The ability to deal with uncertainty is an essential part in feedback control. If the disturbances and the plant dynamics were completely known, there would be no need to use feedback in order to achieve a specified plant response. It is precisely due to uncertainties that feedback is necessary. In the classical controller design methods, plant uncertainties are taken into account by stability margins relating to amplitude and phase. The need to deal with uncertainties in feedback control in a methodical manner, has led to the development of a specific theory of robust control, that provides a structured treatment of the robustness of control systems against plant uncertainties, for both scalar and multivariable plants. The theory also comprises the solution of realistically formulated optimal control problems for uncertain plants. This course describes the main methods of modern robust control theory. Given the nominal plant model, the set of disturbances and the plant uncertainty set, the problem is to design the controller in such a way that some specified control performance is achieved. A minimum requirement is that the closed-loop system should be stable for all plants in the given uncertainty set. The feedback control system is then said to be robustly stable. For open-loop unstable plants the problem of finding a controller such that the closed loop is robustly stable is by itself a well-motivated control problem. This is called the robust stabilization problem. Summarizing, one of the most useful features of a properly designed feedback control system is robustness, i.e., the ability of the closed-loop system to perform satisfactorily despite large variations in the (open-loop) plant dynamics. The aim of this course is to provide methodologies and tools for the analysis and design of robust feedback control systems.

By the end of this course, students will gain the following knowledge and skill: - Knowledge of the concept of a dynamical system together with its mathematical representations such as state equations and transfer functions. - Skill in computing the solution of the system state equations. - Knowledge of the stability properties of dynamical systems - Skill in studying the stability properties of dynamical systems. - Knowledge of the concept of feedback control of dynamical systems. - Knowledge of the main performance requirements of feedback systems. - Knowledge of the main analysis techniques in the time domain and in the frequency domain for the study of stability and performance of feedback control systems. - Skill in analyzing stability and performance of feedback control systems. - Knowledge of the design techniques of feedback controllers based on lead and lag functions. - Skill in designing feedback controllers for single input single output systems through lead, lag and PID functions. - Skill in evaluating the behaviour and performance of controlled systems through numerical simulation. - Knowledge of unstructured uncertainty models. - Skill in deriving weighting functions from performance specifications. - Knowledge of nominal and robust stability; nominal and robust performance. - Skill in the study of nominal and robust stability; nominal and robust performance. - Knowledge of the theory of robust control through H_infinity norm minimization. - Skill in the design of controllers through H_infinity norm minimization. - Knowledge of structured uncertainty modelling and the generalized plant. - Skill in representing structured uncertainty. - Skill in deriving the generalized plant. - Knowledge of robust stability and robust performance analysis through structured singular values. - Skill in the analysis of robust stability and robust performance through structured singular values. - Knowledge of Mu-bounds computation - Basic Knowledge of mu-synthesis. - Skill in the computation of mu-bounds.

Knowledge of differential and integral calculus of vector valued real functions. Basic results of complex numbers, functions of a complex variable, the Laplace transform and a good knowledge of linear algebra and the theory of polynomial and rational functions. Linear system theory. Knowledge of basic feedback control systems analysis and design. Basic knowledge in the design of controllers in the frequency domain. Basic skill of Matlab and Simulink.

Course organization. Prerequisites. Course description. Exam rules. Major topics and course outline (1.5 hr). Control problem formulation and systems representations (3 hr). Internal stability and Bibo stability (1.5 hr). Frequency response representations (Bode plot and polar plot) and Nyquist stability criterion (6 hr). Feedback control systems steady-state response to polynomial references and disturbances, and to sinusoidal disturbances (6 hr). Transient requirements translation and the Nichols chart (1.5 hr). Frequency domain controllers design through loop-shaping (3 hr). Performance specifications and weighting functions (4.5 hr). Unstructured uncertainty models (4.5 hr). Nominal and robust stability; nominal and robust performance (4.5 hr). Robust control through H_infinity norm minimization (6 hr). Structured uncertainty modelling and the generalized plant (6 hr) Robust stability and robust performance analysis through structured singular values (7.5 hr). Mu-bounds computation and introduction to mu-synthesis (3 hr). Exam simulation: guidelines and solution (1.5 hr).

Theoretical and methodological lessons are delivered, on a weekly scheduled basis, by face-to-face instruction in the classroom. Computer laboratory activities are scheduled in order to develop the student’s skill through proper training. Each student is supposed to practice individually with the aid of laboratory work stations. The primary purpose of the laboratory exercises is to apply the methodologies presented in class, through the use of MATLAB, Simulink and the Control System Toolbox. Some exam simulations are presented in the last two weeks of the course.

Selected chapters from: (a) S. Skogestad and I. Postlethwaite, Multivariable feedback Control, John Wiley and Sons, 2010. (b) J. Doyle, B. Francis and A. Tannenbaum, Feedback Control Theory, 1992. (c) K. Zhou and J. C. Doyle, Essentials Of Robust Control, Prentice Hall, 1999. Lecture slides are available as well as laboratory practice handouts.

Slides; Esercitazioni di laboratorio;

Modalità di esame: Elaborato scritto individuale;

Exam: Individual essay;

... Written examination in computer laboratory: based on computer aided analysis and design of a feedback control system, adequately documented through a written report. More precisely, it is required to (A) Understand and translate the design specifications into suitable weighting functions (12 - 16 points); (B) Design a controller that guarantees fulfillment of the assigned requirements, through H_\infinity optimization (6 - 8 points); (C) Report the obtained performance of the designed feedback control system (5 - 7 points); (D) Carry out mu-analysis to study robust stability and/or robust performance of the designed control system against structured uncertainty (5 - 7 points); (E) Provide orderly and clear presentation with legible handwriting (2 points). During the examination, lasting 4 hours, the student may consult the lecture slides provided by the teacher. Detailed instructions and rules are presented in due course.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.