Consider a system with state matrices (a) Use feedback of the form u(t) = Kx(t) + r(t)

Chapter 7, Problem 7.57

(choose chapter or problem)

Consider a system with state matrices (a) Use feedback of the form u(t) = Kx(t) + r(t), where is a nonzero scalar, to move the poles to 3 3j. (b) Choose so that if r is a constant, the system has zero steady-state error; that is y() = r. (c) Show that if F changes to F + F, where F is an arbitrary 2 2 matrix, then your choice of in part (b) will no longer make y() = r. Therefore, the system is not robust under changes to the system parameters in F. (d) The system steady-state error performance can be made robust by augmenting the system with an integrator and using unity feedbackthat is, by setting I = r y, where xI is the state of the integrator. To see this, first use state feedback of the form u = Kx K1x1 so that the poles of the augmented system are at (e) Show that the resulting system will yield y() = r no matter how the matrices F and G are changed, as long as the closed-loop system remains stable. (f) For part (d), use MATLAB (SIMULINK) software to plot the time response of the system to a constant input. Draw Bode plots of the controller, as well as the sensitivity function (S) and the complementary sensitivity function (T).