Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 16-Elec-B2 Advanced Control Systems, December 2019 — a three-hour open-book examination; any non-communicating calculator is permitted. The cover page states “Any four questions constitute a complete paper. Only the first four questions as they appear in your answer book will be marked” and “All questions are of equal value (25%)”. The sitting prints five questions, so each carries 25 marks and a candidate answers four. Question 1 is a seventeen-item multiple-choice block whose per-item weights are printed in the margin as [1], [2] or [3] and total exactly 25. A table of inverse Laplace transforms and a table of Laplace/z transforms are appended. All five questions are worked below, because this set is a study resource rather than a timed sitting.
Reference texts. N. S. Nise, Control Systems Engineering, 8th ed. (Ch. 4 time response, percent overshoot, peak time and settling time; Ch. 6 Routh–Hurwitz stability; Ch. 7 steady-state error and system type; Ch. 8 root-locus sketching rules including break-away/break-in points and angles of departure; Ch. 10 frequency response and stability margins); R. C. Dorf and R. H. Bishop, Modern Control Systems, 13th ed. (Ch. 5, 6, 7, 9); K. Ogata, Modern Control Engineering, 5th ed. (Ch. 5 transient response, Ch. 6 root-locus analysis, Ch. 7 frequency-response analysis); G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, 8th ed. (Ch. 3, 5, 6). These are the references listed for exam code 16-Elec-B2 in the Engineers Canada syllabus, and this paper’s vocabulary (“break-in point”, “intercept of the asymptotes”, “angle of departure”) follows Nise closely.
Check — three source readings checked against the printed paper.
(1) Question 1, item 2. The printed sentence reads “where \(\theta(s)\), \(T(s)\), \(J\) and \(K\) represent torque, angular displacement, moment of inertia and spring constant, respectively”. The paper really does print the first two descriptions in the wrong order. The drawing beside it settles the physics — \(T(t)\) is the applied torque acting on the inertia \(J\), and \(\theta\) is the resulting angular displacement, which is also the only reading under which \(\theta(s)/T(s)\) has the units of a compliance. The answer does not depend on the naming at all, because only the denominator is used.
(2) Question 1, item 16. The margins are read off a printed Bode pair, so they are inherently approximate. The printed Bode pair gives a gain crossover near \(\omega \approx 3\) rad/s where the phase curve sits near \(-116^\circ\), and a phase crossover near \(\omega \approx 10\) rad/s where the magnitude curve is near \(-21\) dB. Both readings point at the same option. The graph is coarse enough that the honest statement is “approximately”, exactly as the question words it.
(3) Question 2. The pole-zero map prints its two crosses to the right of the imaginary axis, at \(s = 1 \pm j1\), and its two circles on the negative real axis at \(-2\) and \(-3\). This is the whole point of the question, because an open-loop plant with two right-half-plane poles is unstable until enough gain is applied, which is the opposite of the usual root-locus habit.
