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.
1
• 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.
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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
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?
4
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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