Answer Happy

Consider the following discrete linear time-invariant system: Xk+1 = Axk+Yk Buk, (0.1) where x ER" is the system state at time k, uk € Rm is the control input at time k, and ½ € {0, 1} is the decision variable at the controller whether it sends uk to the system or not. Consider any initial condition xo ER" which is known to the controller (hence all future state æk, k ≥ 1 are known to the controller). Assume the controller can communicate with the system for d (d E N and d < T) times, which is equivalent to say that T-1 Yk = d. - (0.2) k=0 = A schedule at the controller consists of a sequence of binary-valued variables {(0) : k 0,..., T – 1}. For a given schedule 0, define J(0) as T- J(0) = min Σ (x/Qæk + √k (0)u²z Ruk) + x7Qx™. Uk k=0 Find a closed-form expression of J(0) as well as the optimal control uk's.