5.3 Controls, Transfer Functions, Response, and Feedback

Controls is model recognition

Most FE candidates do not need graduate control theory for this domain — they need to recognize the standard forms fast. A problem may show a transfer function, a block diagram, a step response, a Bode plot, a pole location, or a feedback loop, and ask you to extract one parameter or predict behavior. Your first move is always to name the model.

A transfer function relates output to input in the Laplace domain assuming zero initial conditions: G(s) = Y(s)/U(s). It comes from Laplace-transforming the governing differential equation. The order of the system equals the highest power of s in the denominator (the characteristic polynomial). The poles are the roots of the denominator and govern stability and speed; the zeros are the roots of the numerator and shape the transient. A system is stable when all poles lie in the left half of the s-plane (negative real parts).

The Laplace domain is what makes this tractable: differentiation in time becomes multiplication by s, and integration becomes division by s, so a linear differential equation turns into an algebraic equation in s that you can manipulate, reduce, and read. A few transform pairs recur on the FE — a unit step is 1/s, a unit impulse is 1, and an exponential e^(−at) is 1/(s+a) — and the final value theorem, lim t→∞ y(t) = lim s→0 s·Y(s), lets you read a steady-state value directly from the transform without inverting it.

Recognizing which standard form you are looking at, and which theorem extracts the asked quantity, is most of the battle.

First- and second-order standard forms

Two canonical models cover most FE questions.

First-order: G(s) = K/(τs + 1). The single pole is at s = −1/τ. The step response is exponential, y(t) = K(1 − e^(−t/τ)), reaching 63.2% of final value in one time constant τ, and about 98% in 4τ. Larger τ means a slower system.

Second-order: G(s) = K·ωn² / (s² + 2ζωn·s + ωn²), characterized by natural frequency ωn and damping ratio ζ.

Underdamped systems oscillate at the damped frequency ωd = ωn√(1 − ζ²). Percent overshoot rises as ζ falls. Recognizing ζ from a response plot (overshoot present → underdamped) or from the characteristic equation is a common, fast question.

Block diagrams, feedback, and PID

Reduce a negative-feedback loop with forward path G(s) and feedback path H(s) using the closed-loop formula:

T(s) = G(s) / (1 + G(s)·H(s))

For unity feedback, H = 1, so T(s) = G/(1 + G). Series blocks multiply; parallel blocks add. Feedback's benefits: it reduces sensitivity to plant variation and disturbances and lowers steady-state error. Its risk: too much gain or phase lag erodes gain margin / phase margin and drives the system unstable.

A PID controller sums three actions on the error e(t):

• Proportional (Kp): reacts to present error; raises speed, reduces but does not eliminate steady-state error.
• Integral (Ki): accumulates past error; eliminates steady-state error but adds lag/overshoot.
• Derivative (Kd): anticipates future error from its slope; adds damping, reduces overshoot, sensitive to noise.

Its transfer function is C(s) = Kp + Ki/s + Kd·s. Steady-state error for a step input to a unity-feedback system is e_ss = 1/(1 + Kp_position), and integral action drives that error to zero by adding a pole at the origin (raising the system type). Knowing which term fixes which symptom — offset, sluggishness, or oscillation — answers most FE PID questions.

Stability, poles, and response metrics

Stability is the most-tested controls idea. A linear system is stable only if every pole of its closed-loop transfer function has a negative real part — that is, all poles lie in the left half of the s-plane. A pole on the imaginary axis gives sustained oscillation (marginal stability); any pole in the right half plane gives an unbounded, growing response (unstable). For a characteristic equation you cannot factor quickly, the Routh-Hurwitz criterion tests stability from the coefficients without finding the roots: a necessary condition is that all coefficients be present and of the same sign.

Pole location also predicts speed and shape. The real part sets the decay rate (farther left = faster settling), and the imaginary part sets the oscillation frequency. For an underdamped second-order system, common response metrics are:

Frequency response appears as a Bode plot (magnitude in dB and phase versus log frequency). The gain margin and phase margin measure how much added gain or phase lag the loop tolerates before instability; positive margins mean a stable loop. On the FE you are far more likely to be asked to read a margin or identify a stable pole pattern than to plot a full Bode diagram by hand. The unifying skill is translating between three equivalent pictures — the transfer function, the pole locations, and the time/frequency response — and reading the asked parameter from whichever picture is given.