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ECE Ph.D. Qualifying Exam in Automatic Control (AC), Question 3, 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?} $

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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. $

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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. $

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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} $

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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.} $

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