2. Consider an LTI system under state variable feedback control. Assume that the n-th order system x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) where y(k) R and u(k) R, is both controllable and observable. Let's use the control law u(k) = -Kx(k) +v(k) where K E Rix is a constant gain matrix and v(k) is the new input. The close loop system is now x(k + 1) = A x(k) + Bv(k) y(k) = Cx(k) where Ac = A-BK Show that this system is also controllable; i.e., for any initial condition x(0) and target state x, there exists some finite integer N and control sequence {v(k), k = 0,...,N}, which will achieve x(N) = x . (Hint: consider the control v(k) = u(k) + Kx(k).) 2. Consider an LTI system under state variable feedback control. Assume that the n-th order system x(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k) where y(k) R and u(k) R, is both controllable and observable. Let's use the control law u(k) = -Kx(k) +v(k) where K E Rix is a constant gain matrix and v(k) is the new input. The close loop system is now x(k + 1) = A x(k) + Bv(k) y(k) = Cx(k) where Ac = A-BK Show that this system is also controllable; i.e., for any initial condition x(0) and target state x, there exists some finite integer N and control sequence {v(k), k = 0,...,N}, which will achieve x(N) = x . (Hint: consider the control v(k) = u(k) + Kx(k).)