Consider the system where Xk+1 = and A A = and b = (1-1). Compute the equivalent representation corresponding to a similarity transformation where the first vector of V is an eigenvector of A. Then use this representation to compute a matrix K := (k ka) such that one of the eigenvalues of A-BK is at 0.5. Can you change the other eigenvalue? This problem extends Problem 3. Consider the system Tk+1= Ak + buk, where x R and u R; thus A & Rx3 and b = R1, Let Axk+ Buk 0 -1 1 2 b= 3.0750 8.0250 -0.1500 -1.0250 -2.6750 -0.5500 0.1500 -1.0000 0.0500 = 5 1 Compute a gain matrix K = Rx3 such that, with the feedback law defined by the equation E Uk = -Kk, the eigenvalues associated with the reachable modes are at 0.9 and 0.8.