Clinical Engineering

DANIELA IACOVIELLO
MATTIA MATTIONI

General objectives

The course focuses on the analysis and control of dynamic systems, with particular reference to linear time-invariant systems.

Specific objectives

Knowledge and understanding:
Students will learn the basic methods for the analysis and control of linear time-invariant systems. In particular, students will learn how to characterize a system to be controlled, from the point of view of structural characteristics, and which possible methods can be used for the design of controllers.

Apply knowledge and understanding:
Students will be able to design controllers that ensure the satisfaction of specifications concerning the stability, tracking and rejection of disturbances using frequency domain and eigenvalue assignment methodologies.

Critical and judgment skills:
The student will be able to choose the most suitable control methodology for a specific system based on the control specifications.

Communication skills:
The course activities allow the student to be able to communicate / share the main problems concerning time-invariant linear systems and the possible design choices for the control of such systems.

Learning ability:
The aim of the course is to make students aware on how to deal with analysis and control problems in the context of automatic controls.

By the end of the course, the student will be able to analyze the behavior of linear time-invariant dynamic systems in the time and frequency domains.
Furthermore, the student will be able to design control laws to satisfy specifications on steady-state behavior (e.g., regulation, trajectory tracking, disturbance attenuation and rejection) and on the transient response in the frequency and time domains.

Formal Prerequisites: Geometria, Analisi Matematica II.

Basics on mathematics: real and complex n-valued functions; graphical representations; basics on integral and differential calculus.
Basics on linear algebra and geometry: linear operators, vectorial spaces.
Basics on physics.

Oriented dynamic systems and state representations: from phenomenon to model to abstract system.

Concept of causal dynamic system and their representations: linear, stationary, finite-dimensional systems; implicit and explicit representation; decomposition of the response into free and forced; transition matrix and impulse response matrix and their properties.

Time domain analysis

Natural modes in the free evolution of the state for regular representations; laws of motion and trajectories of natural modes; natural modes in the forced response in state and output and their properties (excitability and observability).

Complex variable domain analysis

The Laplace transform for the analysis of continuous-time systems;
The transfer function and its representations.
The forced response as a system model: the step response and the gain.

Basics on stability theory

Introduction to the stability of equilibrium points of dynamic systems.
Definition of stability for linear systems; conditions and criteria.
Internal stability: the Routh criterion for continuous-time systems.

Frequency domain analysis

Steady state and transient state; steady-state response to canonical inputs. Graphical representations of the harmonic response.
The harmonic response. Representation of the transfer function.
Bode and Polar plots.
Significant parameters of the magnitude of the harmonic response and the step response; connection between time and frequency behavior.
Input-output models and state representations
The realization problem and the calculation of state-space representations from input-output models.
Reachable canonical form realizations and observable canonical form realizations.

Control systems: structure and design specifications

Specifications in the design of a control system.
Feedback, compensation, and mixed control schemes.
Response precision. Limitations on steady-state error.
Disturbance rejection and attenuation.
Transient response specifications and links with the open-loop harmonic response.

Design in the frequency domain

Elementary compensating functions.
Synthesis of compensating functions based on Bode or Nyquist plots.

Design in the Laplace Domain
The root locus.
Stabilization of minimum phase systems via the Root Locus.
Stabilization of non-minimum phase systems.
Direct synthesis for pole assignment.

Structural Properties of LTI systems
Reachability (controllability) and observability: definitions, criteria and the corresponding state decompositions.
The Kalman decomposition and the internal structure of dynamical LTI systems.
The relation with the excitability and observability of natural modes.

Design in the Time Domain

The eigenvalue assignment problem by state feedback.
The reconstructor problem and the asymptotic state observer.
The separation principle.
Criteria for the choice of closed-loop eigenvalues.
Inclusion of the reference signal in state feedback schemes.

Nonlinear Systems and Lyapunov Stability
Stability of equilibria.
The Lyapunov Theorem and the invariance principle.
Stability via the linear approximation and the describing function method.

Examples
Applicative control problems from distinct engineering and science domains to illustrate the analysis and design methodologies.

[1] S. Palani (2022). Automatic Control Systems. Springer Cham.
[2] Franklin, G. F., Powell, J. D., & Emami-Naeini, A. (2010). Feedback control of dynamic systems (Vol. 10). Upper Saddle River, NJ: Pearson.

Classic lectures for theoretical aspects and tutoring sessions for the practical part

Attendance is not mandatory but highly recommended.

Written exam and, if necessary, a supplementary oral exam.
In the latter case, the written exam will count as 75% of the final mark.

Samples of past exams can be accessed via the link below
https://drive.google.com/drive/folders/14wLWnn07Kuc-0iHxbuZ0s88emzy84TtL?usp=drive_link

• (tentatively 1 week, 8 hours) Introduction to Automatic Control: The concept of an oriented dynamic system and feedback. Examples of modeling continuous-time systems in the context of bioengineering.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.


• (tentatively 1 week, 8 hours) Continuous-Time Linear Time-Invariant Dynamic Systems in the Time Domain: Implicit and explicit representations; The set of equivalent representations.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.


• (tentatively 1 week, 8 hours) Time-Domain Analysis and the Free Evolution of Continuous-Time Systems: Simple aperiodic and pseudoperiodic natural modes; characteristic parameters; excitability and observability of natural modes and their study; examples.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.


• (tentatively 1 week, 8 hours) Complex Domain Analysis: The Laplace transform and its properties; the explicit form in 's'.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.


• (tentatively 1 week, 8 hours) Free Evolution in the Laplace Domain: The general case; examples.The Forced Response of Linear Systems and the Transfer Function: Properties and representations; the step response and its characteristic parameters; examples.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.


• (tentatively 1 week, 8 hours) The Realization Problem. Realizations in observable and reachable canonical form. Minimal realizations: Gilbert's method.Introduction to the Stability of Dynamic Systems: Definitions; the internal stability of continuous-time linear and time-invariant systems; the Routh criterion; examples.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.


• (tentatively 1 week, 8 hours) Frequency-Domain Analysis of Continuous-Time Systems: The frequency response and the steady-state response; graphical representations of the frequency response; examples.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.


• (tentatively 1 week, 8 hours) Reachability and Observability of Linear Systems: Definitions; conditions for their study; properties; decompositions; examples.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.


• (tentatively 1 week, 8 hours) Interconnections of Linear Systems in the Time and Laplace Domains: Elementary interconnections and their properties. Feedback Control Systems: Structure and design specifications; feedback, feedforward, and mixed control schemes; response accuracy; limitations on steady-state error; disturbance rejection and attenuation; specifications on the transient response and links with the open-loop frequency response; Examples.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.P. Bolzern, R. Scattolini, N. Schiavoni (2025). Fondamenti di Controlli Automatici (5a edizione). Mc Graw Hill Editore.


• (tentatively 1 week, 8 hours) Frequency Response Design Methods: Elementary compensator functions. Synthesis of compensator functions based on Bode or Nyquist diagrams; examples.Laplace Domain Design Methods: Trial-and-error synthesis; the root locus; direct synthesis; examples.
  • Books: P. Bolzern, R. Scattolini, N. Schiavoni (2025). Fondamenti di Controlli Automatici (5a edizione). Mc Graw Hill Editore.


• (tentatively 1 week, 8 hours) The Eigenvalue Assignment Problem through State Feedback; the stabilization problem through state feedback. The reconstructor problem: the asymptotic state observer; the detectable system case. The problem of stabilization and eigenvalue assignment through output feedback: the separation principle. Criteria for the choice of closed-loop eigenvalues. Inclusion of the reference signal in state feedback schemes; examples.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.


• (tentatively 1 week, 8 hours) Introduction to Lyapunov Stability for Nonlinear Systems: Stability of equilibrium points; the Lyapunov theorem and the Invariance Principle; stability analysis through linear approximation and the describing function.
  • Books: S. Monaco, C. Califano, P. Di Giamberardino, M. Mattioni (2022). Teoria dei Sistemi lineari stazionari a dimensione finita. Società Editrice Esculapio.