In this course, the so-called state space models are discussed and analyzed – an essential tool for describing dynamic systems, also providing the basic concepts of Fisherian statistics and system identification.
The concept of a dynamic system is very general, and the corresponding properties are therefore common to an extremely wide variety of phenomena, whether physical, biological, economic, etc., making the scope of the corresponding theory extremely broad. The first part of the course deals with deterministic systems. After providing various examples of phenomena that can be described using state models, the fundamental concepts of systems are presented. The simplest class of autonomous and linear systems is studied first, followed by concepts of equilibrium points and stability. After addressing the non-autonomous case (presence of inputs), a very important class of systems, called compartmental systems, is described, where state variables are subject to physical constraints of positivity. Moving from a deterministic to a stochastic context, the problem of parametric system identification is introduced, providing basic notions of Fisherian statistics and also addressing the estimation of model complexity. Finally, the last part of the course is dedicated to the study of non-parametric identification and the estimation of inputs for linear dynamic systems.
In the course, an important role is given by the intuitive understanding of the motivations underlying the study of various problems. Abstract theoretical formulations are illustrated through numerous numerical examples, which sometimes precede the proofs of various theorems. The proofs have sometimes been intentionally simplified to make them more easily understandable without sacrificing necessary mathematical rigor.
The concept of a dynamic system is very general, and the corresponding properties are therefore common to an extremely wide variety of phenomena, whether physical, biological, economic, etc., making the scope of the corresponding theory extremely broad. The first part of the course deals with deterministic systems. After providing various examples of phenomena that can be described using state models, the fundamental concepts of systems are presented. The simplest class of autonomous and linear systems is studied first, followed by concepts of equilibrium points and stability. After addressing the non-autonomous case (presence of inputs), a very important class of systems, called compartmental systems, is described, where state variables are subject to physical constraints of positivity. Moving from a deterministic to a stochastic context, the problem of parametric system identification is introduced, providing basic notions of Fisherian statistics and also addressing the estimation of model complexity. Finally, the last part of the course is dedicated to the study of non-parametric identification and the estimation of inputs for linear dynamic systems.
In the course, an important role is given by the intuitive understanding of the motivations underlying the study of various problems. Abstract theoretical formulations are illustrated through numerous numerical examples, which sometimes precede the proofs of various theorems. The proofs have sometimes been intentionally simplified to make them more easily understandable without sacrificing necessary mathematical rigor.
- Teacher: Alessandro Beghi