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24-677 Special Topics: Linear Control Systems
1 Introduction
In this project, you will complete the following goals:
1. Design an lateral optimal controller
2. Consider the relative merits of static LQR control vs. model predictive control for
controlling the nonlinear system
[Remember to submit the write-up, plots, and codes on Gradescope.]
2 Model
We will use the same model from Project 2, which is repeated here for clarity. The errorbased linearized state-space for the lateral dynamics is as follows.
e1 is the distance to the center of gravity of the vehicle from the reference trajectory.
e2 is the orientation error of the vehicle with respect to the reference trajectory.
d
dt
e1
e˙1
e2
e˙2
=
0 1 0 0
0 −
4Cα
mx˙
4Cα
m −
2Cα(lf −lr)
mx˙
0 0 0 1
0 −
2Cα(lf −lr)
Izx˙
2Cα(lf −lr)
Iz
−
2Cα(l
2
f +l
2
r
)
Izx˙
e1
e˙1
e2
e˙2
+
0 0
2Cα
m
0
0 0
2Cαlf
Iz
0
δ
F
+
o
−
2Cα(lf −lr)
mx˙ − x˙
0
−
2Cα(l
2
f +l
2
r
)
Izx˙
ψ˙
des
In lateral vehicle dynamics, ψ˙
des is a time-varying disturbance in the state space equation.
Its value is proportional to the longitudinal speed when the radius of the road is constant.
When deriving the error-based state space model for controller design, ψ˙
des can be safely
assumed to be zero.
d
dt
e1
e˙1
e2
e˙2
=
0 1 0 0
0 −
4Cα
mx˙
4Cα
m −
2Cα(lf −lr)
mx˙
0 0 0 1
0 −
2Cα(lf −lr)
Izx˙
2Cα(lf −lr)
Iz
−
2Cα(l
2
f +l
2
r
)
Izx˙
e1
e˙1
e2
e˙2
+
0 0
2Cα
m
0
0 0
2Cαlf
Iz
0
δ
F
For the longitudinal control:
d
dt
x
x˙
=
0 1
0 0 x
x˙
+
0 0
0
1
m
δ
F
+
0
ψ˙y˙ − fg
Assuming ψ˙ = 0:
d
dt
x
x˙
=
0 1
0 0 x
x˙
+
0 0
0
1
m
δ
F
2
3 Controller Design Approaches
We will look at two different approaches to study the system (of course in addition to these
you can come up with your own to maximize performance!)
• Static LQR control. Linearize the system about the nominal operating speed and
design an infinite horizon LQR controller for the system. Apply this LQR controller
throughout the trajectory.
• Model predictive control. We know that the system will likely slow down around
curves. We can re-linearize the system in terms of the actual operating speed at each
time step and solve a discrete time LQR controller n steps into the future, then output
only the 1st control learned in the sequence.
3.1 Static LQR Control
Start by discretizing the continuous error dynamics at the given time step (the ZOH discretization is probably best). Now we will design the infinite horizon LQR controller. Using
LQR requires manually creating two matrices Q and R. Q works to penalize state variables,
while R penalizes control input. Try to think about what form your Q and R matrices should
take for good performance.
• For Q, large values will heavily restrict changes to the respective states, while small
values will allow the states to easily change.
• Similarly, in R, large values will heavily restrict control input, while small values will
allow the control input to vary widely.
• One idea for tuning is to set the relevant indices of Q and R to
1
(max value of the corresponding state/input)
2
in order to normalize the value. Make sure to experiment outside of this guideline to
determine the best performance.
• There is a relationship between Q and R, though it is subtle. For example, if you
increase weights in Q, you are more heavily penalizing changes in the states, which
will require more control input. This would imply that you should decrease weights in
R to see an effect. Due to this, it may be helpful to keep either Q or R fixed while
varying the other during tuning.
• Diagonal versions of Q and R are often used because they are trivially positive definite
and easy to interpret in terms of the state variables.
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3.2 Model Predictive Control
The LQR controller assumes that the speed is constant – can we remove this constraint?
Consider the following procedure at each time step.
• Using the measured car speed, compute a new linearized model of the (discrete time)
system.
• Compute a finite horizon LQR controller N steps into the future.
• Output the first element of the control sequence.
As an alternative, of course, we could re-linearize the system and recompute an infinite
horizon LQR. This recomputation could be triggered this when the speed has moved sufficiently far away from the operating point. However, there might be advantages to having
an approach that is consistent between all time steps (avoids abrupt changes in control /
computation time).
4 Project 3: Problems [Due April 29th, 2021]
Exercise 1. Like Project 2 we will practice LQR design on an unrelated system before
jumping into the control of the Sprinter. You should solve all problems computationally
using Matlab.
1. For the system
x˙ =
0 1
−10 −7
x +
0
1
u,
find the control u(t) that minimizes the performance measure
J = 10x
2
1
(5) + 1
2
Z 5
0
5x
2
1 + x
2
2 + 0.25u
2
dt.
Plot the control sequence starting from x(0) =
1
1
.
2. Discretize the above system using a step size of T = 0.005 and find a control that
minimizes the performance measure
J = 10x
2
1
(1000) + 1
2
X
999
k=1
5×1(k)
2 + x
2
2
(k) + 0.25u
2
(k).
Plot the control sequence starting from x(0) =
1
1
and compare to the continuous
time result.
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3. Design an infinite time LQR controller for the continuous time system using the cost
function
J =
1
2
Z ∞
0
5x
2
1 + x
2
2 + 0.25u
2
dt.
Plot the control sequence up to 5 seconds starting from x(0) =
1
1
and compare to
the above results.
Exercise 2. Show your approach for designing the two controllers discussed in the prior
section. Specifically, show the following.
1. Show your approach to tuning the Q and R matrices along with the Matlab code you
used to design your controllers. Provide your implementation code in Python as well,
along with a graph that shows your best performance.
2. Show the approach you used to determine N for the model predictive controller. This
should include simulation results as a function of N while holding Q and R constant.
Performance should converge as N increases. Provide your implementation code in
Python, along with a graph that shows your best performance.
Exercise 3. Optimize your Webots performance for both controller types..
You can reuse your longitudinal PID controller from part 1 of this project, or even improve upon it. However, it may require retuning based on observed performance.
Design the two controllers in your controller.py. You can make use of Webots’ builtin code editor, or use your own.
When you complete the track, the scripts will generate a performance plot via matplotlib.
This plot contains a visualization of the car’s trajectory, and also shows the variation of
states with respect to time.
Submit your controller lqr.py and your controller mpc.py and the final completion
plots as described on the title page. Both controllers are required to achieve the following
performance criteria to receive full points:
1. Time to complete the loop = 200 s
2. Maximum deviation from the reference trajectory = 6.5 m
3. Average deviation from the reference trajectory = 2.5 m
4. (NEW!) Average rate of change in steering angle = 0.025 rad/timestep
[10% Bonus]: Complete the loop within 130 s. The maximum deviation and the average
deviation should be within in the allowable performance criteria mentioned above.
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5 Appendix
(Already covered in P1)
Figure 1: Bicycle model[2]
Figure 2: Tire slip-angle[2]
We will make use of a bicycle model for the vehicle, which is a popular model in the
study of vehicle dynamics. Shown in Figure 1, the car is modeled as a two-wheel vehicle
with two degrees of freedom, described separately in longitudinal and lateral dynamics. The
model parameters are defined in Table 2.
5.1 Lateral dynamics
Ignoring road bank angle and applying Newton’s second law of motion along the y-axis:
may = Fyf cos δf + Fyr
where ay =
d
2
y
dt2
inertial
is the inertial acceleration of the vehicle at the center of geometry
in the direction of the y axis, Fyf and Fyr are the lateral tire forces of the front and rear
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wheels, respectively, and δf is the front wheel angle, which will be denoted as δ later. Two
terms contribute to ay: the acceleration ¨y, which is due to motion along the y-axis, and the
centripetal acceleration. Hence:
ay = ¨y + ψ˙x˙
Combining the two equations, the equation for the lateral translational motion of the vehicle
is obtained as:
y¨ = −ψ˙x˙ +
1
m
(Fyf cos δ + Fyr)
Moment balance about the axis yields the equation for the yaw dynamics as
ψI¨
z = lfFyf − lrFyr
The next step is to model the lateral tire forces Fyf and Fyr. Experimental results show that
the lateral tire force of a tire is proportional to the “slip-angle” for small slip-angles when
vehicle’s speed is large enough – i.e. when ˙x ≥ 0.5 m/s. The slip angle of a tire is defined
as the angle between the orientation of the tire and the orientation of the velocity vector of
the vehicle. The slip angle of the front and rear wheel is
αf = δ − θV f
αr = −θV r
where θV p is the angle between the velocity vector and the longitudinal axis of the vehicle,
for p ∈ {f, r}. A linear approximation of the tire forces are given by
Fyf = 2Cα
δ −
y˙ + lfψ˙
x˙
!
Fyr = 2Cα
−
y˙ − lrψ˙
x˙
!
where Cα is called the cornering stiffness of the tires. If ˙x < 0.5 m/s, we just set Fyf and Fyr both to zeros. 5.2 Longitudinal dynamics Similarly, a force balance along the vehicle longitudinal axis yields: x¨ = ψ˙y˙ + ax max = F − Ff Ff = fmg where F is the total tire force along the x-axis, and Ff is the force due to rolling resistance at the tires, and f is the friction coefficient. 7 5.3 Global coordinates In the global frame we have: X˙ = ˙x cos ψ − y˙ sin ψ Y˙ = ˙x sin ψ + ˙y cos ψ 5.4 System equation Gathering all of the equations, if ˙x ≥ 0.5 m/s, we have: y¨ = −ψ˙x˙ + 2Cα m (cos δ δ − y˙ + lfψ˙ x˙ ! − y˙ − lrψ˙ x˙ ) x¨ = ψ˙y˙ + 1 m (F − fmg) ψ¨ = 2lfCα Iz δ − y˙ + lfψ˙ x˙ ! − 2lrCα Iz − y˙ − lrψ˙ x˙ ! X˙ = ˙x cos ψ − y˙ sin ψ Y˙ = ˙x sin ψ + ˙y cos ψ otherwise, since the lateral tire forces are zeros, we only consider the longitudinal model. 5.5 Measurements The observable states are: y = x˙ y˙ ψ˙ X Y ψ 5.6 Physical constraints The system satisfies the constraints that: |δ| 6 π 6 rad F > 0 and F 6 16000 N
x˙ > 10−5 m/s
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Table 1: Model parameters.
Name Description Unit Value
( ˙x, y˙) Vehicle’s velocity along the direction of
vehicle frame
m/s State
(X, Y ) Vehicle’s coordinates in the world
frame
m State
ψ, ψ˙ Body yaw angle, angular speed rad,
rad/s
State
δ or δf Front wheel angle rad State
F Total input force N Input
m Vehicle mass kg 4500
lr Length from rear tire to the center of
mass
m 3.32
lf Length from front tire to the center of
mass
m 1.01
Cα Cornering stiffness of each tire N 20000
Iz Yaw intertia kg mˆ2 29526.2
Fpq Tire force, p ∈ {x, y},q ∈ {f, r} N Depends on input force
f Rolling resistance coefficient N/A 0.028
delT Simulation timestep sec 0.032
5.7 Simulation
Figure 3: Simulation code flow
Several files are provided to you within the controllers/main folder. The main.py script
initializes and instantiates necessary objects, and also contains the controller loop. This loop
runs once each simulation timestep. main.py calls your controller.py’s update method
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on each loop to get new control commands (the desired steering angle, δ, and longitudinal
force, F). The longitudinal force is converted to a throttle input, and then both control
commands are set by Webots internal functions. The additional script util.py contains
functions to help you design and execute the controller. The full codeflow is pictured in
Figure 3.
Please design your controller in the your controller.py file provided for the project part
you’re working on. Specifically, you should be writing code in the update method. Please
do not attempt to change code in other functions or files, as we will only grade the
relevant your controller.py for the programming portion. However, you are free to
add to the CustomController class’s init method (which is executed once when the
CustomController object is instantiated).
5.8 BaseController Background
The CustomController class within each your controller.py file derives from the BaseController class in the base controller.py file. The vehicle itself is equipped with a
Webots-generated GPS, gyroscope, and compass that have no noise or error. These sensors
are started in the BaseController class, and are used to derive the various states of the
vehicle. An explanation on the derivation of each can be found in the table below.
Table 2: State Derivation.
Name Explanation
(X, Y ) From GPS readings
( ˙x, y˙) From the derivative of GPS readings
ψ From the compass readings
ψ˙ From the gyroscope readings
5.9 Trajectory Data
The trajectory is given in buggyTrace.csv. It contains the coordinates of the trajectory as
(x, y). The satellite map of the track is shown in Figure 4.
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Figure 4: Buggy track[3]
6 Reference
1. Rajamani Rajesh. Vehicle Dynamics and Control. Springer Science & Business Media,
2011.
2. Kong Jason, et al. “Kinematic and dynamic vehicle models for autonomous driving
control design.” Intelligent Vehicles Symposium, 2015.
3. cmubuggy.org, https://cmubuggy.org/reference/File:Course_hill1.png
4. “PID Controller – Manual Tuning.” Wikipedia, Wikimedia Foundation, August 30th,
2020. https://en.wikipedia.org/wiki/PID_controller#Manual_tuning
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