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Instructiоns: On а sepаrаte sheet оf paper, answer each оf 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) (10 pts) Represent the system model d 4 ϕ d t 4 + 3 d 3 ϕ d t 3 + 4 d ϕ d t − 6 ϕ = 2 d 3 u d t 3 + 4 d 2 u d t 2 − 5 u , {"version":"1.1","math":"[ frac{d^4 phi}{dt^4}+3frac{d^3 phi}{dt^3}+4frac{d phi}{dt} -6phi=2frac{d^3 u}{dt^3}+4frac{d^2 u}{dt^2} -5u, ]"} in a state-space format. Define the state vector as x = [ ϕ d ϕ d t d 2 ϕ d t 2 d 3 ϕ d t 3 ] ⊤ . {"version":"1.1","math":"[ x=begin{bmatrix}phi & frac{d phi}{dt} & frac{d^2 phi}{dt^2}& frac{d^3 phi}{dt^3}end{bmatrix}^top. ]"} The system input is u and the output y {"version":"1.1","math":"(y)"} is a linear combination of state variables x i . {"version":"1.1","math":"(x_i.)"} 2. (5 pts) Find the transfer function of this system. Problem 2. (15 pts) Find the transfer function matrix relating Y(s){"version":"1.1","math":"( Y(s) )"} and U(s){"version":"1.1","math":"( U(s) )"} of the following system: [ x ˙ 1 ( t ) x ˙ 2 ( t ) ] = [ − 2 0 − 9 1 ] [ x 1 ( t ) x 2 ( t ) ] + [ 1 0 1 1 ] u ( t ) , y ( t ) = [ 0 1 ] [ x 1 ( t ) x 2 ( t ) ] + [ 1 2 ] u ( t ) . {"version":"1.1","math":"begin{eqnarray*} left[begin{array}{c} dot{x}_1(t)\ dot{x}_2(t) end{array}right]&=&left[begin{array}{cc} -2 & 0\ -9 & 1 end{array}right]left[begin{array}{c} {x}_1(t)\ {x}_2(t) end{array}right] + left[begin{array}{cc} 1 & 0\ 1& 1 end{array}right]u(t),\ y(t) &=& left[begin{array}{cc} 0 & 1 end{array}right]left[begin{array}{c} {x}_1(t)\ {x}_2(t) end{array}right] +begin{bmatrix}1&2 end{bmatrix}u(t). end{eqnarray*}"} Here, Y(s){"version":"1.1","math":"( Y(s) )"} and U(s){"version":"1.1","math":"( U(s) )"} are the Laplace transforms of the output y(t){"version":"1.1","math":"( y(t) )"} and input u(t){"version":"1.1","math":"( u(t) )"}, respectively. Problem 3. (25 pts) Let A = [ 1 1 0 − 1 ] . {"version":"1.1","math":"[ A=begin{bmatrix}1 & 1\0 & -1 end{bmatrix}. ]"} (5 pts) Find the eigenvectors of A . {"version":"1.1","math":"(A.)"} (10 pts) Find all the modes of the linear time-invariant system (LTI) continuous-time (CT) system, x ˙ = A x {"version":"1.1","math":"(dot{x}=Ax )"}, and write the solution from initial state x ( 0 ) = [ 1 2 ] ⊤ {"version":"1.1","math":"(x(0)=begin{bmatrix}1 & 2 end{bmatrix}^top )"}as a linear combination of modes. (10 pts) Find all the modes of the linear time-invariant (LTI) discrete-time (DT) system, x [ k + 1 ] = A x [ k ] {"version":"1.1","math":"(x[k+1]=Ax[k] )"}, and write the solution from initial state {"version":"1.1","math":""} x [ 0 ] = [ 3 2 ] ⊤ {"version":"1.1","math":"( x[0]=begin{bmatrix}3 & 2 end{bmatrix}^top )"}as a linear combination of modes. Problem 4. (20 pts) Find the state transition matrix Φ(t,s){"version":"1.1","math":"( Phi(t,s) )"}, s,t≥0{"version":"1.1","math":"( s,tge0 )"}{"version":"1.1","math":"( s,tge0 )"}, for the system: x ˙ ( t ) = [ e − t e − t 0 − e − t ] x ( t ) . {"version":"1.1","math":"begin{align*} dot x(t) = begin{bmatrix} e^{-t} & e^{-t} \ 0 & -e^{-t} end{bmatrix} x(t). end{align*}"} Problem 5. (25 pts) Find a transformation T {"version":"1.1","math":"(T)"} that transforms A = [ 2 4 − 3 0 2 0 0 3 − 1 ] {"version":"1.1","math":"[ A=begin{bmatrix} 2 & 4 & -3\0 & 2 & 0\0 & 3 & -1 end{bmatrix} ]"} into a Jordan canonical form. *** 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 1 Exam. Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam and end the Honorlock session.
Which оf the fоllоwing is а reаson funerаls are important?