Filtering and Optimal Control Single channel
Chair (Coordinator) and Rapporteur: STEFANO BATTILOTTI
Module 1: Module I
- Activity type
- Ingegneria dell'automazione
- SSD
- IINF-04/A
- Year
- 1st year
- Semester
- 2nd semester
- CFU
- 6
- Hours distribution
- 36 classroom hours, 24 training hours
- Lecturers
- STEFANO BATTILOTTI
Module 2: Module II
- Activity type
- Ingegneria dell'automazione
- SSD
- IINF-04/A
- Year
- 1st year
- Semester
- 1st semester
- CFU
- 6
- Hours distribution
- 36 classroom hours, 24 training hours
- Lecturers
- DANIELA IACOVIELLO
Objectives
General goals
The course illustrates the methodologies of optimal control and filtering. The student will be able to formulate, analyze, and search for solutions of estimation, filtering and optimization problems of different nature.
Specific outcomes
- Knowledge and understanding
The Students will lean the methods of filtering and optimal control theory and the main applications in different fields.
- Capability to apply knowledge and understanding
The Students will be able to propose the optimal control and estimation strategies depending on the specific problem at hand.
- Critical and judgment skill
The Students will be able to analyze control problems, proposing and implementing optimal control strategies.
- Communication skills
The course activities will allow the Students to increase their communication capabilities in scientific fields.
- Learning skills
One of the aim of the course is to increase the critical skills of the Students; they will be able to tackle a control problem, starting from the literature analysis, to the optimal control and filtering aspects, up to the implementation.
Learning outcomes
General goals (Mod. I)
The course illustrates the methodologies of optimal estimation and control. The student will be able to formulate, analyze, and search for solutions of optimization problems of different nature by an appropriate use of optimality conditions, with emphasis on optimal estimation and control problems.
Specific outcomes
- Knowledge and understanding
The Students will lean the methods of optimal estimation and control theory and the main applications in different fields.
- Capability to apply knowledge and understanding
The Students will be able to propose the optimal estimation and control strategies depending on the specific problem at hand.
- Critical and judgment skill
The Students will be able to analyze control problems, proposing and implementing optimal estimation and control strategies.
- Communication skills
The course activities will allow the Students to increase their communication capabilities in scientific fields.
- Learning skills
One of the aim of the course is to increase the critical skills of the Students; they will be able to tackle a control problem, starting from the literature analysis, to the optimal control aspects, up to the implementation.
Prerequisites
Linear systems. Mathematics
Programme
Module: Module I
System Identification: Probability theory. Estimates: Definitions and properties. Rao-Cramer lower bound. Optimal estimates: Least squares, maximum likelihood, and Bayesian estimates with minimum error variance. Kalman filtering and prediction. Kalman filter. Steady-state Kalman filter. Recursive least-squares.
Module: Module II
Introduction to optimal Control and motivations
Definitions: local minimum, strict local minimum, global minimum.
Unconstrained optimization: first order necessary conditions; second order conditions
Weierstrass theorem
Constrained optimization; the Lagrangian; first order necessary conditions,
second order sufficient conditions; convexity hypothesis.
Calculus of variations; the Lagrange problem; the Euler equation; the augmented lagrangian;
necessary conditions; necessary and sufficient conditions
Calculus of variations and optimal control; the Hamiltonian function
The Pontryagin minimum principle; necessary conditions; necessary and sufficient conditions
The Hamilton –Jacobi – Bellman equation
The principle of optimality
The regulator problem: the optimal regulator problem on finite time interval; the optimal regulator problem on infinite time interval; the steady state linear optimal regulator problem; the optimal tracking problem; the optimal regulator problem with null final error; the optimal regulator problem with limited control
The minimum time problem; the minimum time problem for steady state system
Singular solutions
The armonic oscillator; the double integrator
LQG problems
Books
Module: Module I
Notes in english
C.Bruni, G. Di Pillo, Metodi Variazionali per il controllo ottimo, Masson , 1993
All the textbooks are available either on-line or in the library.
Module: Module II
Textbooks available in the DIAG library
B.D.O.Anderson, J.B.Moore, Linear Optimal Control, Prentice Hall, 2000
C. Bruni, G. Di Pillo, "Metodi variazionali per il controllo ottimo", Aracne, 2007
L. Evans, An Introduction to Mathematical Optimal control Theory, Berkeley, 1983
How, Jonathan. Principles of Optimal Control, Spring 2008.
(MIT OpenCourseWare: Massachusetts Institute of Technology). License: Creative Commons BY-NC-SA.
D. E. Kirk, "Optimal Control Theory: An Introduction, New York, NY: Dover, 2004
D. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton University Press, 2011
A.Locatelli, "Optimal Control: An Introduction", Birkhäuser, 2001Sc
A.Calogero, Notes on optimal control theory, 2017
PDF documents provided by the Professor
Bibliography
Module: Module I
N/D
Module: Module II
Textbooks available in the DIAG library
B.D.O.Anderson, J.B.Moore, Linear Optimal Control, Prentice Hall, 2000
C. Bruni, G. Di Pillo, "Metodi variazionali per il controllo ottimo", Aracne, 2007
L. Evans, An Introduction to Mathematical Optimal control Theory, Berkeley, 1983
How, Jonathan. Principles of Optimal Control, Spring 2008.
(MIT OpenCourseWare: Massachusetts Institute of Technology). License: Creative Commons BY-NC-SA.
D. E. Kirk, "Optimal Control Theory: An Introduction, New York, NY: Dover, 2004
D. Liberzon, "Calculus of Variations and Optimal Control Theory: A Concise Introduction", Princeton University Press, 2011
A.Locatelli, "Optimal Control: An Introduction", Birkhäuser, 2001Sc
A.Calogero, Notes on optimal control theory, 2017
PDF documents provided by the Professor
Lessons mode
The course takes place in the classroom, alternating theory and application-oriented lessons with examples in different fields
Frequency
free attendance but strongly advised
Exam mode
The two modules consist of a single oral exam.
Example exam questions
Ex. Describe the minimum principle
Ex. Describe the minimum time control problem
Ex. Describe the minimum error variance estimators
Ex. Describe the least squares estimators
Arguments
Module: Module I
- Probability Theory
- Estimates: centering, efficiency and consistency. Cramer-Rao lower bound
- Optimal estimates: least squares, maximal likelihood, minimum error variance.
- Kalman filter. Filtro di Kalman a regime permanente
Module: Module II
- During the course case studies will be described and
implemented. - Introduction to optimal control and motivations
- Definitions of local and global minimum.Unconstrained and
constrained extrema. Convex problem. Weierstrass theorem. Necessary
conditions (about 2 weeks) - Lagrange problem. Weierstrass Erdman conditions;
transversality conditions; Euler-Lagrange equations Exercises (about 2 weeks) - Calculus of variations and optimal control. State equation,
control equation, costate equation.examples and applications (about 2 weeks) - Minimum principle of Pontryagin (about 1 week)
- LQ problems: linear quadratic regulator (about 2 weeks)
- Minimum time problems (about 2 weeks)
- LQG problems: linear quadratic gaussian problems(about 1
week)
Sustainability goals
- Academic year2026/2027
- Degree program to which the course belongsControl Engineering
- Languageeng
- CFU12 CFU, distributed among 2 integrated didactic modules
- Total duration120 hours