Description
1 Introduction
This project will allow you to practice the following skills
• Checking controllability and observability
• Design a state feedback controller
• Design a full state observer
As in Project 1, this project consists of a written component along with a Webots controller
design component.
2 Model
State feedback control is used to drive the states to the origin, but in this case we are following
a trajectory. For this project we will follow the lead of [1] and redefine the state space system
developed in Project 1 to be based on the error in lateral position. Like Project 1, we will
assume that the dynamics are decoupled between the lateral and longitudinal directions,
which allows us to independently design lateral and longitudinal controllers.
We will define the following states to capture the error (see Section 2.5 in [1] for more
detail):
• e1 – the distance of the vehicle mass center from the track centerline. Note that this
was absent in the first Project!
• e2 – the orientation of the vehicle with respect to the road. This is what we controlled
with a PID controller in Project 1.
Using state feedback we will be able to account for BOTH position errors relative to the
center and the heading – this might make state feedback more successful than PID control
for this system.
Assume that the longitudinal velocity Vx is constant and the radius of the road R is
constant and large. Then we can define the desired rate of orientation change as
ψ˙
des =
Vx
R
.
The desired acceleration is then given by
alatdes =
V
2
x
R
= Vxψ˙
des.
Now we can compute the error in lateral acceleration
e¨1 = ¨y + Vxψ˙
| {z }
alat
−
V
2
x
|{z}
R
alatdes
= ¨y + Vx
ψ˙ − ψ˙
des
2
With Vx and a constant we can integrate this easily
e˙1 = ˙y + Vx (ψ − ψdes) (1)
Using the definition of the heading error (e2 = ψ − ψdes) along with the dynamics in Project
1 we can derive the following error dynamics equation.
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
On the sensing side, we typically know the location of the car (GPS) with its heading (compass) and longitudinal velocity (tachometer). In other words, we cannot directly
measure the full state in the error dynamics model.
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3 Project 2: State Space Control [Due April 8, 2021]
Exercise 1. To warm up for the exercises in the project (and help prep for the 2nd exam),
here are several typical homework problems on controllability, observability, controller design,
and observer design.
1. For the following systems determine controllability and observability.
• x˙ =
0 1 0
0 0 1
−1 −3 −3
x +
1
0
0
u, y =
1 1 1
x
• x˙ =
0 1 0
0 0 1
0 2 −1
x +
0 1
1 0
0 0
u, y =
1 0 1
x
• x˙ =
2 1 0 0 0 0 0
0 2 0 0 0 0 0
0 0 2 0 0 0 0
0 0 0 2 0 0 0
0 0 0 0 1 1 0
0 0 0 0 0 1 0
0 0 0 0 0 0 1
x +
2 1 0
2 1 1
1 1 1
3 2 1
−1 0 1
1 0 1
1 0 0
u, y =
2 2 1 3 −1 1 1
1 1 1 2 0 0 0
0 1 1 1 1 1 0
x
2. Find a state space realization for G(s) = 1
s
s+3
s+1
1
s+3
s
s+1
.
3. Consider the discrete time system xk+1 =
1 1 −2
0 1 1
0 0 1
xk +
1
0
1
uk. Design a state
feedback control matrix K such that the closed loop system has all poles at 0.
4. Consider the discrete time system xk+1 =
−2 4
−3 9
x +
0
1
u, y =
3 1
x. Design an
observer matrix L such that the observer has poles at −0.5 ± 0.5j.
Exercise 2. Considering the linearized, error-based state space system for the vehicle in
the Model section above:
1. Check the controllability of the system at the following longitudinal velocities: 2 m/s,
5 m/s and 8 m/s.
2. For longitudinal velocities v from 1 m/s to 40 m/s, plot the following:
(a) log10(
σ1
σn
) versus v (m/s), where σi
is the ith singular value of the controllability
matrix P (i = 1, 2, …, n). (In other words, what is the logarithm of the greatest
singular value divided by the smallest?)
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