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, 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.

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.

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 necessary (but not sufficient) quick check: every coefficient of the characteristic polynomial must be present and have the same sign — a missing or negative coefficient guarantees instability.

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 steady-state error.
• Integral (Ki): accumulates past error and drives steady-state error to zero, but can add overshoot and slow settling.
• Derivative (Kd): anticipates error trend, improves damping and transient response, but amplifies noise.

Steady-state error and margins

Steady-state error 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.

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°). The phase margin is the extra phase lag tolerable at the gain-crossover frequency (where |loop gain| = 1, i.e., 0 dB). Positive margins indicate a stable closed loop, and larger margins mean more robustness; a common design target is a phase margin of roughly 45° to 60°.