Introduction
Many a times, performance of a control system may not be upto the expectation, in which case the performance of the same can be improved by controllers or compensating networks.
1. Insertion of compensating network is nothing but addition of poles and zeros.
2. We can reduce the steady state error by increasing the forward path gain, but it makes the system unstable and oscillatory.
3. Addition of a pole to the open loop transfer function will lead the system towards instability. The speed of the response slows down. But the accuracy of the system increases.
4. Addition of a zero to the open loop transfer function will lead the system towards stability. The speed of the response becomes faster. But the accuracy of the system is reduced.
Compensating Network
1. Cascade Compensation: The compensating network is introduced in forward path in this case. Phase lag/ lead compensators fall into this category.
2. Feedback Compensation: The compensating network is introduced in feedback path in this case.
Phase Lag Compensator
A compensator having the characteristic of a lag network is called a lag compensator. Hence, the poles of this network should be closer to origin than zeroes.
1. Results in a large improvement in steady sate response (i.e. steady state error is reduced). 2. Results in a sluggish response due to reduced bandwidth.
3. It is low pass filter and so high frequency noise signals are attenuated. 4. Acts as an Integrator.
5. Settling time increases. 6. Gain of the system decreases.
_ Fig. Electric lag compensator
1/ sC +
+
General form of lag compensator, s =
.
Frequency of maximum phase lag, = √ = √ T . T = √ Maximum lag angle, ф = t n [
√ ] = sin
(
) = ф ф
Phase Lead compensator
A compensator having the characteristics of a lead network is called a lead compensator. Lead compensator has zero placed more closer to origin than a pole.
1. Lead compensation appreciably improves the transient response.
2. The lead compensation increases the bandwidth, which improves the speed of the response and also reduces the amount of overshoot.
3. A lead compensator is basically a high pass filter and so it amplifies high frequency noise signals.
4. Acts as a differentiator. 5. Settling time decreases. 6. Gain of the system increases.
7. There is no improvement in steady state response. | j | in dB j 0 0 -20 dB/decade √ √ log ф Approximate magnitude plot
Fig. Bode plot of lag compensator Phase plot j
General form of transfer function of lead compensator, s =
= .
Frequency of maximum phase lead, = √ = √ T . T = √ . Also, ф = t n * √ + = sin * + ф ф | j | 20 j 0 0 + 20 dB/decade 20 √ ) √ log ф
Approximate magnitude plot
Fig. Bode plot for lead compensator
1/T jω T T
Fig. Pole-zero plot of a lead network _ + + _ 1/ sC
Comparison of Phase Lag And Phase Lead Compensators
Table. Comparison of characteristics of lead and lag compensators Characteristics Phase-Lead Phase-Lag 1. Circuit Differentiator Integrator
2. ξ n r s s ↑ n r s s ↑↑
3. w n r s s ↑ D r s s ↓
4. T rise , t settling D r s s ↓ n r s s ↑ 5. Phase shift Increases Decreases 6. Phase Margin mprov ↑ R du ↓ 7. Gain cross over
frequency (w ) n r s s ↑ D r s s ↓ 8. Band width Increases Decreases 9. Over shoot Decreases Decreases
10. Gain Decreases Increases
11. Steady state error Increases Decreases 12. Constant ( < > 13. Pole –zero |P|>|Z| |Z|>|P| 14. Wmax √ √ 15. sin ∅m) 16. Time constant R R
Phase Lag – Lead Compensator
A compensator having the characteristics of lag –lead network is called a lag – lead compensator. 1. A lag – lead compensator improves both transient and steady state response.
2. Bandwidth of the system is increased.
The transfer function of lag – lead compensator, s =
; wh r > , 0 < < nd j
Feedback Compensation
In this method, the compensating element is introduced in feedback path of a control system as shown.
Fig. Block diagram of compensated system with tacho generator feedback. After compensation, overall open-loop transfer function
Depending on nature of G(S) and K , damping of response can be controlled. Controllers
A closed loop control system tries to achieve the target output because of the feedback signal. Many a times, the output response achieved is not smooth and also may have steady state error. Thus, the transient and steady state response can be improved by using a control action of transfer function as shown in figure below.
Fig. Usage of controller in a closed loop control system
G(s) c(t) e(t) r(t) Gc(s) G(S) | j | in dB 0 log j 0 00 0 d + 20 dB/dec T T log
Fig. Bode plot of lag – lead compensator - 20 dB/dec
Proportional Controller
Transfer function of a proportional controller is given as, s = K . Proportional controller is usually an amplifier with gain K . It is used to vary the transient response of the control system. One cannot determine the steady state response by changing K . Steady state response depends on the type of the system. However, maximum overshoot is increased in this case.
Integral Controller
Transfer function of a Integral controller is given as, s = K/ s. It is used to decrease the steady state error by increasing the type of the system. However, stability decreases in this case. Derivative Controller
Transfer function of a derivative controller is given as, s = K . s. It is used to increase the stability of the system by adding zeros. steady state error increases, as type of the system decreases in this case.
Proportional + Integral (PI) Controller
Transfer function of PI controller is given as, s = (K + K/ s). It is used to decrease the steady state error without effecting stability, as a pole at origin and a zero are added. In P+I controller, order of a system increases, i.e. it converts a second order system to third order. Proportional + Derivative (PD) Controller
Transfer function of a PD controller is given as, s = (K . s + K ). It is used to increase the stability without effecting the steady state error. Here type of the system is not changed and a zero is added.
Proportional + Integral + Derivative (PID) Controller
Transfer function of a PID controller is given as, s = (K + K/ s + K . s) = ( . . ).
It is used to decrease the steady state error and to increase the stability as one pole at origin and two zeros are added. One zero compensates the pole and other zero will increase the stability. Hence response is faster and highly accurate.