ME451: Speed Control Design Project
ME451: Speed Control Design Project
Week 1: Vehicle Dynamic Modeling
Reference: C.L. Phillips and R.D. Harbor, Feedback Control Systems
Chapter 2.5: Mechanical Translational Systems
Chapter 2, Section 2.14: Linearization
Chapter 4.1, Time Response of First-Order Systems
Chapter 14, Section 14.9: Linearization
Introduction:
The Whirlwind Corporation is engaged in the rapid prototyping of an advance vehicle speed control system. This week you and you colleagues will develop a dynamic analysis model of the test for which the speed control will be prototyped. This test vehicle is our new hybrid electric vehicle with the WindDrive powertrain installed in the Whirlwind Dynamometer system seen in Figure 1. You will have an opportunity to operate it in this facility. The vehicle’s dynamics have already been tested on our test track and the dynamometer test results will be compared to the test track results validate the Whirlwind Dynamometer facility.
[pic]
Figure 1: Dynamometer Control for WindDrive Vehicle Tests
Vehicle Dynamics:
Two opposing forces drive the speed dynamics of a vehicle: the vehicle drag force and engine tractive force (Fig. 2)
[pic]
Figure 2: Forces that Change Vehicle Speed
A mathematical model for vehicle speed dynamics for a vehicle of mass [pic] can be written as
[pic] (1)
where both the vehicle engine tractive force, [pic] and the vehicle drag force, [pic]are functions of the vehicle operating conditions. If we assume that drag is simply a function of vehicle speed while engine tractive force varies with both vehicle speed and throttle, we can write
[pic] (2)
and
[pic] (3)
where [pic] is vehicle speed and [pic] is throttle position.
To develop a new speed control product using modern mathematical techniques, Whirlwind engineers must find the vehicle parameter “[pic]” and the vehicle force functions [pic] and [pic] for the test vehicle. These parameters and functions will provide a model of the test vehicle. The Whirlwind staff expects that this model will be nonlinear. A linearized design model will then be developed at each operating speed for our new speed control system and speed controls developed for those linearized models during week 2 of our design project.
Vehicle Drag Function Model:
The vehicle drag model characterizes all forces that tend to slow the vehicle except the vehicle drivetrain. The Whirlwind Development Group has conducted a series of test track measurements to characterize the vehicle drag force on our test vehicle. These measurements use the vehicle model given in (1)-(3). When coasting in neutral, the engine tractive force on the vehicle [pic] because the vehicle drive is disconnected from the vehicle. Under these conditions, the vehicle model takes the form
[pic] (4)
Using an approximation from the velocity derivative [pic] with rearrangement of the equation terms, the vehicle drag force function [pic] can be measured as
[pic] (5)
where [pic] = acceleration of gravity (ft/sec2) = 32.2 (ft/sec2)
[pic] = vehicle weight (lb) = 3,010 lb (Hybrid Vehicle) + 190 lb (driver)= 3200 lb
[pic] = vehicle speed (ft/sec) at time [pic](sec), and
[pic] = vehicle speed (ft/sec) at time [pic](sec).
All test measurements of speed are taken using the test vehicle’s speedometer that is calibrated in miles/hour (mph). The conversion from units of “mph” to units of “ft/sec” uses the identity 60 mph = 88 ft/sec. Previous test track tests have generated the least mean-squared error results shown in Figure 3. Your first task is to validate the dynamometer system’s response against these results.
[pic]
Figure 3: Test Track Drag Measurements for a Hybrid Electric Vehicle with WindDrive Powertrain
Drag Measurement Procedure
1. Find the dynamometer test system “WindDriveTest” and run it.
2. Set system operating conditions:
gear = “N” neutral, start speed = your desired starting speed
3. “Click” the arrow in the upper left corner of the software window to start dynamometer.
4. Record speeds and times. For a single run, enter at least eight speeds and times in the short form table spanning the speed range 40mph-80 mph.
5. Write a Matlab script or Excel file to compute drag force as a function of velocity.
6. Use Matlab or Excel to find a second-order polynomial regression fit to your data. You may find the function “polyfit” useful. Try “help polyfit” in Matlab. Enter your result in the short form.
7. Use Matlab or Excel to find the root-mean-squared error (e(rms)) of your polynomial fit to the data. Enter your result in the short form.
[pic]
where: [pic]= your polynomial evaluated at each measurement velocity [pic]
[pic]= effective velocity for each force measurement
** If using Matlab you may want to use the command “polyval” for this step.
8. Use Matlab or Excel to plot your force data and the regression line on a single plot. Attach your plot to the short form report. Label your plot appropriately and write your name, group and section number on each page.
Engine Tractive Force Model:
The engine tractive force model characterizes all tractive force from the vehicle drivetrain. The Whirlwind Development Group has again conducted a series of test track measurements to characterize the engine tractive force on our test vehicle. These measurements use the vehicle model given in (1)-(3). When cruising in 5th gear, the engine tractive force [pic] on the vehicle and can be measured as the difference between the total force and the drag force [pic].
[pic] (6)
Using an approximation from the velocity derivative [pic] with rearrangement of the equation terms, the engine tractive function [pic] can be measured as
[pic] (7)
where [pic] = acceleration of gravity (ft/sec2) = 32.2 (ft/sec2)
[pic] = vehicle weight (lb) = 3,010 lb (Hybrid Vehicle) + 190 lb (driver)= 3200 lb
[pic] = vehicle speed (ft/sec) at time [pic](sec),
[pic] = vehicle speed (ft/sec) at time [pic](sec) and
[pic] = the previously measured vehicle drag force model (lbf)
Again, all test measurements of speed are taken using the test vehicle’s speedometer that is calibrated in miles/hour (mph). Previous test track tests have generated the least mean-squared error regression results shown in Figure 4 and 5. Your first task is to validate the dynamometer system’s response against these results.
[pic]
Figure 4: Engine Tractive Power at Rear Wheels
Figure 5: Tractive Force at Rear Wheels
Engine Tractive Force Measurement Procedure
1. Find the dynamometer test system “WindDriveTest” and run it.
2. Set system operating conditions:
gear = “5th”, Throttle = 100%, start speed = your desired starting speed
3. “Click” the arrow in the upper left corner of the software window to start dynamometer.
4. Record speeds and times. For a single run, enter at least eight speeds and times in the short form table spanning the speed range 40mph-80 mph.
5. Write a Matlab script or Excel file to compute tractive force as a function of velocity.
6. Use Matlab or Excel to find a second-order polynomial regression fit to your data. You may find the function “polyfit” useful. Try “help polyfit” in Matlab. Enter your result in the short form.
7. Use Matlab or Excel to plot your force data and the regression line on a single plot. Attach your plot to the short form report. Label your plot appropriately and write your name, group and section number on each page.
8. Repeat steps 1-7 for 0% Throttle.
Assembly of Total Measured Vehicle Model:
Assembly of a vehicle dynamic model proceeds directly using (1)-(3) and the results you have measured for vehicle drag and tractive forces. We will assume that the engine tractive force varies linearly between its values at 100% and 0%. The vehicle dynamic model can now be expressed using your (3) models as
[pic] (8)
where [pic], measured 2nd order regression at 100% throttle
[pic], measured 2nd order regression at 0% throttle
[pic], measured 2nd order regression of drag force
Note that the “[pic]” and “[pic]” terms in equation (8) above perform a linear interpolation between [pic] and [pic]based on throttle position [pic].
The Total Vehicle Model (8) is non-linear and must be linearized about an operating point for use in next week’s control design project. Linearization is accomplished by a Taylor Expansion about an equilibrium operating point ([pic]) for the model. Define the nonlinear force function,
[pic] (9)
In our case, define an operating point velocity [pic]and solve for the throttle position [pic] such that [pic]. At this operating point, the Taylor expansion becomes,
[pic] (10)
where: [pic], the speed deviation from the operating point
[pic], the throttle deviation from the operating point
[pic], the local variation in total force with speed (lbf-sec/ft)
[pic], the local variation in total force with speed (lbf/%)
With this linearization, the linearized vehicle model becomes,
[pic] (11)
Model Assembly and Linearization Procedure:
1. Enter your least mean-squared error regressions for [pic] in the short form report spaces provided.
2. Obtain an operating speed [pic]from your lab consultant and find (graphically) the value of throttle opening [pic]for which [pic]. This value is the operating Throttle value.
3. Linearize [pic]about the operating point [pic]and enter the coefficients [pic] and [pic] in the space provided. Write the linearized vehicle dynamic equations [pic]
4. Compute the linearized system transfer function and identify the vehicle system’s gain and time constant. Comment of your results: Do they make sense? Are they realistic?
Drag Measurement:
1.-5. Drag Test Data:
|Speed (mph) |Speed (ft/s) |Time |[pic] |[pic] |[pic] |[pic] (lbf) |
| | |(sec) |(ft/sec) |( sec) |(ft/sec) | |
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6.-8.
Polynomial regression coefficients:
[pic]= (_________) + (_________) [pic] + (_________) [pic]2
RMS Error
[pic] = ___________________________ lbf
Attach plot as page #1a
5th Gear 100% Throttle Measurement:
|Speed (mph) |Speed (ft/s) |Time |[pic] |[pic] |[pic] |[pic] (lbf) |
| | |(sec) |(ft/sec) |( sec) |(ft/sec) | |
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Polynomial regression coefficients:
[pic]= (_________) + (_________) [pic] + (_________) [pic]2
RMS Error
[pic] = ___________________________ lbf
5th Gear 0% Throttle Measurement:
|Speed (mph) |Speed (ft/s) |Time |[pic] |[pic] |[pic] |[pic] (lbf) |
| | |(sec) |(ft/sec) |( sec) |(ft/sec) | |
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Polynomial regression coefficients:
[pic] = (_________) + (_________) [pic] + (_________) [pic]2
RMS Error
[pic] = ___________________________ lbf
[pic]
Sketch the drag force plot on the plot given above
1. Obtain a operating speed [pic]from your lab consultant and find the value of operating point throttle opening [pic] for which [pic]. Graph your data for drag on the above plot. Your operating point throttle can be estimated.
[pic] = ____________________
2. Graphically linearize [pic]about the operating point [pic]and enter the coefficients [pic] and [pic] in the space provided. Write the linearized vehicle dynamic equation [pic] with numeric values. Attach your work to this sheet as page number 2a.
[pic]
[pic], the local variation in total force with speed (lbf-sec/ft)
[pic], the local variation in total force with throttle (lbf/%)
[pic] = [pic]=
Linearized Vehicle Equation:
3. Find the system Transfer Function, Steady-State Gain and Time Constant
Transfer Functrion, [pic]
Steady-State Gain =
Time Constant, [pic] =
Comment on your modeling results:
-----------------------
Vehicle Drag
Force, FD
Engine Tractive
Force, FE
100% Throttle
Tractive Force (lbf)
Speed feet/second
Hybrid Electric Powertrain Tractive Force Analysis
300
250
200
150
100
50
0
-50
120
110
100
90
[pic]
0% Throttle
80
70
60
10% Throttle
20% Throttle
30% Throttle
40% Throttle
50% Throttle
60% Throttle
70% Throttle
80% Throttle
90% Throttle
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