Control theory Study Guide

📖 Core Concepts Control Theory – Discipline that designs algorithms to drive dynamical systems to desired states while minimizing delay, overshoot, steady‑state error, and ensuring stability. Error Signal – Difference between the measured process variable and the reference (set‑point). Open‑Loop (Feedforward) – Control action does not use the measured output; relies only on a predefined input‑output relationship. Closed‑Loop (Feedback) – Control action depends on the measured output; the error signal is fed back to the controller. Transfer Function – Ratio \(G(s)=\frac{Y(s)}{U(s)}\) that relates Laplace‑domain input \(U(s)\) to output \(Y(s)\) for linear time‑invariant (LTI) systems. State‑Space Model – Vector‑matrix representation \(\dot{x}=Ax+Bu,\; y=Cx+Du\) that captures all internal states needed to predict future behavior. Stability (Pole Location) – Continuous‑time LTI system is asymptotically stable if all poles satisfy \(\operatorname{Re}(p)<0\); discrete‑time if \(|p|<1\). Controllability / Observability – Ability to drive the state to any desired point (controllability) and to reconstruct the state from outputs (observability). 📌 Must Remember Closed‑Loop Advantages – disturbance rejection, reduced sensitivity to parameter variations, stabilization of unstable processes, better reference tracking. PID Controller – \(u(t)=KP e(t)+KI\int e(t)dt+KD\frac{de(t)}{dt}\); most common closed‑loop architecture. BIBO Stability – Bounded input → bounded output for linear systems. Marginal Stability – Poles on the imaginary axis (continuous) or on the unit circle (discrete) → sustained oscillations, no growth/decay. Frequency‑Domain Specs – Gain margin > 6 dB, phase margin > 45°, bandwidth wide enough for desired speed. Time‑Domain Specs – Rise time, peak overshoot, settling time, percent overshoot (often derived from dominant pole locations). Routh‑Hurwitz Criterion – Algebraic test for locating poles in the left half‑plane without solving for them explicitly. Robustness – Performance remains acceptable under plant‑model mismatch or parameter variations. 🔄 Key Processes Model Identification (Offline) Collect input–output data → fit transfer function or state‑space matrices → validate with test data. Controller Design (Classical SISO) Define performance specs → select controller type (e.g., PID, lead‑lag) → draw root‑locus/Bode → adjust gain/phase → verify margins. State‑Space Pole Placement Choose desired pole locations → compute feedback matrix \(K\) such that eigenvalues of \((A-BK)\) match targets. Stability Assessment Compute poles (transfer function) or apply Nyquist/Bode → check \(\operatorname{Re}(p)<0\) (continuous) or \(|p|<1\) (discrete). Robust/Adaptive Loop Implement online parameter identification → update controller gains in real time → apply anti‑wind‑up if actuator limits are reached. 🔍 Key Comparisons Open‑Loop vs Closed‑Loop Open‑Loop: No feedback, simple, cannot reject disturbances. Closed‑Loop: Uses feedback, rejects disturbances, stabilizes unstable plants. Linear vs Nonlinear Systems Linear: Superposition holds; frequency‑domain tools (Bode, Nyquist) apply. Nonlinear: No superposition; require Lyapunov, limit‑cycle, numerical simulation. Frequency‑Domain vs Time‑Domain Analysis Frequency‑Domain: Quick algebra for LTI, gives gain/phase margins, but cannot handle nonlinearities. Time‑Domain (State‑Space): Handles both linear and nonlinear dynamics; suited for MIMO and optimal control. PID vs Feedforward‑plus‑Feedback PID: Pure feedback; simple tuning, good for many SISO plants. Feedforward + Feedback: Adds a predetermined control action to improve tracking when the disturbance is measurable. ⚠️ Common Misunderstandings “All feedback is good.” – Poorly designed feedback can amplify noise or cause instability (wrong gain/phase). “If poles are negative, the system is fast enough.” – Speed also depends on pole magnitude; very fast poles may cause actuator saturation. “BIBO stability ⇒ internal stability.” – BIBO is a property of linear systems; a system can be BIBO stable yet have hidden internal unstable modes if non‑minimum phase. “PID always eliminates steady‑state error.” – Only the integral term guarantees zero steady‑state error for step inputs; improper tuning can lead to wind‑up. 🧠 Mental Models / Intuition Feedback as a “self‑correcting” loop: Think of a driver constantly adjusting the steering wheel based on the car’s deviation from the lane center. Pole placement = “setting the natural frequencies” of the system; moving poles leftward makes the response faster and more damped. Bode plot margins as “safety buffers”: Gain margin = how much you can increase gain before instability; phase margin = how much phase lag you can add. Controllability ⇔ “having enough muscles” to move the body anywhere; observability ⇔ “having enough eyes” to know where you are. 🚩 Exceptions & Edge Cases Marginally Stable Poles on Imaginary Axis – Acceptable only if they are simple (non‑repeated); otherwise any disturbance leads to unbounded growth. Nonminimum‑Phase Zeros – Can cause inverse response; limit achievable performance despite stable poles. Discrete‑Time Stability – Poles inside unit circle; a pole at \(-1\) (continuous \(-\infty\) rad/s) is stable but can cause sign‑alternating response. Actuator Saturation – Even a perfectly designed controller can be rendered ineffective if the control signal exceeds physical limits; use anti‑wind‑up or MPC. 📍 When to Use Which PID – Simple SISO plants with well‑behaved dynamics, where fast implementation is needed. State‑Space Pole Placement – MIMO or SISO systems where full state feedback is available and precise pole locations are required. Root‑Locus / Bode Design – Linear, time‑invariant plants with dominant low‑order dynamics; quick visual tuning. Lyapunov / Sliding‑Mode – Strongly nonlinear or uncertain systems where a formal stability proof is needed. Model Predictive Control (MPC) – Systems with hard constraints on inputs/outputs or multi‑step performance criteria. Robust \(H{\infty}\) Loop‑Shaping – When model uncertainty is significant and a guaranteed margin is required. Adaptive Control – Plant parameters change over time (e.g., payload variations, aging components). 👀 Patterns to Recognize Dominant Pole Approximation – A single pair of poles near the imaginary axis dominates overshoot and settling time. Zero‑Pole Cancellation – Appears as a flat spot in Bode magnitude; often non‑robust if the zero is not exactly matched. Phase Lag at Low Frequencies – Indicates integrator action → good for steady‑state error elimination. Oscillatory Closed‑Loop Response – Look for poles with small negative real parts and non‑zero imaginary parts. High Gain → Reduced Steady‑State Error but Lower Phase Margin – Classic trade‑off visible in Bode plots. 🗂️ Exam Traps “Any negative real pole guarantees fast response.” – Speed also depends on the pole’s distance from the origin; a pole at \(-0.1\) is stable but very slow. Confusing BIBO stability with internal stability – BIBO applies only to linear systems; a nonlinear system can be BIBO‑stable yet have hidden limit cycles. Assuming a PID controller eliminates all steady‑state error regardless of plant type – Pure proportional control cannot; integral term is essential for step disturbances. Choosing feedforward only because the disturbance is measurable – Without feedback, unmodeled dynamics will still cause error. Misreading the Nyquist plot “encirclement count.” – Remember the rule: \(N = Z - P\) (encirclements = poles – zeros of open‑loop inside the contour). --- This guide condenses the most exam‑relevant concepts from the outline. Use it for rapid review, then dive into the detailed sections for deeper understanding.