The objective of the course is to provide the elements of the theory of the control of multivariable dynamic systems in the continuous- and discrete-time domains, using state-feedback and output-feedback.

KNOWLEDGE AND UNDERSTANDING
The student, at the end of the course, should know
• the theory of dynamic systems, with particular emphasis on the notion of a system, its mathematical representations, and its structural properties, along with the concepts of state, equilibrium, stability, and feedback;
• the fundamental concepts underlying the analysis and synthesis of control systems, and the notion of robust control.

APPLYING KNOWLEDGE AND UNDERSTANDING
The student should be able to:
• recognize the characteristics of a dynamic system in phenomena from various fields such as electrical engineering, computer science, electronics, mechanics, chemistry, and biology;
• analyze dynamic systems to investigate their behavior, using both analytical and numerical tools;
• design feedback control systems based on state or output;
• apply the main paradigms of robustness analysis to systems characterized by model uncertainty.

MAKING JUDGEMENTS
The student should be able to evaluate, among several options, how to configure and design the architecture and the controller of a multivariable automatic control system starting from requirements and considering technological constraints.

COMMUNICATION SKILLS
The student should be able to describe in a clear and plain way the functionalities of a control system with the correct use of technical language.

LEARNING SKILLS
The student should be able to read and understand reference textbooks on control of dynamic systems.

The following mathematical topics/tools must be known
• linear systems of equations;
• eigenvalue decomposition;
• derivatives;
• integrals;
• exponentials;
• complex numbers;
• differential and difference equations;
• Laplace and Z-transforms;

1. Introduction.
2. Solutions to Linear Time Invariant systems.
3. Stability of equilibrium. Stability of LTI systems.
4. Structural properties and special forms.
Reachability and controllability. Observability and constructibility. Kalman canonical decomposition.
5. Realization.
6. State feedback. State observer. Separation principle and feedback of the estimated state. Set-point control. Q-design (outline).
7. Robustness
8. Optimization-based control. Open-loop control with constraints.

Panos J. Antsaklis and Anthony N. Michel, Linear Systems, Birkauser, 2006

Lalo Magni, Riccardo Scattolini, Advanced and Multivariable Control, Pitagora Bologna 2014

Hespanha, Joao P. Linear systems theory. Princeton university press, 2018.

1. Introduction. Generalities on systems. Interconnections. Finite-state automata. Regular systems. Linear systems. Equivalent state-space representations. Linearization.

2. Solutions to Linear Time Invariant systems. Impulse response matrix. Modal analysis, continuous time. Modal analysis, discrete time. Transfer functions.

3. Stability.
Movement stability. Stability of equilibrium. Lyapunov theorem. Stability of LTI systems. Lyapunov theorem for linear systems. Stability and transfer functions. Stability of LTI-DT systems. Lyapunov theorem for discrete time linear systems. Stability analysis by linearization. Input-output stability. Conditions for input-output stability in the time domain and in the frequency domain.

4. Structural properties.
Reachability and controllability for continuous time systems. PBH reachability test. Reachability and controllability for discrete time systems. Reachability Gramian. Control with minimum input effort. Linear programming and control. Observability and constructibility for continuous time and discrete time systems. Duality. Kalman canonical decomposition.

5. Realization
State-space realization from an input-output model. Existence and minimality of state-space realizations.

6. Control systems
Generalities. State feedback. State observer. Separation principle and feedback of the estimated state. Set-point control. Parameterization of all stabilizing controllers. Q-design (outline).

7. Robustness. Generalities. Parametric and non-parametric uncertainty. Robustness analysis of real parametric uncertainty.

8. Optimization-based control
Open-loop control with constraints. Minimum time control. Minimum fuel control.

Lectures (85%) and hands-on laboratory (15%).
The lectures are based on teaching material provided by the lecturer on the Moodle Platform. The slides will be made available before each lecture takes place. The slides have been prepared with the goal of being self-contained and no further material is required (i.e., further suggested readings are not mandatory). The lab lectures are interactive lectures in which the students implement (with the help of the instructor, and using their own laptop and Matlab/Simulink) some analysis and control approaches for linear dynamic systems. The starter code for the lab lectures is made available on Moodle as well. The purpose of the lab lectures is improving the understanding of the concepts, rather than learning the tools for designing control systems.
Any arrangements for distance learning will be made in accordance with the guidelines laid down by the University.

The teaching resources for the current and past years' courses are available on Moodle:
https://moodle2.units.it/course/search.php?search=322MI

Final Exam
The final exam consists of a written test lasting one hour and a subsequent oral exam typically lasting between 20 and 30 minutes. The final grade is determined by both the written test (50%) and the oral exam (50%). Both parts are taken during the same exam session.
Written Test
The written test includes up to 2 exercises and one theoretical question. The difficulty level of the exercises is not higher than that of the exercises/examples covered during the course. A collection of exam papers with solutions is available on Moodle.
Oral Exam
The oral exam assesses the student's knowledge and communication skills. During this exam, the student answers one or more questions related to the topics covered in the course. The oral exam may also include a discussion of the written test.
In any type of content produced by the student for admission to or participation in an exam (projects, reports, exercises, tests), the use of Large Language Model tools (such as ChatGPT and the like) must be explicitly declared. This requirement must be met even in the case of partial use.
Regardless of the method of assessment, the teacher reserves the right to further investigate the student's actual contribution with an oral exam for any type of content produced.

This course explores topics closely related to one or more goals of the United Nations 2030 Agenda for Sustainable Development (SDGs).