Expert-level control theory covering PID control, state-space methods, stability analysis, optimal control, robust control, and nonlinear control.
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Transfer function: Laplace domain representation of linear time-invariant systems. PID controller: proportional, integral, derivative terms, most widely deployed controller. Root locus: how closed-loop poles move as gain varies. Bode plot: frequency response magnitude and phase vs frequency. Gain and phase margin: stability margins from Bode plot.
State equation: x-dot = Ax + Bu, y = Cx + Du. Controllability: can reach any state from any initial state with appropriate input. Observability: can infer state from output measurements. Pole placement: choose control gain K so eigenvalues of A minus BK are desired. Kalman filter: optimal state estimator for linear systems with Gaussian noise.
Lyapunov stability: find positive definite V such that V-dot is negative semi-definite. LaSalle invariance: extends Lyapunov for asymptotic stability proof. Input-output stability: BIBO stability, bounded input gives bounded output. Robust stability: system remains stable under bounded model uncertainty.
LQR: linear quadratic regulator, minimizes quadratic cost on state and input. Pontryagin minimum principle: necessary conditions for optimal control. Dynamic programming: Bellman equation, value function approach. MPC: model predictive control, optimize over receding horizon with constraints.
Feedback linearization: cancel nonlinearities, apply linear control to linearized system. Sliding mode: drive state to manifold, robust to disturbances and uncertainty. Adaptive control: update controller parameters online as system changes. Backstepping: systematic design for cascaded nonlinear systems.
| Pitfall | Fix |
|---|---|
| Integral windup | Implement anti-windup in all PID controllers |
| Ignoring actuator saturation | Include saturation in stability analysis |
| Model mismatch in MPC | Robustify MPC or use adaptive approach |
| Derivative kick on setpoint change | Apply derivative on measurement not error |
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Expert-level abstract algebra knowledge. Use when working with groups, rings, fields, modules, Galois theory, representation theory, or algebraic structures. Also use when the user mentions 'group', 'ring', 'field', 'homomorphism', 'isomorphism', 'normal subgroup', 'quotient group', 'Sylow theorem', 'Galois theory', 'polynomial ring', 'ideal', 'module', 'vector space', or 'representation theory'.
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Expert-level acoustics knowledge. Use when working with sound waves, acoustic propagation, resonance, standing waves, Doppler effect, room acoustics, ultrasound, noise control, musical acoustics, or architectural acoustics. Also use when the user mentions 'sound wave', 'decibel', 'frequency', 'resonance', 'standing wave', 'Doppler effect', 'acoustic impedance', 'reverberation', 'ultrasound', 'noise control', 'harmonics', or 'sound intensity'.
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Expert-level aerodynamics covering subsonic and supersonic flow, lift and drag, airfoil theory, boundary layers, compressible flow, and CFD methods.
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Expert-level air quality engineering covering pollutant sources, atmospheric dispersion, air quality standards, emission controls, and air quality modeling.
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Expert-level aircraft structures covering structural design philosophy, wing and fuselage analysis, fatigue and damage tolerance, composite structures, and certification.
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Expert-level algorithms and complexity theory covering sorting, searching, graph algorithms, dynamic programming, greedy algorithms, NP-completeness, and algorithm design paradigms.
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