UNIVERSITI KUALA LUMPUR

MALAYSIA-FRANCE INSTITUTE

CONTROL SYSTEM 2

(LAB REPORT 2)

(Implementation of Lead-Lag Compensator)

NAME : MOHD FADHIL BIN AB.RAHMAN 50213107047

FEZZALIANA DELINA

GROUP : MECHA & IART

PREPARED FOR :

MR. ZULKHAIRI BIN MD ZAN

DUE DATE : 26/3/2010

Experiment 2: Implementation of Lead-Lag Compensator

Objectives:

1. Design the lead/lag compensator to meet the transient response specification of

a second-order plant.

2. Design the lead/lag compensator to meet the steady-state error specification for

a second –order plant.

3. Evaluate the frequency performance of lead/lag compensator

Introduction:

The modification or adjustment of a control system in order to provide a suitable

performance is called compensation. The compensating device may be an electric,

mechanical, a hydraulic, a pneumatic or other type of device or network and is often

called a compensator. Commonly, an electric circuit serves as a compensator in many

control systems. A compensator is an additional component or circuit that is inserted

into a control system to equalize or compensate for a deficient performance. Among the

simplest dynamic structures used in compensator design is the lead-lag compensator.

Using ideal op-amp properties, the compensator’s transfer function can be written as

It has a zero at s = -1/Tc, and a pole at s = -1/atc. Depending on the value of a, the

compensator can be either lead or lag.

Lead Compensator (0 < a <1)

For lead compensator, the zero is always located to the right of the pole in the complex

plane. Figure 2 shows the Bode diagram of a lead compensator, which is basically high

pass filter. The corner frequencies are at z = 1/tc and p = 1/atc. It can be show that the

maximum phase lead angle fm is related to a by

Lag Compensator (a >1)

For lag compensator, the pole is located to the right of the zero in the complex plane.

Figure 3 shows a Bode diagram of the lag compensator. The corner frequencies are at

p

= 1/atc and z = 1/tc.

Table 1: Lead-Lag Compensators

Lead Compensator Lag Compensator

Zero is located to the right of the pole Pole is located to the right of the zero

High pass filter Low pass filter

The primary function is to provide sufficient phase-lead angle to the system to reshape the frequency response

The primary function is to provide attenuation in the high frequency range to given a system sufficient phase margin. The phase lag characteristics are of no consequence in lag compensator.

Improve transient response Improve steady-state error

1.1 Routh-Hurwitz Stability Criterion

The most important problem in linear control system concerns stability. A control

system is stable if and only if all closed-loop poles lie in the left-half s plane. A simple

criterion, known as Routh-Hurwitz stability criterion, is commonly used to determine

stability of a control system. (Refer to Appendix I for a brief discussion on the Routh-

Hurwitz criterion) In the following experiment, students are required to design an

optimum lead/lad compensator for a second-order plant by using Routh-Hurwitz stability

criterion.

Hardware Requirements:

Second-order Plant Module (Plant)

Lead-Lag Compensator Module* (Compensator)

Dual-channel power supply (+12Vdc and –12Vdc)

Function generator

Oscilloscope

Multimeter

Accessories:

BNC cables

T-connectors

Power connectors

*Note: The Compensator Module consists of two stages of lead-lag controller, which

can be selected by setting a toggle switch. In this experiment, STAGE 1 is selected.

Stage Input Feedback

STAGE 1 A1 B1

STAGE 2 A2 B2

Experiment 2-1: Original System without Compensator

Procedure :

1. All cables/wires from the hardware module disconnected.

2. Power supply is turn one and +12 Vdc voltage rating is adjusted for channel-1

and channel-2 for -12 Vdc. The second-order plant hardware module is

connected by power supply.

3. 2 V ppk square pulse is set-up from the function generator at 100 Hz frequency.

4. Function generator to input A is connected of the plant and connect to

oscilloscope to channel-1.

5. Output A of the plant to channel-2 is connected of the oscilloscope.

6. The trimmer R2 was adjusted slowly to get a marginally unstable output. Result

is plotted.

A second order transfer function is given by:

and the characteristic equation is

7. Routh-Hurwitz stability criterion is used to find the value of K for marginally stable

condition.

using Routh Hurwitz :

S2 1

S1 0

S0 0

= _

=

= 0, k = 3

Since k = , and = 10 kΩ

3 =

Therefore, = 20 kΩ

8. Power supply is turn off and all cables/wires disconnect from the hardware module.

Multimeter is used to measure the resistance value R2 of the second-order plant (probe

across pin 2 and pin 6 of the op-amp IC). The value of R1 is 10kΩ. Experimental K

value is determined. your answer is compared with step 7.

From measurement using multimeter, = 11.74 KΩ

From equation k =

=

= 2.17

Experiment 2-2: Lead-Lag Compensator Design

1. The Lead/Lag Compensator of the plant wiil be fed.

The overall system transfer function is

Generally, there are two main concerns in designing a feedback control system. It can

be shown that, in order to have a stable system with minimum steady state errors, the

following conditions must be fulfilled:

I. from Routh-Hurwitz stability criterion analysis:

II. From steady state error analysis:

Based on the above conditions, design a lag compensator by calculating and selecting

suitable values for Rc1, Rc2, C1, and C2. (The available values for C1 and C2 are

0.1mF, 0.01mF, 1nF, and 0.1nF. While the Rc1 and Rc2 can be trimmed from 0 to

20kΩ).

We shall design as follows.

2. The power supply is turn off and all cables/wires from the hardware module is

disconnected. Identify the Lead-Lag Compensator Module. Select the desired value for

Rc1, Rc2, C1, and C2 accordingly. Use the multimeter to confirm the trimmer setting:

VR1 = Rc1

VR2 = Rc2

Rc1 = 11.2 k

Rc2 = 13.4 k

= 0.1µF , = 0.01 µF

3. The power supply is connected to both the plant and the compensator modules.

STAGE 1 of the compensator is connected.

4. The output of the function generator is connected to Input A1 of the compensator.

5. Output A of the plant is connected to Input B1 (Feedback) of the compensator.

6. Output of the controller is connected to Input A of the plant.

7. T-connector is connected to the function generator’s output (system input) to

channel-1 of the oscilloscope.

8. T-connector is connected to Output A of the plant (system output) to channel-2 of the

oscilloscope.

9. The VR1 and VR2 is tuning to yield the best system response. Repeat the design

calculation in step 1 if necessary. Plot the result.

10. The experiment results obtained in Step 9 and Step 6is compared in Experiment 1-

1. Your observation. Is discussed.

11. Step 1 to step 10 for improvement of settling time is repeated by a factor of five with

maximum overshoot less than 12%.

Result :

Before :

C1 = 0.1 µF

C2 = 1 µF

VR1 = 10k

VR2 = 15k

= 2µs V = 0.2 V

After :

VR1 = 15k

VR2 = 1.97k

= 0.5 µs V = 50mV

Before :

C1 = 0.01µF

C2 = 0.1µF

VR1 = 15

VR2 =18

After :

VR1 = 5.43 kΩ

VR2 = 6.75 kΩ

= 1ms

V =0.5 V

Therefore, =

= .

= .

=

Bode plot :

Conclusion

This 2nd order system is marginally stable without lead and lag compensator. To improve transient response and steady state error, lead and lag compensator will be added to uncompensated system.

From the experiment, VR1 and VR2 are found. We get the Rc1 and Rc2’s values which are

VR1 = Rc1 =10k

VR2 = Rc2 =5k

Results

Lag compensator

From table, we chose C1 = 0.1µ, C2=0.1n

Therefore,

Kc = - 1000, Tc = 0.001, α = 5

= .

Zero: -1000

Poles: -200,-0.15 31.62j

Therefore, Pole is located to the right of the zero is verified from this equation.

Bode plot for lag compensator

Lead compensator

Original Phase margin

Phase margin = 0.387 deg

Calculation for lead compensator:

% OS = 12%

Zeta (ζ) = 0.56

Øm = = 56.41°

PM from graph = 0.387°

Ømax = 56.41° - 0.387° + 10 = 66°

=

0.914 =

0.914+0.914 = 1 -

1.914 = 0.086

= 0.045

Therefore we chose = 0.05 (values from the table)

For = 0.05, we choose C1 = 1n, C2 = 0.1n, Rc1 =10k, Rc2 =5k

Therefore,

Tc = , Kc = -10 = 0.05

= .

Zero: -100k

Poles: -2000k, -0.15 31.62

Therefore, for lead compensator, pole is located to the left of the zero is verified from this equation.

Bode plot for Lead Compensator;

Conclusion

For lead compensator, zero is located to the right of the pole. It is a low pass filter that primary function is to provide sufficient phase-lead angle to the system to reshape the frequency response.

Lead compensator is to improve transient response.

For lag compensator, pole is located to the right of the zero. It is a low pass filter that primary function is to provide attenuation in the high frequency range to given a system sufficient phase margin.

The phase lag characteristic is or no consequence in lag compensator. Lag compensator is to improve the steady-state error.