Description
1. Last week, you found that the rotational motion of a control moment gyro on a spacecraft
could be described in state-space form as
x˙ =
−b/m
x +
1/m
u
y =
1
x
where the state x is angular velocity, the input u is applied torque, and
m = 2 b = 1
are parameters. In what follows, please assume the input has the form
u = −3x + kreferencer + d
where r is a known reference signal and d is an unknown disturbance load. You may assume
that both r and d are constant.
(a) Reference tracking. Suppose d = 0. Please do the following:
• Find kreference so that y = r in steady-state.
• Rewrite the system as
x˙ = Aclx + Bclr
y = Cclx
for an appropriate choice of Acl, Bcl, and Ccl.
• Find and plot the step response of this system in MATLAB using “step.”
• Find the step response of this system by hand. In other words, express y(t) in terms
of scalar exponential functions if x(0) = 0 and r = 1. Use MATLAB to evaluate
this expression, plotting it on the same figure as before.
• Find the rise time, settling time, overshoot, and steady-state value of the step response from your MATLAB plot. Your answers can be approximate.
(b) Disturbance rejection. Suppose r = 0. Please do the following:
• Rewrite the system as
x˙ = Aclx + Bcld
y = Cclx
for an appropriate choice of Acl, Bcl, and Ccl.
• Find and plot the step response of this system in MATLAB using “step.”
• Find the step response of this system by hand. In other words, express y(t) in terms
of scalar exponential functions if x(0) = 0 and d = 1. Use MATLAB to evaluate
this expression, plotting it on the same figure as before.
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• Find the steady-state error in response to a unit disturbance load. In other words,
find the steady-state value of y(t)−r if x(0) = 0 and d = 1. Do so by hand, verifying
your result with the plot.
(c) Disturbance rejection with integral action. Again, suppose r = 0. But this time, consider
the alternative input
u = −3x + kreferencer + d − kintegralv
where kintegral = 34 and where we define
v˙ = y − r.
Please do the following:
• Define
z =
x
v
.
Rewrite the system as
z˙ = Aclz + Bcld
y = Cclz
for an appropriate choice of Acl, Bcl, and Ccl.
• Is this system asymptotically stable? (Please do all computation by hand.)
• Find and plot the step response of this system in MATLAB using “step.”
• Find the step response of this system by hand. In other words, express y(t) in terms
of scalar exponential functions if
z(0) =
0
0
and d = 1. Use MATLAB to evaluate this expression, plotting it on the same figure
as before.
• Find the steady-state error in response to a unit disturbance load. In other words,
find the steady-state value of y(t) − r if
z(0) =
0
0
and d = 1. Do so by hand, verifying your result with the plot.
• What differences are there between the results here and the results in (b)? Why?
2
2. Last week, you found that the rotational motion of an antenna on a spacecraft could be
described in state-space form as
x˙ =
0 1
0 −b/m
x +
0
1/m
u
y =
1 0
x
where the state elements are angle (x1) and angular velocity (x2), the input u is an applied
torque, and
m = 0.1 b = 0.5
are parameters. In what follows, please assume the input has the form
u = −
5 1
x + kreferencer + d
where r is a known reference signal and d is an unknown disturbance load. You may assume
that both r and d are constant.
(a) Reference tracking. Suppose d = 0. Please do the following:
• Find kreference so that y = r in steady-state.
• Rewrite the system as
x˙ = Aclx + Bclr
y = Cclx
for an appropriate choice of Acl, Bcl, and Ccl.
• Find and plot the step response of this system in MATLAB using “step.”
• Find the step response of this system by hand. In other words, express y(t) in terms
of scalar exponential functions if
x(0) =
0
0
and r = 1. Use MATLAB to evaluate this expression, plotting it on the same figure
as before.
• Find the rise time, settling time, overshoot, and steady-state value of the step response from your MATLAB plot. Your answers can be approximate.
(b) Disturbance rejection. Suppose r = 0. Please do the following:
• Rewrite the system as
x˙ = Aclx + Bcld
y = Cclx
for an appropriate choice of Acl, Bcl, and Ccl.
• Find and plot the step response of this system in MATLAB using “step.”
• Find the step response of this system by hand. In other words, express y(t) in terms
of scalar exponential functions if
x(0) =
0
0
and d = 1. Use MATLAB to evaluate this expression, plotting it on the same figure
as before.
• Find the steady-state error in response to a unit disturbance load. In other words,
find the steady-state value of y(t) − r if
x(0) =
0
0
and d = 1. Do so by hand, verifying your result with the plot.
(c) Disturbance rejection with integral action. Again, suppose r = 0. But this time, consider
the alternative input
u = −
5 1
x + kreferencer + d − kintegralv
where kintegral = 10 and where we define
v˙ = y − r.
Please do the following:
• Define
z =
x
v
.
Rewrite the system as
z˙ = Aclz + Bcld
y = Cclz
for an appropriate choice of Acl, Bcl, and Ccl.
• Is this system asymptotically stable? (You may use MATLAB for the computation.)
• Find and plot the step response of this system in MATLAB using “step.”
• Find the steady-state error in response to a unit disturbance load. In other words,
find the steady-state value of y(t) − r if
z(0) =
0
0
and d = 1. Do so by hand, verifying your result with the plot.
• What differences are there between the results here and the results in (b)? Why?
3. Last week, you found that the rotational motion of an axisymmetric spacecraft about its yaw
and roll axes could be described in state-space form as
x˙ =
0 λ
−λ 0
x +
1
0
u
y =
1 0
x
where the state elements x1 and x2 are the angular velocities about yaw and roll axes, the
input u is an applied torque, and the parameter λ = 9 is the relative spin rate. In what
follows, please assume the input has the form
u = −
6 −1
x + kreferencer + d
where r is a known reference signal and d is an unknown disturbance load. You may assume
that both r and d are constant.
(a) Reference tracking. Suppose d = 0. Please do the following:
• Prove that there exists no choice of kreference for which y = r in steady-state.
HINT: try to find kreference in the normal way and see what happens.
• Consider the alternative output
y =
0 1
x.
Find kreference so that y = r in steady-state.
Please continue to use this new output for the rest of the problem, including parts (a), (b), and (c).
• Rewrite the system as
x˙ = Aclx + Bclr
y = Cclx
for an appropriate choice of Acl, Bcl, and Ccl.
• Find and plot the step response of this system in MATLAB using “step.”
• Find the step response of this system by hand. In other words, express y(t) in terms
of scalar exponential functions if
x(0) =
0
0
and r = 1. Use MATLAB to evaluate this expression, plotting it on the same figure
as before.
• Find the rise time, settling time, overshoot, and steady-state value of the step response from your MATLAB plot. Your answers can be approximate.
(b) Disturbance rejection. Suppose r = 0. Please do the following:
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• Rewrite the system as
x˙ = Aclx + Bcld
y = Cclx
for an appropriate choice of Acl, Bcl, and Ccl.
• Find and plot the step response of this system in MATLAB using “step.”
• Find the step response of this system by hand. In other words, express y(t) in terms
of scalar exponential functions if
x(0) =
0
0
and d = 1. Use MATLAB to evaluate this expression, plotting it on the same figure
as before.
• Find the steady-state error in response to a unit disturbance load. In other words,
find the steady-state value of y(t) − r if
x(0) =
0
0
and d = 1. Do so by hand, verifying your result with the plot.
(c) Disturbance rejection with integral action. Again, suppose r = 0. But this time, consider
the alternative input
u = −
6 −1
x + kreferencer + d − kintegralv
where kintegral = −5 and where we define
v˙ = y − r.
Please do the following:
• Define
z =
x
v
.
Rewrite the system as
z˙ = Aclz + Bcld
y = Cclz
for an appropriate choice of Acl, Bcl, and Ccl.
• Is this system asymptotically stable? (You may use MATLAB for the computation.)
• Find and plot the step response of this system in MATLAB using “step.”
• Find the steady-state error in response to a unit disturbance load. In other words,
find the steady-state value of y(t) − r if
z(0) =
0
0
and d = 1. Do so by hand, verifying your result with the plot.
• What differences are there between the results here and the results in (b)? Why?
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