Course Details for A.Y. 2016/2017

Course Objectives

The goal of this course is to introduce students to the analysis of control systems, which are dynamic systems characterized by input and output variables, and state variables. On successful completion of this module, the students should be able to understand the meaning and the use of dynamic models, to use the models for the computation of the output of systems as a function of the input, to analyze the frequency response of a system, to study the stability property of a system before and after the closure of a control-loop, to analyze the structural properties of reachability and observability of the system state.

Course Content

• Systems and models: examples of electrical, mechanical, thermic, hydraulic systems; definition of causal systems, of state and of input and output functions. The state-transition function and the output transformation.
• State-space models of finite-dimensional linear systems: time-domain analysis; unforced and forced system response; the Dirac impulse. The state-transition matrix; the state and output impulse response matrices. Implicit and explicit representations. Time-invariant systems: solution of a system of linear differential equations. Matrix exponential: definition as a series, computation using the spectral decomposition. Convolution integrals and sums. Discretization of continuous-time systems. Coordinate transformations and similar matrices.
• time-domain  modal analysis: natural modes in the unforced state response. State trajectories: geometric path and time path. Stability, asymptotic stability and instability of natural modes. Natural modes and forced state and output response. Excitability and observability of modes.
• Systems analysis using transforms: Laplace transform and z-transform. The use of convolution, derivation and delay theorems in the systems analysis. The use of transforms in the modal analysis and in the computation of the matrix exponential. The use of transforms in the computation of the forced response. Connections between Laplace, Fourier and z-transforms. 
• Steady state and transient in the system response: the case of exponential and sinusoidal inputs. Harmonic response and frequency analysis. Graphic representations (amplitude and phase) of transfer functions. Decibel, semilog charts and Bode diagrams. Methods for drawing by hand of Bode diagrams. The polar graph of transfer functions (Nyquist plot).
• Stability theory: definition of equilibrium point; simple and asymptotic stability of an equilibrium point. Stability of linear time-invariant systems. The Lyapunov criterion. Positive (negative) definite and semi-definite functions. Quadratic forms. Sylvester criterion. Stability analysis by means of linear approximation: the system Jacobian. Lyapunov equation. Routh criterion. Stability of interconnected systems: Nyquist criterion.
• Reachability and observability:  indistinguishable and unobservable states; the observability matrix and the observability Gramian. The Cayley-Hamilton theorem. Reachable and controllable states. Invariance of unobservable and reachable subspaces. Structural decompositions and Kalman decomposition of systems. Controller and observer canonical forms of systems. State-space realization of transfer functions.

Learning Outcomes (Dublin Descriptors)

On successful completion of this course, the student should

Prerequisites and Learning Activities

The students must know the basic notions of calculus,  analytic geometry, matrix theory and vector analysis.

Assessment Methods and Criteria

Written and oral examination.

Textbooks

• A. Giua, C. Seatzu, Analisi dei sistemi dinamici , Springer Verlag. 2009. In alternativa a Grasselli/Menini/Galeani - Analisi dei sistemi dinamici    
• O.M. Grasselli, L. Menini, S. Galeani, Sistemi dinamici , Hoepli. 2007. In alternativa a Giua/Seatzu - Analisi dei sistemi dinamici    
• A. Ruberti, A. Isidori, Teoria dei Sistemi , Boringhieri . 1979. Testo per consultazione (accessibile in biblioteca)    
• A. Ruberti, A. Isidori, Teoria della stabilità , Siderea. 1977.    

Internet Resources

Course page updates

• A.Y. 2012/2013
• A.Y. 2013/2014
• A.Y. 2014/2015
• A.Y. 2015/2016
• A.Y. 2016/2017
• A.Y. 2017/2018
• A.Y. 2018/2019