Instructions: On a separate sheet of paper, answer each of…
Instructions: 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. (20 pts) For the dynamical system model, = + u , {“version”:”1.1″,”math”:”\=\left+\leftu, \]”} (10 pts) find a state transformation z = T ( x ) {“version”:”1.1″,”math”:”\( z= T(x)\)”} that brings the system into the controller form; (10 pts) represent the system in the z {“version”:”1.1″,”math”:”\(z\)”}-coordinates. Problem 2. (20 pts) For the dynamical system model, = + u , y = x 2 , {“version”:”1.1″,”math”:”\begin{eqnarray*} \left&=&\left+\leftu,\\ y&=&x_2, \end{eqnarray*}”} (10 pts) find a state transformation z = T ^ ( x ) {“version”:”1.1″,”math”:”\(z=\hat{T}(x)\) “} that brings the system into the observer form; (10 pts) represent the system in the z {“version”:”1.1″,”math”:”\(z\)”}-coordinates. Problem 3. (20 pts) For the uncertain dynamical system model, x ˙ = A x + B ( u + Φ m ) , {“version”:”1.1″,”math”:”\ “}where A = , B = , and ‖ Φ m ‖ ≤ 10. {“version”:”1.1″,”math”:”\begin{eqnarray*}A&=&\left,\,\, B=\left,\\ &\mbox{and}& \|{\Phi}_m\|\le 10. \end{eqnarray*}”} (10 pts) design a sliding surface σ ( x ) = S x = 0 {“version”:”1.1″,”math”:”\(\sigma(x)= S x= 0\)”} so that the nominal system, x ˙ = A x + B u {“version”:”1.1″,”math”:”\( \dot{ x}= A x+ B u \)”}, restricted to this surface has its pole located at − 3 {“version”:”1.1″,”math”:”\(-3\)”}; (10 pts) design a sliding-mode state-feedback stabilizing controller for the uncertain system so that the system in sliding has its pole at − 3 {“version”:”1.1″,”math”:”\(-3\)”}. Problem 4. (20 pts) Let A ∈ R m × m {“version”:”1.1″,”math”:”\(A\in \mathbb{R}^{m\times m}\)”} and B ∈ R n × n {“version”:”1.1″,”math”:”\( B \in \mathbb{R}^{n \times n}\)”}. Express det ( A ⊗ B ) {“version”:”1.1″,”math”:”\(\det(A\otimes B)\)”} in terms of det A {“version”:”1.1″,”math”:”\(\det A\) “} and det B {“version”:”1.1″,”math”:”\(\det B\)”} where the symbol ⊗ {“version”:”1.1″,”math”:”\(\otimes\)”} denotes the Kronecker product. (You may find the identities, ( A ⊗ C ) ( D ⊗ B ) = A D ⊗ C B {“version”:”1.1″,”math”:”\(( A\otimes C)(D\otimes B)=AD\otimes CB\)”} and det ( A ⊗ I r ) = ( det A ) r {“version”:”1.1″,”math”:”\(\det (A\otimes I_r)=\left(\det A\right)^r\)”}, to be useful in your derivation.) Then employ the obtained formula to evaluate det ( A ⊗ B ) {“version”:”1.1″,”math”:”\(\det(A\otimes B)\)”} for the case when A = and B = . {“version”:”1.1″,”math”:”\ \quad \mbox{and} \quad B=\left. \]”} Problem 5. (20 pts) For the uncertain dynamical system model, \begin{eqnarray*} \dot{x} &=&\leftx +\left\left(u+\xi(t) \right)\\ &=& A x+b(u+\xi(t))\\ y&=& \leftx= c x, \end{eqnarray*} where \( \xi=\xi(t)\) represents the uncertainty in the system model, \(|\xi(t)|\le 7\), design a sliding-mode observer. The eigenvalues of the observer matrix, \(A_o= A- l c\), should be located at \(-1/2\pm j\sqrt{3}/2\). While solving the Lyapunov matrix equation, \( A_o^{\top} P+ P A_o=- Q\), take \. \] *** 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 and submit your work: Midterm Exam 2 Click Submit Quiz to end the exam. End the Examity session.