2. Consider the following continuous time second-order linear time invariant system 2:C8] = 1 (:0)+()uce dx (t) y(t) = x(6) Assume that control law is a state variable feedback u(t) = -Kx(t) that minimizes the following infinite horizon cost functional J = {y?(t) + Ru?()}dt 00 (a) Determine if (A,C) is observable and {A,B) is controllable. (b) Draw the symmetric root locus for RE (0,0). (C) Determine the asymptotic values of the eigenvalues of the close loop matrix A-BK as R 00 (d) Determine the asymptotic values of the eigenvalues of the close loop matrix A - BK as R0. (e) Determine: i. the value of R. ii. the state feedback gain K, when one of the eigenvalues of the close loop matrix A - BK is la = -2. 1 Note: You should be able to solve this problem using basic root locus laws, without having to solve a Riccati equation. 2. Consider the following continuous time second-order linear time invariant system 2:C8] = 1 (:0)+()uce dx (t) y(t) = x(6) Assume that control law is a state variable feedback u(t) = -Kx(t) that minimizes the following infinite horizon cost functional J = {y?(t) + Ru?()}dt 00 (a) Determine if (A,C) is observable and {A,B) is controllable. (b) Draw the symmetric root locus for RE (0,0). (C) Determine the asymptotic values of the eigenvalues of the close loop matrix A-BK as R 00 (d) Determine the asymptotic values of the eigenvalues of the close loop matrix A - BK as R0. (e) Determine: i. the value of R. ii. the state feedback gain K, when one of the eigenvalues of the close loop matrix A - BK is la = -2. 1 Note: You should be able to solve this problem using basic root locus laws, without having to solve a Riccati equation