chapter 7
TRANSCRIPT
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