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22-Elec-B2 Advanced Control Systems: December 2015

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

  1. Question 1 Near-instability, input-disturbance rejection, and a 90° phase-margin design
  2. Question 2 State-space realisation and pole placement with reference scaling
  3. Question 3 Least-squares identification of a second-order discrete model
  4. Question 4 Controllability, observability and BIBO stability of a parametrised system
  5. Question 5 Sampled-data loop: ZOH equivalent, Jury stability range and a deadbeat redesign
  6. Question 6 Transport delay under proportional control: margins, step error and redesign

Start with Question 1 →

Paper format. National Examinations, December 2015 — 07-Elec-B2 Advanced Control Systems. Three hours, closed book; tables of Laplace and z-transforms are supplied as pages 4 and 5 of the paper. Six questions are printed, “any four questions constitute a complete paper” and “all questions are of equal value”, i.e. 25 marks each. All six are solved here, because the set is a study resource rather than a sitting.

Reference texts. G. F. Franklin, J. D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems (frequency response and stability margins, Ch. 6; state-space design and pole placement, Ch. 7; digital control and the ZOH equivalent, Ch. 8). K. Ogata, Modern Control Engineering (Routh and root locus, Ch. 5–6; controllability and observability, Ch. 9). N. S. Nise, Control Systems Engineering (steady-state error and system type, Ch. 7). A. V. Oppenheim and A. S. Willsky, Signals and Systems (z-transform and the unit-circle stability test, Ch. 10). L. Ljung, System Identification: Theory for the User (least-squares ARX estimation and its convergence conditions, Ch. 7–8).

Sign conventions used throughout. The error is always $e = r - y$. On Question 1 the block diagram shows the disturbance subtracted at the plant input (a minus at $d$, a plus at $u$), so the plant sees $u - d$; this is read from the printed figure, not assumed. Phase margins are quoted at the gain crossover and gain margins at the phase crossover, both from the exact transfer functions rather than from asymptotic sketches. “At least 6 dB” is applied literally as a gain-margin ratio of $10^{6/20} = 1.9953$; the customary 2:1 shorthand is quoted alongside where it differs.