The Analysis of a Speed Control Using P-I Control

Control Systems ME 451Dominic Waldorf

Section 006 Group C Wednesday 7:00 PM

Dr. Jongeun Choi and TA Nilay KantMarch 22, 2016

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Abstract:

Speed control using P-I control is an important method in the understanding of

control system. The experiment assisted in proving the theoretical pure proportional

control, integral control, and proportional and integral control values to the experimental

values. The task was accomplished by deriving an equation from the block diagram of

the system. Once the transfer function was calculated, the equation was manipulated to

give the kpc, kic, Ts, ki, and kp values for each of the different methods of response. The

experiment also assisted in understanding what happens when those values are changed.

The experiment was performed using a DCMCT motor unit and the Labview

software on the desktop. There were four different experiments performed. For each

experiment the values were adjusted, giving different response for each experiment.

The optimal results were given when manually tuning the parameters to improve

the settling time. The goal is to choose the kp and ki values so there is no steady state

error, there is no overshoot, and the 2% settling time is less than or equal to .25 seconds.

This was done during the proportional and integral control when bsp was set to zero. The

old values were ki=2.25, kp=.19, and Ts=.28. The new/optimal values were ki=1.6356,

kp=.148, and Ts=.66.

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Table of Contents

Nomenclature Listing............................................................................................................. 1

Introduction.............................................................................................................................. 2

Theory and Analysis............................................................................................................... 2

Experimental Equipment and Procedure........................................................................4

Results......................................................................................................................................... 6

Discussion............................................................................................................................... 11

Conclusion............................................................................................................................... 12

Reference................................................................................................................................. 13

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Nomenclature Listing

bsp set-point constantK Gainkp Proportional Gain Contstant

kpcCritically Damped Proportional gain constant

kpu Gain required for marginal instabilityki integral gain constantkic critically damped integral gain constantTs settling timeTu period of oscillation at marginal instabilitytc output low pass filter time constant

τ time constant

Table 1: variable definition

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Introduction

Speed control using a P-I control is a very useful tool for understanding the

operations of a DCMCT motor unit control system. The experiment performed is very

important to for Mechanical Engineers to understand because it helps give the engineer a

firm grasp on how to manipulate parameters to achieve the desired results. These results

can help better tune controllers such as cruise control in a car or even an airplane. If the

controller is better tuned and able to adjust to different external interactions, it will

improve the safety of the vehicle. Not only will it improve safety, but it will also

improve the performance of the engine. The goal of the experiment is to have no steady

state error, no overshoot, and the 2% settling time less than or equal to .25 seconds.

Theory and Analysis

The PI control to the DC motor plant was used in the experiment. A block

diagram modeled the system. The block diagram was used to derive the transfer function

which is as follow:

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Figure 1: Block Diagram of Transfer Function

H (s )= K∗kp∗bsp∗s+K∗kiτ∗tc∗s3+ ( tc+τ )∗s2+ ( K∗kp+1 )∗s+K∗ki

(1)

After each of the control methods were given certain parameters based on the desired

outcome. These values are the theoretical values for the experiment. These equations

were derived in the pre-lab on the experiment.

Proportional Control:

(ki = 0) kpc=¿ (2)

Integral Control:

(kp=0)

kic= 14 K∗(tc+τ ) (3)

Ts=8∗( tc+τ ) (4)Proportional and Integral Control:

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(bsp = 0)

Ts=.25 (5)

ki=256∗tc+ τK (6)

kp=1k∗(2∗( tc+ τ )∗√ K∗ki

tc+τ−1) (7)

(bsp = 1)

kp≈ .4 kpu (8)

ki ≈ kp.8∗Tu

≈ .5∗kpuTu (9)

Experimental Equipment and Procedure

For the experiment, a DCMCT motor unit (figure 1) was used and Labview was

used to input and adjust the variables to display the results.

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Figure 1: DCMCT motor unit

The experiment began by powering up the DCMCT motor unit and downloading

the Labview zip file. After, a small value of kp was as added to make sure that the motor

responded. In order to obtain a zero steady state error, no overshoot, and a 2% settling

time, the value of Ts was set to be less than or equal to .25. The largest voltage the motor

can receive is plus or minus 15 volts, so it is important to know that the motor will cut off

if it exceeds this parameter.

The first experiment performed was the Pure Proportional Control. The

reference signal was set to amplitude of 25 rad/s, a frequency of .6 Hz, and an offset of

50 rad/s. The simulated transfer function was set to K ≈ 18 and τ ≈ 0.085. The filter tc

was set to .03 to eliminate unwanted noise. The integral gain (ki) is set to zero and the set

point (bsp) to 1. The kp value was then set to .01 and increase by .01 V*s/rad until a

second order response for the tachometer was reached. Then the kpc theoretical value

had to be calculated and compared to the actual kpc value. The theoretical value was

calculated using equation (1). Then the kp value at critical instability kpu and the Tu

period had to be recorded.

The second experiment performed was the Pure Integral Control. The

proportional gain was set to zero. The integral gain was the swept from 0 V*s/rad to 2.5

V*s/rad. Overshot and steady state error then had to be described as ki was increased.

Then the kic theoretical had to be calculated using equation (2). The theoretical value of

kic was then compared to the actual kic value. The theoretical settling time had to be

calculated using equation (3) and compared to the actual settling time.

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The third experiment performed was Proportional and Integral Control with

bsp = 0. The ki and kp gain coefficients were calculated using equations (5) and (6)

respectively. These values were plugged into the program and a response was given.

The Ts estimated value from equation (4) and the actual Ts value were compared. After,

these parameters were adjusted manually in order to achieve a better response.

The fourth experiment performed was Proportional and Integral Control with

bsp = 1. The gain parameters were given from equations (8) and (9). They were input

into the ZN values when running the experiement. The response was plotted and the

overshoot and settling time was recorded.

Results

Pure Proportional Control:

The pure proportional control experiment left the ki value at zero in the transfer

function. As the kp value was increased, the RPMs of the wheel increased as well. The

wheel also had a fluctuation in which it would spin fast then slow down repeatedly. At

the beginning the wheel was slightly sticking which gave a different actual response than

the simulated. Using equation (2) gave the kp value equaling .016476. The actual value

from the experiment was 0.06. The first graph (Figure 2) was when kp=kpc(actual). The

second graph (Figure 3) was when kp=kpu which was 0.4 and the Tu value was 3.34

seconds.

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Figure 2: Speed as a function of time at kp=kpc(actual)

Figure 3: Speed as a function of time at kp=kpu

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Pure Integral Control:

Pure integral control gave a different response than the pure proportion control.

The kp constant was set to zero in the transfer function. After, the ki value was increased

and a response occurred. At around ki=.1 the wheel accelerates rather quickly, then

slows down. As ki continues to increase more and more, it begins to oscillate back and

forth, faster and faster. The simulated response stays at a constant 15 rad/s. The kic

value and Ts values were calculated using equations (3) and (4). The calculated value of

Kic was .12077 and the actual value was .15 which gave 19.5% error. The Ts value was

calculated to be .92 seconds and the actual value was .87 seconds giving 5.4% error. The

ki actual value was input into the Labview software which gave the following response in

figure 4.

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Figure 4: Speed as a function of time at ki=kic(actual)

Proportional and Integral Control:

Proportional and integral control is a combination of both proportional and

integral control. First, the response was graphed after calculating ki and kp with bsp

equal to zero. The ki and kp values were calculated using equations (5), (6), and (7). The

ki value was calculated to be 1.635555 and kp was calculated to be .1489. These values

were plugged into the software and are shown in figure 5. Then Ts(actual) was compared

to Ts(theoretical). The theoretical Ts was .25 and the actual Ts was .66. Then the ki and

kp values were manually tuned. The manually tuned values were ki=2.25, kp=.19, and

Ts=.28. These parameters improved the settling time by .38 seconds.

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Figure 5: Speed as a function of time at bsp=0, ki=1.6356, Ts=.25, and kp=.1489

The bsp value was then set to 1. The values of the two k values were calculated

using the ZN method and observations were made. The kp value was calculated using

equation (8) and the ki value was calculated using equation (9). The estimated

coefficients were somewhat similar but it had a smaller settling time. The kp value was

calculated to be .16 and ki was calculated to be .02395. These values were plugged into

the software and the results are shown in figure 6. The overshoot was calculated to be

1.0983 and the settling time of .07 seconds. There was no saturation because of the large

change in response with variable change. There is more overshoot as the set point

weighting factor increases. The recorded settling time was 0.1.

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Figure 6: Speed as a function of time at bsp=1, kp=.16, and ki=.02395

Discussion

The function of a DCMCT motor unit is important to understanding control

systems. As demonstrated in the experiment, there is a different response for the pure

proportional control, integral control, and the combined proportional and integral control.

These differences are important to understand when setting the parameters because

different parameters give different responses.

For proportion control, the steady state error was low, along with low overshoot

and minimal settling time for the 2% steady state error. The integral control had the

highest steady state error with no overshoot and a large settling time for the 2% steady

state error. The integral and proportional control with the bsp=0 had a medium steady

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state error with low overshoot, and medium settling time for the 2% steady state error.

When the bsp was set to 1 it had the highest steady state error with lots of overshoot and

small settling time for the 2% steady state error.

The optimal values recorded in the experiment to be as ki = 2.25, kp = .19, and Ts

= .28 seconds. These optimal results improved the settling time by .38 seconds. This

was achieved by manually tuning the integral and proportional control method. There

was low steady state error, low overshoot, and a small settling time.

Conclusion

In summary, the operation of the DCMCT motor helps give a better

understanding of control systems. Knowing how different inputs changes the response is

important when trying to tune a system. This experiment has many different

applications. The main application is for motorized vehicles. Components such as the

cruise control are assisted using this method. The goal was achieved to choose the kp

and ki values so that there is no steady state error, there is no overshoot, and the 2%

settling time is less than or equal to .25 seconds. It was found to use the manually tuned

parameters in the proportional and integral control method.

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Reference

Control Systems Pre-lab number 6 for Speed Control

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