Instrumentation and Control
Instrumentation and Control tests sensors and actuators, Laplace-domain models, PID control, stability, and closed-loop performance.
Core rules and concepts
- A first-order system G=K/(tau s+1) reaches 63.2% of its final value at one time constant tau; a second-order system is characterized by natural frequency omega_n and damping ratio zeta.
- PID output u=K_p e+K_i integral e dt+K_d de/dt; integral action removes steady-state error, while derivative action adds damping but amplifies noise.
- Closed-loop transfer function for unity feedback is T=G/(1+G); stability requires all poles in the left half-plane.
- Routh-Hurwitz gives stability from the characteristic polynomial; Bode plots give gain and phase margins.
- The final-value theorem gives steady-state error for stable systems, but only for inputs whose transforms satisfy its conditions.
Worked example
For plant G=10/(s+2) with unity feedback, T=G/(1+G)=10/(s+2+10)=10/(s+12). For a unit step input, steady-state output is T(0)=10/12=0.833. Steady-state error is 1-0.833=0.167, or 16.7%. A proportional gain increase would reduce this error but could reduce stability margin. Integral action would drive the error to zero for a step if the closed loop remains stable.
Traps
Using positive feedback by mistake, applying the final-value theorem to an unstable system, ignoring actuator saturation, and differentiating a noisy measurement without filtering are common. Check units and whether error is defined as r-y or y-r.