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[[Category:CNSIP]]
 
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[[Category:problem solving]]
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[[Category:automatic control]]
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[[Category:optimization]]
  
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= [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] in Automatic Control (AC),  Question 3, August 2011=
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<center>
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<font size= 4>
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[[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]]
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</font size>
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<font size= 4>
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Automatic Control (AC)
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Question 3: Optimization
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</font size>
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August 2011
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</center>
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----
 
----
 
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==Question==
 
==Question==
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'''Part 1. (20 pts)'''
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'''Part 1. ''' 20 pts
  
  
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue} \text{ Consider the optimization problem, }</math></span></font>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>\text{maximize} -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2}</math>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>\text{subject to  }  x_{1}\geq0, x_{2}\geq0</math><font color="#ff0000" face="serif" size="4"><br></font>
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'''<math>\color{blue}\left( \text{i} \right) \text{ Characterize feasible directions at the point } x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right]</math>'''<br>
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<math>\color{blue}\left( \text{ii} \right) \text{Write down the second-order necessary condition for } x^{*} \text{. Does the point } x^{*} \text{ satisfy this condition?}</math><br>
  
 
:'''Click [[ECE-QE_AC3-2011_solusion-1|here]] to view student [[ECE-QE_AC3-2011_solusion-1|answers and discussions]]'''
 
:'''Click [[ECE-QE_AC3-2011_solusion-1|here]] to view student [[ECE-QE_AC3-2011_solusion-1|answers and discussions]]'''
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'''Part 2.'''
 
'''Part 2.'''
  
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}  \text{ Use the simplex method to solve the problem, }</math></span></font>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="texhtml">maximize &nbsp; &nbsp; &nbsp; &nbsp;''x''<sub>1</sub> + ''x''<sub>2</sub></span>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>\text{subject to  }  x_{1}-x_{2}\leq2</math><font color="#ff0000" face="serif" size="4"><br></font>'''&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<math>x_{1}+x_{2}\leq6</math>''' &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<math>x_{1},-x_{2}\geq0.</math>
  
  
 
:'''Click [[ECE-QE_AC3-2011_solusion-2|here]] to view student [[ECE-QE_AC3-2011_solusion-2|answers and discussions]]'''
 
:'''Click [[ECE-QE_AC3-2011_solusion-2|here]] to view student [[ECE-QE_AC3-2011_solusion-2|answers and discussions]]'''
 
----
 
----
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'''Part 3.'''
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'''Part 3.''' (20 pts)
  
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}\text{ Solve the following linear program, }</math></span></font>
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<span class="texhtml">maximize &nbsp; &nbsp;−&nbsp;''x''<sub>1</sub> − 3''x''<sub>2</sub> + 4''x''<sub>3</sub></span><br>
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<span class="texhtml"><sub></sub></span>subject to &nbsp; &nbsp;<span class="texhtml">''x''<sub>1</sub> + 2''x''<sub>2</sub> − ''x''<sub>3</sub> = 5</span>
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<span class="texhtml">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2''x''<sub>1</sub> + 3''x''<sub>2</sub> − ''x''<sub>3</sub> = 6</span>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>x_{1} \text{ free, } x_{2}\geq0, x_{3}\leq0.</math>
  
  
 
:'''Click [[ECE-QE_AC3-2011_solusion-3|here]] to view student [[ECE-QE_AC3-2011_solusion-3|answers and discussions]]'''
 
:'''Click [[ECE-QE_AC3-2011_solusion-3|here]] to view student [[ECE-QE_AC3-2011_solusion-3|answers and discussions]]'''
 
----
 
----
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'''Part 4.'''
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'''Part 4.''' (20 pts)
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 +
&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue} \text{ Consider the following model of a discrete-time system,  }</math></span></font>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<math>x\left ( k+1 \right )=2x\left ( k \right )+u\left ( k \right ), x\left ( 0 \right )=0, 0\leq k\leq 2</math><br>
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<math>\color{blue}\text{Use the Lagrange multiplier approach to calculate the optimal control sequence}</math><br>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>\left \{ u\left ( 0 \right ),u\left ( 1 \right ), u\left ( 2 \right ) \right \}</math>
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<math>\color{blue}\text{that transfers the initial state } x\left( 0 \right) \text{ to } x\left( 3 \right)=7 \text{ while minimizing the performance index}</math><br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>J=\frac{1}{2}\sum\limits_{k=0}^2 u\left ( k \right )^{2}</math><br>
  
  
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'''Part 5.''' (20 pts)
 
'''Part 5.''' (20 pts)
  
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue} \text{ Consider the following optimization problem, }</math></span></font>
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<font color="#ff0000">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</font><math>\text{optimize} \left(x_{1}-2\right)^{2}+\left(x_{2}-1\right)^{2}</math>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<math>\text{subject to  }  x_{2}- x_{1}^{2}\geq0</math>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>2-x_{1}-x_{2}\geq0</math>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>x_{1}\geq0.</math>
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<math>\color{blue} \text{The point }  x^{*}=\begin{bmatrix}
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0 & 0
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\end{bmatrix}^{T} \text{ satisfies the KKT conditions.}</math>
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<math>\color{blue}\left( \text{i} \right) \text{Does } x^{*} \text{ satisfy the FONC for minimum or maximum? Where are the KKT multipliers?}</math>
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<math>\color{blue}\left( \text{ii} \right) \text{Does } x^{*} \text{ satisfy SOSC? Carefully justify your answer.}</math><br>
  
 
:'''Click [[ECE-QE_AC3-2011_solusion-5|here]] to view student [[ECE-QE_AC3-2011_solusion-5|answers and discussions]]'''
 
:'''Click [[ECE-QE_AC3-2011_solusion-5|here]] to view student [[ECE-QE_AC3-2011_solusion-5|answers and discussions]]'''
 
----
 
----
 
[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]]
 
[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]]

Latest revision as of 09:17, 13 September 2013


ECE Ph.D. Qualifying Exam

Automatic Control (AC)

Question 3: Optimization

August 2011

Question

Part 1. 20 pts


 $ \color{blue} \text{ Consider the optimization problem, } $

               $ \text{maximize} -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2} $

               $ \text{subject to } x_{1}\geq0, x_{2}\geq0 $

$ \color{blue}\left( \text{i} \right) \text{ Characterize feasible directions at the point } x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right] $

$ \color{blue}\left( \text{ii} \right) \text{Write down the second-order necessary condition for } x^{*} \text{. Does the point } x^{*} \text{ satisfy this condition?} $

Click here to view student answers and discussions

Part 2.


 $ \color{blue} \text{ Use the simplex method to solve the problem, } $

               maximize        x1 + x2

               $ \text{subject to } x_{1}-x_{2}\leq2 $
                                        $ x_{1}+x_{2}\leq6 $                                         

                                        $ x_{1},-x_{2}\geq0. $


Click here to view student answers and discussions

Part 3. (20 pts)


 $ \color{blue}\text{ Solve the following linear program, } $

maximize    − x1 − 3x2 + 4x3

subject to    x1 + 2x2 − x3 = 5

                   2x1 + 3x2 − x3 = 6

                   $ x_{1} \text{ free, } x_{2}\geq0, x_{3}\leq0. $


Click here to view student answers and discussions

Part 4. (20 pts)

 $ \color{blue} \text{ Consider the following model of a discrete-time system, } $

                    $ x\left ( k+1 \right )=2x\left ( k \right )+u\left ( k \right ), x\left ( 0 \right )=0, 0\leq k\leq 2 $

$ \color{blue}\text{Use the Lagrange multiplier approach to calculate the optimal control sequence} $

                   $ \left \{ u\left ( 0 \right ),u\left ( 1 \right ), u\left ( 2 \right ) \right \} $

$ \color{blue}\text{that transfers the initial state } x\left( 0 \right) \text{ to } x\left( 3 \right)=7 \text{ while minimizing the performance index} $
                   $ J=\frac{1}{2}\sum\limits_{k=0}^2 u\left ( k \right )^{2} $


Click here to view student answers and discussions

Part 5. (20 pts)

 $ \color{blue} \text{ Consider the following optimization problem, } $

                            $ \text{optimize} \left(x_{1}-2\right)^{2}+\left(x_{2}-1\right)^{2} $

                        $ \text{subject to } x_{2}- x_{1}^{2}\geq0 $

                                                 $ 2-x_{1}-x_{2}\geq0 $

                                                 $ x_{1}\geq0. $

$ \color{blue} \text{The point } x^{*}=\begin{bmatrix} 0 & 0 \end{bmatrix}^{T} \text{ satisfies the KKT conditions.} $

$ \color{blue}\left( \text{i} \right) \text{Does } x^{*} \text{ satisfy the FONC for minimum or maximum? Where are the KKT multipliers?} $

$ \color{blue}\left( \text{ii} \right) \text{Does } x^{*} \text{ satisfy SOSC? Carefully justify your answer.} $

Click here to view student answers and discussions

Back to ECE Qualifying Exams (QE) page

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