


Politecnico di Torino  
Academic Year 2012/13  
02CVGMQ Mathematical Control Theory 

1st degree and Bachelorlevel of the Bologna process in Mathematics For Engineering  Torino 





Subject fundamentals
The notion of ''system'' is broadly considered a cornerstone of the rational approach to applied sciences, both from the epistemological and technological point of view. It is based on the construction and the investigation of mathematical models, for which advances mathematics are an indispensable tool. For an applied mathematician, especially interested in engineering problems, the ability of interpreting the phenomena of the real word in term of ''system'' is fundamental.
This course aim to introduce in a rigorous and rational framework the basic notions and the principal results of the mathematical theory of linear systems and their control. Both the analysis and the design aspects are part of the training program. 
Expected learning outcomes
At the end of the course, the student will be able to compute the controllability and abservability indices of a system described by a linear model, and to investigate its stability properties. The student will be also able to compute, in simple situations, static or dynamic feedback laws, depending either on the state or the output, with the aim to improve the system performances. The student will be able to study
simple optimization problem ﾍs associated to a linear system, and to determine their solutions in feedback form. 
Prerequisites / Assumed knowledge
Differential and integral calculus for functions of one or more variables. Linear algebra. Elements of functional analysis.

Contents
Abstract notion of system. Basic facts about systems of ordinary differential equation, homogeneous and with forcing term, with constant coefficient. Linear equivalence. Jordan form, companion form. Stability and asymptotic stability of the equilibrium position. Liapunov functions, Liapunov first and second theorem, matrix Liapunov equation.
Linear systems with inputs. Reachable sets. Controllability and observability. Kalman form. Stabilization. Brunowsky form. Asymptotic observers. BIBO stability. Pontrjagin maximum principle. Optimal time problem, quadratic regulator. 
Delivery modes
Exercice sessions will take pace in classroom. Application of
theoretical results to some concrete problem will be illustrated. Simple simulations using MATLAB are possible. 
Texts, readings, handouts and other learning resources
Handnotes are prepared by the teacher, and are available on ``Portale della didattica del Politecnico'' for the students of the course.

Assessment and grading criteria
Short written exercise followed by an oral.

