Study notes

Qwestrum Engineering360 · Chemical Engineering · Process Dynamics & Control

Stability of Process Loops

A control loop is stable if all roots of its characteristic equation lie in the left half-plane; this is checked with the Routh-Hurwitz test, root locus, or the Bode/Nyquist gain and phase margins.

Exam tip: keep SI units consistent end-to-end, write the governing relation symbolically before substituting, and sanity-check magnitude and sign.

Key formulas & points

Skim these first — then read the full notes below.

$Open-loop transfer G_{OL} = G_{c} G_{p} G_{m}$ (controller × process × measurement)
A wrong controller sign or positive feedback destabilizes the loop
Phase margin above ~45° is a common robustness target

Topic details

Introduction

This Coughanowr topic determines whether a tuned loop will stay stable. You form the open-loop transfer function, test stability by Routh-Hurwitz on the characteristic polynomial or graphically by root locus, and quantify robustness with the gain and phase margins read from Bode plots or the Nyquist encirclement of the critical point.

Key relations & formulas

1 + G_{OL}(s) = 0

(closed-loop characteristic equation)

GM = 1/|G_{OL}(j\omega_{180})|; PM = 180^{§K1§} + ∠G_{OL}(j\omega_{c})

(stability margins)

Formulas (Indian textbook notation)

$Routh-Hurwitz \frac{array}{root} locus: all closed-loop poles in the left half-plane$

Notation and sign conventions

Relation 1 —

1 + G_{OL}

1 + G_{OL}(s) = 0

(closed-loop characteristic equation)

Write this relation with symbols exactly as in Process Systems Analysis & Control — Coughanowr & LeBlanc before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Relation 2 —

GM = 1/|G_{OL}

GM = 1/|G_{OL}(j\omega_{180})|; PM = 180^{§K1§} + ∠G_{OL}(j\omega_{c})

(stability margins)

Relation 3 —

Routh-Hurwitz \frac{array}{root} locus: all closed-loop poles in the left half-plane

Formulas (Indian textbook notation)

$Routh-Hurwitz \frac{array}{root} locus: all closed-loop poles in the left half-plane$

Concept in depth

Feedback can destabilise: if the loop delays a signal by 180° while its gain still exceeds one, the correction reinforces rather than cancels the error, and oscillations grow. The characteristic equation 1 + G_OL = 0 captures this, and its roots must have negative real parts for decay. The gain margin says how much more gain the loop tolerates before the critical frequency reaches unity gain, and the phase margin says how much extra phase lag it tolerates — both measure distance from the (−1, 0) point on the Nyquist plot. Dead time is especially dangerous because it adds phase lag without reducing gain.

Assumptions and validity limits

State assumptions explicitly before using any relation for stability of process loops — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.

Wrong assumptions invalidate the entire solution even when the formula is correct. In Process Dynamics & Control viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Process Dynamics & Control papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to stability of process loops.
4. Use equation 1:

1 + G_{OL}

.
5. Use equation 2:

GM = 1/|G_{OL}

.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Stability of Process Loops appears in DCS and plant automation. In Indian chemical curricula this topic is tested because it connects theory to dynamic models and loop tuning.

GATE and semester exams often combine stability of process loops with earlier units — revise prerequisites before attempting mixed problems.

Industry interview panels sometimes ask: "Where did you use stability of process loops?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

Students forget to include the measurement and valve dynamics in G_OL, misread gain and phase margins (or their frequencies), and treat a marginally stable loop as safe. Errors in building the Routh array (sign or first-column zeros) are frequent.

Quick revision checklist

Before attempting stability of process loops problems, confirm you can:

Open-loop transfer G_{OL} = G_{c} G_{p} G_{m}

(controller × process × measurement)
2. A wrong controller sign or positive feedback destabilizes the loop
3. Phase margin above ~45° is a common robustness target

Revise the solved examples in Process Systems Analysis & Control — Coughanowr & LeBlanc and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Gain margin from open-loop gain

Problem

At the phase-crossover frequency (phase = −180°), the open-loop magnitude |G_OL| = 0.4. Find the gain margin.

Solution

GM = 1/|G_OL| = 1/0.4 = 2.5 (or 20log₁₀2.5 = 8 dB). The loop gain could rise 2.5× before instability, a modest but positive margin.

Conceptual check — Stability of Process Loops

Problem

In a Process Dynamics & Control semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of stability of process loops." What should a complete answer include?

Exams & GATE

Coughanowr Ch. 11 — sketch the Nyquist encirclement of (−1, 0).

📖 Standard books (India)

Process Systems Analysis & Control — Coughanowr & LeBlanc
Read: Syllabus unit
Dynamic modelling and control loops

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📚Key formulas & points