ME451 LabSpeed Control Design
ME451: Speed Control Design Project
Week 2: Speed Control Design
Reference: C.L. Phillips and R.D.Harbor, Feedback Control Systems
Chapter 7:Root-Locus Analysis and Design
Section 10: PID Design and Section 11: Analytical PID Design
Introduction:
The Whirlwind Corporation is engaged in the rapid prototyping of an advance vehicle speed control system. Last week you and you colleagues developed a dynamic analysis model of a test vehicle on 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. This week you will develop a speed control design, prototype the design and test it. The vehicle’s speed control performance will then be evaluated on the Whirlwind Dynamometer facility.
The Vehicle Dynamics Model
Two opposing forces drive the speed dynamics of a vehicle: the vehicle drag force and engine tractive force (Fig. 1)
Figure 1: Forces that Change Vehicle Speed
A mathematical model for vehicle speed dynamics for a vehicle of mass can be written as
(1)
where both the vehicle engine tractive force, and the vehicle drag force, 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
(2)
and
(3)
where is vehicle speed and is throttle position.
To develop a new speed control product using modern mathematical techniques, Whirlwind engineers have identified the vehicle mass “” and the vehicle force functions and for the test vehicle. The vehicle mass is simply a function of the weight of the vehicle plus occupants.
(4)
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 used the vehicle model given in (1)-(3).
(5)
where velocity has units ft/sec. Drag force has units lbf and is a function of velocity only. The engine tractive force model characterizes all tractive force from the vehicle drivetrain. The Whirlwind Development Group 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).
(6)
where drag force has units of lbf and is a function of both velocity (ft/sec) and throttle opening (%). When the full non-linear model is assembled, it takes the form
(7)
Collecting terms yields the final form of the non-linear hybrid vehicle model.
(8)
where = vehicle speed (ft/sec), and
= Throttle opening (%)
Model Linearization and Operating Point
A linearized model is required for most control system design and analysis. The nonlinear system model above can be linearized about an operating point defined by a combination of speed and throttle . Following the standard linearization procedure about an equilibrium point,
1.Given either or , find the other operating point coordinate such that the system is in equilibrium, i.e. all derivatives = 0. Here the velocity derivative is equal to the total force and at equilibrium, the total force on the vehicle is zero.
(9)
2.At this equilibrium operating point, perform a Taylor expansion of the nonlinear force model
(10)
3.Free constants in the above equation are now removed by redefining the model coordinates and observing that, at equilibrium = 0 through step 1 above.
(11)
where: and are deviations from the operating point.
4.The partial derivatives become constant parameters when evaluated at the operating point . Neglecting the higher order terms, the linearized force
(12)
where:,
5.Finally, we note that the time derivative of the deviation from equilibrium equals the time derivative of the original variable,
(13)
so that we find
(14), (15)
The linear transfer function for this system gives the ratio of speed deviation to throttle deviation. Taking the Laplace transform of (15) and rearranging yields
(16)
Speed Control Design:
The system schematic for the Whirlwind Speed Control System is shown in Figure 2. This system includes all nonlinear effects and allows the control to regulate the speed around the operating point by manipulating throttle-opening deviation . The schematic in this form includes both an input for operating point vehicle speed and an input for operating point throttle opening . The form shown in Figure 2 is useful for building a speed control and/or non-linear simulation of the system but not as useful for control design with linear design methods. Figure 3 shows the same system after modification to deviation (linearized) variables. This system has an output of vehicle speed deviation from a desired operating point and a reference input representing desired deviation from the operating point,. Both the system error and the throttle deviation output from the controller are identical in both figures. The linearized schematic, Fig. 3, allows our linearized model to be used in the controller design.
The block diagram of the linearized system (Figure 3) is shown below with controller transfer function and system open-loop transfer function. Remember that all variables are deviations from steady state speed and throttle values. To maintain 60 mph, = 0 ft/sec
The purpose of the closed-loop system is to improve the transient and steady-state speed regulation of the vehicle as compared to its open-loop (non-feedback controlled) behavior. As a baseline, let’s calculate some “open-loop” values for the vehicle given its transfer function as shown in (16). Assuming a “small” 5 mph deviation in vehicle speed, the initial open-loop initial acceleration is generated from the open-loop throttle deviation required for a= 5 mph = 7.3 ft/sec speed deviation. This throttle deviation is computed from the inverse of the steady state gain using the final-value theorem
(17)
The initial acceleration generated by the open-loop system can be calculated using the initial-value theorem and applying an input step of magnitude
(18)
thus the open-loop initial acceleration is a linear function of the desired change in speed. Is this true of the closed-loop system? Is your answer dependent on controller design? Remember, both open-loop and closed-loop systems are linear. A reasonable estimate of a comfortable “g”level for vehicle passengers might allow ten times that acceleration. Let’s design a zero steady-state error speed control with 10 times the speed of response of the open-loop system.
Root Locus Design:
To achieve zero steady-state error requires a type 1 control with at least PI action.
This controller has one pole s = 0 and one zero at s = -
From the closed-loop system block diagram,
The “open-loop” transfer function needed to draw the root locus has a numerator polynomial and denominator polynomial . For a stable system (), a root locus for the closed loop system can be drawn using the Matlab command “rlocus” and a typical root locus shape is shown in Figure 4 for . This root locus shows that as increases, the closed-loop poles converge from their open-loop values, 0 and , to a repeated real root then diverge to complex-conjugate roots with second-order response and finally become strictly real again. A good choice of root location is not the fast, purely real, poles obtained at high gain but rather the lower gain, slightly oscillatory root location (You should think about this and ask your Laboratory Consultant whether your idea for why is correct). The root locus allows computation of from the graph for each value of used to draw it.
Figure 4: Root Locus Design Example
Once and are determined, the Matlab routine “step” can be used to compute the predicted system step response to a unit () input. Since this is a linear problem, we can scale the output time response result to the magnitude of the input you computed earlier simply by multiplying by the magnitude of . A typical, underdamped step response would have the shape shown in Figure 5.
Figure 5: Step Response Example
Why the overshoot? (It’s underdamped) Is the overshoot a good thing? (Yes, it allows a quick response to near the steady state value).
Equivalent results (with less insight) can be achieved using the strictly analytical results given in your text for analytical design.
Whirlwind Speed ControlDesign
Specifications:
1. A system settling time of 10 seconds.
2. A zero system steady-state error.
3. A reasonable acceleration for a cruise control.
4. A damping of ζ = 0.707
Design a PI, PD, or PID controlto meet the above objectives and show that your design meets them.
Test your speed controldesign on the Whirlwind dynamometer system and show that it has acceptable response.
1Revised: 11/12/07
Speed Control Design ProjectWeek 2Vehicle Dynamic Modeling
Name:Date:
1.a) Enter your model from last week’s modeling lab below
Operating Point Speed = ft/secOperating Point Throttle = %
Differential Equation Model:
Transfer Function Model:
Time Constant = secSteady-State Gain = (ft/sec)/%
b) Use the polynomials provided by your Lab Consultant to find an “exact” linearized model at your operating speed.
Operating Point Speed = ft/secOperating Point Throttle = %
Differential Equation Model:
Transfer Function Model:
Time Constant = secSteady-State Gain = (ft/sec)/%
How do the two models compare?
Confirm your model with your Lab Consultant.
2. Compute the open-loop vehicle acceleration (ft/sec2) resulting from an open-loop change in speed of 5 mph. What steady-state change in throttle is required? Enter your results below.
Open-loop vehicle acceleration, = ft/sec2
Required Throttle change = %
Question:
Is the derivative of the velocity deviation equal to the actual vehicle acceleration?
3.Design a PID Controller to meet the Whirlwind design specifications.
Enter the maximum acceleration, ft/sec2
= = =
Document your work and attach to this sheet.
- Draft a short work order with the above information and present it to the lab technician. The lab technician will then run the Whirlwind Dynamometer system with your controller values, and present to you the output data.
- Demonstrate,in the form of a one page report, that your output data meets all design specifications.
1Revised: 11/12/07