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Control Engineering Basics

Learn with multiple-choice questions, including calculations: open-loop and feedback control, feedforward control, block diagrams and closed-loop transfer functions, the Laplace transform and transfer functions, the final value theorem, the time constant and step response of first-order systems, the damping ratio and natural angular frequency of second-order systems, overshoot and settling time, steady-state error and system type, poles and stability, the Routh–Hurwitz stability criterion, the Nyquist stability criterion, Bode plots and straight-line approximations, time delay, gain margin and phase margin, P, PI and PID control, the Ziegler–Nichols ultimate sensitivity method, integral windup, derivative kick, sampling and the zero-order hold in digital control, the s-plane and z-plane, cascade control, the control loop of a DC-DC converter and the basics of sequence control.

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The temperature of an air conditioner, the speed of a motor, the output voltage of a power supply, the position of a hard-disk head: many quantities around us are held constant by feedback control, which compares a sensor measurement with a setpoint and adjusts the manipulated variable. This topic covers classical control, the foundation for designing and analyzing such loops.

It starts with the differences between open-loop, feedback and feedforward control, block diagrams and closed-loop transfer functions, and how negative feedback suppresses variation, then covers the Laplace transform, transfer functions and the final value theorem. It continues by working through, with calculations, the time constant and step response of first-order systems, the damping ratio and natural angular frequency of second-order systems, overshoot and settling time, how pole positions relate to the response, and steady-state error and system type. For stability it covers the pole condition for stability, the Routh–Hurwitz stability criterion and the Nyquist stability criterion, and for frequency response it covers Bode plots and straight-line approximations, integrators, time delay, reading the gain margin and phase margin, and the relation between phase margin and overshoot. The second half examines what each action of P, PI and PID control does, the Ziegler–Nichols ultimate sensitivity method, integral windup and its countermeasures, derivative kick, discrete-time PI calculations on a microcontroller, the phase lag due to the zero-order hold and the correspondence between the s-plane and the z-plane, and touches on cascade control of motors, the control loop of a DC-DC converter and the basics of sequence control. It also serves as an introduction to automatic control as covered in the machinery subject of Japan's Class 3 Chief Electrical Engineer examination (Denken 3) and in the circuit theory and control engineering subject of Korea's Engineer Electricity examination (jeongi gisa). The values in the calculation questions, including those of the wrong choices, have been checked by computing them in Python (NumPy). The gain of negative feedback and the phase margin and compensation of op-amps are covered in "Analog and electronic circuits"; RC time constants in "Circuit and electromagnetism basics"; the z-transform, difference equations and the bilinear transform in "Digital Signal Processing Basics"; PWM and timer interrupts in "Embedded Systems & Firmware"; fixed point and saturating arithmetic in "Computer Arithmetic"; and process control in semiconductor fabs (APC) in "Yield and quality control".

All Control Engineering Basics questions (40)
  1. Which of the following correctly describes feedback control (closed-loop control)?
  2. Which of the following is the correct reason for combining feedforward control with feedback control?
  3. In a negative-feedback block diagram, the forward path has the transfer function G(s) and the feedback path has the transfer function H(s). Which of the following is the closed-loop transfer function Y(s)/R(s) from the setpoint R(s) to the output Y(s)?
  4. A negative-feedback system has a forward gain G = 100 and a feedback gain H = 0.1. When G halves to 50 because a component degrades, which of the following is closest to the fractional decrease in the closed-loop gain?
  5. With all initial values zero, which of the following is the correct pair of operations that represent differentiation and integration of a time function in the s domain after the Laplace transform?
  6. A step of size 1 is applied to a system with the transfer function G(s) = 5/(s + 2). Which of the following is the final value of the output (its value as t → ∞)?
  7. Which of the following correctly describes the transfer function of a linear time-invariant system?
  8. A step of size 1 is applied at time 0 to a first-order system with the transfer function G(s) = 2/(1 + 0.5s). Which of the following is closest to the output at t = 1 s?
  9. When a step of size 2 was applied to a first-order system, the output finally settled at 6 and reached about 63% of its final value 0.4 s after the input. Which of the following is the transfer function of this system?
  10. Which of the following is the range of the damping ratio ζ for which the step response of the second-order system G(s) = ωn²/(s² + 2ζωn s + ωn²) overshoots the final value and approaches it while oscillating?
  11. Which of the following is the correct pair of natural angular frequency ωn and damping ratio ζ for the second-order system with the transfer function G(s) = 25/(s² + 4s + 25)?
  12. A step is applied to a second-order system (standard form without zeros) with a damping ratio ζ = 0.5. Which of the following is closest to the overshoot (the maximum excursion beyond the final value, as a fraction of the final value)?
  13. Which of the following is the settling time, until the step response of the system G(s) = 25/(s² + 4s + 25) stays within ±2% of its final value, estimated with the approximation ts ≈ 4/(ζωn)?
  14. A second-order system without zeros has its poles at s = −σ ± jωd. Which of the following correctly describes how the step response changes when only ωd is increased while σ is kept the same?
  15. In a unity-feedback system the forward transfer function is G(s) = 9/(1 + 2s). Which of the following is the steady-state error for a step setpoint of size 1?
  16. In a unity-feedback system with a stable closed loop, the forward transfer function G(s) contains one integrator 1/s (a type 1 system). Which of the following correctly describes the steady-state error?
  17. Which of the following is the condition for a continuous-time linear system to be stable (its impulse response approaches zero with time)?
  18. Which of the following closed-loop systems, given by their characteristic equations, is stable?
  19. A unity-feedback system has the loop transfer function K/(s(s + 1)(s + 2)). Which of the following is the range of K for which the closed-loop system is stable?
  20. Which of the following correctly describes the Nyquist stability criterion?
  21. Which of the following correctly describes the Bode plot of the first-order system G(s) = 1/(1 + sT)?
  22. Which of the following correctly describes the Bode plot of an integrator G(s) = 1/s?
  23. For G(s) = 10/((1 + s/100)(1 + s/1000)), which of the following is the gain at the angular frequency ω = 10⁴ rad/s, found with the straight-line approximation of the Bode plot?
  24. Which of the following correctly describes the frequency characteristics of a time-delay element e^(−Ls) with delay time L, and its effect on a control system whose loop transfer function contains it?
  25. Which of the following correctly describes the gain margin and phase margin read from the Bode plot of the loop transfer function?
  26. Which of the following is closest to the gain margin of the loop transfer function L(s) = 2/(s(s + 1)(s + 2))?
  27. Raising the gain of the loop transfer function reduced the phase margin from 60° to 30°. Which of the following correctly describes the change in the closed-loop step response?
  28. A plant that contains no integrator (a type 0 system) is under proportional (P) control. Which of the following correctly describes the general change when the proportional gain Kp is increased?
  29. Which of the following is the correct pair of the main purpose of adding integral (I) action in PI control and its side effect?
  30. Which of the following is the correct pair of a property of derivative (D) action in PID control and a measure taken when implementing it?
  31. In PID control, which of the following prevents the manipulated variable from jumping sharply at the instant the setpoint is changed as a step (derivative kick)?
  32. In the Ziegler–Nichols ultimate sensitivity method, the gain is raised under proportional control alone until a sustained oscillation appears, and from that gain Ku and the period of oscillation Tu the PID parameters are set to Kp = 0.6Ku, Ti = 0.5Tu and Td = 0.125Tu. Which of the following are the values for Ku = 8 and Tu = 2 s?
  33. Which of the following correctly describes integral windup?
  34. Positional PI control is implemented on a microcontroller, with the duty cycle of a PWM signal (0 to 100%) as the manipulated variable. Which of the following is a correct countermeasure against integral windup?
  35. A digital PI controller with a sampling period Ts = 1 ms calculates the integral with the rectangular approximation Σe·Ts. With an integral gain Ki = 10 s⁻¹, which of the following is the increase in the integral term when an error e = 0.2 lasts for 50 samples?
  36. In digital control with a sampling period Ts = 1 ms, the zero-order hold of the D/A conversion acts as a time delay of about Ts/2. With a gain crossover frequency of 100 Hz, which of the following is closest to the phase lag caused by this delay?
  37. A continuous-time first-order system with its pole at s = −100 s⁻¹ is discretized with a sampling period Ts = 1 ms. Which of the following is closest to the discrete-time pole given by the relation z = e^(sTs)?
  38. A motor is controlled with cascade control, with a current (torque) loop inside and a speed loop outside. Which of the following is the correct design approach?
  39. The output-voltage feedback loop of a buck DC-DC converter with a switching frequency of 500 kHz is being designed. Which of the following is the usual guideline for choosing the gain crossover frequency?
  40. Which of the following correctly describes sequence control?

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