Mathematics / advanced
Control, estimation, and optimal decisions
Analyze how inputs shape dynamical systems through controllability, observability, feedback, and optimal control.
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Module 01 advanced
Control, estimation, and optimal decisions
Analyze how inputs shape dynamical systems through controllability, observability, feedback, and optimal control.
- 01
State-space models and input–output maps
Separate internal state, forcing and measurement while deriving a causal response from the matrix exponential.
27 min - 02
Controllability and reachability Gramians
Connect algebraic rank with finite-time state transfer and quantify the input energy required in each direction.
31 min - 03
Observability and minimal realizations
Recover internal information from output histories and identify the modes a transfer experiment cannot reveal.
29 min - 04
State feedback and pole placement
Design a closed-loop characteristic polynomial while separating assignable modes, stabilization and implementation limits.
29 min - 05
Lyapunov equations and stability margins
Build an energy certificate for a Hurwitz matrix and quantify a sufficient perturbation bound in that chosen metric.
31 min - 06
Linear-quadratic regulation and Riccati equations
Derive a feedback law from a quadratic value function and select the stabilizing Riccati solution with explicit hypotheses.
34 min - 07
Kalman filtering and estimation error
Combine a dynamical prediction with a noisy observation while distinguishing covariance bookkeeping from actual model accuracy.
32 min - 08
Pontryagin’s maximum principle
Derive state–costate necessary conditions while keeping sign conventions, endpoint rules and normality explicit.
34 min - 09
Hamilton–Jacobi–Bellman equations
Turn remaining optimal cost into a feedback equation and verify a policy by integrating a pointwise inequality.
34 min - 10
Robust control and the small-gain theorem
Bound uncertain feedback amplification using causal gains while retaining well-posedness and internal-stability qualifications.
33 min