Final Extended Homework

Problem 1: Constrained, Free Final Time Optimal Control

Generate the free final time optimal control for the system x^· = u with the initial state zero with the final state constraint x(tfinal) = 1. In Simulink use a time step of 0.05 seconds and initial guess at the control of 0.1 for all time and initial guess of final time of 1 sec.

a) Impose a control (input) constraint of +/-1 and set the lower bound on final time at 0.1 seconds and the MAXIMUM value of tfinal to be 10 seconds. Find the optimal control when the performance index is J = t_final + ₀∫^tfinal (u² - x)dt. Plot the final state and control time histories and show the final cost value.

b) Keep the input and final time constraints the same but change the performance to J = t_final + ₀∫^tfinal (u² - 2.0x)dt. Plot the final state and control time histories and show the final cost value.

c) Now impose a tfinal maximum constraint of 20 sec and find the optimal control, leaving the minimum value of tfinal to be 0.1 seconds. Plot the final state and control time histories and show the final cost value.

Problem 2: Full-State Feedback, Observers, and Kalman Filter State Estimation

Modify the Simulink diagram you developed for Homework 6 to implement the following dynamics and initial conditions: x^·· + 0.7 x^· + x = u, x^·(0) = 1, x(0) = 0, and have the output be y = x. Set the random noise variance to 0.1.

a) Develop the feedback controller using the LQR function with Q identity and R identity. Using the true full-state feedback, plot the state and control time histories along with the output time history. Leave the controller gain fixed for the next parts.

b) Develop an observer using Q=10*eye and R identity. Plot the observer state estimates. If they look close to the true states (and they should), then use those estimated states in the feedback controller. Plot the true states and the control time history. Leave the observer gain fixed for the next part.

c) Implement the Kalman filter to estimate the states. First, use Q as identity and R as identity. Compare the Kalman filter state estimates when using the true states in the feedback control. Assuming they look reasonable, close the loop using the Kalman filter state estimates in the feedback and plot the time histories of the states and controls.

d) Finally, after using both the observer-reconstructed states and the Kalman filter-reconstructed states, which of the two produce the best results when used in the feedback control (i.e. which estimate the states better and which has a better quality control signal).

Attachment:- Assignment Files.rar