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. (15 pts) Determine if the following discrete-time system, x=x+u=Ax+bu,{“version”:”1.1″,”math”:”\begin{eqnarray*} x &=&\left x+\leftu\\ &=& A x+ b u, \end{eqnarray*}”} is reachable, controllable or neither. Carefully justify your answer. Problem 2. (15 pts) For the discrete-time dynamical system, x=Ax=xy=cx=x,{“version”:”1.1″,”math”:” \begin{eqnarray*} x &=& A x = \left x \\ y &=& c x =\left x, \end{eqnarray*}”} find the initial state x{“version”:”1.1″,”math”:”\( x \)”} such that y=2{“version”:”1.1″,”math”:”\( y=2 \) “} and y=6{“version”:”1.1″,”math”:”\( y=6 \)”}. Problem 3. (15 pts) Let x˙=Ax+bu=x+u,y=cx+du=x−3u.{“version”:”1.1″,”math”:”\begin{eqnarray*} \dot{ x} &=& A x + b u =\left x + \leftu,\\ y &=& c x + du= \left x – 3u. \end{eqnarray*}”} (5 pts) Find the state transformation that transforms the pair (A,b){“version”:”1.1″,”math”:”\( ( A, b) \)”} into the controller form. (10 pts) Find the representation of the dynamical system model in the new coordinates. Problem 4. (20 pts) Construct a state-space model for the transfer function G(s)=.{“version”:”1.1″,”math”:”G(s)=\left. “} Problem 5. (20 pts) 1. (10 pts) For what range of the parameter γ{“version”:”1.1″,”math”:”\( \gamma \)”} the quadratic form x⊤x{“version”:”1.1″,”math”:”x^{\top}\left x “} is negative semi-definite? 2. (10 pts) For what range of the parameter γ{“version”:”1.1″,”math”:”\( \gamma \)”} this quadratic form is positive semi-definite? Problem 6. (15 pts) For the system modeled by x˙=Ax+bu=x+u,{“version”:”1.1″,”math”:”\begin{eqnarray*} \dot{ x}&=& A x+ b u\\ &=&\left x+\leftu, \end{eqnarray*}”} (10 pts) 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±j2{“version”:”1.1″,”math”:”\(-1 \pm j2 \)”}; (5 pts) Let y=cx+du=x−3u.{“version”:”1.1″,”math”:”\( y = c x+du=\left x -3 u. \)”} Find the transfer function,Y(s)R(s){“version”:”1.1″,”math”:”\( \frac{Y(s)}{R(s)} \)”} , of the closed-loop system. Problem 7. (20 pts) Find the transfer function of the system described by the following state-space equations, x˙(t)=x(t)+u(t)y(t)=x(t)+u(t).{“version”:”1.1″,”math”:”\begin{eqnarray*} \dot{x}(t) &=&\leftx(t) + \leftu(t)\\ y(t) &=& \left x(t) + \leftu(t). \end{eqnarray*}”} Problem 8. (15 pts) For the following linear time-varying (LTV) system model, x˙(t)=x(t);{“version”:”1.1″,”math”:”\dot{ x}(t)=\left x(t); “} (10 pts) Find the state transition matrix Φ(t,τ){“version”:”1.1″,”math”:”\( \Phi(t,\tau)\)”}; (5 pts) Let x(2)=⊤{“version”:”1.1″,”math”:”\( x(2)=\left^{\top}\)”}. Find x(1){“version”:”1.1″,”math”:”\( x(1) \)”}. Problem 9. (15 pts) Solve the equation x˙(t)=(cost)x(t),x(0)=5,{“version”:”1.1″,”math”:”\”} then find x(π){“version”:”1.1″,”math”:”\( x(\pi)\)”}. Problem 10. (15 pts) Recall that an equilibrium state of a dynamical system is a state of rest, that is, the system starting from that state stays there thereafter. Determine the equilibrium state of the following continuous-time (CT) system, x˙(t)=x(t)−.{“version”:”1.1″,”math”:”\dot{x}(t)=\leftx(t)-\left. “} Problem 11. (15 pts) For the system described by the following state-space equations, x˙(t)=x(t)+u(t)],{“version”:”1.1″,”math”:”\dot{x}(t) =\left x(t) + \leftu(t)],”} construct a state-feedback controller u=−kx{“version”:”1.1″,”math”:”\(u=- k x\) “} and determine the parameter a{“version”:”1.1″,”math”:”\(a \) “} such that the closed-loop system poles are all at −2{“version”:”1.1″,”math”:”\(-2\)”}. Problem 12. (20 pts) Consider the following model of a discrete-time (DT)dynamical system: x=Ax+bu=x+uy=cx+du=x−3u.{“version”:”1.1″,”math”:”\begin{eqnarray*} x &=& A x + bu=\left x + \leftu\\ y &=& c x+du=\left x -3 u. \end{eqnarray*}”} (10 pts) Design an asymptotic state observer for the above system with the observer poles located at −3{“version”:”1.1″,”math”:”\( -3 \)”} and −4{“version”:”1.1″,”math”:”\( -4\)”}. Write the equations of the observer dynamics. (10 pts) Denote the observer state by x~{“version”:”1.1″,”math”:”\(\tilde{x}\)”}. Let u=−kx~+r{“version”:”1.1″,”math”:”\( u=- k \tilde{x} + r\)”}. Determine the feedback gain k{“version”:”1.1″,”math”:”\( k \)”} such that the controller’s poles are at −1{“version”:”1.1″,”math”:”\( -1 \)”} and −2{“version”:”1.1″,”math”:”\( -2 \)”}, that is, det=(z+1)(z+2).{“version”:”1.1″,”math”:”\=(z+1)(z+2).\]”}Find the transfer function, YR{“version”:”1.1″,”math”:”\( \frac{Y}{R} \)”}, of the closed-loop system driven by the combined observer controller compensator. *** Congratulations, you are almost done with the Final Exam. 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: Final Exam Submit your exam to the assignment Final Exam. Return to this window and click the button below to agree to the honor statement. 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