Description
1. The relationship between the applied torque τ (in units of N · m) and the angular velocity ω
(in units of rad/s) of a control moment gyro on a spacecraft is
mω˙ + bω = τ,
where
m = 2 kg · m2
is the moment of inertia, and
b = 1 kg · m2
/s is the coefficient of viscous friction.
Our goal in this problem is to spin down the gyro—i.e., for ω(t) → 0 as t → ∞.
(a) Put this system in state-space form. Define states, inputs, and outputs as follows:
x =
ω
, u =
τ
, y =
ω
.
(b) Suppose we choose the input u(t) = 0 for all t ≥ 0. Is the system asymptotically stable?
(c) Suppose we choose the input u(t) = −3x(t) for all t ≥ 0. Is the system asymptotically
stable?
(d) Suppose x(0) = 1. For each of the inputs considered in (b) and (c), write an expression
for y(t) in terms of the scalar exponential function, and use MATLAB to evaluate this
expression for t ∈ [0, 10]. Plot both results on the same figure. What is the difference?
2. The relationship between the applied torque τ (in units of N · m) and the angle θ (in units of
rad) of an antenna on a spacecraft is
m¨θ + b
˙θ = τ,
where
m = 0.1 kg · m2
is the moment of inertia, and
b = 0.5 kg · m2
/s is the coefficient of viscous friction.
Our goal here is to keep the antenna angle zero—i.e., for θ(t) → 0 as t → ∞.
(a) Put this system in state-space form. Define states, inputs, and outputs as follows:
x =
θ
˙θ
, u =
τ
, y =
θ
(b) Suppose we choose the input u(t) = 0 for all t ≥ 0. Is the system asymptotically stable?
(c) Suppose we choose the input
u(t) = −
5 1
x(t)
for all t ≥ 0. Is the system asymptotically stable?
1
(d) Suppose
x(0) =
0
4
For each of the inputs considered in (b) and (c), please do the following:
• Write an expression for y(t) in terms of the matrix exponential function, and use
MATLAB to evaluate this expression for t ∈ [0, 2].
• Write an expression for y(t) in terms of scalar exponential functions—i.e., diagonalize
to simplify the matrix exponential—and use MATLAB to evaluate this expression
for t ∈ [0, 2]. If you arrive at an expression with complex exponentials, please rewrite
them in terms of sines and cosines.
Plot all four results—two for the input of (b), two for the input of (c)—on the same
figure. What differences do you see?
3. Euler’s equations describe the rotational motion of a rigid body subject to applied torques.
When written with respect to principle axes (where J1, J2, J3 are the principal moments of
inertia), these equations can be written
τ1 = J1ω˙ 1 − (J2 − J3) ω2ω3
τ2 = J2ω˙ 2 − (J3 − J1) ω3ω1
τ3 = J3ω˙ 3 − (J1 − J2) ω1ω2.
Consider an axisymmetric rigid body, for which
J1 = J2 = Jt
and
J3 = Ja.
The subscripts t and a indicate the transverse and axial moments of inertia, respectively.
Suppose τ2 = τ3 = 0, so we restrict the applied torque to be about only one of the transverse
axes. Then, we have
τ1 = Jtω˙ 1 − (Jt − Ja) ω2ω3
0 = Jtω˙ 2 + (Jt − Ja) ω1ω3
0 = Jaω˙ 3.
The third equation implies that ω3 is constant. We call it the spin rate and denote it by
n = ω3.
What remains is
τ1 = Jtω˙ 1 − (Jt − Ja) nω2
0 = Jtω˙ 2 + (Jt − Ja) nω1.
If we define the relative spin rate as
λ =
Jt − Ja
Jt
n,
2
then we can simplify our equations of motion further to
(τ1/Jt) = ˙ω1 − λω2
0 = ˙ω2 + λω1.
(1)
In what follows, we will assume that Jt = 5000 kg·m2
, Ja = 2000 kg·m2
, and n = 15 rad/sec.
Our goal is to eliminate roll and pitch motion—i.e., for ω1(t) → 0 and ω2(t) → 0 as t → ∞.
(a) Put this system in state-space form. Define states, inputs, and outputs as follows:
x =
ω1
ω2
, u =
τ1/Jt
, y =
ω1
(b) Suppose we choose the input u(t) = 0 for all t ≥ 0. Is the system asymptotically stable?
(c) Suppose we choose the input
u(t) = −
1 6
x(t)
for all t ≥ 0. Is the system asymptotically stable?
(d) Suppose
x(0) =
1
−4
For each of the inputs considered in (b) and (c), please do the following:
• Write an expression for y(t) in terms of the matrix exponential function, and use
MATLAB to evaluate this expression for t ∈ [0, 2].
• Write an expression for y(t) in terms of scalar exponential functions—i.e., diagonalize
to simplify the matrix exponential—and use MATLAB to evaluate this expression
for t ∈ [0, 2]. If you arrive at an expression with complex exponentials, please rewrite
them in terms of sines and cosines.
Plot all four results—two for the input of (b), two for the input of (c)—on the same
figure. What differences do you see?
4. The approximate relationship between the force f applied laterally to the base of an upright
rocket (e.g., with a gimbaled thruster) and the pitch angle θ of this rocket is
¨θ =
mg`
Jt
θ −
γ
Jt
˙θ +
`
Jt
f,
where we will assume:
• m = 1 is the mass of the rocket,
• g = 10 is the acceleration of gravity,
• ` = 5/2 is the length of the rocket,
• Jt = 5 is the transverse moment of inertia of the rocket,
• γ = 20 is a coefficient of viscous friction.
The goal is to keep the rocket upright—i.e., for θ(t) → 0 as t → ∞.
3
(a) Put this system in state-space form. Define states, inputs, and outputs as follows:
x =
θ
˙θ
, u =
f
, y =
θ
(b) Suppose we choose the input u(t) = 0 for all t ≥ 0. Is the system asymptotically stable?
(c) Suppose we choose the input
u(t) = −
28 4
x(t)
for all t ≥ 0. Is the system asymptotically stable?
(d) Suppose
x(0) =
1
0
For each of the inputs considered in (b) and (c), please do the following:
• Write an expression for y(t) in terms of the matrix exponential function, and use
MATLAB to evaluate this expression for t ∈ [0, 3].
• Write an expression for y(t) in terms of scalar exponential functions—i.e., diagonalize
to simplify the matrix exponential—and use MATLAB to evaluate this expression
for t ∈ [0, 3]. If you arrive at an expression with complex exponentials, please rewrite
them in terms of sines and cosines.
Plot all four results—two for the input of (b), two for the input of (c)—on the same
figure. What differences do you see?
HINT: You may have trouble solving this problem at first, since you will likely find that
the closed-loop “A” matrix Acl is not diagonalizable for the input proposed in part (c).
Instead, try the coordinate transformation x = V z, where V is the matrix produced in
MATLAB as follows:
[V, J] = jordan(Acl)
Then, use the fact that
e
Jt =
e
ht teht
0 e
ht
whenever
J =
h 1
0 h
Note that J is an example of a matrix in so-called “Jordan form.”
4
5. Extra Credit (worth +25% — no partial credit, but you may iterate with Tim, Donald, or
Akshay until your answer is acceptable). In this problem, you will explore some properties of
the matrix exponential and its application to analysis of volume-preserving flows.
(a) Suppose that A ∈ R
n×n
is diagonalizable and that it has eigenvalues λ1, . . . , λn. Prove
that the eigenvalues of e
A are e
λ1
, . . . , eλ
n
Hints:
• the eigenvalues of M ∈ R
n×n
satisfy det(λI − M) = 0
• if M ∈ R
n×n
is invertible, then det(M−1
) = (det M)
−1
(b) Consider a set Z0 ⊆ R
n and a state-space system
x˙ = Ax,
where A ∈ R
n×n
is diagonalizable. We can propagate the set Z0 along the flow induced
by the system as
Z(t) = e
AtZ0 =
e
Atz : z ∈ Z0
Denote the volume of the set Z0 by vol (Z0). We say that the system preserves volume
if vol (Z(t)) = vol (Z0) for all t. Is it possible for a stable system to preserve volume?
Prove your answer.
Hints:
• if M ∈ R
n×n and Z ⊆ R
n
, then
vol (MZ) = |det M| vol (Z),
where
MZ = {Mz : z ∈ Z}
• if M ∈ R
n×n has eigenvalues λ1, . . . , λn, then det M = λ1 · · · λn
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