1. Consider U₁ = 20 m/s. Denote z = [y vyr]T, u = 8, y = y, and the external disturbance as d = rd- Specify A, B, C, and D. Discuss reasonable limits on the inputs and state variables. Calculate the system transfer function. Verify that the system eigenvalues and transfer function poles are the same. Draw a block diagram of the state-space model using four integrators. [3 points] 2. Investigate the controllability and observability of the system. Provide the controllability and observability matri- ces and the rank of each matrix. [1 point] 3. Specify a set of performance criteria for your system, including transient response time, settling time, and maxi- mum overshoot. Using the open-loop poles of the system, explain your rationale for the selected design criteria. [1 point] 4. Provide a set of desirable closed-loop poles that satisfy your design criteria. Explain your logic. Design the state feedback and provide K. Verify that the eigenvalues of A - BK are the same as the desirable closed-loop poles. Provide the following figures. a) pole-zero diagram of the open loop system. b) pole-zero diagram of the closed- loop system. c) Evolution of the state variables and output of the closed-loop system versus time. d) Evolution of the control input of the closed-loop system versus time. Explain your observations and the differences between the open loop and closed-loop cases. [3 points] 5. Design an output regulator such that the output of the system settles at a nonzero reference yd. Provide the design details and your calculations to obtain and I. Plot the variation of the output and reference versus time. [2 points] 1 6. Design a servo system such that the output of your system settles at a nonzero reference ya. Provide the augmented state-space model, the desirable closed-loop poles for the servo system, and the values of K and Kr. Plot the variation of output and reference versus time. [2 points] 7. Consider a reasonable step disturbance in the input of the system. Provide your rationale about the size of the step disturbance. Plot and compare the output of your system for Task 5 and 6 in the presence of disturbance. What differences do you observe between the two cases? Explain your answer. [1 points]

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