LINEAR SYSTEM THEORY

The course aims to drive the students in the acquisition of the basic knowledge and methodology in modeling analysis and control of Linear Time Invariant (LTI) systems.

At the end of the course the student will have acquired the "Knowledge and understanding"

• on the internal and external representation of an LTI system

• on the structural properties of an LTI system (stability, controllability and observability)

At the end of the course the student will be able to "Applying knowledge and understanding"

• to compute the response of an LTI system to an external stimulus both in the time and frequency domains

• to recognize the conditions in which it is possible to design the controller and the state estimator for a LTI system and know how to design them.

The course includes lectures for the acquisition and knowledge required and the exercises aimed at their applications.

Should teaching be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programme planned and outlined in the syllabus.

The cultural propaedeuticities required for understanding the contents covered during the course are:

• treatment of functions (study of a function, derivation and integration)
• treatment of matrices (diagonalization, transposition and inversion of a matrix, calculation of the eigenvectors and eigenvalues, calculation of the rank and of the characteristic polynomial)
• knowledge of Physics I relating to the laws of the dynamics of rigid bodies and their applications
• knowledge of Physics II and Electrotechnics related to LRC series / parallel circuits, Kirchhoff's laws and their applications.

The formal propaedeuticities required are related to the passing of the courses of Analysis I and Linear Algebra and Geometry.

1. LTI Systems Internal Representation: Definition of a Dynamical System - Systems Classification - Linear Time Invariant (LTI) Systems in continuous and discrete time – State space representation - State equation solution by Lagrange - State transition matrix - Equivalent systems and similarity transformations – System modes in continuous and discrete time. Multi-input/Multi-output systems.

2. LTI Input/Output Systems Representation: Mathematical note on the Laplace and Zeta operator – Relations between the S plane and Z plane (Tustin transformation) - Transfer function - Systems connection and block diagrams – System realizations and minimality - Jordan canonical form.

3. Stability for LTI Systems: Definition of Stability - Internal stability and eigenvalues - Input/output stability and poles (BIBO stability) – Asymptotic stability criteria (Routh and Hurwitz) – Stability in connected systems – Stability and relation between the S plane and Z plane.

4. Input/Output Response in Time Domain : Pulse and step response for first order and second order systems – Step response and pole placement - Step response and zero.

5. Input/Output Response in Frequency Domain: Harmonic function - Frequency response for LTI System - Bode diagrams – Characteristics of the frequency response for first and second order systems – Minimum phase systems.

6. Structural Properties for LTI Systems: Controllability - State feedback controller design for eigenvalues placement - Control canonical form - System decomposition based on the controllability – Observability - Asymptotic observer design (the state estimator) - Control canonical form - System decomposition based on the observability - Principle of separation of estimation and control - Kalman decomposition and transfer function.

T1) G. F. Franklin et al, Feedback control of dynamic systems, Addison Wesley

T2)  R.C.Dorf and R.H. Bishop, Modern Control Systems, Addison Wesley

The exam consists of a written test and an oral test. The written test consists of a maximum number of two exercises. The correct execution of an exercise refers to the assessment of minimum skills and allows access to the oral exam. The final evaluation will be determined by averaging the evaluation of both tests. If the conditions so require, the verification of learning can also be carried out online (Learning assessment may also be carried out online, should the conditions require it.)

If the conditions allow it, for students attending the course there are two ongoing tests which will take place respectively during the teaching suspension period and at the end of the course. Passing the first test is a necessary condition to be able to access the second. Each test consists of a maximum of four exercises. The correct execution of an exercise refers to the assessment of minimum skills. Passing both tests in itinere allows you to take the oral exam directly. The result of the ongoing tests remains valid for the entire academic year. If the two tests are passed with an overall score greater than or equal to 25/30, the oral test is not compulsory but at the student's discretion.

The evaluation during the oral interview will be based on: knowledge of the contents, the relevance of the answers with respect to the questions asked, the property of technical language, and the ability to make connections between the contents of the program.