C Two matrices A and B are orthogonal if A" B = 0, i.e., if the columns of A and B are orthogonal to each other. Show that P and I - P are orthogonal. 3 Recall the inverted pendulum on a cart from HW5 Problem 7. m Q M The nonlinear equations of motion for this system are: (M+ m)p - mlocos0 = -cp - ml02 sin 0 + F, (J + m12)0 - mlpcos0 = -70 + mgl sine. a For simplicity, take c = y = 0. Linearizing around the equilibrium point eq = the state space formulation has plant matrix A and input matrix B: 0 0 A = 0 m212g/1 0 0 , B= 0 Mimglu 0 0 [ml/u where u = MtJt - m-12, Mt = M + m, and Jt = J + ml. Show that this linearized system is controllable. b Consider the linearized system with nonzero c and y: 0 0 0 0 0 0 A = B = - CJt/u - yml / u Jtu Mimgl/u -cml/u YMt/ H. ml / u