Control + estimation¶

What you will develop¶

Design closed loops; analyze stability and robustness; infer hidden state; reason about controllability, observability and tradeoffs.

Before you begin¶

Recorded prerequisites:

Study and practice¶

Progress by the work you can do, not by a fixed calendar.

Practice sequence 41¶

Spine: Åström & Murray, Feedback Systems, 2e

Reading: Ch. 1 Introduction; Ch. 2 Feedback Principles

Know: Understand feedback/feedforward architectures, disturbance rejection, tracking and fundamental tradeoffs.

Reconstruct: Derive proportional-feedback closed-loop transfer on a simple linear plant.

Do: Control a simulated thermal chamber under disturbances using open loop, feedforward and feedback; compare.

Defend: Why can negative feedback destabilize a system?

Gate: draw loop, label signals/units, derive closed-loop relation from first principles.

Source: source

Practice sequence 42¶

Spine: Åström & Murray, Feedback Systems, 2e

Reading: Ch. 3 examples; Ch. 4 dynamic behavior / state-space core

Know: Build state-space models; analyze equilibria, linearization, modes and response.

Reconstruct: Derive state transition for linear system and local linearization around an equilibrium.

Do: Identify a low-order state model from simulated step-response data and compare prediction to held-out input.

Defend: What state variables are physically meaningful vs merely sufficient coordinates?

Gate: model a novel physical system from diagram to state equations.

Source: source

Practice sequence 43¶

Spine: Åström & Murray

Reading: State feedback / reachability / observability chapters and exercises

Know: Reason about controllability, observability and state feedback; understand unreachable/unseen modes.

Reconstruct: Derive controllability and observability matrices for a small LTI system and interpret rank.

Do: Design actuator/sensor placement for a toy spacecraft or bioreactor to recover lost controllability/observability.

Defend: Can a system be perfectly stable yet uncontrollable in the way you care about?

Gate: diagnose an unseen system's actuator/sensor limitations before designing gains.

Source: source

Practice sequence 44¶

Spine: Åström & Murray

Reading: Frequency response, loop analysis, robustness and design tradeoffs chapters

Know: Use frequency-domain reasoning for bandwidth, margins, disturbance/noise rejection and robustness.

Reconstruct: Derive sinusoidal steady-state response and explain gain/phase margins conceptually.

Do: Tune a controller where increasing bandwidth improves tracking but amplifies sensor noise and excites unmodeled dynamics.

Defend: What does robustness mean when the model class itself is wrong?

Gate: Bode/response diagnosis with one intentionally hidden unmodeled pole.

Source: source

Practice sequence 45¶

Spine: Åström & Murray + estimation synthesis

Reading: Implementation + state estimation/Kalman-filter supplement

Know: Estimate hidden state from noisy measurements; integrate model, sensor and controller into an end-to-end loop.

Reconstruct: Derive scalar Kalman update as precision-weighted fusion; connect predict/update to Bayes.

Do: Build an inverted-pendulum/cart-pole or equivalent simulated system with noisy sensors, state estimator and stabilizing controller.

Defend: Why is a controller only as good as the state information it can infer?

Gate: Module defense: 90 minutes, unknown plant -> model -> observability -> estimator -> controller -> robustness tests.

Source: source

Exit gate¶

Closed-book: 150 min: derive closed loop; state model; controllability/observability; frequency tradeoff; scalar Kalman update.

Novel problem: Unknown plant: propose model, sensors, actuators, estimator and feedback architecture before tuning.

Artifact: Simulated closed-loop system under disturbance, noise, saturation, delay and model mismatch.

Defend: Defend stability, robustness, bandwidth, sensor information and actuator limits.

Pass criterion: Pass if stable nominal performance survives at least three adversarial perturbations or failures are correctly predicted.

Transfer problems¶

Try these before consulting solutions or asking for the complete answer.

Closed loop: Derive closed-loop transfer for proportional control and identify disturbance/noise paths.
Stability: Find gain range that stabilizes a simple plant and show one destabilizing delay.
State model: Derive state equations from a mechanical/electrical diagram.
Controllability: Identify unreachable modes and propose actuator change.
Observability: Identify hidden modes and propose sensor change.
Estimator: Fuse model prediction and noisy measurement using scalar Kalman logic.
Bandwidth: Show tracking vs noise-rejection tradeoff as controller bandwidth changes.
Saturation: Demonstrate integrator windup or saturation failure and mitigation.
Robustness: Perturb plant parameters/unmodeled poles and map stability/performance degradation.
Integrated: Unknown plant: choose sensors/actuators, identify model, estimate state, stabilize and test disturbances.