Polimi · 2025/26
Model Identification and Data Analysis — Module 2
From state-space models and their impulse response through subspace (4SID) identification, the Kalman filter's Riccati recursion and its asymptotic gain to minimum variance control of an ARMAX plant — the second module of the Politecnico di Milano Model Identification and Data Analysis course, rebuilt as an interactive, exam-focused study guide.
9 chapters~5 h reading 76 past-exam questions
After this course you can
- Convert a discrete-time linear system between its state-space, transfer-function and impulse-response forms, and identify which internal modes the input/output view cannot see.
- Identify a state-space model from measured data with the 4SID algorithm — choosing the system order from the Hankel rank — and reproduce the same identification in MATLAB.
- Estimate a plant's frequency response from single-sinusoid experiments, designing the frequency grid and amplitude schedule that make the estimate reliable.
- Build a Kalman filter from a given model, solve its Riccati recursion, and determine graphically whether a steady-state solution exists when the asymptotic theorems do not apply.
- Estimate un-measured states and unknown physical parameters, choosing between a model-based filter and a black-box software sensor on the trade-offs that separate them.
- Design a minimum variance controller for an ARMAX plant, verify every assumption it rests on, repair those that fail, and recognise the case where no repair is possible.
Syllabus
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Prerequisites: Notation, Signals & the Algebra of the Delay Operator
What MIDA2 reads, writes and computes with from its first slide and never stops to teach — summation, set and quantifier notation and the conditioning bar; the plant, its sampling interval and the linear combinations that define it; polynomial long division in the delay operator; white noise, stochastic processes and AR(1); least squares, the prediction error and argmin; and white-box modelling. Twelve of the seventeen are MIDA1's, and the other five are taught by nobody.
50 minutes reading. -
Three Representations of a Discrete-Time System
A discrete-time linear system can be written three ways — state space, transfer function, impulse response — and the six conversions between them are not equally honest. Two of them lose information, and what they lose is exactly the modes that observability and reachability tests detect: the hidden, sometimes unstable, states the input/output view cannot see.
high exam weight. 27 minutes reading. 13 past-exam questions. -
Subspace Identification — the 4SID Algorithm
Given nothing but a measured impulse response, recover a state-space model. The Hankel factorisation makes this constructive rather than an optimisation: grow Hankel matrices until the rank stops climbing, split the result into observability and reachability factors, and read the matrices off. Then do it again with noise, where the rank test collapses and the SVD takes over.
high exam weight. 35 minutes reading. 15 past-exam questions. -
Identification in the Frequency Domain
The other black-box route. Instead of one experiment and a Hankel matrix, run H single-sinusoid experiments and turn each into one point of the frequency response, then fit a transfer function to those points. The dataset is manufactured rather than collected — and the payoff is a model whose error is small exactly where a control design will look.
medium exam weight. 26 minutes reading. 2 past-exam questions. -
The Kalman Filter
A state you cannot measure, recovered from one you can. The Kalman filter runs a replica of the plant alongside the real thing and corrects it with the output error, using a gain that is not tuned by hand but computed in closed form from a Riccati recursion. That recursion is autonomous — it runs offline, before any measurement arrives — which means the accuracy of a software sensor can be quoted before the sensor exists.
high exam weight. 34 minutes reading. 10 past-exam questions. -
Steady State, the ARE, and When the Theorems Fail
The time-varying Kalman gain is awkward to certify and expensive to run, so in practice everyone uses its limit. Finding that limit means solving the Riccati equation at equilibrium, choosing between its roots, and proving the recursion actually gets there. Two theorems settle all of that at a glance — and the exam has spent nine sittings asking about systems where they do not apply.
high exam weight. 42 minutes reading. 13 past-exam questions. -
Black-Box Software Sensing
Seen from outside, a Kalman filter is nothing but a map from the measured signals to the unmeasured state. That map can be identified from data instead of constructed from a model — which trades away the need for physics and buys, in return, a requirement that is easy to miss: during training you must own the very sensor you intend to delete.
low exam weight. 23 minutes reading. 2 past-exam questions. -
The Extended Kalman Filter and Gray-Box Identification
Two ideas that turn out to be one. Linearising a nonlinear model at every step gives the extended Kalman filter; promoting an unknown physical parameter to a state lets the same filter identify it. The second trick forces the first — appending a parameter to the state multiplies two unknowns together, so a linear plant stops being linear the moment you try this.
low exam weight. 29 minutes reading. 2 past-exam questions. -
Minimum Variance Control
The course's last task, and its heaviest exam topic. Given an ARMAX plant, the controller that minimises the variance of the tracking error is a short formula built from one polynomial division — but it only exists if seven hypotheses hold, and half the marks in the archive are awarded for checking them, repairing the ones that fail, and knowing which failure cannot be repaired at all.
high exam weight. 52 minutes reading. 19 past-exam questions.