The treatment in 16B so far has treated closed-loop control as being about holding a system steady at some desired operating point, by placing the eigenvalues of the state transition matrix. This control used something proportional to the actual present state to apply a control signal designed to bring the eigenvalues in the region of stability. Meanwhile, the idea of controllability itself was more general and allowed us to make an open-loop trajectory that went pretty much anywhere. This problem is about combining these two ideas together to make feedback control more practical - how we can get a system to more-or-less closely follow a desired traject y, even though it might not start exactly where we wanted to start and in principle could be affected y small disturbances throughout. In this question, we will also see that everything that you discrete-time can also be used to do closed-loop control in Consider the specific 2-dimensional: system ve learned to do closed-loop control in tinuous time. d x(t) = Ax(t) + bu(t) + (t) B2+H dt E (t) + u(t) + w(t) where u(t) is a scalar valued continuous control input and (a) In an ideal noiseless scenario, the desired control sign sired trajectory x* (t) that satisfies the following dynamics: =x* (t) = Ax* (t) + __^* (t). d (8) The presence of the bounded noise term (t) makes the actual state x(t) deviate from the desired x* (t) and follow (7) instead. In the following subparts we will analyze how we can adjust the desired control signal u* (t) in (8) to the control input u in (7) so that the deviation in the state caused by w(t) remains bounded. Represent the state as (t) xt (t) + Ax(t) and u(t)= *(t) + Au(t). Using (7) and (8), show that we can represent the evolution of the trajectory deviation Ax(t) as a function of the con- trol deviation Au(t) and the bounded disturbance (tas: (7) t) is a bounded disturbance (noise). u* (t) makes the system follow the de- d dfX(t) = (t) + b^u(t) + (t). (HINT: Write out equation (7) in terms of x* (t), Ax(t), u* (t) and Au(t).)) (9)