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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) Consider the following discrete-time system, x [ k + 1 ] = [ a − 2 0 0 3 ] x [ k ] + [ 0 1 ] u [ k ] = A x [ k ] + b u [ k ] , {"version":"1.1","math":"begin{eqnarray*} x[k+1] &=&left[begin{array}{cc} a-2 & 0\ 0 & 3 end{array}right] x[k]+left[begin{array}{c} 0\ 1 end{array}right]u[k]\ &=& A x[k]+ b u[k], end{eqnarray*}"}where a ∈ R {"version":"1.1","math":"( ain mathbb{R})"} is a parameter. Is this system reachable? For what value of the parameter a {"version":"1.1","math":"( a)"} the system is controllable? Justify your answers. Problem 2. (15 pts) For the discrete-time dynamical system, x [ k + 1 ] = A x [ k ] = [ 0 2 0 0 ] x [ k ] y [ k ] = c x [ k ] = [ 1 0 ] x [ k ] , {"version":"1.1","math":" begin{eqnarray*} x[k+1] &=& A x[k] = left[begin{array}{cc} 0 & 2\ 0 & 0 end{array}right] x[k] \ y[k] &=& c x[k] =left[begin{array}{cc} 1 & 0 end{array}right] x[k], end{eqnarray*}"} find the initial state x[0]{"version":"1.1","math":"( x[0] )"} such that y[0]=2{"version":"1.1","math":"( y[0]=2 ) "} and y[1]=6{"version":"1.1","math":"( y[1]=6 )"}. Problem 3. (15 pts) Let x [ k + 1 ] = A x [ k ] + b u [ k ] = [ 0 0 1 0 ] x [ k ] + [ 1 0 ] u [ k ] , y [ k ] = c x [ k ] + d u [ k ] = [ 1 1 ] x [ k ] − 3 u [ k ] . {"version":"1.1","math":"begin{eqnarray*} x[k+1] &=& A x[k] + b u[k] =left[begin{array}{cc} 0 & 0\ 1 & 0 end{array}right] x[k] + left[begin{array}{c} 1\ 0 end{array}right]u[k],\ y[k] &=& c x[k] + du[k]= left[begin{array}{cc} 1 & 1 end{array}right] x[k] - 3u[k]. 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 ) = [ 4 s 2 2 s 2 + 6 s + 4 5 s + 2 0 1 ] . {"version":"1.1","math":"G(s)=left[begin{array}{cc} frac{4s^2}{2s^2+6s+4} & frac{5}{s+2} \ 0 & 1end{array}right]. "} Problem 5. (20 pts) 1. (10 pts) For what range of the parameter γ{"version":"1.1","math":"( gamma )"} the quadratic form x ⊤ [ 1 0 − 2 γ 4 ] x {"version":"1.1","math":"x^{top}left[begin{array}{cc}1 & 0\ -2gamma &4 end{array}right] x "} is positive semi-definite? 2. (10 pts) For what range of the parameter γ{"version":"1.1","math":"( gamma )"} this quadratic form is positive definite? Problem 6. (15 pts) For the system modeled by x ˙ = A x + b u = [ 0 0 1 2 ] x + [ 1 0 ] u , {"version":"1.1","math":"begin{eqnarray*} dot{ x}&=& A x+ b u\ &=&left[begin{array}{cc} 0 & 0\ 1 & 2 end{array}right] x+left[begin{array}{c} 1\ 0 end{array}right]u, 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 − 2 ± j {"version":"1.1","math":"(-2 pm j )"}; (5 pts) Let y = c x = [ 0 1 ] x . {"version":"1.1","math":"( y = c x=left[ begin{array}{cc} 0 & 1 end{array} right] x. )"} 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 ) = [ 0 1 0 0 1 0 0 1 0 2 0 0 0 1 0 0 0 0 0 3 0 0 0 0 1 ] x ( t ) + [ 1 0 0 1 0 0 0 0 0 0 ] u ( t ) y ( t ) = [ 2 1 0 7 2 7 0 1 2 1 ] x ( t ) + [ 1 0 0 2 ] u ( t ) . {"version":"1.1","math":"begin{eqnarray*} dot{x}(t) &=&left[begin{array}{cc|ccc} 0 & 1 & 0 & 0 & 1\ 0 & 0 & 1 & 0 & 2 \hline 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 3 \ 0 & 0 & 0 & 0 & 1 end{array}right]x(t) + left[begin{array}{cc} 1 &0 \ 0 & 1\hline 0 & 0\ 0 & 0\ 0 & 0 end{array}right]u(t)\ y(t) &=& left[begin{array}{cc|ccc} 2 & 1 & 0 & 7 & 2 \ 7 & 0 & 1 & 2 & 1 end{array}right] x(t) + left[begin{array}{c} 1 & 0\ 0 & 2 end{array}right]u(t). end{eqnarray*}"} Problem 8. (15 pts) Consider the following nonlinear system, x ˙ 1 = − x 1 + ( x 2 − 1 ) 2 x ˙ 2 = x 1 2 − x 2 + 1. {"version":"1.1","math":"begin{eqnarray*} dot{ x}_1 &=&-x_1+(x_2-1)^2\ dot{x}_2 &=& x_1^2-x_2+1. end{eqnarray*}"} (10 pts) Find the equilibrium states and linearize the nonlinear system about these equilibrium states; (5 pts) Determine local stability of the equilibrium states. Problem 9. (15 pts) Consider the discrete-time (DT) LTI system x [ k + 1 ] = [ − 0.5 1 0 0.5 ] x [ k ] = A x [ k ] . {"version":"1.1","math":"[ x[k+1]=begin{bmatrix}-0.5 & 1\0 & 0.5end{bmatrix}x[k]=Ax[k]. ]"} Does this system have a quadratic Lyapunov function? If yes, find one. If no, explain why not. Problem 10. (15 pts) Consider the continuous-time (CT) LTI system, x ˙ ( t ) = [ − 0.5 1 0 − 0.5 ] x ( t ) = A x ( t ) . {"version":"1.1","math":"dot{x}(t)=left[begin{array}{cc} -0.5 & 1\ 0 & -0.5 end{array}right]x(t)=Ax(t). "}Does this system have a quadratic Lyapunov function? If yes, find one. If no, explain why not. Problem 11. (15 pts) For the system described by the following state-space equations, x ˙ ( t ) = [ 0 1 2 0 0 0 0 1 0 0 a 1 0 0 − 1 0 ] x ( t ) + [ 0 1 0 0 ] u ( t ) , {"version":"1.1","math":"dot{x}(t) =left[begin{array}{cc|cc} 0 & 1 & 2 & 0\ 0 & 0 & 0 & 1\hline 0 & 0 & a & 1\ 0 & 0 & -1 & 0 end{array}right] x(t) + left[begin{array}{c} 0 \ 1\hline 0 \ 0 end{array}right]u(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 − 1 {"version":"1.1","math":"(-1)"}. Problem 12. (20 pts) Consider the following model of a continuous-time (CT)dynamical system: x ˙ = A x + b u = [ 0 0 1 2 ] x + [ 1 0 ] u y = c x + d u = [ 0 1 ] x − 2 u . {"version":"1.1","math":"begin{eqnarray*} dot{x} &=& A x + bu=left[ begin{array}{cc} 0 & 0\ 1 & 2 end{array} right] x + left[ begin{array}{c} 1\ 0 end{array} right]u\ y &=& c x+du=left[ begin{array}{cc} 0 & 1 end{array} right] x -2 u. end{eqnarray*}"} (10 pts) Design an asymptotic state observer for the above system with the observer poles located at − 2 {"version":"1.1","math":"( -2 )"} and − 3 {"version":"1.1","math":"( -3)"}. 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 , − 1 } {"version":"1.1","math":"( {-1,, -1} )"} , that is, det [ s I 2 − A + b k ] = ( s + 1 ) 2 . {"version":"1.1","math":"[ detleft[s I_2- A+ b kright]=(s+1)^2.]"}Find the transfer function, Y ( s ) R ( s ) {"version":"1.1","math":"( frac{Y(s)}{R(s)} )"}, 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. 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