Measurements, Instrumentation, and Controls

Transfer functions, poles, and zeros

Open-loop vs closed-loop

The characteristic equation and stability

PID control action

Steady-state error and system type

Relative stability: gain and phase margins

Exam strategy

Key equations

Worked examples

Common pitfalls

• •Confusing the open-loop product with the closed-loop ratio: stability is decided by $1+G_1G_2H=0$ (the open-loop product set to $-1$), but the response and DC gain come from the closed-loop ratio $G_1G_2/(1+G_1G_2H)$. Use the right object for the question.
• •Counting all poles as the system type. The type is only the number of open-loop poles at the origin (pure $1/s$ integrators). $G=K/[(s+1)(s+5)]$ is type 0, not type 2.
• •Putting the $+1$ in the wrong error formula. Step error is $1/(1+K_p)$; ramp error is $1/K_v$
• •Assuming all-positive coefficients guarantees stability. For order $\ge3$ that is necessary but not sufficient — apply Routh-Hurwitz (e.g. $a_2a_1>a_0$
• •Sign at the summing junction: negative feedback yields the $1+G_{\text{loop}}$ denominator; a positive-feedback sign error gives $1-G_{\text{loop}}$ and a spurious unstable pole.
• •Applying the Final Value Theorem to an unstable or marginally stable loop. It is valid only when every closed-loop pole has a negative real part; otherwise the limit is meaningless.
• •Mixing degrees and radians, or sign, in the margins: phase margin is referenced to $-180^\circ$ and the gain margin in dB is positive for a stable design.

References

• NCEES FE Reference Handbook — Instrumentation, Measurement, and Control (Control Systems)
• Nise, Control Systems Engineering — Block-diagram reduction, system type/error constants, Routh-Hurwitz, gain and phase margins.
• Ogata, Modern Control Engineering — Transfer functions, closed-loop poles, and stability analysis.