2.3-1. Digital control of harmonic oscillator. A harmonic oscillator is described by x₁ = x2 x2=-wx+u. (1) a. Discretize the plant using a sampling period of T. b. With the discretized plant, associate a performance index of 1 N-1 1 = {[51 (xx)² + $2 (x²)²] + ½ Σ [91(x)²+92(x)²+ruz], 2 k=0 (2) where the state is x = [xx]. Write scalar equations for a digital optimal controller. c. Write a MATLAB subroutine to simulate the plant dynamics, and use the time response program Isim.m to obtain zero-input state trajectories. d. Write a MATLAB subroutine to compute and store the optimal control gains and to update the control uk given the current state xk. Write a MATLAB driver program to obtain time response plots for the optimal controller.