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EECS 16B
Designing Information Systems and Devices II
UC Berkeley
Fall 2023
Homework 9
This homework is due on Saturday, October 28, 2023, at 11:59PM.
Self-grades and HW Resubmissions are due on the following Saturday,
November 4th, 2023, at 11:59PM.
1. Open-Loop and Closed-Loop Control
In last week’s lab-related System ID problem, we built SIXT33N’s motor control circuitry and devel-
oped a linear model for the velocity of each wheel. We are one step away from our goal: to have
SIXT33N drive in a straight line! We will see how to use the model we developed in the System ID
problem to control SIXT33N’s trajectory to be a straight line.
More specifically, in this problem, we will explore how to use open-loop and closed-loop control to
drive the trajectory of your car in a straight line.
Part 1: Open-Loop Control
An open-loop controller is one in which the input is predetermined using your system model and
the goal, and not adjusted at all during operation. To design an open-loop controller for your car,
you would set the PWM duty-cycle value of the left and right wheels (inputs
u
L
[
i
]
and
u
R
[
i
]
) such
that the predicted velocity of both wheels is your target wheel velocity (
v
t
). You can calculate these
inputs from the target velocity
v
t
and the
θ
L
,
θ
R
,
β
L
,
β
R
values you learned from data. In the System
ID problem and lab, we have modeled the velocity of the left and right wheels as
v
L
[
i
] =
d
L
[
i
+
1
]
−
d
L
[
i
] =
θ
L
u
L
[
i
]
−
β
L
;
(1)
v
R
[
i
] =
d
R
[
i
+
1
]
−
d
R
[
i
] =
θ
R
u
R
[
i
]
−
β
R
(2)
where
d
L
,
R
[
i
]
represent the distance traveled by each wheel.
(a) Find the open-loop control that would give us
v
L
[
i
] =
v
R
[
i
] =
v
t
.
That is,
solve the model
(Equations (
1
) and (
2
)) for the inputs
u
L
[
i
]
and
u
R
[
i
]
that make the velocities
v
L
[
i
] =
v
R
[
i
] =
v
t
.
1
EECS 16B Homework 9
2023-10-21 17:17:12-07:00
In practice, the
θ
L
,
θ
R
,
β
L
,
β
R
parameters are learned from noisy data, and so can be wrong. This means
that we will calculate the velocities for the two wheels incorrectly.
When the velocities of the two
wheels disagree, the car will go in a circle instead of a straight line. Thus, to make the car go in a
straight line, we need the distances traveled by both wheels to be the same at each timestep.
This prompts us to simplify our model. Instead of having two state variables
⃗
v
L
and
⃗
v
R
, we can just
have a state variable determining how far we are from the desired behavior of going in a line – a state
which we will want to drive to 0.
This prompts us to define our state variable
δ
to be the
difference
in the distance traveled by the left
wheel and the right wheel at a given timestep:
δ
[
i
]
:
=
d
L
[
i
]
−
d
R
[
i
]
(3)
We want to find a scalar discrete-time model for
δ
[
i
]
of the form
δ
[
i
+
1
] =
λ
OL
δ
[
i
] +
f
(
u
L
[
i
]
,
u
R
[
i
])
.
(4)
Here
λ
OL
is a scalar and
f
(
u
L
[
i
]
,
u
R
[
i
])
is the control input to the system (as a function of
u
L
[
i
]
and
u
R
[
i
]
).
(b) Suppose we apply the open-loop control inputs
u
L
[
i
]
,
u
R
[
i
]
to the original system.
Using Equa-
tions (
1
) and (
2
), write
δ
[
i
+
1
]
in terms of
δ
[
i
]
, in the form of Equation (
4
). What is the eigen-
value
λ
OL
of the model in Equation (
4
)?
Would the model in Equation (
4
) be stable with
open-loop control if it also had a disturbance term?
(HINT: For open-loop control, we set the velocities to v
L
[
i
] =
v
R
[
i
] =
v
t
.
What happens when we
substitute that into Equations
(
1
)
and
(
2
)
and then apply the definition of
δ
[
i
]
and
δ
[
i
+
1
]
?)
Part 2: Closed-Loop Control
Now, in order to make the car drive straight, we must implement closed-loop control – that is, control
inputs that depend on the current state and are calculated dynamically – and use feedback in real time.
(c)
If we want the car to drive straight starting from some timestep
i
start
>
0
, i.e.,
v
L
[
i
] =
v
R
[
i
]
for
i
≥
i
start
, what condition does this impose on
δ
[
i
]
for
i
≥
i
start
?
(d)
How is the condition you found in the previous part different from the condition:
δ
[
i
] =
0,
i
≥
i
start
?
(5)
Assume that
i
start
>
0, and that
d
L
[
0
] =
0,
d
R
[
0
] =
0.
This is a subtlety that is worth noting and often requires one to adjust things in real systems.
(e) From here, assume that we have reset the distance travelled counters at the beginning of this
maneuver so that
δ
[
0
] =
0.
We will now implement a feedback controller by selecting two
dimensionless positive coefficients,
f
L
and
f
R
, such that the closed loop system is stable with
eigenvalue
λ
CL
. To implement closed-loop feedback control, we want to adjust
v
L
[
i
]
and
v
R
[
i
]
at
each timestep by an amount that’s proportional to
δ
[
i
]
. Not only do we want our wheel velocities
to be some target velocity
v
t
, we also wish to drive
δ
[
i
]
towards zero. This is in order to have the
car drive straight along the initial direction it was pointed in when it started moving. If
δ
[
i
]
is
positive, the left wheel has traveled more distance than the right wheel, so relatively speaking,
we can slow down the left wheel and speed up the right wheel to cancel this difference (i.e., drive
it to zero) in the next few timesteps. The action of such a control is captured by the following
© UCB EECS 16B, Fall 2023.
All Rights Reserved. This may not be publicly shared without explicit permission.
2
EECS 16B Homework 9
2023-10-21 17:17:12-07:00
velocities.
v
L
[
i
] =
v
t
−
f
L
δ
[
i
]
;
(6)
v
R
[
i
] =
v
t
+
f
R
δ
[
i
]
.
(7)
Give expressions for
u
L
[
i
]
and
u
R
[
i
]
as a function of
v
t
,
δ
[
i
]
,
f
L
,
f
R
, and our system parameters
θ
L
.
θ
R
,
β
L
,
β
R
, to achieve the velocities above.
(f) Using the control inputs
u
L
[
i
]
and
u
R
[
i
]
found in part (e),
write the closed-loop system equation
for
δ
[
i
+
1
]
as a function of
δ
[
i
]
. What is the closed-loop eigenvalue
λ
CL
for this system in
terms of
λ
OL
,
f
L
, and
f
R
?
(g)
What is the condition on
f
L
and
f
R
for the closed-loop system in the previous part to be stable
in the presence of disturbance?
Stability in this case means that
δ
is bounded and will not go arbitrarily high. In fact, if our calculated
β
and
θ
are perfectly accurate, then
δ
[
i
]
→
0, so the car will (eventually) drive straight!
One question remains – what if our calculated
β
and
θ
are
not
perfectly accurate? The answer turns
out to be that there is some small steady-state discrepancy that your
δ
will converge to. You will see
how to quantify this in next week’s homework.
© UCB EECS 16B, Fall 2023.
All Rights Reserved. This may not be publicly shared without explicit permission.
3
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EECS 16B Homework 9
2023-10-21 17:17:12-07:00
2. Cayley-Hamilton and Controllability Matrix
(a) We can define the
characteristic polynomial
of a matrix
A
∈
R
n
×
n
as
p
A
(
λ
) =
λ
n
+
c
n
−
1
λ
n
−
1
+
· · ·
+
c
1
λ
+
c
0
λ
0
(8)
where each
c
i
∈
R
is a constant. The characteristic polynomial has roots that are the eigenvalues
of
A
. That is, we can equivalently define
p
A
(
λ
) =
det
{
λ
I
−
A
}
(9)
We say that any of the eigenvalues of
A
“satisfy” the characteristic polynomial in that
p
A
(
λ
i
) =
0
(10)
where
λ
i
is the
i
th eigenvalue of
A
. Now, let
A
be a diagonalizable matrix, where we may write
A
=
V
Q
V
−
1
.
Prove that
A
satisfies its own characteristic polynomial.
In other words, prove
that
p
A
(
A
) =
0
n
×
n
, where 0
n
×
n
is a
n
×
n
matrix of zeros.
(HINT: It is not correct to simply plug in
λ
=
A into
det
{
λ
I
−
A
}
.)
(b) Now, consider some vector
⃗
b
∈
R
n
.
Using the result from the previous part, show that
A
n
⃗
b
is
linearly dependent on
A
n
−
1
⃗
b
,
A
n
−
2
⃗
b
, . . . ,
A
⃗
b
,
⃗
b
.
(c) Instead of setting
⃗
b
to be a vector, let it be a matrix
B
∈
R
n
×
m
.
Now, show that the columns of
A
n
B
are linearly dependent on the columns of
A
n
−
1
B
,
A
n
−
2
B
, . . . ,
AB
,
B
.
(HINT: If we were to write B
=
h
⃗
b
1
⃗
b
2
· · ·
⃗
b
m
i
where each column is n-dimensional, we can write
A
i
B
=
h
A
i
⃗
b
1
A
i
⃗
b
2
· · ·
A
i
⃗
b
m
i
. Make sure you convince yourself of this.)
(d) Consider a discrete time system of the form
⃗
x
[
i
+
1
] =
A
⃗
x
[
i
] +
B
⃗
u
[
i
]
(11)
where
A
∈
R
n
×
n
and
B
∈
R
n
×
m
. The controllability matrix for this discrete time system is given
by
C
=
h
A
n
−
1
B
A
n
−
2
B
· · ·
AB
B
i
(12)
Conclude that the rank of your controllability matrix will not change if, instead, you made
your controllability matrix
h
A
n
B
A
n
−
1
B
· · ·
AB
B
i
(i.e., you prepended
A
n
B
to your
original controllability matrix).
© UCB EECS 16B, Fall 2023.
All Rights Reserved. This may not be publicly shared without explicit permission.
4
EECS 16B Homework 9
2023-10-21 17:17:12-07:00
3. CCF Transformation and Controllability
(a) Consider the following discrete time system
⃗
x
[
i
+
1
] =
A
⃗
x
[
i
] +
B
⃗
u
[
i
]
(13)
Suppose we define a change of basis operation given by
M
⃗
z
[
i
] =
⃗
x
[
i
]
⇐⇒
⃗
z
[
i
] =
M
−
1
⃗
x
[
i
]
. This
yields a new discrete time system of the form
⃗
z
[
i
+
1
] =
e
A
⃗
z
[
i
] +
e
B
⃗
u
[
i
]
(14)
for some
e
A
and
e
B
defined in terms of
M
,
A
, and
B
.
What is the controllability matrix for the
system in eq. (
14
), in terms of
M
,
A
, and
B
?
(b) Consider the change of basis given by
⃗
z
[
i
] =
T
−
1
⃗
x
[
i
]
where, under this change of basis transfor-
mation, we have the following discrete time system
⃗
z
[
i
+
1
] =
A
CCF
⃗
z
[
i
] +
B
CCF
⃗
u
[
i
]
(15)
Using the result from the previous part, determine an expression for
T
in terms of
C
, the
controllability matrix of the original system in eq. (
13
), and
C
CCF
, the controllability matrix of
the system in eq. (
15
).
(c) We know that the controllability matrix for a system in CCF will always be full rank.
Using this,
prove that you can find a transformation matrix
T
as in the previous part if and only if your
original system is controllable.
(HINT: To prove this, you can first show that, if such a T exists, then
your original system is controllable. Then, you can show that, if your original system is controllable,
there will exist such a transformation matrix T.) (HINT: Recall that T must be invertible (equivalently,
full rank) in order for it to be a valid transformation matrix. You may use without proof the fact that
rank
(
AB
) =
min
(
rank
(
A
)
, rank
(
B
))
.)
(d) Consider the following discrete-time dynamics model:
⃗
x
[
i
+
1
] =
"
1
1
0
1
#
|
{z
}
A
⃗
x
[
i
] +
"
0
1
#
|{z}
⃗
b
⃗
u
[
i
]
(16)
Find the transformation matrix
T
such that the dynamics model for
⃗
z
[
i
] =
T
−
1
⃗
x
[
i
]
is in CCF.
You may use a calculator/computer to perform any computations, if you wish.
(HINT: First, find the characteristic polynomial of A. Use this to determine what A
CCF
and
⃗
b
CCF
should
be, and then use this to determined
C
CCF
.)
© UCB EECS 16B, Fall 2023.
All Rights Reserved. This may not be publicly shared without explicit permission.
5
EECS 16B Homework 9
2023-10-21 17:17:12-07:00
4. Gram-Schmidt Basics
(a)
Use Gram-Schmidt to find a matrix
U
whose columns form an orthonormal basis for the col-
umn space of
V
.
V
=
1
1
1
0
1
1
0
1
1
0
0
1
0
0
1
(17)
(b) Show that you get the same resulting vector when you project
⃗
w
=
1
−
1
0
−
1
0
onto the columns of
V
as you do when you project onto the columns of
U
, i.e.
show that
V
(
V
⊤
V
)
−
1
V
⊤
⃗
w
=
U
(
U
⊤
U
)
−
1
U
⊤
⃗
w
.
(18)
Feel free to use
NumPy
for the projection onto the columns of
V
, but compute the projection onto
the columns of
U
by hand. Comment on whether projecting upon the
V
or
U
basis is computa-
tionally more efficient.
(HINT: Which of these matrices allow us to circumvent the matrix inversion in
the projection formula?)
Contributors:
• Bozhi Yin.
• Kaitlyn Chan.
• Yi-Hsuan Shih.
• Vladimir Stojanovic.
• Moses Won.
• Druv Pai.
• Anish Muthali.
© UCB EECS 16B, Fall 2023.
All Rights Reserved. This may not be publicly shared without explicit permission.
6
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- a. Currently, you serve as a robotic engineer by a company, and your boss want you to design a robot in spatial space to satisfy the following conditions: i. Max reachable distance of the robot is 1 meter. ii. Only using revolute joints, connecting links and rigid load bearing links (the length of link is 0.5m). iii. The degree of freedom (Dof) of the robot is 2. What is your design, how many joints and links will you use? b. The payload of your design is 10 kg (each load bearing link can hold 5 kg, 5 kg+5 kg=10kg, the weight and payload of the connecting links will be ignored), but your boss wants your designed robots to handle 20 kg payload. How will you update your current design, and how many joints and links will you use in the new design? What is the Dof of the new designed robot.arrow_forward(PID solution with the requirements and simple explanation. Thx) INSTRUCTION: Given three bare processes develop a control system using feedbackand feedforward concept/principle. Your output is a process and instrumentationdiagram (P & ID) using ISA's Instrument Identification and Symbols standards withexplanation. PROCESS 2: The liquid level inside the tank is regulated at a value of 3.0 m on a condition ofcontinuous liquid inflow and outflow. Requirements: Two feedback solutions One feedforward solutionarrow_forwardSystem Specification Task 1 is to complete the development of a simple simulated 2-wheel drive mobile robot platform. You will need to research suitable actuators and platform designs. From review of your research then produce a specification and model of the platform to hold the proposed processor unit (myRIO), and actuators together (your chosen motors from selection available) with a battery (Hardware datasheets on BB).arrow_forward
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