COURSE LANGUAGE: English

YEAR OF THE DEGREE PROGRAMME (I, II, III): II

SEMESTER (I, II, ANNUAL): I

CFU: 9

REQUIRED PRELIMINARY COURSES (IF MENTIONED IN THE COURSE STRUCTURE “REGOLAMENTO”) 
Geometry and Linear Algebra, Calculus II, Physics I.

PREREQUISITES (IF APPLICABLE) 
Basic matrix calculus (elementary operations, computation of determinants and inverses). Foundational knowledge of differential equation theory. Basic understanding of dynamics of mechanical and thermal systems. 

LEARNING GOALS 
The course aims to provide students with: the fundamentals of mathematical modeling of natural and/or artificial systems in continuous and discrete time; analytical techniques for systems described by input–state–output and input–output mathematical models, with particular emphasis on linear time-invariant systems; and the main methods for analyzing feedback systems. In addition, the course introduces students to the use of standard software tools for system analysis and simulation. 

EXPECTED LEARNING OUTCOMES (DUBLIN DESCRIPTORS) 

Knowledge and understanding 
The training program is designed to equip students with the methodological tools needed to describe simple engineering systems through appropriate mathematical models, to derive small-signal models of nonlinear systems, and to characterize the time response of linear systems in the time, complex, and frequency domains. To this end, students will be introduced to the fundamental concepts of the Laplace transform, the Z-transform, and the Fourier transform, gaining the ability to perform domain transformations using standard tables. Furthermore, the course will cover the main parameters that characterize linear systems in both the time and frequency domains. 

Applying knowledge and understanding 
At the end of the course, students will be able to: 

• Obtain the input–state–output or input–output representation of simple dynamic systems; 
• Evaluate, in the time domain, the response of finite-dimensional linear time-invariant systems; 
• Identify the natural modes of evolution of a linear system; 
• Analyze the stability of the equilibrium points of a dynamic system; 
• Define the Laplace and Z transforms of continuous-time and discrete-time signals; 
• Compute the Laplace and Z transforms of elementary signals using standard tables; 
• Perform the inverse Laplace and inverse Z transforms of rational functions through partial-fraction decomposition; 
• Determine the transfer function of linear systems and use it to evaluate the system response to canonical inputs; 
• Analyze block diagrams composed of multiple subsystems and derive their overall model; 
• Plot the Bode diagrams of a system and determine its bandwidth; 
• Use the Matlab/Simulink environment to numerically assess system properties and responses to assigned signals. 

COURSE CONTENT/SYLLABUS 

• Review of matrix algebra: elementary operations on matrices and vectors; eigenvalues and eigenvectors of a matrix; vector spaces; Banach and Hilbert spaces; p-norms for matrices and vectors. 
• Dynamic systems: input, state, and output variables; input–state–output and input–output representations; classification of dynamic systems. 
• Modeling fundamentals and examples of mathematical models. 
• Nonlinear systems: equilibrium points of nonlinear systems; linearization around a trajectory and around an equilibrium point. 
• Analysis of linear time-invariant systems in continuous and discrete time: the superposition principle; free and forced responses; computation of the state transition matrix via diagonalization; natural modes. 
• Discretization techniques for continuous-time systems; sampled-data systems: sampler and zero-order hold (ZOH); sampled-data representation of finite-dimensional linear systems. 
• Stability of equilibrium points: simple and asymptotic stability, instability; examples of stability analysis for nonlinear systems (e.g., the pendulum); introduction to Lyapunov theory; stability of linear systems, Routh criterion, application of the Routh criterion to discrete-time systems; input–output stability of linear systems. 
• Definition and properties of the Laplace transform of a signal. 
• Computation of the Laplace transform of canonical signals; use of tables to compute the inverse Laplace transform of an analytic function; inverse Laplace transform of rational functions. 
• Definition and properties of the Z-transform of a signal. 
• Computation of the Z-transform of canonical signals; use of tables to compute the inverse Z-transform of an analytic function; inverse Z-transform of rational functions. 
• Analysis of continuous-time linear time-invariant systems using the Laplace transform: transfer matrix, impulse and step responses, characteristic parameters of the step response; natural modes interpreted through the Laplace transform. 
• Analysis of discrete-time linear time-invariant systems using the Z-transform: transfer matrix, impulse and step responses; natural modes interpreted through the Z-transform. 
• Definition and properties of the Fourier series and Fourier transform for different classes of signals; frequency-domain analysis of linear systems and harmonic response matrix. 
• Construction of Bode diagrams for linear systems; system bandwidth and classification of linear systems based on bandwidth characteristics. 
• Decomposition of a linear system response into steady-state and transient components; steady-state response to polynomial, sinusoidal, and periodic inputs. 
• Interconnected systems and block diagrams: series, parallel, and feedback interconnections; representation of interconnected systems; introduction to the stability of interconnected systems. 
• Realization theory for single-input single-output (SISO) systems: canonical forms of observability and controllability. 
• The Matlab/Simulink environment for the simulation of dynamic systems and the analysis of structural properties. 

READINGS/BIBLIOGRAPHY 

1. Franklin, D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, 8^th Edition, Person.

Additional textbooks and/or lecture notes recommended by the instructor.  

TEACHING METHODS 
The instructor will employ: a) lectures for approximately 80% of the total class hours, and b) in-class exercises using the MATLAB/Simulink tool (https://www.mathworks.com/) for about 20% of the total class hours. 

EXAMINATION/EVALUATION CRITERIA 

Exam type:

• Written and oral.
• Project discussion.

In case of a written exam, questions refer to:

• Numerical exercises.
• Open answers.

The written examination is intended to assess the student’s ability to compute the response of a linear system to assigned input signals, to plot Bode diagrams, and to analyze the stability properties of interconnected systems. 
The oral examination, which follows the written test, consists of a discussion on the theoretical topics covered in the lectures and on simple MATLAB/Simulink assignments, aimed at verifying the student’s understanding of the concepts and contents of the course program. 

Evaluation pattern: 
The result of the written examination is a prerequisite for admission to the oral examination. Passing the written test alone is not sufficient to pass the course.