Answer Happy

Q3) [30 points) Consider the following state equation: :-[ ]-+CH- = u, y = = [1 0] x .
(a) Design the gain matrix K of a state feedback control law u = -Kx so that the closed- loop system has on = 1 rad/sec, 5 = 0.707. (b) Design the gain matrix L of a full-order state observer so that the state estimate error dynamics have on = 5 rad/sec, = 0.5. (c) Use your answers to define an observer-based compensator (that is, write the equations for i, u) that achieves both objectives. (d) In MATLAB, use the ode450 function to integrate the equations of the 2n- dimensional state equation that combines the closed-loop state equation and the observer- based compensator. Let G= 1. Integrate these equations over the time interval t = [04] sec, given an initial state x(0) =[10 –10]?, an initial state estimate f(0) = [0 0,7, and the reference input r(t) = sin(t). Plot xı(t) and xz(t) and their estimates în and iz over time. Q4) (30 points) The simple pendulum has the following equation of motion: ᎫᎾ + mgl sin = T, 2 where J is the pendulum's moment of inertia about the point of rotation and T is the applied torque. = (a) Let the state variables be x = and the input to the system be u = T. Put the equation of motion into the nonlinear state space form, i = f(x,u). (b) Suppose that u = 0. Find the 2 equilibrium points Xeq,1 and Xeq.2 of the system. (c) Linearize the nonlinear state space model about ř = (a) and about the input û = 0 to obtain the linear state space model is = Axs + Bus, where xs = x - † and ug = u - ū. Do this by deriving the Jacobian matrices A and B. (d) Suppose that there is no applied torque (T=0) and the system starts at rest with the pendulum angle at 0= 1/2. Use the MATLAB function ode45 to solve both the four Ist- order nonlinear equations of motion and the 1st-order linearized equations of motion from part (c). Use the parameters J= 1 kgm2, m = 1 kg, and 1= 1 m. Plot 0 and È over time from the nonlinear and linearized models.