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.https://www.polistudy.me/enhttps://www.polistudy.me/mida2/prerequisites/https://www.polistudy.me/mida2/prerequisites/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.summation and product notationset notationconditioning barfirst-order logiclinear combinationdynamical systemsampling timedelay operatorlong divisionpolynomial long divisiondivision with remainderimpulse responsewhite noisestochastic processAR(1) processleast squaresprediction errorobjective functiongradientparametric identificationexperiment designwhite-box modellinghttps://www.polistudy.me/mida2/system-representations/https://www.polistudy.me/mida2/system-representations/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.state-space representationtransfer functionimpulse responsetransformation trianglechange of basissimilarity transformationobservabilityreachabilityKalman decompositionminimal realizationcontrollable canonical formHankel matrixpole-zero cancellationhidden modeasymptotic stabilitystrictly properdirect feedthroughhttps://www.polistudy.me/mida2/subspace-identification/https://www.polistudy.me/mida2/subspace-identification/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.4SIDHankel factorizationrank plateausystem order estimationextended observability matrixextended reachability matrixshift invariancesingular value decompositiontruncationleast-squares pseudo-inversen4sididdatapersistent excitationhttps://www.polistudy.me/mida2/frequency-domain-identification/https://www.polistudy.me/mida2/frequency-domain-identification/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.frequency-domain identificationexcitation frequency gridamplitude scheduleactuator power budgetfrequency response theoremlinearity testsinusoid parameterisationnormal equationsfrequency-response pointnon-quadratic performance indexiterative optimisationfrequency weightingclosed-loop identificationNyquist frequencyhttps://www.polistudy.me/mida2/kalman-filter/https://www.polistudy.me/mida2/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.software sensingKalman filterprocess noisemeasurement noisenoise covariancescross-covariancedifference Riccati equationKalman gaininnovationerror covarianceone-step predictorfiltermulti-step predictionexogenous inputcoloured noisemodel extensionshaping filterhttps://www.polistudy.me/mida2/asymptotic-kalman-filter/https://www.polistudy.me/mida2/asymptotic-kalman-filter/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.asymptotic Kalman filteralgebraic Riccati equationsteady-state covarianceasymptotic Kalman gainfilter stabilityasymptotic theoremssufficient conditionobservabilityreachability of the noise channelgraphical DRE analysisMöbius mapfixed pointattracting and repelling equilibriasubsystem decouplingfictitious outputhttps://www.polistudy.me/mida2/blackbox-sw-sensing/https://www.polistudy.me/mida2/blackbox-sw-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.black-box software sensingMIMO LTI mapground-truth datasetFIR architectureIIR architecturestatic nonlinearityteacher forcingopen-loop trainingclosed-loop deploymentrecurrent neural networkmeta-architecturephysics-derived regressorsun-measurable stateshttps://www.polistudy.me/mida2/graybox-identification/https://www.polistudy.me/mida2/graybox-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.extended Kalman filterlinearizationJacobiangray-box identificationstate extensionfictitious parameter noiserandom walktuning knobconvergence speed versus steady-state varianceidentifiabilitysensors-versus-unknowns budgetsimulation-error optimisationhttps://www.polistudy.me/mida2/minimum-variance-control/https://www.polistudy.me/mida2/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.discretizationzero-order holdsampling intervalaliasinghidden zerosnon-minimum-phaseARMAXminimum variance controlMVC assumptionscanonical representationall-pass filterspectral equivalencek-step-ahead predictorpolynomial divisioncontrol lawclosed-loop characteristic polynomialde-biasinggeneralised minimum variance control