6.3 Feedback Control and PID

6.3 Feedback Control and PID

Quick Answer: Regulatory PID drives error to zero with P (present), I (accumulated), and D (rate). P-only cannot finish a self-regulating load change. Do not crank Kc into a dead-time-dominant oscillator, and do not put a tight level loop on a tank whose job is to surge.

Regulatory feedback is the ordinary PE loop: measure PV, compare to setpoint (SP), move the MV so load upsets (feed composition, upstream pressure, ambient) do not walk the product off spec. Feedforward, cascade, and ratio are later tools. This section is the single-loop PID you must debug on a drawing and in a paragraph of symptoms.

What P, I, and D actually do

• Proportional (P). Controller output (CO) moves with present error e = SP − PV (sign depends on action). Larger controller gain Kc means a bigger CO move per unit error. Equivalently, proportional band PB(%) = 100 / Kc when Kc is dimensionless in %/% . A 50% band is Kc = 2. Wide band = meek controller. P is fast and it does not remember; on a self-regulating plant a sustained load leaves a leftover error (offset).
• Integral (I). CO keeps moving as long as error is nonzero. That is what kills offset. Integral time Ti (minutes per repeat) is how long the I term takes to repeat the P move. Reset rate is the reciprocal, repeats per minute. Smaller Ti (more repeats per minute) is stronger integral — and easier windup and oscillation. Integral adds lag; it is the term that turns a once-stable P loop into a hunter if you get greedy.
• Derivative (D). CO responds to rate of change. It is a brake on a fast PV run-away and a kick on a fast approach. It amplifies noise. Many flow loops run D = 0; many temperature loops use a little D. Derivative on PV differentiates the measurement, not the error, so a setpoint step does not produce a spike. Derivative on error does produce that spike (derivative kick) every time an operator drops in a new SP.

Qualitative algorithm forms (do not treat a brand name as NCEES gospel):

• Independent / parallel / “ideal ISA”: P, I, and D terms are additive; changing Ti does not rescale the D term the way series form does.
• Series / interacting: the form analog pneumatic controllers used; I and D share a factor so the knobs interact. A PE item that says “interacting form” wants you to know the knobs are not independent, not to recopy a Ziegler–Nichols table.

If the question does not name a form, reason in jobs: offset → add or speed I; sluggish but stable → more Kc or less PB; noisy or kick on SP → D on PV or less Td.

Worked numbers: P-only offset

Self-regulating exchanger, Kp = 2.0 °C/%, P-only controller Kc = 4.0 (%/%). Closed-loop gain factor is 1 + Kc Kp = 1 + 8 = 9. A load (fouling, inlet-temperature rise) that would have moved the PV 18 °C in open loop still moves it in closed loop, but only

$\text{offset} \approx \frac{\Delta PV_{\text{open-loop}}}{1 + K_c K_p} = \frac{18}{9} = 2.0\,^{\circ}\text{C}.$

The loop is not “broken”; P-only cannot finish that load. Adding integral drives the 2 °C residual to zero, given enough time and no saturation. If someone claims P-only has zero offset because “the valve moved,” they confused a CO change with PV = SP.

Integral windup and anti-reset

When the valve is at 0% or 100%, extra integral still accumulates error. When the load finally returns, CO has to unwind across a huge hidden sum — the PV overshoots badly. That is reset windup. Anti-reset (anti-windup) stops the crime: freeze I while the MV is saturated, or back-calculate I so the PI sum equals the saturated output (external reset feedback). Bumpless schemes use the same bookkeeping.

Derivative kick

Operator raises SP 10 °C in one click. Error jumps 10 °C. If D sees error, CO spikes as if the PV had crashed. Derivative on PV (or setpoint filtering) avoids that. If a question describes a valve slam only at SP changes, with a quiet loop during load upsets, look at D on error — not at Kc first.

Reverse vs direct acting

Direct acting: increasing PV increases CO (a cooling-water valve on a hot PV: hotter → more coolant). Reverse acting: increasing PV decreases CO (steam on a temperature loop: hotter → less steam).

Fail-open vs fail-closed is a valve choice; action is a controller choice. They must agree so that “more CO” moves the process the way the sign of error requires. Wrong action is unstable even at tiny Kc: the loop drives away from SP. Air-to-open with reverse action on steam is the usual heat-up story; do not memorize a single pairing as universal — read the fail position on the P&ID and then set action.

Bumpless transfer

Switching auto → manual → auto without a CO jump means the unused mode tracks. In manual, the PID bias or integral is forced to the current valve position (and SP may track PV). When you return to auto, error is small and CO is already where the valve is. If a PE item shows a 20% valve kick at the instant of A/M transfer, tracking was not implemented or SP was left miles from PV.

Gap / deadband (intro)

Gap (deadband) around SP: inside the band, the controller does not chase. Outside the band, it drives the PV back toward the band. That is how you let a surge tank breathe. It is not the same as valve stiction, and it is not an excuse to ignore a reactor temperature.

Worked design: why tight level on a surge tank is wrong

A 10,000-gal feed surge tank is in the plant because upstream flow is noisy. The vessel’s job is to let level wander — 30% to 70% — so the outlet flow to the reactor stays nearly steady. An LIC with large Kc and fast integral treats every inch of level as an error. The outlet valve then copies the inlet disturbance, the reactor feed becomes as noisy as the tank inlet, and you paid for a tank you refused to use. Correct PE answer: averaging level, wide proportional band, slow or no integral, or explicit gap control. Tight LIC belongs on a steam drum that must not uncover tubes, not on a surge drum.

Symptom vs which term to move

This is diagnostic, not a vendor recipe. NCEES is not asking you to recite Ziegler–Nichols constants.

Exam trap: increasing Kc on a dead-time-dominant loop

Dead time does not obey Kc. During θ the controller is acting on old news. Raising gain makes larger, later mistakes — classic growing oscillation at a period near 2θ to 4θ. IMC-style reasoning (without worshiping a formula) says Kc scales like τ / (Kp (θ + τc)): when θ dominates, Kc must shrink. If the step test in 6.2 gave θ = 3 min and τ = 1 min, you do not “tune through it” with more proportional action. You wait (slower I), maybe add a dead-time compensator in design life, and on the exam you cut Kc when the loop is already ringing.