math concept
4 topics use this
Math concept
Control Theory
Core equation
$$u(t) = K_p e + K_i \int e\,dt + K_d \dot{e}$$
Control theory designs feedback laws to make dynamical systems behave as desired. PID controllers, state-space methods, and optimal control are its pillars — used in robotics, aerospace, chemical engineering, and climate models.
Transfer functions and the Laplace domain
A linear time-invariant (LTI) system is characterised by its transfer function $G(s) = Y(s)/U(s)$, the ratio of output to input in the Laplace domain. The closed-loop transfer function with controller $C(s)$ is:
\[T(s) = \frac{C(s)G(s)}{1 + C(s)G(s)}\]PID control
The PID controller computes control input from error $e(t) = r(t) - y(t)$:
\[u(t) = K_p e + K_i\int_0^t e\,d\tau + K_d\dot{e}\]- $K_p$ (proportional): reacts to current error
- $K_i$ (integral): eliminates steady-state error
- $K_d$ (derivative): damps oscillations
State-space representation
\[\dot{\mathbf{x}} = A\mathbf{x} + B\mathbf{u}, \qquad \mathbf{y} = C\mathbf{x} + D\mathbf{u}\]Controllability (can we steer to any state?) requires rank of $[B, AB, \ldots, A^{n-1}B]$ to be $n$. Observability (can we infer state from outputs?) requires rank of $[C^\top, A^\top C^\top, \ldots]$ to be $n$.
Fields that use this concept
Engineering & CS
Robotics
Kalman Filter SLAM
Simultaneous localization and mapping using the Extended Kalman Filter to jointly estimate robot pose and landmark positions.
Optimal Control in Robotics
Optimal control finds inputs that minimize a cost functional over a trajectory, from LQR for linear systems to MPC and iLQR for nonlinear robots.
PID Control
The proportional-integral-derivative controller — the most widely deployed feedback control law in engineering.
Engineering & CS
Signal processing