Answer Happy

Problem 2 (20 %): Consider the system equation x(k+1) = Ax(k) + Bu(k) 0 1 A 1 0 1 -1 -1 1 1 1 1 1 -1 1 1 -1 0 1 B (a) Using Matlab, check the stability of x(k+1) = Ax(k); (b) using Matlab, check the controllability of x(k+1) = Ax(k)+Bu(k); (c) find u(k), k=0,1,2, 3, which make x(k) from x(0) = [1,0,0,0]" tox(4) = [0,0,0,1]", and calculate E4 = (u(0))2 + (u(1))2+(u(2))2 + (u(3))2 (as the control energy which is in fact minimal, as compared with any other controls u(0),u(1),u(2),u(3) to do the same job); and (d) (i) (for ECE4501) formulate the equation for solving u(k), k=0,1,2,3,4, which can control the system state vector from x(0) = [1,0,0,0]T to x(5) = [0,0,0,1)", and find a set of such solutions uſi),i= 0,1,2,3,4 (in this case, the solutions are not unique and some solutions can be such that E5 = (u(0))2 + (u(1))2 + (u(2))2 + (u(3))2 + (u(4))2 < E4 in (c) above); or (d) (ii) (for ECE6501) work out Part (i) above, find the control input u(k), k = 0,1,2,3,4, which can control the system state vector from x(0) = [1,0,0,0]" to x(5) = [0,0,0,1)+, such that Es = (u(0))2 + (u(1))2 + (u(2))2 + (u(3))2 + (u(4))2 is minimized (and compare it with E4 in Part (c)).