Which sоciоlоgist’s аnаlysis of bureаucracies has been applied to schools in the United States because of the growing number of students enrolled in schools and the greater degree of specialization required within a technologically complex society?
Instructiоns: This is а clоsed-nоte, closed-book exаm. On а 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) Consider the following nonlinear dynamical system model:x˙=f(x)+G(x)u=[41+x2x1x2]+[x21]u.{"version":"1.1","math":"begin{eqnarray*} dot{x}&=&f(x) + G(x)u\ &=& left[begin{array}{c} frac{4}{1+x_2}\ x_1x_2 end{array}right] + left[begin{array}{c} x_2\ 1 end{array}right] u. end{eqnarray*}"} (10 pts) Find the equilibrium states corresponding to the constant input u=−2{"version":"1.1","math":"( u=-2)"}. (10 pts) Construct Taylor's linearized state-space models for small deviations about the obtained equilibria. Problem 2. (20 pts) (15 pts) Find the linear state-feedback control law that minimizesJ=∫0∞(14x(t)2+9u(t)2)dt{"version":"1.1","math":"[ J=int_{0}^{infty} left(frac{1}{4}x(t)^2+9u(t)^2right)dt ]"}subject tox˙(t)=2x(t)+3u(t),x(0)=1.{"version":"1.1","math":"[ dot{x}(t)=sqrt{2}x(t)+3u(t),quad x(0)=1. ]"} (5 pts) Find the value of the performance index for the closed-loop system driven by the optimal controller. Problem 3. (20 pts) Construct u=u(t){"version":"1.1","math":"( u=u(t) )"} that minimizesJ(u)=12∫01u(t)2dt{"version":"1.1","math":"[ J(u)=frac{1}{2}int_0^1 u(t)^2dt ]"}subject tox˙=[0100]x+[01]u,x(0)=[00],x(1)=[13].{"version":"1.1","math":"[ dot{x}=left[begin{array}{cc} 0 & 1\ 0 & 0 end{array}right]x + left[begin{array}{c} 0\ 1 end{array}right]u,,,, x(0)=left[begin{array}{c} 0\ 0 end{array}right],,,, x(1)=left[begin{array}{c} 1\ 3 end{array}right]. ]"} Problem 4. (20 pts) Use dynamic programming to find u[0]{"version":"1.1","math":"( u[0])"} and u[1]{"version":"1.1","math":"( u[1])"} that minimize the performance index,J=(x[2]−1)2+2∑k=01u[k]2{"version":"1.1","math":"[ J=(x[2]-1)^2+2sum_{k=0}^1 u[k]^2 ]"}subject tox[k+1]=bu[k],x[0]=10,{"version":"1.1","math":"[ x[k+1]=b u[k],quad x[0]=10, ]"}where b≠0{"version":"1.1","math":" ( bne 0)"}. Note that there are no constraints on u[k]{"version":"1.1","math":"( u[k])"}. Also, find the value of J∗{"version":"1.1","math":"( J^* )"}. Problem 5. (20 pts) EvaluateJ0=3∑k=0∞‖x[k]‖22{"version":"1.1","math":"[ J_0=3sum_{k=0}^infty |x[k]|_2^2 ]"}on the trajectories of the systemx[k+1]=[−0.5000.5]x[k],x[0]=[01].{"version":"1.1","math":"[ x[k+1]=left[begin{array}{cc} -0.5 & 0\ 0 & 0.5 end{array}right]x[k],quad x[0]=left[begin{array}{c} 0\ 1 end{array}right]. ]"} Problem 6. (20 pts) Given the following model of a dynamical system:x˙=2u1+2u2,x(0)=3,{"version":"1.1","math":"[ dot{x}=2u_1+2u_2,qquad x(0)=3, ]"}and the associated performance indexJ=∫0∞(x2+ru12+ru22)dt,{"version":"1.1","math":"[ J= int_{0}^{infty}left(x^2+ru_1^2+ru_2^2right)dt, ]"}where r>0{"version":"1.1","math":"( r>0)"} is a parameter. (10 pts) Find the solution to the algebraic Riccati equation corresponding to the linear state-feedback optimal controller. (5 pts) Write the equation of the closed-loop system driven by the optimal controller. (5 pts) Find the value of J{"version":"1.1","math":"( J )"} for the optimal closed-loop system. Problem 7. (20 pts) Given the following model of a dynamical system: x ˙ 1 = x 2 + a x ˙ 2 = u , {"version":"1.1","math":"begin{eqnarray*} dot{x}_1 &=& x_2 + a\ dot{x}_2 &=& u, end{eqnarray*}"}where a ∈ R {"version":"1.1","math":"(ain mathbb{R})"} and | u | ≤ b , b > 0. {"version":"1.1","math":"[ |u| le b,quad b>0.]"}The performance index to be minimized isJ=∫0tfdt.{"version":"1.1","math":"[ J=int_{0}^{t_f}dt. ]"}Find the state-feedback control law u=u(x1,x2){"version":"1.1","math":"( u=u(x_1,x_2))"} that minimizes J{"version":"1.1","math":"( J)"} and drives the system from a given initial condition x(0)=[x1(0),x2(0)]⊤{"version":"1.1","math":"( x(0)=[x_1(0),x_2(0)]^top )"} to the final state x(tf)=0{"version":"1.1","math":"( x(t_f)= 0 )"}. Proceed as indicated below. (5 pts) Derive the equations of the optimal trajectories. (5 pts) Derive the equation of the switching curve. (10 pts) Write the expression for the optimal state-feedback controller. Problem 8. (20 pts) Let A∈Rm×m{"version":"1.1","math":"( Ain mathbb{R}^{mtimes m})"} and B∈Rn×n{"version":"1.1","math":"( B in mathbb{R}^{n times n})"}. Express det(A⊗B){"version":"1.1","math":"(det(Aotimes B))"} in terms of detA{"version":"1.1","math":"( det A )"} and detB{"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)=AD⊗CB{"version":"1.1","math":"( (Aotimes C)(Dotimes B)=ADotimes C B )"} and det(A⊗Ir)=(detA)r{"version":"1.1","math":"( det (Aotimes I_r)=left( det Aright)^r)"} to be useful in your derivation.) Then employ the obtained formula to evaluate det(A⊗B){"version":"1.1","math":"( det(Aotimes B))"} for the case whenA=[102−2]andB=[26−80−10023].{"version":"1.1","math":"[ A=left[begin{array}{cc} 1 & 0\ 2 &-2 end{array}right] quad mbox{and} quad B=left[begin{array}{ccc} 2 & 6 & -8\ 0 & -1 & 0\ 0 & 2 & 3 end{array}right]. ]"} Problem 9. (20 pts) For the nonlinear system model of Problem 1, that is, the model[x˙1x˙2]=[41+x2+x2ux1x2+u],{"version":"1.1","math":"[ left[begin{array}{c} dot{x}_1\ dot{x}_2 end{array}right]=left[begin{array}{c} frac{4}{1+x_2} + x_2u\ x_1x_2+u end{array}right], ]"}find the equilibrium state xe{"version":"1.1","math":"( x_e )"} corresponding to ue=−2{"version":"1.1","math":"( u_e=-2)"} and such that xe1=−1{"version":"1.1","math":"( x_{e1}=-1)"}. Then, construct a linear in x{"version":"1.1","math":"( x )"} and u{"version":"1.1","math":"( u )"} model describing the system operation about (xe,ue){"version":"1.1","math":"( (x_e, u_e))"}. Problem 10. (20 pts) Let A = [ 1 2 3 0 ] . {"version":"1.1","math":"[ A=begin{bmatrix} 1 & 2 & 3 & 0end{bmatrix}. ]"}a. (3 pts) Find spark ( A ) {"version":"1.1","math":"(mbox{spark} (A) )"} b. (3 pts) Find ‖ x ‖ 0 {"version":"1.1","math":"( |x|_0)"} of the sparsest, non-zero, solution to A x = 0 {"version":"1.1","math":"(Ax=0 )"} c. (4 pts) Find a sparsest solution of A x = 0 {"version":"1.1","math":"( Ax=0)"} Let B = [ 1 2 − 2 3 3 2 − 1 1 2 − 3 ] . {"version":"1.1","math":"[ B=begin{bmatrix} 1 & 2 & -2 & 3 & 3\ 2 &-1 & 1& 2& -3end{bmatrix}. ]"}d. (3 pts) Find spark ( B ) {"version":"1.1","math":"(mbox{spark} (B) )"} e. (3 pts) Find ‖ x ‖ 0 {"version":"1.1","math":"( |x|_0 )"} of the sparsest, non-zero, solution to B x = 0 {"version":"1.1","math":"(Bx=0 )"} f. (4 pts) Find a sparsest solution to B x = 0 {"version":"1.1","math":"(Bx=0)"}. *** Congratulations, you are almost done with 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.). 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