Chapter 7. Lead Compensator Design

1. Objectives of Experiment

• To learn Lead Compensator Design on the basis of the root locus theory.

• To monitor the change that is shown as applying the lead compensator to the

pendulum motor.

• To perform the location control of the pendulum motor using the lead

compensator.

2. Compensator

The most important thing for the design of the controller to control one process

or a system is to select the controller that fits to the process. Selecting the

appropriate controller that is suitable for the process is no less than completing

90% of the design work. The simplest method to design control is to offset the

bad pole point or the zero point of the process using the controller, but this

method is actually impossible to execute. It is because the pole point and the

zero point of the transfer function that are acquired from the actual process are

not the accurate value but the ball park figure, and the order of the actual

process may be higher. In addition, the location of the pole point or the zero

point is changed in accordance with the change of the variables in the actual

process. Therefore, it is recommendable to design the controller to move the root

to the desired location instead of directly offsetting the pole point and the zero

point of the process. It is the root locus graph to suggest the standard for the

selection of the controller that is important like this.

The first phase that adjusts the system in order to get the satisfying result is to

set up the gains on the root locus. However, it is not sufficient to adjust only

gains when you change the system to satisfy the defined specification. As

increasing the gains, you can generally improve the steady state movement, but

as a result, the stability becomes worse or unstable. At this time, it is required to

redesign the system(reconstruct or add devices or components) in order to make

the system operated as required. The device to be added for the purpose of

satisfying the specification is the compensator. The compensator compensates

the insufficient performance of the original system.

In Chapter 7 and 8, we will understand how to design in order to put the closed

loop roots on the desired location as defining the pole point or the zero points of

the controller and changing the original root locus and also check the response as

comparing the simulation and the experiment results.

3. How to Design Lead Compensator

3.1 Meaning of Lead Compensator Lead Compensator or Lead Controller is the same terminology and let's call it

C(s). This consists of one pole point and one zero point and can be

expressed as shown in the formula (5.8). In the lead compint <IMG

src=".\PIC20D.gif" width=14px height=16px > and one zero point <IMG

src=".\PIC20E.gif" width=14px height=16px > and can be expressed as shown

in the formula (5.8). In

= , (7.1)

It tends to draw the root locus toward the zero point because the pole

point(p) of the lead compensator is nearer to the axis of the imaginary number

than the zero point(z). Therefore, you can reduce the overshoot and accelerate

the peak time or the settling time if using the lead compensator. The lead

compensator is basically the high-pass filter. The locations of the pole point and

the zero point in S-plane of the lead compensator are as follows.

[Figure 7.1] Pole Point and Zero Point of Lead Controller

3.2 Lead Compensator Design Designing the lead compensator using the root locus graph is very effective

when the specification is given as the quantity of time area, that is to say, the

damping ratio of the required major closed loop poles, the number of original

non-attenuation vibration, the maximum overshoot, the rise time and the settling

time. The design is done as follows.

[Step 1] Indicate the location and the area that the pole is put in accordance

with the given capacity specification on the s-plane.

[Step 2] Indicate the pole point and the zero point of on the s-plane and

draw the root rocus graph. Here, check if you can make the closed loop pole

that is required only with the open loop gains adjustment. If it is impossible,

execute the following procedure.

[Step 3] Put the zero point of the lead compensator right under the location of

the required pole point.

[Step 4] Calculate the angle on the desired location of the pole point(the total of

each pole point and the zero point is -180) and get the pole point of the lead

compensator. That is to say, calculate by -180=-angle of the pole point + angle

of the zero point.

[Step 5] Get as calculating the distance between each pole point and zero

point that you want.

[Step 6] Draw the root locus graph using the values that are designed by

CEMTool. In addition, check the step responses. If the step response does not

satisfy the performance specification, design again from the procedure 3.

4. Lead Compensator Design of Pendulum Motor

Let's examine what is the change of the motor response as designing the lead

compensator of the pendulum motor. Firstly, the motor response when the lead

compensator does not exist is as shown in [Figure 7.2].

[Figure 7.2] Step Response of Pendulum Motor that is not compensated

Design the lead compensator to make the overshoot of the step response of the

pendulum motor be within 10% and to make the steady state error be less than

2% within one second of the settling time.

Reference Files : pch7_1.m (X:\CEMTool\Experiment\Pendulum\pch7_1.m)

[Step 1] Selection of Dominant Root

If you substitute L=5, in the formula (6.6) and (6.7), > 0.69 and

>54. Therefore, the dominant root exists in the shaded area as shown

in [Figure 7.3] and decide the dominant root in this area. (-7 ± j5)

[Figure 7.3] Location of Required Root

, = -35 ± j35

[Step 2] Drawing Root Locus of Pendulum Motor Draw the root locus of the pendulum motor, and you can get the drawing as

shown in [Figure 7.4]. (refer to Chapter 6)

[Figure 7.4] Root Locus Block Diagram of Pendulum System Motor

[Step 3] Getting Zero Point of Lead Compensator The zero point z of the lead compensator is right under the desired pole

point(pole point of the dominant root), so z=-35.

[Step 4] Getting Pole Point of Lead Compensator The total of the angles of each pole point and zero point shall be -180.

Therefore, calculate as follows.

Here, are the angle between the dominant root and each angle of the

system. Add as much as the number of system roots. is the angle between the

angle of the lead compensator and the dominant angle. In addition, is the

angle between the dominant angle and the zero point.

[Step 5] Getting K of Lead Compensator Calculate the distance from each pole point and zero point to the dominant root

as follows,

==

This becomes of the lead compensator. Therefore, the lead compensator is

designed as follows

= ,

If you execute these procedure above after executing pch7_1.m file and entering

one of the pole pints(-35 ±35i in this text), you can get the same result.

5. Simulation

Reference Files : pch7_1.blk (X:\CEMTool\Experiment\Pendulum\pch7_1.blk)

pch7_1.m (X:\CEMTool\Experiment\Pendulum\pch7_1.m)

Add the lead compensator that you design in the paragraph 4 to the pendulum

motor models and let's examine the change from the response before adding the

lead compensator. Firstly, get the pole point(p), the zero point(z) and the

gains(K) as executing the reference file pch7_1.m. At this time, if you enter the

list command in CEMTool, z, p and K shall be checked. Check z, p and K and open

pch7_1.blk file. Then, you can check the file as shown in [Figure 7.5] that the

designed lead compensator is applied to the transfer function of the pendulum

motor.

This file(Reference File pch7_1.blk) is the block file that makes the lead

compensator as shown in the formula (7.1) as the macro block using the transfer

function block(rounded part on the figure), adds it to the block file that requires

the step response of the pendulum motor and gives the step response. This lead

compensator block consists of the macro block(refer to SIMTool for the details)

and the parameters of the lead compensator block, gains K, pole point p and zero

point z, can be entered. Select the macro block of the lead compensator with a

mouse and 'View Inside Macro Block' on the menu that is displayed as pressing

the right button of the mouse. Then, you can see the connection inside. For the

simulation, set up as the initial location of the motor starts from 0 and reaches to

30cm after 1 sec. And execute the simulation at the interval of 0.001 sec. during

5 sec.

[Figure 7.5] Step Response Simulation of Pendulum Motor that Lead Compensator is applied

(pch7_1.blk)

[Figure 7.6] is the result after executing the file that is configured as shown in

[Figure 7.5]. The result is saved in the variable sim_lead.

[Figure 7.6] Simulation Response that Lead Compensator is applied

6. Experiment

Check the simulation result that we checked in the previous paragraph by the

experiment. We will compare the step response of the pendulum motor that the

lead compensator is applied with the simulation result. Set up the step input as

the initial location of the motor starts from 0 and reaches to 30cm after 1 sec.

※ All simulation in the paragraph 5 shall be completed before the execution of

the reference files. If the simulation in the paragraph 5 is not executed, the

experiment in this paragraph cannot be executed.

Reference Files : pch7_1.m (X:\CEMTool\Experiment\Pendulum\pch7_1.m)

pch7_2.blk (X:\CEMTool\Experiment\Pendulum\pch7_1.blk)

initial.blk (X:\CEMTool\Experiment\Pendulum\initial.blk)

pend_io.m (X:\CEMTool\Experiment\Pendulum\pend_io.m)

comp_lead.m (X:\CEMTool\Experiment\Pendulum\comp_lead.m)

[Step 1] Power Off and Pendulum Separation of Pendulum System First, check if the power of the pendulum system is off and while the power is

off, separate the pendulum of the system.

[Step 2] Experiment Block Configuration

[Figure 7.7] Experiment Block that Lead Compensator is applied (pch7_2.blk)

Configure the experiment blocks as shown in [Figure 7.7] that is the reference

file pch8_2.blk using SIMTool to get the step response of the pendulum motor

that the lead compensator is applied. Connect the lead compensator block that is

used in the simulation as shown in the rounded area on [Figure 7.7] to the input

part of the pendulum motor through the analogue output. Feedback the pulse

signal that is generated from the encoder block as converting it as the length unit

that is pulse to cm. Therefore, the cart location of the pendulum is sent back in

the length unit. The scope block is connected. Then, you can check the result in

the length unit. Connect the out block to the scope block to save the experiment

result in the variable exp_lead.

[Step 3] Block Setup After completing the experiment block configuration, configure each block.

Configure the step block like the simulation as the cart location moves 30cm after

1 sec. Set up the encoder block to receive the signal of four multiplies through

the channel number 0. Set up the gain block as 1.27/4000 with the name Pulse

to cm. This aims to convert the pulse signal of the encoder into the length unit as

considering that the cart moves 1.27cm when the motor rotates once. Configure

the gains as K, the zero point as z and the pole point as p as double clicking the

lead compensator block. Design the lead compensator as executing the reference

file pch7_1.m and it is applied to the defined value of the lead compensator block.

Set up the scope block as the minimum value is 0 and the maximum value is 30

like the simulation. Set up the out block to save the experiment result in the

variable exp_lead. Get the coefficient of the lead compensator as executing

pch7_1.m file using the dominant root to make the coefficient of the lead

compensator that is used in the simulation in the paragraph 5 after the

configuration

[Step 4] Hardware Setup and C-Code Generation Check if the hardware is set up as RG-DSPIO01 in 'Hardware Interface' window

that is displayed as selecting 'AUTOTool- Parameter‘ in SIMTool menu. Set up the

execution time in the setup window that is displayed as pressing 'Parameter

Setup' button of the window above. Set up the starting time as 0 sec., the ending

time as 5 sec. and the sampling time as 0.001 sec. After completing the

execution time setup, press 'C-Code Generation and Compile' in 'Hardware

Interface' window, and convert the block that is configured by SIMTool to C-Code.

Transfer it to DSP board. After transferring to DSP board, DOS window is

displayed.

[Step 5] Power on and Initialization of Pendulum System If DOS window is displayed, put Mode switch in the electric part of the

pendulum system to Manual mode and turn the power of the pendulum system

on. Initialize the cart location as pressing INITIALIZE button on MOVE. Open the

reference file initial.blk and press 'Execution-Execution' button on SIMTool menu

or the execution icon. Then, convert Mode switch of the pendulum system to

CEMTool mode.

[Step 6] Result Check After completing the transfer to DSP board and fixing the cart location in the

center as the pendulum system is initialized , close DOS window and press

'Execution' button in 'Hardware Interface window'. Then, the scope block is

connected to make you check the displacement of the cart as shown in [Figure

7.7]. Therefore, the graph window as shown in [Figure 7.8] is displayed.

[Figure 7.8] Experiment Response that Lead Compensator is applied

[Step 7] Comparison to Simulation Result Execute comp_lead.m file in CEMTool command window to compare the

simulation result to the experiment result. However, it shall be executed only if

you do not execute other simulation or experiment right after completing the

simulation or the experiment. The graph that the simulation result and the

experiment result are drawn is as shown in [Figure 7.9]. The blue line is the

experiment result and the red line is the simulation result. As a result of

comparing the experiment result to the simulation result, we can recognize that

two results are very similar.

[Figure 7.9] Comparison of Simulation and Experiment Result