ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019

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Exam Rules:
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• You may use one (1) page of notes (1 front side and 1 back side)

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• You may NOT use MATLAB or any equivalent processing software
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• You MUST work on this exam by yourself with NO input from anyone

• You MUST show ALL of your work!!! NO partial credit will be given for
submissions that do not show all of the work.



BEFORE YOU BEGIN: Please check to see that you have all six pages! There are four
numbered problems. Please read each problem CAREFULLY!

ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019

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1. (10 pts) An open-loop LTI plant is constructed by cascading two dynamic systems
together, where the output of dynamic system 1 is the input to dynamic system 2:
()() = )) + 2 + ) .̇1̇)2 = 311 1))1 ))5 31)5 + .11)12
= [11 1)] 31)5 + [11]

Construct the open-loop state-space representation for the complete cascaded
system with as the input and as the output, by defining the A, B, C, and D matrices.
Hint: ALGEBRA!!!

ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019

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̇ = =− − − D + E
001H = [1 0 0] + [0]

2. (10 points) Use the eigenvector matrix and its inverse (below) to transform the state-
space system above into modal (i.e. diagonal) form. Inspect your result to determine
which modes (or mode) are uncontrollable and which are unobservable. Explain which
mode is dominant and why.
= LM E 0 −5 0−1 0 22 0 1H O1 = E 0 −1 2−1 0 00 2 1H



ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019

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()() = 3s + 2

An optimal (LQR) state-feedback gain () will be implemented for the first order SISO
dynamic system above. The LQR cost function is given by:
() = T 3LM ) + )5WX

The pre-computed LQR solution for the optimal state-feedback gain () as a function of
the relative cost weighting factor LZ[M for this dynamic system is given below. Note:
Evaluating the function below is equivalent to calling the Matlab LQR function with
and weighting matrices to generate the output.
LM = () + 2) > 0

3a. (5 points) Derive an expression for the closed loop pole as a function of the state-
feedback control gain. Hint: Convert the transfer function above to state-space first.













3b. (3 points) Determine is the LQR optimal state-feedback gain when the output
weighting is 15 times larger than the control weighting? What is the closed-loop pole
associated with this gain?






3c. (2 points) Determine the relative cost weighting factor LZ[M associated with a closed-
loop LQR optimal pole at -5 ?



ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019

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A multi-input open-loop plant is given by the following state-space realization with
control input u, process noise w, and measurement noise q:



An output-feedback control system has been designed that includes full state feedback
control, integral control for reference tracking, and state estimation. The following state-
space realization was constructed to simulate this complete closed-loop system. In this
model, perfect knowledge of the open-loop matrices has been assumed.





!x
2×1[ ]
=A x
2×1[ ]
+B u
2×1[ ]
+F w
1×1[ ]
y
1×1[ ]
=C x
2×1[ ]
+D u
2×1[ ]
+ θ
1×1[ ]
!ˆx
!x I
!x
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
4.33 −11.7 −0.97 −8.5 8.5
−0.9 0.61 0.43 1.5 −1.5
−5 −0.27 0.94 −3 3
5.8 5.7 −0.97 −10 −9
−1.4 8.1 0.43 2 −9
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
xˆ
x I
x
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
+
0 0 −2.83
0 0 0.5
1 0 −1
0 3 0
0 2 0
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
r
w
θ
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
u
yˆ
y
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=
2.54 0.13 −0.47 0 0
2.9 2.8 −0.48 0 0
8 −2.7 −0.94 0 0
5 0.27 −0.94 3 −3
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
xˆ
x I
x
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
+
0 0 0
0 0 0
0 0 0
0 0 1
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
r
w
θ
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019

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4a. (3 pts) Determine the open-loop A, C, and F matrices.









4b. (3 pts) Determine the full state feedback ( ) and the integral control ( ) gains.







4c. (2 pts) Determine the state estimation gains (K).







4d. (2 pts) Determine the open-loop D matrix.


G0 GI