Facoltà di Ingegneria - Guida degli insegnamenti (Syllabus)

The student should know the basic notions of calculus, linear algebra, numerical analysis, and the basic notions of Control Engineering : state space representations of linear, time invariant, finite dimensional dynamical systems, both discrete time and continuous time. He should be able to compute the response to standard inputs. He should be able to analyze stability and existence of steady-state response. He should be able to analyze control problems and to synthesize possible solutions. He should be able to apply identification techniques in order to derive models from experimental data and to validate the models.

The aim of the course is to provide advanced knowledge concerning methods, techniques and tools for modelling, simulation and analysis of dynamic systems performance. In particular the construction of dynamic models and a thorough theoretical and practical knowledge of numerical analysis methods for the solution of the fundamental problems in control theory.

Students develop design skills concerning simulation procedures and systems, and the ability to evaluate the results and choose the most appropriate algorithms and simulation software for the particular application. Students acquire the knowledge and the necessary practice in the use of simulation software and the ability to work in the lab and they learn to produce technical reports.

Through guided exercises, design projects and lab activities the students develop: the ability to learn by assessing the completeness and adequacy of their preparation; the independence of judgment in analysing data and/or contrasting situations that require their own analysis; the communication skills in formulating and properly describe the solutions to the problems under consideration.

1. Analysis of the error. Representations of the numbers in the computer . 2. Nonlinear equations. The bisection method. The Newton-Raphson method. The multi variate secant method. The fixed point scheme. Algebraic equations. The Sturm's sequence . The Bairstow method. 3. Systems of nonlinear equation. Generalized Fixed point scheme. Generalized Newton Raphson scheme. 4. Systems of linear equation. Rouchè-Capelli Theorem. Gauss method. LU decomposition. 5. Norms of vectors and matrices. Condition index. Estimate k(A). Reachable accuracy. 6. Orthogonal matrices and their properties. Overdetermined systems and least square solutions. Normal equations and QR solution. 7. Gershgorin theorem. Diagonalizable matrices. Generalized eigenvectors, Jordan canonical form. Real Jordan canonical form. Exponential of a matrix. 8. Difference equations. Differential equations. Initial value problems; Euler's method; Crank-Nicolson, Heun, Runge-Kutta methods. Linear multistep methods. Predictor-corrector methods; stability theory; stiff systems. Consistency, zero-stability and convergence. Relative and absolute stability. Systems of first order ODE. 9. Modelling of linear and nonlinear dynamical systems. 10. Modelling of discrete event dynamical systems. 11. Simulation environments and software (Matlab/Simulink, , Virtual Reality Toolbox,...) 12. Design and realization of software simulators.

The learning evaluation will consist of a written test divided into two parts , each one to be completed in an hour. The first part consists of four questions of a theoretical nature, on the topics discussed in class and contained in the materials provided to the students. The second part, that takes place immediately after the first, consists of three problems to be solved with the use of Matlab . It Each student must also complete a practical project on one of the topics discussed in class and present a report on this activity. The project can also be done in groups , with a maximum of three students . In this case , the discussion of the project must take place with the participation of all students in the context of the same group. In the case of a negative result of one of the tests, the student can repeat only that part, provided this is done within the same academic year.

Correctness, completeness and clarity in answering the questions in the theory test. Accuracy and completeness in solving the exercises. As for the project, the student must prove that he can apply the concepts learned in the course, to properly use the tools and appropriate technologies and to write a clear technical report.

The first test consists of 4 groups of questions on the various parts of the program, each group contains a question which is assigned a score between 0 and 10, and a question which is assigned a score between 0 and 6. The student must answer a question in each group, choosing two questions for 10 points and two for 6 points. The second test consists of three questions, each of which is assigned a score between 0 and 10. A test is considered sufficient if the score is greater or equal to 15. The practical project is assigned a score from 0 to 30 and is sufficient only if the score is greater or equal to 18.

The overall grade is given by the arithmetic mean, rounded up to the whole, the sum of the scores obtained respectively in the test and in the project if all are sufficient. The overall grade required to pass the exam is 18 points. Otherwise the overall grade is Not sufficient . The student who in addition to getting a score greater than or equal to 30 has demonstrated complete mastery of the topics addressed, and clarity of exposition will have a 30 e lode.

Analisi Numerica, A.M. Perdon Pitagora Editrice 2006 Lectures slides and exercises can be found on the web site ESSE3Web

- Ingegneria Informatica e dell'Automazione (Corso di Laurea Magistrale (DM 270/04))

**Università Politecnica delle Marche**

P.zza Roma 22, 60121 Ancona

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