Control Systems

Control systems calculators and formulas for feedback control and PID tuning. Covers proportional, integral, and derivative gain calculations for stable system response.

10Formulas

All levels Difficulty

Updated Oct 2026

← All Robotics & Controls (34) Control Systems (10)

Fundamental

Rise Time (2nd-Order System, Approx.)

t_r = 1.8 / omega_n

Fundamental

Peak Time (2nd-Order System)

t_p = pi / (omega_n * sqrt(1 - zeta^2))

Fundamental

Percent Overshoot (2nd-Order System)

%OS = exp(-zeta*pi / sqrt(1 - zeta^2)) * 100

Fundamental

Settling Time (2nd-Order System)

t_s = 4 / (zeta * omega_n)

Fundamental

Closed-Loop Gain (Negative Feedback)

G_cl = G / (1 + G*H)

Fundamental

Derivative Control Output

u_d = Kd * (de/dt)

Fundamental

Integral Control Output

u_i = Ki * integral(e dt)

Fundamental

Proportional Control Output

u = Kp * e

Fundamental

Closed-Loop Transfer Function

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

Fundamental

PID Controller Output

u(t) = K_p·e(t) + K_i∫e(t)dt + K_d·de/dt