obtain

(11.6)

Thus, Eqn. (11.5) gives the frequency at which the phase angle of the system is maximum and Eqn. (11.6) gives the maximum phase angle of the lag compensator.

11.3.2 Electrical Representation of the Lag Compensator

A simple lag compensator using resistor and capacitor is shown in Fig. 11.5.

Fig. 11.5 ∣ A simple lag compensator

Applying Kirchoff's voltage law to the above circuit, we obtain

(11.7)

and (11.8)

Taking Laplace transform on both sides of Eqs. (11.7) and (11.8), we obtain

(11.9)

and (11.10)

Substituting from Eqs. (11.10) to (11.9), we obtain

Therefore, the transfer function of the above circuit is

Rearranging the above equation, we obtain

(11.11)

Comparing Eqn. (11.1) and Eqn. (11.11), we obtain

, and .

11.3.3 Effects of Lag Compensator

The following are the effects of adding lag compensator to a given system are:

1. The lag compensator attenuates the high-frequency noise signals in the control loop.
2. It increases the steady-state error constants of a system.
3. Gain crossover frequency of a compensated system gets lowered.
4. Bandwidth of the compensated system decreases.
5. Maximum peak overshoot, rise time and settling time of the system increases.
6. The system becomes more sensitive to the parameter variations.
7. As it acts like a proportional integral controller, it makes the system less stable.
8. Transient response of the compensated system becomes slower.

11.3.4 Design of Lag Compensator

The objective of designing the lag compensator is to determine the values of and for an uncompensated system based on the desired system requirements. The design of lag compensator is based on the frequency domain specifications or time-domain specifications. The Bode plot is used for designing the lag compensator based on the frequency domain specifications, whereas the root locus technique is used for designing the lag compensator based on the time-domain specifications. Once the values of and are determined, the transfer function of the lag compensator can be obtained. The transfer function of the compensated system is . If the Bode plot or root locus technique is plotted for the compensated system, it will satisfy the desired system requirements.

11.3.5 Design of Lag Compensator Using Bode Plot

Consider the open-loop transfer function of the uncompensated system . The objective is to design a lag compensator for so that the compensated system will satisfy the desired system requirements. The steps for determining the transfer function of the compensated system using Bode plot are explained below:

Step 1:If the open-loop transfer function of the system has a variable , then or .

Step 2:Depending on the input and TYPE of the system, the variable present in the transfer function of the uncompensated system is determined based on either the steady-state error or the static error constant of the system.

The static error constants of the system are:

Position error constant,

Velocity error constant,

Acceleration error constant,

The relation between the steady-state error and static error constant based on the TYPE of the system and input applied to the system can be referred to Table 5.5 of Chapter 5.

Step 3:Construct the Bode plot for with gain obtained in the previous step and determine the frequency domain specifications of the system (i.e., phase margin, gain margin, phase crossover frequency and gain crossover frequency).

Step 4:Let the desired phase margin of the system be . With a tolerance , determine as

where to .

Step 5:Determine the new gain crossover frequency of the system for the phase margin . Let it be .

Step 6:Determine the magnitude A of the system in dB from the magnitude plot corresponding to .

Step 7:As the magnitude of system at the gain crossover frequency must be zero, the Bode plot must be either increased or decreased by dB.

Step 8:The value of in the transfer function of the lag compensator will be determined as

Here is when the magnitude plot is to be increased and is when the magnitude plot is to be decreased.

Step 9:The value of in the transfer function of the lag compensator will be determined by using the equation, .

Step 10: Thus, the transfer function of the lag compensator will be determined as

Step 11: The transfer function of the compensated system will be . If the Bode plot for the compensated system is drawn, it will satisfy the desired system requirements.

The flow chart for determining the parameters present in the transfer function of the lag compensator using Bode plot is shown in Fig. 11.6.

Fig. 11.6 ∣ Flow chart for designing the lag compensator using Bode plot

Solution

1. Let and desired phase margin .
2. The value of is determined by using as

   

   Solving the above equation, we obtain .

3. The Bode plot for H(s) = 1 is drawn.
   1. Given
   2. Substituting and in the above equation, we obtain

      

   3. The corner frequency existing in the given system is

      rad/sec

      To sketch the magnitude plot:

   4. The changes in slope at different corner frequencies are given in Table E11.1(a).

      Table E11.1(a) ∣ Determination of change in slope at different corner frequencies

      
   5. Assume the lower frequency as ω_l = 0.1 rad/sec and higher frequency as = 20 rad/sec.
   6. The values of gain at different frequencies are determined and given in Table E11.1(b).

      Table E11.1(b) ∣ Gain at different frequencies

      
   7. The magnitude plot of the given system is plotted using Table E11.1(b) and is shown in Fig. E11.1.

      To sketch the phase plot:

   8. The phase angle of the given loop transfer function as a function of frequency is obtained as

      

   9. The phase angle at different frequencies is obtained using the above equation and the values are tabulated as shown in Table E11.1(c).

      Table E11.1(c) ∣ Phase angle of a system for different frequencies

      
   10. The phase plot of a given system is plotted with the help of Table E11.1(c) and is shown in Fig. E11.1(a).
       

       Fig. E11.1(a)

   11. The frequency domain specifications of the given system are:

       Gain crossover frequency, rad/sec.

       Phase crossover frequency, rad/sec.

       Gain margin, dB

       Phase margin,

       It can be noted that the uncompensated system is stable, but the phase margin of the system is less than the desired phase margin which is .

4. Let .
5. The new gain crossover frequency for is = 0.92 rad/sec.
6. The magnitude of the system corresponding to is A = 27 dB.
7. The magnitude plot of the uncompensated system is decreased by 27 dB so that the gain at is zero dB.
8. The value of in the lag compensator will be determined as

   

   i. e.,

9. The value of in the transfer function of the lag compensator is determined as

   

   i. e.,

10. Thus, the transfer function of the lag compensator is

    

11. Thus, the transfer function of the compensated system is

11.3.6 Design of Lag Compensator Using Root Locus Technique

When a system is desired to meet the static error constant alongwith other time-domain specifications such as peak overshoot, rise time, settling time, damping ratio of the system and undamped natural frequency of oscillation, then lag compensator will be designed using root locus technique. The step-by-step procedure for designing the lag compensator using root locus technique is discussed below:

Step 1:The root locus of an uncompensated system with the loop transfer function is constructed.

Step 2:Determine using .

Step 3:Draw a line from origin with an angle from the negative real axis and determine the point at which it cuts the root locus of the uncompensated system. Let that point be the dominant pole of the closed-loop system .

Step 4:If the uncompensated system has a gain , the gain is determined by using the formula , or else, we can proceed to the next step.

Step 5:The static error constant of the uncompensated system is determined by

Position error constant,

Velocity error constant,

Acceleration error constant,

Let the static error constant determined for the system be .

Step 6:Determine the factor by which the static error constant is to be increased is determined by using

Step 7:Select the zero and pole of the lag compensator which lie very close to the origin such that the pole lies right to zero of the compensator. Let z and p be the zero and pole of the compensator. The pole and zero are chosen such that z = 10p.

Step 8:The transfer function of the lag compensator is obtained as .

Step 9:The transfer function of the compensated system is obtained as .

Step 10:The root locus of the compensated system is drawn and with the help of the damping ratio , the dominant pole of the compensated system is determined. The new dominant pole obtained is .

Step 11:The constant is determined by using .

Step 12:If the static error constant of the compensated system is determined, it will satisfy the desired specifications.

Step 13:If the static error constant does not satisfy the desired specifications, then alternative values of z and p are chosen and step 7–12 will be continued; otherwise, we can stop the procedure.

Thus, the transfer function of the lag compensator using root locus technique is ­determined.

The flow chart for designing the lag compensator using root locus technique is shown in Fig. 11.7.

Fig. 11.7 ∣ Flow chart for designing the lag compensator

Solution:

1. The root locus for an uncompensated system is drawn.
   1. For the given system , the poles are at 0, –1 and –4 i.e., and zeros does not exist i.e., .
   2. Since , the number of branches of the root loci for the given system is .
   3. The details of the asymptotes are:
      1. Number of asymptotes for the given system .
      2. Angles of asymptotes

         for

         Therefore,

      3. Centroid

         

   4. As all the poles and zeros are real values, there is no necessity to calculate the angle of departure and angle of arrival.
   5. To determine the number of branches existing on the real axis.

      If we look from the pole p = –1, the total number of poles and zeros existing on the right of –1 is one (odd number). Therefore, a branch of root loci exists between –1 and 0.

      Similarly, if we look from the point at , the total number of poles and zero existing on the right of the point is three (odd number). Therefore, a branch of root loci exists between and –4.

      Hence, two branches of root loci exist on the real axis for the given system.

   6. The breakaway/break-in points for the given system are determined by

      (1)

      For the given system, the characteristic equation is

      (2)

      i.e.,

      Therefore,

      Differentiating the above equation with respect to and using Eqn. (1), we obtain

      

      

      Solving for s, we obtain

      

      (i) For , K is 0.8794. Since is positive, the point is a breakaway point. (ii) For , K is –6.064. Since is negative, the point is not a breakaway point.

      Hence, only one breakaway point exists for the given system.

   7. The point at which the branch of root loci intersects the imaginary axis is ­determined as follows:

      Using Eqn. (2), the characteristic equation for the given system is

      

      i.e., (3)

      Routh array for the above equation is

      

      To determine the point at which the root loci crosses the imaginary axis, the first element in the third row must be zero i.e., . Therefore, .

      As K is a positive real value, the root locus crosses the imaginary axis and the point at which it crosses imaginary axis is obtained by substituting K in Eqn. (3) and solving for s.

      The solutions for the cubic equation are .

      Hence, the point in the imaginary axis where the root loci crosses is .

      The complete root locus for the system is shown in Fig. E11.2.

   

   Fig. E11.2

2. Using and settling time , the dominant closed-loop poles are

   (since ).

3. The gain K at dominant closed-loop poles is obtained using the magnitude condition,

   

   

   

   i. e., .

   Therefore, the transfer function of the compensated system is .

4. The static error constant for the uncompensated system is

   .

5. The factor by which the static error constant to be increased is determined by using

   Factor

6. Let the zero of the compensator be at 0.1 and the pole of the compensator be at 0.01.
7. Therefore, the transfer function of the lag compensator is

   .

8. 8. Thus, the transfer function of the compensated system is

   

   where

11.4 Lead Compensator

The lead compensator is one that has a simple pole and a simple zero in the left half of the s-plane with the zero nearer to the origin. The term lead in the lead compensator refers that the output voltage leads the input voltage and the phase angle of the numerator of the ­transfer function is greater than that of denominator. The general transfer function of the lead ­compensator is given by

, (11.12)

where and are constants.

The pole-zero configuration of the lead compensator is shown in Fig. 11.8.

Fig. 11.8 ∣ Pole-zero configuration of G_1e(s)

The corner frequencies present in the lead compensator whose transfer function is given by Eqn. (11.12) are at and . The Bode plot and polar plot of the lead compensator are shown in Figs. 11.9(a) and 11.9(b) respectively.

(a)

(b)

Fig. 11.9 ∣ Plots of lead compensator

The Bode plot shown in Fig. 11.9(a) is plotted with and . It is inferred that (i) magnitude of lead compensator is low at low frequencies and (ii) magnitude of lead compensator is zero at high frequencies. Hence, the lead compensator behaves like a high-pass filter. The magnitude and phase plots for different values of are shown in Figs. 11.10(a) and 11.10(b) respectively.

(a)

(b)

Fig. 11.10 ∣ Bode plot for different values of (a) magnitude plot and (b) phase plot

11.4.1 Determination of Maximum Phase Angle Φ_m

The modified transfer function of the lead compensator is

(11.13)

The magnitude and phase angle of the lead compensator are

(11.14)

and (11.15)

The maximum phase angle occurs at

Differentiating Eqn. (11.15) with respect to , we obtain

Solving the above equation, we obtain

(11.16)

Substituting the above equation in Eqn. (11.15), we obtain

(11.17)

Thus, Eqn. (11.16) gives the frequency at which the phase angle of the system is maximum and Eqn. (11.17) gives the maximum phase angle of the lag compensator.

11.4.2 Electrical Representation of the Lead Compensator

A simple lead compensator using resistor and capacitor is shown in Fig. 11.11.

Fig. 11.11 ∣ A simple lead compensator

Applying Kirchoff's current law to the above circuit, we obtain

(11.18)

Taking Laplace transform on both sides, we obtain

(11.19)

Therefore, the transfer function of the above circuit is

(11.20)

Comparing Eqs. (11.12) and (11.20), we obtain

and

11.4.3 Effects of Lead Compensator

The effects of adding lead compensator to the given system are:

1. Damping of the closed-loop system increases since a dominant zero is added to the system.
2. Peak overshoot of the system, rise time and settling time of the system decrease and as a result of which the transient response of the system gets improved.
3. Gain margin and phase margin of the system get increased.
4. Improves the relative stability of the system.
5. Increases the bandwidth of the system that corresponds to the faster time response.

11.4.4 Limitations of Lead Compensator

The limitations of adding lead compensator to the given system are:

1. Single-phase lead compensator can provide a maximum phase lead of . If a phase lead of more than is required, multistage compensator must be used.
2. There is always a possibility of reaching the conditionally stable condition even though the desired system requirements are achieved.

11.4.5 Design of Lead Compensator

The objective of designing the lead compensator is to determine the values of and for an uncompensated system based on the desired system requirements. The design of lead compensator can be either based on the frequency domain specifications or time- domain ­specifications. The Bode plot is used for designing the lead compensator based on the ­frequency domain specifications and root locus technique is used for designing the lead ­compensator based on the time-domain specifications. Once and are ­determined, the transfer function of the lead compensator can be obtained. The ­transfer function of the compensated system . If the Bode plot or root locus technique is plotted for the compensated system, it will satisfy the desired system requirements.

11.4.6 Design of Lead Compensator Using Bode Plot

Let the open-loop transfer function of the uncompensated system be . The ­objective is to design a lead compensator for so that the compensated system will ­satisfy the desired system requirements. The steps for determining the transfer function of the compensated system using Bode plot are explained below:

Step 1:If the open-loop transfer function of the system has a variable , then ; otherwise .

Step 2:Depending on the input and TYPE of the system, the variable present in the open-loop transfer function of the uncompensated system is determined based on either the steady-state error or the static error ­constant of the system.

The static error constants of the system are

Position error constant,

Velocity error constant,

Acceleration error constant,

The relation between the steady-state error and static error constant based on the TYPE of the system and input applied to the system can be referred to Table 5.5 of Chapter 5.

Step 3:Construct the Bode plot for the uncompensated system with gain obtained in the previous step and determine the frequency domain specifications of the system (i.e., phase margin, gain margin, phase crossover frequency and gain crossover frequency). Let the phase margin of the uncompensated system be .

Step 4:Let the desired phase margin of the system be . With a tolerance , determine as

where to .

Step 5:Determine using .

Step 6:Determine in dB. Let it be .

Step 7:Determine the frequency from the magnitude plot of the uncompensated system for the magnitude of dB. Let this frequency be the new gain crossover frequency .

Step 8:Determine using .

Step 9: Determine using .

Step 10: Thus, the transfer function of the lead compensator will be determined as

Step 11: The transfer function of the compensated system will be . If the Bode plot for the compensated ­system is drawn, it will satisfy the desired system requirements.

Flow chart for designing the lead compensator using Bode plot

The flow chart for determining the parameters present in the transfer function of the lead compensator using Bode plot is shown in Fig. 11.12.

Fig. 11.12 ∣ Flow chart for designing the lead compensator using Bode plot

Solution:

1. Let .
2. The value of is determined by using as

   

   Solving the above equation, we obtain .

3. 3. The Bode plot for H(s) = 1 is drawn.
   1. Given
   2. Substituting in the above equation, we obtain

      

   3. The corner frequency existing in the given system is

      rad/sec

      To sketch the magnitude plot:

   4. The changes in slope at different corner frequencies are given in Table E11.3(a).

      Table E11.3(a) ∣ Determination of change in slope at different corner frequencies

      
   5. Assume the lower frequency as ω_l = 0.1 rad/sec and higher frequency as ω_h = 20 rad/sec.
   6. The values of gain at different frequencies are determined and given in Table E11.3(b).

      Table E11.3(b) ∣ Gain at different frequencies

      
   7. The magnitude plot of the given system is plotted using Table E11.3(b) and is shown in Fig. E11.3.

      To sketch the phase plot:

   8. (h) The phase angle of the given loop transfer function as a function of frequency is obtained as

      

   9. The phase angle at different frequencies is obtained using the above equation and the values are tabulated as shown in Table E11.3(c).

      Table E11.3(c) ∣ Phase angle of the system for different frequencies

      
   10. (j) The phase plot of the given system is plotted with the help of Table E11.3(c) and is shown in Fig. E11.3.
       

       Fig. E11.3

   11. (k) The frequency domain specifications of the given system are

       Gain crossover frequency, rad/sec

       Phase crossover frequency, rad/sec

       Gain margin, dB and

       Phase margin,

       It can be noted that the uncompensated system is stable, but the phase margin of the system is less than the desired phase margin.

4. With tolerance value , is calculated as

   i. e., .

5. The value of in the lead compensator is determined by using, .

   i.e.,

   

   Solving the above equation, we obtain

   

6. Let dB
7. The new gain crossover frequency corresponding to B from the magnitude plot of the uncompensated system is 5 rad/sec.
8. Using , determine T as .

   i. e., .

9. The value of is determined as i.e., .
10. The transfer function of the lead compensator is

    

11. Thus, the transfer function of the compensated system is

11.4.7 Design of Lead Compensator Using Root Locus Technique

The desired time-domain specifications that can be specified for designing the lead compensator are peak overshoot, settling time, rise time, damping ratio and undamped natural frequency of the system. The step-by-step procedure for designing the lead compensator for an uncompensated system whose open-loop transfer function is given by using root locus technique is given below:

Step 1: Determine the damping ratio and undamped natural frequency of the system based on the given time-domain specifications.

Step 2: Determine the dominant closed-loop poles of the system using or . Let and be the dominant closed-loop poles of the system that is marked in s-plane as shown in Fig. 11.13(a).

Fig. 11.13(a)

Step 3: Determine the angle of the loop transfer function at any one of the dominant pole i.e., either or . Let the angle obtained be degrees i.e.,

Step 4: The angle obtained in the previous step, i.e., should be an odd multiple of . If it is not so, some values of angle can be added or subtracted from to make it an odd multiple of . The angle by which has to be added or subtracted is obtained using . The value of is always less than and if is greater than , then multiple lead compensators can be used.

Step 5: Draw a line parallel to the X-axis from point to and also join the point with the origin as shown in Fig. 11.13(b).

Fig. 11.13(b)

Step 6: Determine and using draw a line from the point that bisects the negative X-axis at point as shown in Fig. 11.13(b).

Step 7: Draw two lines and from point P₁ such that , which is shown in Fig. 11.13(c).

Fig. 11.13(c)

Step 8: The points and in the negative real axis corresponds to the pole and zero of the lead compensator respectively.

Step 9: The constants in the lead compensator ( and ) are determined by using and .

Step 10: Using the constants determined in the previous step, the transfer function of the lead compensator is obtained as

.

Step 11: The transfer function of the compensated system is obtained using

Step 12: The constant of the system is determined by using .

Step 13: If the root locus of the compensated system is drawn, we can see that the root locus of the compensated system will pass through the dominant closed-loop poles of the system.

The flow chart for designing the lead compensator using the root locus technique is shown in Fig. 11.14.

Fig. 11.14 ∣ Flow chart for designing the lead compensator

Solution

1. Refer to the solution of Example 11.2 and follow the steps to obtain the root locus of the system.

   The complete root locus for the system is shown in Fig. E11.4.

2. Using and undamped natural frequency = 2, the dominant closed-loop poles are .
   

   Fig. E11.4

3. The angle of the system at dominant closed-loop pole is

   

4. Since the angle obtained in the previous step is not an odd multiple of , the angle to be contributed by lead compensator is given by .
5. A line parallel to the X-axis from point to and also the point is joined with the origin.
6. The angle is determined as and using , a line from the point is drawn which bisects the negative X-axis at point .
7. Two lines and are drawn from point such that
8. The points C and D are the poles and zeros of the lead compensator respectively. Since the zero is at and there exists pole-zero cancellation, the zero of lead compensator is taken slightly left of point –1. Let it be . Therefore, and
9. The constants in the lead compensator are determined by using and Therefore, and .
10. Therefore, the transfer function of lead compensator is
11. Thus, the transfer function of the compensated system is .
12. Using the magnitude condition, , the constant is determined as

    

    .

    Thus, the transfer function of the compensated system is

    

11.5 Lag–Lead Compensator

In the previous sections, the unique advantages, disadvantages and limitations of the lead compensator and lag compensator have been discussed. Lead compensator will improve the rise time and damping and also affects natural frequency of the system and lag compensator will improve the damping of the system, but it also increases the rise time and settling time. Therefore, lag–lead compensator is a combination of lag and lead compensators which is used for a system to gain the individual advantages of each compensator. Also, the use of both the compensators is necessary for some systems as the desired result cannot be achieved when the compensators are used alone. In general, in lag–lead compensator, the lead compensator is used for achieving a shorter rise time and higher bandwidth and lag compensator is used for achieving good damping for a system.

The general transfer function of the lag–lead compensator is given by

(11.21)

where are constants and

The term in Eqn. (11.21) which produces the effect of lead compensator is

The term in Eqn. (11.21) that produces the effect of lag compensator is

The pole-zero configuration of the lag–lead compensator is shown in Fig. 11.15.

Fig. 11.15 ∣ Pole-zero configuration of G_la–le (s)

The Bode plot and polar plot of the lag–lead compensator is shown in Figs. 11.16(a) and 11.16(b) respectively.

(b)

Fig. 11.16 ∣ Plots of lag–lead compensator

The Bode plot shown in Fig. 11.16(a) is plotted with and . The conclusion inferred from the Bode plot of the lag compensator shown in Fig. 11.16(a) is that magnitude of lag–lead compensator is zero at low and high frequencies.

11.5.1 Electrical Representation of the Lag–Lead Compensator

A simple lag-lead compensator using resistor and capacitor is shown in Fig. 11.17.

Fig. 11.17 ∣ A simple lag–lead compensator

Applying Kirchoff's current law to the above circuit as shown in Fig. 11.17, we obtain

(11.21)

The output voltage of the circuit is given by

(11.22)

Taking Laplace transform on both sides of the above equations, we obtain

(11.23)

(11.24)

Substituting Eqn. (11.24) in Eqn. (11.23), we obtain

Simplifying the above equation, we obtain

Therefore, the transfer function of the above circuit is

Therefore,

(11.25)

Rearranging Eqn. (11.21), we obtain

(11.26)

Comparing Eqn. (11.26) with Eqn. (11.25), we get

, ,

From the above representation, it can be noted that the values of and should be same for the lag–lead compensator.

11.5.2 Effects of Lag–Lead Compensator

The following are the effects of adding lag–lead compensator to the given system:

1. Used when both the fast response and good static accuracy for a system is required.
2. Increases low-frequency gain that improves steady state.
3. Increases the bandwidth of the system that results in faster response of the system.

11.5.3 Design of Lag–Lead Compensator

The objective of designing the lag–lead compensator is to determine the values of constants ( and ) present in the transfer function of a compensator for a uncompensated ­system based on the desired system requirements. The design of lag–lead compensator can be either based on the frequency domain specifications or based on time-domain ­specifications. The Bode plot is used for designing the lag–lead compensator based on the frequency domain specifications and root locus technique is used for designing the lag-lead compensator based on the time-domain specifications. Once the constant is determined, the transfer function of the lag–lead compensator can be obtained. The transfer function of the compensated system is ; and if the Bode plot or root locus technique is plotted for the compensated system, it will satisfy the desired system requirements.

11.5.4 Design of Lag–Lead Compensator Using Bode Plot

Consider the open-loop transfer function of the uncompensated system be . The ­objective is to design a lag–lead compensator for so that the compensated system will satisfy the desired system requirements. The step by step procedure for determining the transfer function of the compensated system using Bode plot is explained below:

Step 1:If the open-loop transfer function of the system has a variable , then ; otherwise .

Step 2:Depending on the input and TYPE of the system, the variable present in the open-loop transfer function of the uncompensated system is determined based on either the steady-state error or the static error constant of the system.

The static error constants of the system are

Position error constant,

Velocity error constant,

Acceleration error constant,

The relation between the steady-state error and static error constants based on the TYPE of the system and input applied to the system can be referred to Table 5.5 of Chapter 5.

Step 3:Construct the Bode plot for the uncompensated system with gain obtained in the previous step and determine the frequency domain specifications of the ­system (i.e., phase margin, gain margin, phase crossover frequency and gain crossover frequency). Let the phase margin, gain margin, phase crossover frequency and gain crossover frequency of the uncompensated system be and .

Step 4:Let the desired phase margin of the system be .

Step 5:Determine using and .

Step 6:Determine the frequency of the system at which the phase plot of the uncompensated system is . Let the frequency be the new gain crossover frequency .

Step 7:Determine the constant, .

Step 8:Determine the transfer function of the lag compensator as .

Step 9:Determine the magnitude of uncompensated system at . Let it be dB.

Step 10:Draw a line from the point () so that it bisects the dB line and dB line and determine the frequencies corresponding to the intersection points. Let rad/sec and rad/sec be the frequencies at which the line intersects dB line and dB line respectively.

Step 11:Determine the constants and using and .

Step 12:Thus, the transfer function of the lag–lead compensator will be determined as

Step 10:The transfer function of the compensated system will be . If the Bode plot for the compensated system is drawn, it will satisfy the desired system requirements.

Flow chart for designing the lag–lead compensator using Bode plot

The flow chart for determining the parameters present in the transfer function of the lag–lead compensator using Bode plot is shown in Fig. 11.18.

Fig. 11.18 ∣ Flow chart for designing the lag–lead compensator using Bode plot

Solution:

1. Let .
2. The gain is determined by using as

   

   Solving the above equation, we obtain .

3. The Bode plot for H(s) = 1 is drawn.
   1. Given
   2. Substituting in the above equation, we obtain

      

   3. The corner frequencies existing in the given system are

      rad/sec and rad/sec.

      To sketch the magnitude plot:

   4. The changes in slope at different corner frequencies are given in Table E11.5(a).

      Table E11.5(a) ∣ Determination of change in slope at different corner frequencies

      
   5. Assume the lower frequency as = 0.1 rad/sec and higher frequency as = 20 rad/sec.
   6. The gain at different frequencies are determined and given in Table E11.5(b).
   7. The magnitude plot of the given system is plotted using Table E11.1(b) and is shown in Fig. E11.5.

      Table E11.5(b) ∣ Gain at different frequencies

      

      To sketch the phase plot:

   8. The phase angle of the given loop transfer function as a function of frequency is obtained as

      

   9. The phase angle at different frequencies is obtained using the above equation and the values are tabulated as shown in Table E11.5(c).

      Table E11.5(c) ∣ Phase angle of the system for different frequencies

      
   10. The phase plot of the given system is plotted with the help of Table E11.5(c) and is shown in Fig. E11.5.
       

       Fig. E11.5

   11. The frequency domain specifications of the given system are:

       Gain crossover frequency, rad/sec.

       Phase crossover frequency, rad/sec.

       Gain margin, dB

       Phase margin, .

       It is to be noted that the uncompensated system is stable, but the phase margin of the system is less than the desired phase margin which is .

4. The desired phase margin for the compensated system is .
5. Using and , is determined as 7.54.
6. Let
7. The time constant of lag compensator is determined as

   

8. Therefore, the transfer function of lag compensator is .
9. The magnitude of the system at dB.
10. If the slope of +20 dB/decade is drawn from (1.4 rad/sec, –14 dB), the slope cuts the 0 dB line and –20 dB line at rad/sec and at rad/sec respectively.
11. Thus, the transfer function of the lead portion of the lag–lead compensator is

    

12. Therefore, the transfer function of the lag–lead compensator is

    

11.5.5 Design of Lag–Lead Compensator Using Root Locus Technique

The transfer function of the lag–lead compensator is given by , and . The design of lag–lead compensator using root locus technique is based on the relation between β and γ. Therefore, two different cases exist in designing the lag–lead compensator for a system.

Case 1: When

The step-by-step procedure for designing the lag–lead compensator using root locus ­technique when is discussed below.

Step 1:Determine the damping ratio and undamped natural frequency of the system based on desired time-domain specifications of the system.

Step 2:Determine the dominant pole of the system using . Let the dominant poles of the system be and .

Step 3:Follow the procedure shown in Fig. 11.14 in determining the constants of lead compensator, i.e., and .

Step 4:Determine the constant using the condition .

Step 5:Using the static error constant and constants determined in the previous steps, the value of is determined.

Step 6:Determine using

Step 7:Now, determine the transfer function of the lag–lead compensator .

Step 8:The transfer function of the compensated system is .

Flow chart for designing the lag–lead compensator using root locus technique when

The step-by-step procedure for designing the lag–lead compensator using root locus ­technique when is shown in Fig. 11.19.

Fig. 11.19 ∣ Flow chart for designing lag-lead compensator when

Case 2: When

The step-by-step procedure for designing the lag–lead compensator using root locus ­technique when is discussed below:

Step 1:Determine the damping ratio and undamped natural frequency of the system based on desired-time domain specifications of the system.

Step 2:Determine the dominant pole of the system using . Let the dominant poles of the system be and .

Step 3:Using the formula for static error constant, determine the value of constant .

Step 4:Determine the angle of the loop transfer function at any one of the ­dominant pole, i.e., either or . Let the angle obtained be degrees.

Step 5:The angle obtained in the previous step, i.e., should be an odd multiple of . If it is not so some value of angle can be added or subtracted from to make it an odd multiple of . The angle by which has to be added or subtracted is obtained using the formula, . The value of is always less than and if is greater than , then multiple lead compensators can be used.

Step 6:Determine the length of points OA and OB such that and . The points in the s-plane are shown in Fig. 11.20.

Fig. 11.20

Step 7:Determine the constants and using the formula and .

Step 8:Determine using .

Step 9:The transfer function of the lag–lead compensator is determined as .

Flow chart for designing the lag–lead compensator using root locus technique when

The step-by-step procedure for designing the lag–lead compensator using root locus technique when is shown in Fig. 11.21.

Fig. 11.21 ∣ Flow chart for designing lag-lead compensator when

Review Questions

1. What is the principle of compensation? What are the types of compensation?
2. Draw the electric network that could be used for phase lag compensation. Derive its transfer function and hence obtain Bode plot.
3. Derive the transfer function of phase lag network and sketch the frequency response curves to the same.
4. Under what circumstances a lag compensator is preferred? Why?
5. What are the steps involved in designing a phase-lag compensator?
6. What are the advantages of the lag compensator?
7. Give the diagram to lead compensator.
8. How are lag and lead compensators realized using electric circuits?
9. Under what circumstances is a lead compensator preferred? Why?
10. Show how a phase lead compensation network can be designed.
11. Derive the transfer function of phase lead network and obtain the necessary expressions for system compensation with lead network.
12. Enumerate the design steps involved in the phase lead compensation.
13. Compare lead, lag and lag-lead compensation.
14. Write a short note on series compensation.
15. Explain the features of feedback compensation.
16. Write short notes on the feedback compensation.
17. Show the realization of a lag–lead compensator using electrical circuit.
18. Write a note on series compensation using lag-lead networks.
19. What is meant by feedback compensation?
20. Consider a TYPE 1 system with open loop transfer function Design a lag compensator to meet the following specifications:

    Damping ratio

    Settling time = 10 sec

    Velocity error constant

21. For a unity feedback system with open-loop gain design a compensator to meet the following:

    Damping ratio

    Settling time

    Velocity error co-efficient

22. Determine the transfer function of a lead compensator that will provide a phase lead of and gain of 8 db at
23. A unity feedback network system has . Design a compensator ­network for the system so that the static velocity error constant and the phase margin is at least .
24. Design a lead compensator for a unity-feedback open-loop transfer function to meet the following specifications:

    Damping ratio

    Settling time

    and Velocity error constant

25. Design a lag compensator for the system with unity feedback to achieve the following specifications:

    Velocity error constant

    and phase margin

26. A unity feedback system with the open-loop transfer function is required to have (i) velocity error constant and (ii) phase margin . Design a suitable phase lag compensator to meet the above specifications.
27. Explain the steps involved in the time-domain design of phase lead ­compensation.
28. The open-loop transfer function of a system is given as . Design a lead compensation for the system to meet the following specifications:

    (i) Setting time 4 sec and (ii) peak over shoot for step input %.

29. A unity feedback system with forward path transfer function is to have (i) the velocity error constant and (ii) phase margin . Design a suitable phase-lag compensator to meet the above specifications.
30. A unity feedback system with is to be compensated to have a phase margin of without sacrificing the velocity error constant. Design a suitable lag network and compute the component values for a suitable impedance level.
31. Design a compensator for a unity feedback system with open-loop gain given by so that the static velocity error constant is and the dominant closed-loop poles are located at .
32. For the system whose open-loop transfer function is , ­design a compensator so that the damping ratio is 0.5 and undamped natural frequency is 2.
33. Design a compensator for the system with to have a phase ­margin of .