4.3 Control Systems

Open-loop versus closed-loop

The Control Systems area is one of the larger Electrical-and-Computer topics at 6–9 questions. Start every problem by classifying the architecture. Open-loop control issues a command and trusts it — a toaster timer runs regardless of how brown the bread gets. Closed-loop (feedback) control measures the output, compares it to a reference at a summing junction to form an error signal e = r − (measured output), and adjusts the command to drive the error toward zero.

Feedback's benefits are the usual exam talking points: it reduces sensitivity to disturbances, improves tracking of the reference, and lowers sensitivity to plant variation. Its cost is the risk of instability if loop gain and phase are mismanaged — more gain is not automatically better, and excess gain or phase lag can push closed-loop poles into the right half-plane.

The canonical block diagram is: a reference r enters a summing junction, the error e feeds a controller and plant (forward path G), the output y is sensed by a feedback element H, and the sensed signal is subtracted at the summer. The loop gain is the product around the loop, G(s)H(s); both its magnitude and phase determine whether the feedback is corrective or, at the wrong frequency, regenerative. Recognizing this structure and which block is which is the first step in nearly every control item.

Block-diagram reduction

Most control items reduce to combining blocks into one transfer function. For a unity negative-feedback loop with forward-path gain G(s):

T(s) = G(s) / (1 + G(s))

For non-unity feedback with a sensor/feedback block H(s):

T(s) = G(s) / (1 + G(s)H(s))

The denominator set to zero, 1 + G(s)H(s) = 0, is the characteristic equation; its roots are the closed-loop poles. Sign matters: positive feedback gives 1 − G(s)H(s) and destabilizes quickly. Series blocks multiply (G1·G2); parallel blocks add (G1 + G2).

Stability and Routh–Hurwitz

A closed-loop system is stable when all roots of the characteristic equation lie in the left half-plane. The Routh–Hurwitz criterion lets you decide this without factoring the polynomial: build the Routh array from the characteristic-equation coefficients, and the number of sign changes in the first column equals the number of right-half-plane poles. No sign changes means a stable system. A fast necessary (but not sufficient) check: every coefficient of the characteristic polynomial must be present and have the same sign — a missing or negative coefficient guarantees instability.

Worked Routh check: For characteristic equation s³ + 6s² + 11s + 6 = 0, all coefficients are present and positive. Building the first column of the Routh array gives 1, 6, (6·11 − 1·6)/6 = 10, and 6 — all positive, no sign changes, so the system is stable (its poles are in fact s = −1, −2, −3).

PID control

The PID controller is the workhorse and a frequent FE topic. Its output is the sum of three actions on the error e(t):

u(t) = Kp·e(t) + Ki·∫e(t)dt + Kd·de(t)/dt

• Proportional (Kp): responds to present error; raising it speeds response but can increase overshoot and leave a steady-state offset.
• Integral (Ki): accumulates past error and drives steady-state error to zero, but can add overshoot and slow settling.
• Derivative (Kd): anticipates the error trend, improves damping and transient response, but amplifies noise.

Steady-state error and the final value theorem

Steady-state error e_ss is the leftover difference between reference and output as t → ∞. It depends on the input type (step, ramp) and the system type — the number of pure integrators (poles at the origin) in the open loop. A type-1 system (one integrator) tracks a step with zero error; integral control adds an integrator to kill step error. The final value theorem, e_ss = lim_{s→0} s·E(s), computes it directly (valid only if the closed loop is stable).

Worked final-value example: A unity-feedback system has open loop G(s) = 20/(s + 4) and a unit-step input R(s) = 1/s. The error is E(s) = R(s)/(1 + G(s)) = (1/s)/(1 + 20/(s+4)). Then e_ss = lim_{s→0} s·E(s) = lim_{s→0} 1/(1 + 20/(s+4)) = 1/(1 + 20/4) = 1/6 ≈ 0.167. Because G has no integrator (type 0), a finite step error remains; adding integral action would make it zero.

Relative stability — gain and phase margins

Relative stability is read from the Bode plot. The gain margin is how much loop gain can increase before instability, measured at the phase-crossover frequency (where phase = −180°). , 0 dB). Positive margins indicate a stable closed loop, larger margins mean more robustness, and a common design target is a phase margin of roughly 45° to 60°. Negative margins mean the closed loop is unstable.

For a quick second-order intuition, a larger damping ratio ζ corresponds to a larger phase margin and less overshoot, tying the time-domain and frequency-domain views together — recognizing that link lets you answer many control items without a full calculation.