LabManual Control Design

7/29/2019 LabManual Control Design

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ME 4053 MECHANICAL SYSTEMS LABORATORY

Controller Parameter Formulas

In the System Identification portion of the controls experiment, a linear, first-order approximation of the

motor/amplifier system was obtained. This model is completely characterized by two constants: The motor constant,

K m, and the motor time constant, Tm. In this exercise, you are to design three types of classical controllers for themotor: Proportional (P), Proportional plus Derivative (PD), and Proportional plus Integral plus Derivative (PID)

controllers. Formulas for the gains in the various controller types considered here are presented in any reasonablycomplete text on control theory. These formulas will be explained further in the second lecture that accompanies this

experiment. To prepare for the second week of the experiment, merely compute the required gains according to the

formulas presented below. In the calculations, use the time constant and motor constant obtained in Week 1.

In addition to a system model, control design is based upon control specifications. In this lab, we will use

two primary control specs: Maximum percent overshoot, M p, and settling time, ts. Each laboratory section will be

assigned these values prior to the closed-loop control experiment.

Proportional Controller. For the proportional controller regulating a second order system with a step input,

the proportional gain can be specified to meet a maximum overshoot specification. The time domain solution shows

that the required damping ratio, ζ , can be determined from the fractional overshoot, M p, from the relationship

(1)

Thus equation (1) can be solved for the damping ratio, ζ, given the control spec M p. Once ζ is determined, the gain

of the proportional controller can be found from the following formula:

(2)

Use Equation (2) to compute the proportional controller gain and enter this value into Table 1. Note that there is noway to independently specify the settling time of the closed-loop system because the P-controller has only a singlecontrol gain.

Proportional plus Derivative Controller. For the PD-controller regulating a second order system with a step

input, the two control system gains can be specified to at least approximately meet both a maximum overshoot

specification and a maximum settling time specification. The time domain solution shows that the desired damping

ratio, ζ , can still be determined from the fractional overshoot, M p, by Equation (1). Further analysis shows that the

appropriate natural frequency can be determined from the desired settling time, t ds by

(3)

The required proportional gain and derivative gain can then be determined from the following formulas

(4)

and

(5)




Calculate these gains for your assigned values of M p and ts and record your computed gains in Table 1.

PID Controller. When regulating a second order plant, PID control results in an overall third-order system.

For the typical underdamped or oscillating system, two poles will be complex, and the third will be real. The

proportional plus integral plus derivative controller has three free design parameters. The third degree of freedom

can be used to ensure that the third system pole, which is at s3 = −σ , will be far to the left in the s plane of the two

complex poles. The two complex poles will then tend to dominate the dynamics of the system, and the system willhave approximately second order behavior. As before, Equations (1) and (3) are used to find the required damping

ratio and natural frequency, respectively. A multiplier parameter, n, is introduced to represent the added degree of

freedom. Each lab section will be assigned three values for the multiplier: n1, n2, and n3. Corresponding to each

value of n, a PID controller will be designed and tested.

The location of the third closed-loop system pole is given by the relation:

(6)

Then, the desired gains can be computed with the following formulas:

(7)

(8)

(9)

Calculate these gains for the assigned values of M p, ts, and ni, i =1,2,3, and record the results in Table 1.

Calculations and Simulations to be Completed as Homework

For each control strategy, the appropriate gains should be computed and entered into a table such as Table

1. This table should be written or pasted into your research notebook to document the work for this week. As an

expedient, the tables may be taped into the research notebook. It is never acceptable to staple exhibits into a research

notebook.

Next create a SIMULINK model as illustrated in Figure 1.Refer to the Appendix for suggestions and

instructions on completing the simulation assignment.

Use the model to simulate P and PD controllers with the calculated gains. Paste or tape the plots of these

simulation results into your notebook.

In the case of the PID controller of the non-linear model, simulate the response with gains computed for n equal to n1, n2, and n3. Select the gain setting with the best response for minimizing the following controller score,

Score = settling time (sec) × max overshoot (percent)

After selecting the best value of n as a starting point, adjust or tweak the controller by altering the values of the gains

to further minimize the controller score. Be prepared to run the actual physical experiment with the adjusted

controller parameters during the second week.




Please begin the homework assignment in the lab and at least build and test the simulation model in the lab.

Be sure to insert the following results into your research notebook:

o Completed Table 1 with comments

o Completed Table 2

o Plot of simulation of proportional controller with calculated gains for both linear and non-linear model (Refer to the Appendix)

o Plot of simulation of PD controller with calculated gains for both linear and non-linear model

o Plot of simulation of all three PID controllers with calculated gains for both linear and non-linear

model

o Plot of simulation of PID controller with adjusted gains for non-linear model

Submit the notebook for inspection and grading at the start of the second lab week.

Table 1. Calculated Control Parameters and Simulated Response for Linear Model

(Deadbands Set to Zero)

Description Proportionalgain

Integralgain

Derivativegain

MaxOvershoot

SettlingTime

Controller score

K p

volts/rev

K i

volts/(rev-

sec)

K d

volts/(rev/sec)(%) ( sec)

% × sec

P

PD

PID

PID

PID

Table 2. Calculated Control Parameters and Simulated Response for Non-Linear Model

(Deadbands Set to Observed Values)

Description Proportional

gain

Integral

gain

Derivative

gain

Max

Overshoot

Settling

Time

Controller

score

K p

volts/rev

K i

volts/(rev-

sec)

K d

volts/(rev/sec) (%) ( sec) % × sec

P

PD

PID

PID

PID

PID adjusted

Note that the gains in Table 2 are exactly identical to the gains in Table 1.




Figure 1. SIMULINK/Quanser System to Implement Parameter Identification Experiment

Figure 2. Location and Parameters for the Blocks Used in Parameter Identification




Figure 3. SIMULINK Model Used to Verify Parameter Identification

Figure 4. Location and Parameters for the Blocks Used in Parameter Verification




Figure 5. Simulation Model for P, PD, and PID Controllers Including Dead Band and Saturation




Appendix A. Guidelines and Examples for Control Experiment Homework

Figure 5 above shows a model for the brushless DC motor system used in the ME4053 controls lab. The

model includes an amplifier dead-zone and input saturation.

Two sets of simulation are to be performed as part of the homework. For the first set, the amplifier dead-zone is set to zero to simulate a linear system. The second set of simulations include the non-linearities of the system

as determined by the limits of the dead-zone measured during the first week. The response from this simulation

should mimic the response of the actual motor system as tested in the second week of the controls lab.

First construct the SIMULINK block diagram needed for the simulation of the P, PD, and PID control of

the DC motor system. Now prepare the run the simulations by entering the following motor and controller

parameters:

Motor parameters:

K m, T m

Dead zone parameters:Simulation Set 1 (Linear System):

Start of dead zone = 0

End of dead zone = 0Simulation Set 2 (Non-linear System):

Start of dead zone = D1 = negative limit

End of dead zone = D2 = positive limit

Saturation parameters:

Simulation Set 1 (Linear System):

Upper limit = 1e6

Lower limit = -1e6Simulation Set 2 (Non-linear System):

Upper limit = 10

Lower limit = -10

Controller parameters:Set the following calculated or adjusted control parameters:

K p, K i, K d

Simulate the closed-loop response of the system for a particular reference signal. The reference signal is a plot of the

desired flywheel angle (rev) vs time. Each semester, the exact description of this desired angle will be specified.

Note that when executing the simulations, K d and K i are set zero for the case of the proportional controller.Similarly K i is set to zero for the case of the proportional plus derivative controller.

Before actually executing the simulations, select a convenient and efficient simulation method using the

following settings:

Select “Simulation” drop-down menuSelect “Parameters”

Set “Stop Time” to the assigned value

Set “Solver options”

Type: Variable-step, ode23s (stiff/Mod. Rosenbrock)

or ode45 (Dormand-Prince)

Finally select “OK” and run your simulations as described and practiced before.




For each of the required simulations, the output to “DESPOS” will be the time sequence recording the

desired signal the motor is commanded to follow, and the output to “POS” will be recorded as the actual response of

the motor. If it is necessary to lengthen the simulation to get a more complete picture of the response, increase the

stop time of the simulation. To set the stop time, use the “Simulation” drop down menu and the “Parameters” option.

This change will simulate more response cycles. The arrays containing the data for “DESPOS” and “POS” will be

automatically sent as outputs to the MATLAB workspace. Use the commands learned in the MATLAB tutorial to plot these responses.

Plot the simulation results using the following commands in the MATLAB workspace:

>>plot(Time,DesPos,’r’,Time,Pos,’b’),grid on,hold on,

>>Title(‘PID simulation of a brushless DC motor’)

>>xlabel(‘Time (sec)’),ylabel(‘Position (rev)’)

These plots are the records and documentation of the required simulations.

For the position control of the motor, the optimal response is to have zero steady state error (within

measurement precision), the smallest feasible maximum overshoot, and the smallest feasible settling time. Inspecteach plot by zooming in to evaluate the following controller score that is used as the figure of merit of the design of

the controller:

Score = settling time (sec) × max overshoot (percent)

In practice, a control designer typically tweaks the calculated controller gains to achieve the optimal response of the

actual system, which may have second order features that cannot be anticipated precisely. Consequently, prepare for

the actual controller development next week by tweaking the PID controller gains for the simulation of the system

with dead band to obtain a response with zero steady state error and a minimum product of the maximum overshootand settling time.

Figure A.1 below shows typical simulation results for the non-linear model. Figures A.2 and A.3 show

simulation results and experimental results for the proportional and PID controllers respectively. The simulation

results should be prepared as part of the homework assignment. The experimental results will be obtained during the

second week of the experiment.

Figure A.1. Simulation Results for PID Control of Non-Linear Model

With the following quantitative results:




Maximum percent overshoot:

Settling Time: t s = 6.93 s

(Settling time is determined based on the 2% criteria. Therefore, the settling time is the last time the error is larger than 2%)

Controller Score = 14.7 % × 6.93 s = 102 % × s(Record the controller scores of the linear model simulation in table 2 and those of the non-liner model simulation in

table 3 above)

Figure A.2. Comparison of Simulation and Experimental results for PID Control of Non-Linear Model

Figure A.3. Comparison of Simulation and Experimental Results for Proportional Control of Non-Linear

Model

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