MathMods :: Joint MSc

Additional Info

• ECTS credits: 6
• University: University of L'Aquila
• Semester: 1
• Lecturer 1: Alessandro D'Innocenzo
• Objectives:

   

  System Identification and data analysis module
  a) The objective of this module is to initiate the students to the study of data analysis and stochastic estimation theory, with focus on dynamical system identification and state estimation by filtering theory. After the completion of this course a student will be able to formulate and analyze problems of estimation and identification of dynamical models from noisy measurements, proposing various possible solutions and defining their statistical properties. Moreover, the student will be able to address problems in data analysis and compression, and in pattern recognition. The notions acquired in this course will increase the student's capability of modeling, simulation, control design and data analysis.
  b) At the end of this course the student:
  - will know methods and fundamental results of stochastic estimation theory;
  - will know the main methodologies of dynamical system estimation with noisy measurements;
  - will have knowledge of main state estimation and filtering techniques for linear and nonlinear systems;
  - will be able to write simulation programs to evaluate the accuracy of models estimated from noisy measurement of a dynamical system;
  - will be able to write simulation programs to evaluate the accuracy of dynamical system state estimation, and to estimate a model for a dynamical system from input-output data.
  - will be able to evaluate which estimation technique is more suitable for a given problem in the field of stochastic system estimation;
  - will be able to read and understand advanced scientific textbooks and articles on the topics of the course
  Machine Learning Module:
  The objective of this module is to provide the motivations, definitions and techniques for the acquisition and manipulation of data from networked automation systems, in order to derive, through Machine Learning techniques, predictive models, and use these models to manage resources in a dynamic and optimal way.
  Upon successful completion of this module, the student should have knowledge of techniques of ML and optimal control, be able to choose the best technique for building a predictive model of an automation system, and use it to solve an optimal control problem with respect to certain specifications and constraints.

• Topics:

   

  System Identification and data analysis module
  - Basics of linear algebra: vectors, matrices, subspaces. Four fundamental subspaces of a matrix, pseudo-inverse, projections. Fundamentals of probability theory: events, random variables, probability distributions and moments; standardizing random variables; multivariate Gaussian distributions; conditional probability and conditioning of variables; independency of events and of random variables. Computing the conditional expectation of Gaussian random variables; the Hilbert space of finite-variance random variables; conditional expectation and projection.
  - The Singular Value Decomposition of a matrix and its relationship with pseudo-inverse and the four fundamental subspaces. QR factorization. First applications to data analysis and data compression: low-rank approximation of matrices and Eckart–Young–Mirsky theorem.
  - Approximate and least-norm solutions to overdetermined and underdetermined linear systems of equations via ordinary least squares. Geometric interpretation of least squares. Applications to data fitting: learning linear and nonlinear functions from data through regression.
  - Estimation theory: Basic facts on estimation, real phenomena and their models, data collection, training and testing datasets, model validation. Minimum variance estimate and conditional expectation in the general case and in the Gaussian case; optimal and sub-optimal estimation by means of projections; Maximum a Posteriori Estimation; Maximum Likelihood Estimation. Estimation of distribution parameters; Gaussian examples. Markov estimator.
  - Short review of linear dynamical systems: time-invariant models; impulse responses and transfer matrices. Structural properties (observability, controllability), stability.
  - Stochastic dynamical systems and Kalman Filter: white noise and signal-generating model; linear discrete-time stochastic systems; state and output innovation processes; the Kalman Filter as the optimal estimator; recursive computation of the error covariances and filter gain (Riccati equations). Optimal predictor and optimal smoothing with the extended state. Brief notes on the filtering of continuous-time stochastic systems with sampled observations. Notes on the steady-state solution of Riccati equations and to the stationary filter. State estimation of nonlinear systems: extended Kalman filter.
  Notes on the parameter estimation for stochastic systems with maximum likelihood.
  - Classical results in system identification: overview of prediction error methods (PEM) for models in input-output form (Box-Jenkins, ARMAX, ARX models). Least-squares solution to PEM for ARX models.
  - Subspace identification: the Ho-Kalman method for deterministic realization of impulse responses. Ho-Kalman revisited (general inputs, measurement noise). Subspace methods: the MOESP algorithm for noisy systems. Model order reduction via truncated realization.
  Machine Learning Module:
  This module covers the fundamentals of data analysis, Machine Learning and Model Predictive Control techniques for monitoring and managing networked automation systems. First of all we will provide the basic notions of techniques, based on Machine Learning, exploited to extract a predictive model of a system. Therefore, the basic notions of optimal control theory will be provided, such as the definition and interpretation of an optimization problem with quadratic cost function and affine and quadratic constraints, the knowledge of existing techniques and tools for the solution and their computational complexity, and the integration of the predictive models described above as constraints of such optimization problems. Finally, it will be shown with simulation examples in Matlab how to apply the techniques introduced in the course to modeling and control problems of networked automation systems. In detail, the module is organized as follows:
  Introduction on data collection and pre-processing, on model classes, cost functions, exercises on Matlab.
  Elements of convex optimization, recalls of optimal control, (in-depth analysis of) Model Predictive Control, exercises on Matlab.
  Introduction to Machine Learning, perceptron, Support Vectors Machines.
  Regression models and techniques: AutoRegressive eXogenous (ARX) models, Regression trees, Random forests, exercises on Matlab.
  Elements of Artificial Neural Networks, exercises on Matlab.
  Identification and MPC algorithms based on models obtained from Machine Learning techniques, Matlab exercises.
  Matlab exercises with application on real data-sets: construction of predictive models and control loops.

• Prerequisites:

   

  Mathematics: probability theory, matrix analysis, integro-differential calculus. Computer science: basics of computer programming. Systems and Control Theory: basics on linear and nonlinear control systems
  Machine Learning module:
  Systems theory, Analysis and processing of signals.