Feedback, Time Response and Stability

Section 5 of the GATE Instrumentation Engineering (IN) paper. Every loop this paper measures and drives — a temperature bath, a flow valve, a servo — is a feedback system, and this section is the mathematics that decides whether the loop settles, oscillates or runs away. Start with the block diagram and the signal flow graph, both ways of writing the same transfer function; then read what a step input reveals about the system in the time domain (how fast, how much overshoot, how much steady-state error) and in the frequency domain (Bode, Nyquist, Routh — three separate routes to the same yes-or-no question of stability); then state-space, the representation that scales to more than one input and output; and finally what actually turns the mathematics into motion — actuators, PID tuning, compensators, and the supervisory layer of DCS and PLC that industrial instrumentation runs on.

1. Feedback and the signal flow graph

A feedback system compares its actual output against a reference and acts on the difference, the error. Negative feedback — subtracting the output from the reference — is what makes a system self-correcting: it trades away some open-loop gain for reduced sensitivity to disturbances and to variation in the plant itself, and this trade is the entire reason feedback control exists rather than open-loop control. A signal flow graph says the same thing a block diagram does, as nodes (signals) and directed, weighted branches (gains) instead of boxes and arrows, and it is solved by Mason's gain formula rather than by block-diagram reduction.

🧠 Mason's gain formula, read in order
Overall gain = (sum over every forward path P_k of P_k * cofactor Delta_k) / Delta, where Delta = 1 - (sum of individual loop gains) + (sum of products of two non-touching loop gains) - (products of three non-touching loops) + ... and Delta_k is Delta with every loop touching path k deleted. The sign pattern alternates; the commonest slip is stopping after the single-loop term when two loops in the graph never share a node.

Two loops are non-touching if they share no common node. This is exactly the condition that lets Delta pick up a second-order term, and forgetting to check it is the single most common arithmetic error in applying the formula — a graph with three loops where only two pairs touch still contributes one non-touching product to Delta.

2. Time response: transient and steady state

A standard second-order system's step response is read off two numbers: the damping ratio zeta and the natural frequency omega_n. Percentage overshoot depends on zeta alone; settling time depends on zeta*omega_n alone (the real part of the poles); and the system is underdamped for 0 < zeta < 1 (oscillatory decay), critically damped at zeta = 1 (fastest response with no overshoot) and overdamped for zeta > 1 (slower, no overshoot, no oscillation).

Steady-state error by system type and input
System type (poles at origin)Step inputRamp inputParabolic input
Type 0finite (1/(1+Kp))infiniteinfinite
Type 1zerofinite (1/Kv)infinite
Type 2zerozerofinite (1/Ka)
🎯 Why the error goes to zero one step late
Each pole the loop gain carries at the origin is one free integration, and an integrator's whole job is to drive its input — here, the error — to zero in steady state against a constant demand. A Type 1 system already integrates once, which is enough to erase a constant (step) error but not a linearly growing (ramp) one; it takes a second integration, Type 2, to erase that.

3. Three routes to the same stability question

  • Routh's criterion works on the closed-loop characteristic equation's coefficients alone — no plot needed — and a sign change in the first column of the array signals a right-half-plane pole for each change.
  • The Nyquist criterion counts encirclements of the point (-1, 0) by the open-loop frequency response as the frequency sweeps the whole imaginary axis, and relates that count to closed-loop right-half-plane poles through the number of open-loop right-half-plane poles — it is the one method that also certifies stability for a system with open-loop poles in the right half-plane.
  • The Bode plot reads stability off the phase and gain margins: gain margin is how much the gain can rise before the phase-crossover point reaches instability, phase margin is how much extra phase lag the system can absorb at the gain-crossover frequency before the same happens — both must be positive for a stable minimum-phase system, and a larger phase margin means a better-damped closed loop.

The root locus is a fourth view of the same question, plotted differently: it traces every closed-loop pole location as one gain parameter K varies from zero to infinity, so a design that must keep zeta above some floor, or settling time below some ceiling, is read directly as a region the locus must stay inside — no frequency sweep, no coefficient array, just the s-plane.

⚠️ Gain margin in dB is not a ratio
Gain margin is usually reported in decibels as -20*log10(|G(jw)|) at the phase-crossover frequency, and a POSITIVE dB value means the system is stable with room to spare. Reading a gain margin of 0 dB as safe, rather than as the boundary of instability, is the standard slip; 0 dB means the loop gain already equals 1 exactly where the phase is -180 degrees.

4. State-space, actuators and industrial control

State-space writes a system as x-dot = Ax + Bu, y = Cx + Du: a first-order vector differential equation plus an output map, in place of one high-order scalar one. Its advantage over a single transfer function is exactly the case a transfer function cannot describe cleanly — multiple inputs and outputs — and it also exposes internal behaviour (like an unstable mode that happens to cancel against a zero and vanish from the transfer function) that a transfer function alone can hide. A time-delay system, e^(-sT) in the loop, is the other case a plain rational transfer function cannot represent exactly; it is handled either by a Pade approximation or by reading the delay's own phase contribution directly on the Bode and Nyquist plots.

The actuator turns a controller's decision into motion or flow: control and servo valves for fluid loops, servo and stepper motors for position and speed loops. On the controller side, the industrial hierarchy runs from on-off (cheapest, always some cycling around setpoint) through P (offset that never fully closes), PI (adds integral action to erase that offset, at some cost to speed), PID (adds derivative action to anticipate and damp, at some cost to noise sensitivity), up to cascade (an inner fast loop corrects a disturbance before it reaches the outer loop), feedforward (acts on a measured disturbance directly, before it shows up as an error) and ratio control (holds one flow at a fixed proportion of another, the standard blending-loop structure).

ℹ️ Lead, lag and the supervisory layer
A lead compensator adds phase near the crossover to speed the response and raise phase margin; a lag compensator sacrifices some phase to boost low-frequency gain and shrink steady-state error; a lead-lag does both, in two separate frequency bands. Above the individual loop, a Distributed Control System coordinates many such loops across a plant and a Programmable Logic Controller handles discrete on/off and sequencing logic — supervisory control is the layer that sets setpoints and coordinates both, rather than closing any loop itself.

Key takeaways

  • Mason's gain formula needs Delta's non-touching-loop terms, not just the single-loop sum — check which loops share a node before writing Delta down.
  • Overshoot depends only on zeta; settling time depends only on zeta*omega_n — read the two separately, never swap them.
  • One free integrator per system type erases one polynomial order of steady-state error: Type 1 kills a step error, Type 2 also kills a ramp error.
  • Routh needs only the characteristic polynomial's coefficients; Nyquist alone certifies stability when the open loop itself has right-half-plane poles; Bode reads it off two margins that must both be positive.
  • PID adds derivative action to PI for anticipation, at the cost of noise sensitivity; cascade, feedforward and ratio are structures above single-loop PID, not replacements for it.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. In Mason's gain formula, two loops of a signal flow graph are called non-touching when they

    1. share no common node
    2. have equal loop gain
    3. lie on the same forward path
    4. both pass through the output node
    Show answer

    Answer: A — share no common node

    Non-touching means no shared node, and it is exactly this condition that lets Delta pick up a product-of-two-loop-gains term. Equal gain, sharing a forward path or both touching the output are unrelated properties and none of them determines whether the loops are touching.

  2. A unity-feedback system has open-loop transfer function K/(s(s+2)). Its steady-state error to a unit ramp input is

    1. zero
    2. 2/K
    3. infinite
    4. 1/K
    Show answer

    Answer: B — 2/K

    One pole at the origin makes this Type 1, so the ramp error is finite and equal to 1/Kv where Kv = lim s*G(s) as s tends to 0 = K/2. So the error is 1/(K/2) = 2/K. Answering zero confuses this with a Type 2 system's ramp response, and answering infinite confuses it with a Type 0 system's.

  3. A second-order system has damping ratio zeta = 1. Its step response is

    1. oscillatory with decaying amplitude
    2. critically damped, fastest non-oscillatory response
    3. overdamped and slower than critical damping
    4. undamped, sustained oscillation
    Show answer

    Answer: B — critically damped, fastest non-oscillatory response

    zeta = 1 is exactly the critically damped case: the fastest response that still has no overshoot. zeta between 0 and 1 is underdamped (oscillatory), zeta greater than 1 is overdamped (slower still), and zeta = 0 is undamped.

  4. The Nyquist stability criterion is preferred over the Bode plot specifically when

    1. the system has no poles in the right half-plane
    2. the open-loop system itself has poles in the right half-plane
    3. only gain margin is required, not phase margin
    4. the system is a pure time delay with no poles at all
    Show answer

    Answer: B — the open-loop system itself has poles in the right half-plane

    The Nyquist criterion relates encirclements of -1 to closed-loop right-half-plane poles through the KNOWN number of open-loop right-half-plane poles, so it remains valid and answers the question even when the open loop is itself unstable — a case an ordinary Bode-margin reading does not cleanly handle.

  5. Which of the following are true of gain margin and phase margin?

    1. Both must be positive for a stable minimum-phase closed loop.
    2. 0 dB gain margin means the system is safely stable.
    3. A larger phase margin generally means a better-damped closed loop.
    4. Gain margin is read at the gain-crossover frequency.
    Show answer

    Answer: A — Both must be positive for a stable minimum-phase closed loop.; C — A larger phase margin generally means a better-damped closed loop.

    Both margins positive is the stability condition for a minimum-phase system, and a larger phase margin does correlate with better damping. 0 dB gain margin is the boundary of instability, not safety. And gain margin is read at the PHASE-crossover frequency (where phase is -180 degrees); phase margin is read at the gain-crossover frequency (where gain is 0 dB) — swapping the two crossover points is the usual error.

  6. The root locus of a system plots, as gain K varies from 0 to infinity, the

    1. open-loop zero locations only
    2. closed-loop pole locations
    3. phase margin as a function of frequency
    4. steady-state error as a function of K
    Show answer

    Answer: B — closed-loop pole locations

    The root locus traces where the closed-loop poles sit in the s-plane as K sweeps from zero (where they start at the open-loop poles) to infinity (where they end at the open-loop zeros or go to infinity). It says nothing directly about frequency response margins or steady-state error, though a designer reads both off the locus indirectly.

  7. Compared to a transfer function, the state-space representation's principal advantage is that it

    1. always gives a lower-order model
    2. extends naturally to multiple inputs and outputs and exposes internal states
    3. eliminates the need for a stability criterion
    4. applies only to time-invariant single-input systems
    Show answer

    Answer: B — extends naturally to multiple inputs and outputs and exposes internal states

    State-space handles multiple inputs and outputs naturally, which a single scalar transfer function cannot, and it can reveal an internal unstable or unobservable mode that happens to cancel against a zero and so is invisible in the transfer function. It does not shrink model order by itself, does not remove the need to check stability, and is not restricted to single-input systems — the opposite is true.

  8. Adding derivative action to a PI controller, making it PID, primarily

    1. removes steady-state error entirely by itself
    2. anticipates the rate of change of error, improving damping, at the cost of noise sensitivity
    3. converts the loop from Type 1 to Type 2
    4. is only usable in cascade control structures
    Show answer

    Answer: B — anticipates the rate of change of error, improving damping, at the cost of noise sensitivity

    Derivative action acts on the rate of change of the error, giving an anticipatory, damping effect that reduces overshoot — its price is amplifying high-frequency measurement noise, which is why it is usually filtered in practice. Integral action, not derivative, is what removes steady-state error and adds a system type; PID is used in single loops as freely as in cascade structures.

  9. A lead compensator is added to a control loop chiefly to

    1. boost low-frequency gain and shrink steady-state error
    2. add phase near crossover, speeding response and raising phase margin
    3. attenuate all frequencies equally
    4. replace the need for a PID controller entirely
    Show answer

    Answer: B — add phase near crossover, speeding response and raising phase margin

    A lead compensator's job is to add positive phase in the crossover region, which speeds the transient response and increases phase margin. Boosting low-frequency gain to shrink steady-state error is the LAG compensator's job, the opposite tradeoff; a lead-lag combines both in separate bands, and neither replaces PID outright.

  10. In the industrial control hierarchy, a Programmable Logic Controller (PLC) is chiefly used for

    1. continuous PID loop tuning
    2. discrete on/off and sequencing logic
    3. generating the Bode plot of a plant
    4. computing the root locus in real time
    Show answer

    Answer: B — discrete on/off and sequencing logic

    A PLC's traditional strength is discrete, sequential and on/off logic — interlocks, sequencing, batch steps — while a DCS is built around continuous control of many analog loops; the line has blurred with modern hardware, but the syllabus distinction is exactly this discrete-versus-continuous split. Plotting a Bode diagram or root locus is off-line analysis work, not a plant-floor controller's job.