Consider the third-order continuous-time LTI system with A = 208 0 0 0 3 0 -8 -6

Consider the third-order continuous-time LTI system

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with A = 208 0 0 0 3 0 -8 -6 x = Ax + Bu y = Cx 0 B = 0, and C= [1 0 0]. Using = 8 0 0 0 6 0 0 4 3 R = 1.5 (a) First design a LQ controller for this continuous time-system using the MATLAB function lqr. Let the optimal controller gain vector be K. Simulate the closed-loop system x = (A - BK)x with X(0) = [2 0 -2]. Sample the closed-loop output response y(t) with sampling time T = 0.2 s. (b) Next, discretize the continuous-time system directly with sampling time T = 0.2 s and design a LQ controller for this discrete-time LTI system with the same Q and R as in part (a). Compare the closed-loop output response with the sampled continuous-time output obtained in part (a). (c) Increase Q and R by a factor of 10 and repeat both the continuous-time and discrete-time designs. Describe how the respective output responses change.