Instructions: This is a closed-note, closed-book exam. On…
Instructions: This is a closed-note, closed-book exam. On a separate sheet of paper, answer each of the exam problems shown below. Write your answers clearly. Unless otherwise stated, you will need to justify your answers to get the full credit. Problem 1. (10 pts) For the continuous-time model, x ˙ = x + u , {“version”:”1.1″,”math”:”\x+\leftu, \]”}construct (5 pts) the Euler discrete-time model for the sampling period T s = 3 {“version”:”1.1″,”math”:”\( T_s=3 \)”}; (5 pts) the exact discrete-time model for the sampling period T s = 3 {“version”:”1.1″,”math”:”\( T_s=3 \)”} Problem 2. (15 pts) Consider the following symmetric matrix, Q = = . {“version”:”1.1″,”math”:”\=\left. \]”} (5 pts) Compute the Schur complement, Δ 11 {“version”:”1.1″,”math”:”\(\Delta_{11}\)”}, of Q 11 {“version”:”1.1″,”math”:”\( Q_{11} \)”}; (10 pts) Use the result of Part 1 to determine if Q{“version”:”1.1″,”math”:”\( Q \)”} is positive definite, positive semi-definite, negative definite, negative semi-definite, or indefinite? Justify your answer. Problem 3. (20 pts) For the discrete-time model,x=x+uy=x,{“version”:”1.1″,”math”:”\begin{eqnarray*} x &=& \left x+\leftu\\ y &=& \left x, \end{eqnarray*}”}when implementing a model predictive controller (MPC), we impose the constraints on the output of the form − 4 ≤ u ≤ 3. {“version”:”1.1″,”math”:”\\le 3. \]”}Suppose that the prediction horizon Np=2{“version”:”1.1″,”math”:”\(N_p=2 \)”}. (10 pts) Express the above constraints in your MPC implementation in terms of Δ U {“version”:”1.1″,”math”:”\( \Delta U\)”} when using the augmented model of the plant; (10 pts) Express the above constraints in your MPC implementation in terms of U {“version”:”1.1″,”math”:”\(U\)”} when using the non-augmented model of the plant. Problem 4. (15 pts) For the following discrete-time system, x = x + 2 u , x = − 3 , 0 ≤ k ≤ 1 , {“version”:”1.1″,”math”:”\=x+2u,\quad x=-3,\quad 0\le k\le 1, \]”}find the optimal control sequence { u , u } {“version”:”1.1″,”math”:”\, u\} \]”}that transfers the initial state x{“version”:”1.1″,”math”:”\(x\)”} to x = 7 {“version”:”1.1″,”math”:”\(x=7 \)”} while minimizing the performance index J = 1 2 ∑ k = 0 1 u 2 = 1 2 u ⊤ u . {“version”:”1.1″,”math”:”\^2=\frac{1}{2} u^{\top} u. \]”} Problem 5. (20 pts) Find u = u ( t ) {“version”:”1.1″,”math”:”\( u=u(t) \)”} that minimizes J ( u ) = 1 2 ∫ 0 2 u 2 ( t ) d t , {“version”:”1.1″,”math”:”\”}subject to x ˙ ( t ) = x ( t ) + u ( t ) , x ( 0 ) = , x ( 2 ) = . {“version”:”1.1″,”math”:”\x(t) + \leftu(t),\quad x(0)=\left, \quad x(2)=\left. \]”} Problem 6. (20 pts) For the optimization problem, maximize − x 1 2 − 4 x 2 2 subject to x 1 2 + 2 x 2 2 − 4 ≥ 0 , {“version”:”1.1″,”math”:”\”}find all the points that satisfy the KKT conditions. *** Congratulations, you are almost done with Midterm Exam 2. DO NOT end the Examity session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to Gradescope: Midterm Exam 2 Click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Examity session.