Question

Problem 2) 30 points-LQR Controller Development For the simulation of Problem 1, linearize the system where CL is the input and h, altitude, is the output/measurement. An example was shown in class on how to linearize this model. Take those state-space matrices and develop a controller to reject disturbances in the measured altitude. Note that the C matrix will be a 1x3 in size, but to do full-state feedback, you'll need to have a second state-space model where C is identity and D is the appropriate size vector of zeros - see the below Simulink model screenshot for an example. Make the initial altitude error 100 feet. A) First, compute the eigenvalues of the open loop system and note if the system is stable or unstable. Verify that the system is controllable using the commands rank and ctrb. (5 points) B) Develop a feedback controller gain, K, using the lqr command in MATLAB. First, use $Q = 0.01 \times eye(3)$ and $R = 1$. Show the closed loop eigenvalues and note it the system is stable. Plot the altitude error time history (note, it should start and 100 and go toward zero) and the delta-control time history. (10 points) C) Develop another feedback controller gain, K, using $Q = eye(3)$ and $R = 0.01$. Show the closed loop eigenvalues and note it the system is stable. Plot the altitude error time history and the delta-control time history. (10 points) D) Comment on the difference in responses between parts B and C (speed of responses). E) Note, though, that the lift coefficient, CL, is typically limited to small values in magnitude (around 0.2 or so in size). Make the value of Q small enough to limit the size of the initial value of delta CL to 0.3 in magnitude, State the value of Q you used and show the state and control time histories in a 15 second simulation? (10 points) 0 just_zero control, delta_CL $x = Ax + Bu$ $y = Cx + Du$ State-Space_from_linsys1 delta_h, altitude error $x = Ax + Bu$ $y = Cx + Du$ State-Space_C_identity_D_zero k'u 3 Gain_from_LQR

          Problem 2) 30 points-LQR Controller Development
For the simulation of Problem 1, linearize the system where CL is the input and h,
altitude, is the output/measurement. An example was shown in class on how to linearize
this model. Take those state-space matrices and develop a controller to reject
disturbances in the measured altitude. Note that the C matrix will be a 1x3 in size, but to
do full-state feedback, you'll need to have a second state-space model where C is identity
and D is the appropriate size vector of zeros - see the below Simulink model screenshot
for an example. Make the initial altitude error 100 feet.
A) First, compute the eigenvalues of the open loop system and note if the system is
stable or unstable. Verify that the system is controllable using the commands rank
and ctrb. (5 points)
B) Develop a feedback controller gain, K, using the lqr command in MATLAB.
First, use $Q = 0.01 \times eye(3)$ and $R = 1$. Show the closed loop eigenvalues and note it
the system is stable. Plot the altitude error time history (note, it should start and
100 and go toward zero) and the delta-control time history. (10 points)
C) Develop another feedback controller gain, K, using $Q = eye(3)$ and $R = 0.01$. Show
the closed loop eigenvalues and note it the system is stable. Plot the altitude error
time history and the delta-control time history. (10 points)
D) Comment on the difference in responses between parts B and C (speed of
responses).
E) Note, though, that the lift coefficient, CL, is typically limited to small values in
magnitude (around 0.2 or so in size). Make the value of Q small enough to limit
the size of the initial value of delta CL to 0.3 in magnitude, State the value of Q
you used and show the state and control time histories in a 15 second simulation?
(10 points)
0
just_zero
control, delta_CL
$x = Ax + Bu$
$y = Cx + Du$
State-Space_from_linsys1 delta_h, altitude error
$x = Ax + Bu$
$y = Cx + Du$
State-Space_C_identity_D_zero
k'u
3
Gain_from_LQR

Problem 2) 30 points-LQR Controller Development
For the simulation of Problem 1, linearize the system where CL is the input and h,
altitude, is the output/measurement. An example was shown in class on how to linearize
this model. Take those state-space matrices and develop a controller to reject
disturbances in the measured altitude. Note that the C matrix will be a 1x3 in size, but to
do full-state feedback, you'll need to have a second state-space model where C is identity
and D is the appropriate size vector of zeros - see the below Simulink model screenshot
for an example. Make the initial altitude error 100 feet.
A) First, compute the eigenvalues of the open loop system and note if the system is
stable or unstable. Verify that the system is controllable using the commands rank
and ctrb. (5 points)
B) Develop a feedback controller gain, K, using the lqr command in MATLAB.
First, use Q = 0.01 × eye(3) and R = 1. Show the closed loop eigenvalues and note it
the system is stable. Plot the altitude error time history (note, it should start and
100 and go toward zero) and the delta-control time history. (10 points)
C) Develop another feedback controller gain, K, using Q = eye(3) and R = 0.01. Show
the closed loop eigenvalues and note it the system is stable. Plot the altitude error
time history and the delta-control time history. (10 points)
D) Comment on the difference in responses between parts B and C (speed of
responses).
E) Note, though, that the lift coefficient, CL, is typically limited to small values in
magnitude (around 0.2 or so in size). Make the value of Q small enough to limit
the size of the initial value of delta CL to 0.3 in magnitude, State the value of Q
you used and show the state and control time histories in a 15 second simulation?
(10 points)
0
justzero
control, deltaCL
x = Ax + Bu
y = Cx + Du
State-Spacefromlinsys1 deltah, altitude error
x = Ax + Bu
y = Cx + Du
State-SpaceCidentityDzero
k'u
3
GainfromLQR

Added by Craig M.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Umar Sohail Qureshi

11/08/2021

Step 1

Determine if the system is stable or unstable based on the eigenvalues. Show more…

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Thanks for your feedback!

Problem 2) 30 points - LQR Controller Development For the simulation of Problem 1, linearize the system where CL is the input and h, altitude, is the output/measurement. An example was shown in class on how to linearize this model. Take those state-space matrices and develop a controller to reject disturbances in the measured altitude. Note that the C matrix will be a 1x3 in size, but to do full-state feedback, you'll need to have a second state-space model where C is identity and D is the appropriate size vector of zeros - see the below Simulink model screenshot for an example. Make the initial altitude error 100 feet. A) First, compute the eigenvalues of the open-loop system and note if the system is stable or unstable. Verify that the system is controllable using the commands rank and ctrb. (5 points) B) Develop a feedback controller gain, K, using the lqr command in MATLAB. First, use Q=0.01*eye(3) and R=1. Show the closed-loop eigenvalues and note if the system is stable. Plot the altitude error time history (note, it should start at 100 and go toward zero) and the delta-control time history. (10 points) C) Develop another feedback controller gain, K, using Q=eye(3) and R=0.01. Show the closed-loop eigenvalues and note if the system is stable. Plot the altitude error time history and the delta-control time history. (10 points) D) Comment on the difference in responses between parts B and C (speed of responses). E) Note, though, that the lift coefficient, CL, is typically limited to small values in magnitude (around 0.2 or so in size). Make the value of Q small enough to limit the size of the initial value of delta CL to 0.3 in magnitude. State the value of Q you used and show the state and control time histories in a 15-second simulation? Problem 2) 30 points - LQR Controller Development For the simulation of Problem 1, linearize the system where CL is the input and h, altitude, is the output/measurement. An example was shown in class on how to linearize this model. Take those state-space matrices and develop a controller to reject disturbances in the measured altitude. Note that the C matrix will be a 1x3 in size, but to do full-state feedback, you'll need to have a second state-space model where C is identity and D is the appropriate size vector of zeros - see the below Simulink model screenshot for an example. Make the initial altitude error 100 feet. A) First, compute the eigenvalues of the open-loop system and note if the system is stable or unstable. Verify that the system is controllable using the commands rank and ctrb. (5 points) B Develop a feedback controller gain, K, using the lqr command in MATLAB First, use Q=0.01*eye(3) and R=1. Show the closed-loop eigenvalues and note if the system is stable, Plot the altitude error time history (note, it should start at 100 and go toward zero) and the delta-control time history. 10 points C Develop another feedback controller gain, K, using Q=eye(3) and R=0.01. Show the closed-loop eigenvalues and note if the system is stable. Plot the altitude error time history and the delta-control time history. (10 points) D Comment on the difference in responses between parts B and C (speed of responses). E Note, though, that the lift coefficient, CL, is typically limited to small values in magnitude (around 0.2 or so in size). Make the value of Q small enough to limit the size of the initial value of delta CL to 0.3 in magnitude, State the value of Q you used and show the state and control time histories in a 15-second simulation? control, delta_CL ng+xy= y=C+Du just_zero State-Space_from_linsys1 delta_h.altitude error Ax+Bu3 y=Cx+Du State-Space_C_identity_D_zero Gain_from_LQR