* = Ax+Bu AB For a state-space representation or G(9) y = Cx+Du C D a) The controllability Grammian is defined as Le = f*ABB'ell'edt and the observability Grammian is defined as Lo = f* A'C' CeAldt. Prove that ||G|3 = CL.C' = B'L,B. b) Prove that the controllability Grammian(Le) and the observability Grammian(L.) satisfy the Lyapunov equations in Eqns. 1-2, respectively. AL+LA+BB' = 0 (1) A'L. +LA+C'C=0 (2) c) Develop a Matlab program code (LMI programming is strongly suggested) to calculate Le and L. for an arbitrary (but reasonable) transfer function. Also, calculate the transfer function 11:|2 norm by using the residue theorem. Compare the results. * = Ax+Bu AB For a state-space representation or G(9) y = Cx+Du C D a) The controllability Grammian is defined as Le = f*ABB'ell'edt and the observability Grammian is defined as Lo = f* A'C' CeAldt. Prove that ||G|3 = CL.C' = B'L,B. b) Prove that the controllability Grammian(Le) and the observability Grammian(L.) satisfy the Lyapunov equations in Eqns. 1-2, respectively. AL+LA+BB' = 0 (1) A'L. +LA+C'C=0 (2) c) Develop a Matlab program code (LMI programming is strongly suggested) to calculate Le and L. for an arbitrary (but reasonable) transfer function. Also, calculate the transfer function 11:|2 norm by using the residue theorem. Compare the results