Controls, Instrumentation, and Uncertainty Cases

Treat the loop as a system

A control or instrumentation prompt describes a chain: physical variable, sensor, signal conditioning, controller, actuator, plant, feedback, and disturbance. Name those pieces before calculating. A temperature controller may use a thermocouple, amplifier, digital controller, control valve, heat exchanger, and outlet-temperature feedback. A speed controller may use an encoder, motor drive, rotating inertia, and load-torque disturbance.

FE controls questions are largely recognition based. A transfer function relates output to input in the Laplace domain for zero initial conditions. 2% of final value in one tau and is essentially settled by about 4*tau. A second-order system adds natural frequency omega_n and damping ratio zeta: zeta < 1 is underdamped (overshoot and ringing), zeta = 1 is critically damped, zeta > 1 is overdamped. Feedback reshapes the closed-loop response and can cut steady-state error, but a pole with a positive real part means an unstable response.

Worked problem: first-order step response

A thermocouple-amplifier chain has transfer function G(s) = 5/(2s + 1). Reading the standard form K/(taus + 1) gives K = 5 and tau = 2 s. After a unit step, the output rises toward 5; at t = tau = 2 s it has reached 0.6325 = 3.16, and it is about 98% settled by 4tau = 8 s. If the prompt asks for the value at t = 4 s (= 2tau), the response fraction is 1 - e^(-2) = 0.865, so output = 0.865*5 = 4.32.

Measurement vocabulary and uncertainty

Instrumentation items are vocabulary traps. Accuracy is closeness to the true value; precision is repeatability; resolution is the smallest detectable increment; sensitivity is output change per input change; bias is a systematic offset. A transducer reading 103 kPa against a 100 kPa standard with tight scatter is biased, not imprecise. A device can be precise but inaccurate; a display can have fine resolution yet poor accuracy if the sensor or calibration is weak.

Uncertainty propagation depends on the operation. For a sum or difference of independent quantities, combine absolute uncertainties by root-sum-square: u_R = sqrt(u_a^2 + u_b^2). For a product or quotient, combine relative uncertainties: (u_R/R) = sqrt((u_a/a)^2 + (u_b/b)^2). For a power, R = x^n gives (u_R/R) = |n|(u_x/x). Worked case: power P = VI with V = 12.0 V ±2% and I = 3.0 A ±3%. Relative uncertainty in P = sqrt(0.02^2 + 0.03^2) = sqrt(0.0004 + 0.0009) = sqrt(0.0013) = 0.0361 ≈ 3.6%, so P = 36.0 W ±1.30 W.

Sensors, sampling, and signal conditioning

Sensor choice follows the measured variable and environment. Thermocouples tolerate wide ranges but need cold-junction compensation; RTDs are accurate over moderate ranges; strain-gage load cells need bridge circuits and amplification; orifice and Venturi meters infer flow from pressure drop; encoders measure position and speed digitally. Signal conditioning amplifies small outputs, filters unwanted frequencies, isolates, linearizes, and digitizes. Sampling must capture the highest meaningful frequency: the Nyquist minimum is twice that frequency, and practical systems add margin to avoid aliasing.

A vibration signal with content to 450 Hz needs sampling of at least 900 Hz.

Worked problem: closed-loop gain and second-order parameters

A unity-feedback loop has forward transfer function G(s) and feedback H(s) = 1. The closed-loop transfer function is T(s) = G/(1 + GH). If G(s) = K/(s(s + 4)), the closed-loop denominator is s^2 + 4s + K. Comparing to the standard form s^2 + 2zetaomega_ns + omega_n^2, we read omega_n = sqrt(K) and 2zeta*omega_n = 4, so zeta = 2/sqrt(K). For K = 4, omega_n = 2 rad/s and zeta = 1 (critically damped — no overshoot). 5 (underdamped — the step response overshoots and rings).

This shows the FE-tested trade-off: raising loop gain speeds the response and shrinks steady-state error but reduces damping, eventually causing overshoot and, if a pole crosses into the right half-plane, instability.

Worked problem: full-scale versus reading uncertainty

A pressure gauge is rated ±0.5% of full scale with a full-scale span of 500 kPa, and it reads 120 kPa. The absolute uncertainty is fixed by the span: 0.005 * 500 = ±2.5 kPa, regardless of the reading. As a percent of the reading, that is 2.5/120 = 2.1% — much larger than 0.5%. Had the spec said ±0.5% of reading, the uncertainty would be only 0.005 * 120 = ±0.6 kPa. The FE distractor applies the 0.5% to the reading rather than the span. Always check whether a tolerance is percent-of-full-scale (constant absolute error) or percent-of-reading (constant relative error) before reporting an uncertainty.

Uncertainty-propagation quick rules

• Sums/differences (R = A ± B): combine absolute uncertainties in quadrature, u_R = √(u_A² + u_B²).
• Products/quotients (R = AB or A/B): combine relative uncertainties in quadrature, (u_R/R) = √((u_A/A)² + (u_B/B)²).
• Percent of full scale gives a constant absolute error; percent of reading gives a constant relative error.