CFU: 6
Prerequisites
Analysis of multivariable systems. Structural properties of linear systems. Classical design of controllers.
Preliminary Courses
None.
Learning Goals
The aim of the course is to provide the student with the main methodologies for the design of advanced control systems for multivariable linear systems.
Expected Learning Outcomes
Knowledge and understanding
The training course aims to provide students with the methodological tools for the design of control systems for multivariable systems. To this end, the main design techniques based on the state space representation of the plants will be introduced, such as the eigenvalue allocation technique, the optimal control, and the optimal H-infinite control. The problem of estimating the state of a system through an observer (Kalmanfilter) and that of order reduction will also be addressed.
Applying knowledge and understanding
At the end of the course, students will be able to design controllers for multivariable systems using the tools made available by the Matlab/Simulink software, and to evaluate the performance and robustness ensured by the designed controller.
Course Content - Syllabus
- References to time-invariant linear systems.
Modal decomposition of linear systems in the time domain (diagonalization of the dynamic matrix and evaluation of the transition matrix); reachability, controllability and observability subspaces; the reachability, controllability and observability Gramians; Kalman canonical forms; H-infinity norm of a linear time-invariant system. - Stability theory.
Stability of equilibrium points. Lyapunov's Method. Lyapunov equations for linear systems. - Control of multivariable systems.
Analysis of multivariable systems: open loop systems, closed loop systems. Nyquist's criterion for multivariable systems. The small gain theorem. Stabilizability and detectability of linear systems; state feedback pole assignment, the Ackerman formula for SISO system; Heymann's lemma; observer theory and design; separation principle and pole assignment via output feedback. - Optimal Control.
Basics on static optimization with equality constraints. Dynamic optimization problems; the Hamilton-Jacobi-Bellman equation; the maximum principle for problems with and without terminal constraints; solution of minimum time optimal control problems; linear quadratic regulators; Riccati differential equations; infinite-time horizon problems and Riccati algebraic equations; closed-loop stability properties of the linear quadratic regulator; Robustness of the LQ optimal controller.
Stochastic optimal control: the LQG problem and the Kalman filter. Requirements in the frequency domain and the choice of weights for LQG control. LQG control with integral action. - H-infinity optimal control
Formulation of the standard H-infinity problem. Robust control: Robust stability and performance analysis. Design of robust controllers. - Order reduction techniques.
Balanced realizations for the order reduction the plant and/or of the controller.
Readings/Bibliography
L. Magni, R. Scattolini, Advance and Multivariable Control, Pitagora Editrice Bologna.
Other books and notes suggested by the teacher.
Teaching Method
The teacher will use: a) lectures for about 70% of the total hours, b) classroom exercises through the use of the Matlab/Simulink software (www.mathworks.com) for about 30% of the total hours.
Examination/Evaluation criteria
Exam type
Only oral exam which will consist of a discussion on the theoretical topics covered in the course and on the presentation of a project developed by the student.
Evaluation pattern
In the oral exam the student must demonstrate to master the theoretical topics developed in the course, he must also be able to clearly illustrate the guidelines used in the development of his project.