The University of New South Wales School of Electrical Engineering and Telecommunications ELEC4632 Computer Control Systems Final Examination, Term 3, 2020 1. Time allowed : 2 hours 2. Reading time : 10 minutes 3. Scanning time : 15 minutes 4. Total exam duration: 9:30am – 11:55am (Sydney time). 5. The paper contains 6 questions (60 marks in total). 6. Each question has the value indicated. 7. This exam contributes 60% to the course final mark. 8. Candidates should attempt all questions. 9. This is an open book examination. 10. This paper contains 4 pages. 11. Please combine answers in one pdf file, name the file by zID first name family name, and click the submit button before the due time. 1 This study source was downloaded by 100000831262840 from CourseHero.com on 11-07-2021 06:26:13 GMT -06:00 https://www.coursehero.com/file/81668718/Final-Exam-20pdf/ Th is s tud y r eso urc e w as sha red vi a C ou rse He ro. com NO collaborations allowed and the work submitted must be your own. During the exam you may NOT be in contact with anyone else via any form of communication media (email, messag- ing, phone, video conferencing, etc). Violating this will be consid- ered an academic misconduct and disciplinary action will be taken against anyone who is proven to have violated this rule. ANSWERS MUST BE WRITTEN IN INK. EXCEPT WHERE THEY ARE EXPRESSLY REQUIRED, PENCILS MAY ONLY BE USED FOR DRAWING, SKETCHING OR GRAPHICAL WORK. 2 This study source was downloaded by 100000831262840 from CourseHero.com on 11-07-2021 06:26:13 GMT -06:00 https://www.coursehero.com/file/81668718/Final-Exam-20pdf/ Th is s tud y r eso urc e w as sha red vi a C ou rse He ro. com Question 1 (10 marks) Sample the continuous-time system x˙(t) = ( 0 4 0 −1 ) x(t) + ( 0 1 ) u(t− 1.35), y(t) = ( 1 −2 ) x(t) using the sampling interval h = 0.3. Question 2 (10 marks) A discrete-time control system is described by x(k + 1) = 1.5 0 3a1− a −2a2 19 −3.0 0 1.7 x(k), y(k) = 0.5 + a 2 0 3.1 8− a a− 10 1 0.4 0 3− a x(k). where the parameter a varies from −∞ to∞. Determine for which values of a this system is observable. Question 3 (10 marks) Given the system x(k + 1) = ( 1.2 0.4 0.6 0.3 ) x(k) + ( 2 0 ) u(k), y(k) = ( −2 3 ) x(k). Design (a) deadbeat state-feedback control law; (b) state estimator with the poles 0.0 and −0.25. 3 This study source was downloaded by 100000831262840 from CourseHero.com on 11-07-2021 06:26:13 GMT -06:00 https://www.coursehero.com/file/81668718/Final-Exam-20pdf/ Th is s tud y r eso urc e w as sha red vi a C ou rse He ro. com Question 4 (10 marks) (a) The characteristic equation of a discrete-time control system is given by z2 + (3K − 1)z +K = 0 where K varies from −∞ to ∞. Determine the range of K for stability. (b) The characteristic equation of a discrete-time control system is given by z3 + 1.4z2 + 0.75z + 1.05 = 0. Determine the number of characteristic roots outside the unit disk without calculating them. Is this system stable? Question 5 (10 marks) Consider the following nonlinear discrete-time system of the form x1(k + 1) = x1(k) + x2(k) 2 − 1, x2(k + 1) = 3x1(k)2 + sin(pix2(k)) + x2(k). Find all singular points of this system and study their stability. Question 6 (10 marks) Consider the optimal control problem for the system x(k + 1) = −3x(k) + u(k) with initial condition x(0) = 0.8, the cost function J = ∞∑ k=0 x2(k)→ min and the constraint |u(k)| ≤ 2. Determine the optimal control strategy and the optimal value of the cost function. 4 This study source was downloaded by 100000831262840 from CourseHero.com on 11-07-2021 06:26:13 GMT -06:00 https://www.coursehero.com/file/81668718/Final-Exam-20pdf/ Th is s tud y r eso urc wa s sha red vi a C ou rse He ro. com Powered by TCPDF (www.tcpdf.org)
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