ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019 Page 1 of 6 Sign the Honor Statement: I pledge that I have neither given nor received unauthorized aid on this exam: Please SIGN your name: Please PRINT your name: Exam Rules: • You may use a scientific calculator that is NOT programmable • You may use one (1) page of notes (1 front side and 1 back side) • You may NOT use a laptop computer, PDA, Blackberry, iPhone, or Cell phone • You may NOT use any text book or reference book • You may NOT use MATLAB or any equivalent processing software • You may NOT discuss any part of this exam with anyone • You MUST work on this exam by yourself with NO input from anyone • You MUST show ALL of your work!!! NO partial credit will be given for submissions that do not show all of the work. BEFORE YOU BEGIN: Please check to see that you have all six pages! There are four numbered problems. Please read each problem CAREFULLY! ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019 Page 2 of 6 1. (10 pts) An open-loop LTI plant is constructed by cascading two dynamic systems together, where the output of dynamic system 1 is the input to dynamic system 2: ()() = )) + 2 + ) .̇1̇)2 = 311 1))1 ))5 31)5 + .11)12 = [11 1)] 31)5 + [11] Construct the open-loop state-space representation for the complete cascaded system with as the input and as the output, by defining the A, B, C, and D matrices. Hint: ALGEBRA!!! ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019 Page 3 of 6 ̇ = =− − − D + E 001H = [1 0 0] + [0] 2. (10 points) Use the eigenvector matrix and its inverse (below) to transform the state- space system above into modal (i.e. diagonal) form. Inspect your result to determine which modes (or mode) are uncontrollable and which are unobservable. Explain which mode is dominant and why. = LM E 0 −5 0−1 0 22 0 1H O1 = E 0 −1 2−1 0 00 2 1H ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019 Page 4 of 6 ()() = 3s + 2 An optimal (LQR) state-feedback gain () will be implemented for the first order SISO dynamic system above. The LQR cost function is given by: () = T 3LM ) + )5WX The pre-computed LQR solution for the optimal state-feedback gain () as a function of the relative cost weighting factor LZ[M for this dynamic system is given below. Note: Evaluating the function below is equivalent to calling the Matlab LQR function with and weighting matrices to generate the output. LM = () + 2) > 0 3a. (5 points) Derive an expression for the closed loop pole as a function of the state- feedback control gain. Hint: Convert the transfer function above to state-space first. 3b. (3 points) Determine is the LQR optimal state-feedback gain when the output weighting is 15 times larger than the control weighting? What is the closed-loop pole associated with this gain? 3c. (2 points) Determine the relative cost weighting factor LZ[M associated with a closed- loop LQR optimal pole at -5 ? ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019 Page 5 of 6 A multi-input open-loop plant is given by the following state-space realization with control input u, process noise w, and measurement noise q: An output-feedback control system has been designed that includes full state feedback control, integral control for reference tracking, and state estimation. The following state- space realization was constructed to simulate this complete closed-loop system. In this model, perfect knowledge of the open-loop matrices has been assumed. !x 2×1[ ] =A x 2×1[ ] +B u 2×1[ ] +F w 1×1[ ] y 1×1[ ] =C x 2×1[ ] +D u 2×1[ ] + θ 1×1[ ] !ˆx !x I !x ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 4.33 −11.7 −0.97 −8.5 8.5 −0.9 0.61 0.43 1.5 −1.5 −5 −0.27 0.94 −3 3 5.8 5.7 −0.97 −10 −9 −1.4 8.1 0.43 2 −9 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ xˆ x I x ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ + 0 0 −2.83 0 0 0.5 1 0 −1 0 3 0 0 2 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ r w θ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ u yˆ y ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 2.54 0.13 −0.47 0 0 2.9 2.8 −0.48 0 0 8 −2.7 −0.94 0 0 5 0.27 −0.94 3 −3 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ xˆ x I x ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ + 0 0 0 0 0 0 0 0 0 0 0 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ r w θ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ME 5554 / AOE 5754 / ECE 5754 FINAL EXAM – FALL 2019 Page 6 of 6 4a. (3 pts) Determine the open-loop A, C, and F matrices. 4b. (3 pts) Determine the full state feedback ( ) and the integral control ( ) gains. 4c. (2 pts) Determine the state estimation gains (K). 4d. (2 pts) Determine the open-loop D matrix. G0 GI
