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) Find the equilibrium pair (xe,ue){“version”:”1.1″,”math”:”\((x_e, u_e)\)”} corresponding to u e = 2 {“version”:”1.1″,”math”:”\(u_e=2\)”} for the following nonlinear model, = y = x 1 2 + x 2 u . {“version”:”1.1″,”math”:”\begin{eqnarray*} \left&=&\left\\ y&=& x_1^2+x_2u. \end{eqnarray*}”} Problem 2. (10 pts) Linearize the nonlinear model, = y = x 1 2 + x 2 u , {“version”:”1.1″,”math”:”\begin{eqnarray*} \left&=&\left\\ y&=& x_1^2+x_2u, \end{eqnarray*}”}about the equilibrium found in the previous problem. Problem 3. (10 pts) For the system modeled by x ˙ = A x + b u = x + u , {“version”:”1.1″,”math”:”\begin{eqnarray*} \dot{x}&=&A x+ b u\\ &=&\left x+\leftu, \end{eqnarray*}”}construct a state-feedback control law, u=−kx+r{“version”:”1.1″,”math”:”\(u=- k x+r\)”}, such that the closed-loop system poles are located at −1{“version”:”1.1″,”math”:”\(-1\)”} and −2{“version”:”1.1″,”math”:”\(-2\)”}. Problem 4. (15 pts) Design an asymptotic observer for the plant, x ˙ = A x + b u = x + u , y = c x + d u = x + 3 u . {“version”:”1.1″,”math”:”\begin{eqnarray*} \dot{x}&=&A x+ b u = \left x+\leftu,\\ y&=& c x+du = \left x + 3u. \end{eqnarray*}”}The observer poles are to be located at −3{“version”:”1.1″,”math”:”\(-3\)”} and −4{“version”:”1.1″,”math”:”\(-4\)”}. Write down the equations of your observer. Problem 5. (15 pts) Is the following quadratic form, f = x ⊤ Q x = x ⊤ x , {“version”:”1.1″,”math”:”\ x, \]”}positive definite, positive semi-definite, negative definite, negativesemi-definite, or indefinite? Carefully justify your answer. Problem 6. (20 pts) EvaluateJ0=∫0∞y(t)2dt{“version”:”1.1″,”math”:”\”}subject to x ˙ = x , x ( 0 ) = y = x . {“version”:”1.1″,”math”:”\begin{eqnarray*} \dot{x}&=&\leftx, \quad x(0)=\left\\ y&=& \leftx. \end{eqnarray*}”} Problem 7. (10 pts) Determine the weight q {“version”:”1.1″,”math”:”\(q\)”} so that the pole of the system x ˙ ( t ) = x ( t ) + 2 u ( t ) , x ( 0 ) = 1 , {“version”:”1.1″,”math”:”\dot{x}(t)=x(t)+2u(t),\quad x(0)=1, “} driven by the optimal linear state-feedback controller, u = − k x {“version”:”1.1″,”math”:”\(u=-kx\)”}, that minimizes J = ∫ 0 ∞ ( q x ( t ) 2 + 3 u ( t ) 2 ) d t {“version”:”1.1″,”math”:”J=\int_0^{\infty}\left(qx(t)^2+3u(t)^2\right) dt”}is located at − 3 {“version”:”1.1″,”math”:”\(-3\)”}. Problem 8. (10 pts) Determine the optimal state-feedback controller, u=−kx{“version”:”1.1″,”math”:”\(u=-kx\)”}, that minimizes J=∫0∞u(t)2dt{“version”:”1.1″,”math”:”\”} subject to x ˙ ( t ) = x ( t ) + 2 u ( t ) , x ( 0 ) = 2 , {“version”:”1.1″,”math”:”\”} and determine the optimal value of J{“version”:”1.1″,”math”:”\(J\)”}. *** Congratulations, you are almost done with Midterm Exam 1. 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 1 Submit your exam to the assignment Midterm Exam 1. Click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Examity session.