MODELLISTICA E OTTIMIZZAZIONE

Knowledge and Understanding

At the end of the course, students will have acquired an in-depth knowledge of the main mathematical and computational tools for the modelling, analysis, and control of systems and processes of relevance to management engineering.

In particular, students will be able to:

• understand the main concepts and tools of mathematical modelling of linear dynamic systems in both continuous-time and discrete-time domains;

• understand the fundamentals of the representation, analysis, and order reduction of dynamic models using techniques based on singular value decomposition and balanced realizations;

• understand the principles of optimal control;

• understand the fundamentals of robust control and controller design in the presence of uncertainty;

• understand the fundamental principles of modelling using neural networks and the main methodologies for system identification from data;

• understand the main methods for the formulation and solution of linear programming problems;

• understand the fundamentals of binary and integer programming and the main algorithms used for their solution.

Applying Knowledge and Understanding

At the end of the course, students will be able to apply the acquired knowledge to the formulation and solution of modelling, control, and optimization problems.

In particular, students will be able to:

• formulate mathematical models of systems and processes based on the available information;

• analyse a dynamic model and assess its relevant characteristics for subsequent control system design;

• apply model reduction techniques based on Hankel singular values;

• formulate and solve optimal control problems;

• analyse control problems in the presence of uncertainty and apply robust control methodologies;

• develop neural models from experimental data and time series using dedicated software tools;

• formulate decision-making and resource allocation problems using linear, binary, and integer programming models;

• select and apply appropriate algorithmic methods for solving optimization problems;

• use MATLAB for modelling, numerical analysis, simulation, and the solution of optimization problems.

Making Judgements

At the end of the course, students will be able to critically assess the characteristics of a modelling or optimization problem and independently select the most appropriate methodology according to the objectives, available information, and constraints of the problem.

In particular, students will be able to:

• compare different approaches to modelling the same system and select the most appropriate time domain;

• assess the level of complexity required to adequately represent a process;

• select a model reduction technique based on the characteristics of the system and the objectives of the analysis;

• assess the trade-offs between performance, robustness, and complexity in controller design;

• assess the suitability of a model obtained using artificial intelligence techniques with respect to the available data;

• select an appropriate formulation and solution technique for an optimization problem;

• critically interpret the results obtained using numerical methods and software tools.

Communication Skills

At the end of the course, students will be able to describe and discuss the problems addressed, the assumptions underlying the adopted models, the methodologies employed, and the results obtained using appropriate technical terminology.

In particular, students will be able to:

• present the mathematical formulation of a problem;

• justify the choice of a particular model or solution method;

• interpret and discuss numerical and simulation results;

• communicate the results of an analysis to stakeholders with different levels of technical expertise.

Learning Skills

The course develops students' ability to independently address new modelling and optimization problems by selecting appropriate mathematical, algorithmic, and computational tools.

At the end of the course, students will be able to independently use the scientific literature and technical documentation relating to the methods studied and further investigate modelling, control, and optimization methodologies that are not directly addressed during the lectures.

Students are expected to have basic knowledge of:

• linear algebra;

• differential and integral calculus.

Basic knowledge of programming and MATLAB is also useful.

Knowledge of linear algebra is considered a fundamental prerequisite for fully understanding the course content. Programming and MATLAB skills, on the other hand, are useful but not essential, as the necessary tools and techniques will be introduced and used during the practical sessions.

Attendance at lectures is mandatory.

Students are required to attend at least 70% of the course lectures, in accordance with the Teaching Regulations of the Master's Degree Programme in Management Engineering.

Attendance is particularly important for the practical and application-oriented activities, during which students develop skills in the use of MATLAB and in the formulation and solution of modelling and optimization problems.

Enrollment in the course on the University of Catania's Studium platform is also mandatory.

The course covers methodologies for modelling, analysis, control, and optimization, with particular reference to problems of relevance to engineering and process management.

Linear Programming (Prof. Buscarino)

Mathematical formulation of linear programming problems. Feasible region and optimal solution. Standard form. Simplex method. Geometric and analytical interpretation. Fundamental theorems of linear programming. Solution analysis and interpretation of results.

Binary and Integer Programming (Prof. Buscarino)

Formulation of binary programming problems. Branch and Bound techniques. Formulation of integer programming problems. Optimality conditions and main algorithmic techniques for solving nonlinear problems.

Modelling of Dynamic Systems (Prof. Guastella)

Review of linear dynamic systems. State-space representation. Fundamental properties of linear systems. Continuous-time and discrete-time representations. Fundamental concepts and terminology.

Control of Dynamic Systems (Prof. Guastella)

Equilibrium and stability. Controllability and observability. State-feedback linear regulator and asymptotic observer.

Singular Value Decomposition (Prof. Buscarino)

Review of matrix algebra. Singular Value Decomposition. Geometric and numerical interpretation of singular values. Applications to system representation and analysis.

Principal Component Analysis and Model Reduction (Prof. Buscarino)

Principal Component Analysis (PCA). Dimensionality reduction. Open-loop balanced realization. Controllability and observability Gramians. Hankel singular values. Interpretation of system characteristic values and criteria for model reduction.

Optimal Control (Prof. Buscarino)

Formulation of optimal control problems. Cost functionals. Linear-quadratic optimal control. Riccati equation. Optimal controller synthesis. Analysis of the properties and performance of the controlled system. Implementation using MATLAB.

Control in the Presence of Uncertainty (Prof. Buscarino)

Motivations and challenges of robust control. Model uncertainty. Robustness and performance. Introduction to the main robust control methodologies. Performance analysis of a controlled system in the presence of uncertainty.

Data-Driven and Neural Network Modelling (Prof. Buscarino)

Introduction to data-driven modelling. Least-squares method. Methods for estimating model order. Structure and operation of neural networks. Supervised learning. Data preparation and representation. Modelling of dynamic systems using neural networks. Use of time series. Model training, validation, and performance evaluation. Implementation using MATLAB.

1. F. S. Hillier, G. J. Lieberman, Introduction to Operations Research, McGraw-Hill, 11th edition, 2021, ISBN 9781259872990.
2. A. Giua, C. Seatzu. Analisi dei sistemi dinamici. Springer Science & Business Media, 2009.
3. L. Fortuna, M. Frasca, A. Buscarino, Optimal and Robust Control – Advanced Topics with MATLAB, CRC Press, 2021, ISBN 9781032053004

Supplementary teaching materials, including lecture notes, slides, exercises, and materials related to the MATLAB practical sessions, will be made available through the Studium platform.

Additional materials may be provided during the course in relation to the topics covered and the practical activities proposed.