Process Control Quiz

General Form:

$$G(s) = \frac{Y(s)}{U(s)} = \frac{b_ms^m + \cdots + b_1s + b_0}{a_ns^n + \cdots + a_1s + a_0}$$

First Order System:

$$G(s) = \frac{K}{\tau s + 1}$$

Second Order System:

$$G(s) = \frac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}$$

Rise Time (10-90%):

$$t_r \approx \frac{2.2}{\omega_n}$$ for 0.3 < ζ < 0.8

Peak Time:

$$t_p = \frac{\pi}{\omega_d}$$ where ω_d = ω_n√(1-ζ²)

Percent Overshoot:

$$\%OS = 100e^{-\zeta\pi/\sqrt{1-\zeta^2}}$$

Settling Time (2%):

$$t_s \approx \frac{4}{\zeta\omega_n}$$

PID Equation:

$$u(t) = K_p e(t) + K_i \int_0^t e(\tau) d\tau + K_d \frac{de(t)}{dt}$$

Ziegler-Nichols Tuning (Open-Loop):

Routh-Hurwitz Criterion:

For a polynomial: a_nsⁿ + a_n-1s^n-1 + ... + a₀

Number of roots with positive real parts = number of sign changes in first column of Routh array

Nyquist Criterion:

Z = N + P

Z = closed-loop poles in RHP

N = net encirclements of -1 point

P = open-loop poles in RHP