Control Theory Fundamentals

Mathematical foundations and derivations behind RoboControl algorithms.

PID Control Theory

The proportional-integral-derivative controller is fundamental to robotics.

Continuous Time Formulation

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

Discrete Implementation

Trapezoidal integration for accurate discrete approximation.

State Space Representation

Modern control approach using matrix notation.

$$\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}$$ $$\mathbf{y} = \mathbf{C}\mathbf{x} + \mathbf{D}\mathbf{u}$$

Where x is the state vector, u is the input vector, y is the output vector, and A, B, C, D are system matrices.

Stability Analysis

Ensure your control system converges to desired states.

Lyapunov Stability

System is asymptotically stable if $V(\mathbf{x}) > 0$ and $\frac{dV}{dt} < 0$.

Pole Placement

Design feedback gains to place closed-loop poles in desired locations.

Frequency Domain Analysis

Bode plots and frequency response for robustness evaluation.

Stability Margins

• Gain Margin: > 3 dB (typical)
• Phase Margin: > 45° (typical)
• Bandwidth: > 2x fastest state frequency

Kalman Filtering

Optimal sensor fusion for state estimation.

Predict:
$$\hat{\mathbf{x}}_{k|k-1} = \mathbf{A}\hat{\mathbf{x}}_{k-1|k-1} + \mathbf{B}\mathbf{u}_{k-1}$$ $$\mathbf{P}_{k|k-1} = \mathbf{A}\mathbf{P}_{k-1|k-1}\mathbf{A}^T + \mathbf{Q}$$

Update:
$$\mathbf{K}_k = \mathbf{P}_{k|k-1}\mathbf{C}^T (\mathbf{C}\mathbf{P}_{k|k-1}\mathbf{C}^T + \mathbf{R})^{-1}$$ $$\hat{\mathbf{x}}_{k|k} = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k(\mathbf{z}_k - \mathbf{C}\hat{\mathbf{x}}_{k|k-1})$$ $$\mathbf{P}_{k|k} = (\mathbf{I} - \mathbf{K}_k\mathbf{C})\mathbf{P}_{k|k-1}$$