Multivariate Control

• Objectives of the course:

This course is part of the master in Automatic Control and Robotics.

The goal of the subject is to show the students the feedback control techniques for multivariable systems from an input-output perspective (external description). A particular emphasis is placed in the use of feedback to modify the system dynamics and to reduce its sensitivity in front of disturbances and model uncertainties. The course presents the theory for multivariable (MIMO) systems but it also stresses its application to SISO (single input – single output) systems.

• Description of the course:

1. INTRODUCTION
2. BASIC FACTS OF LINEAR SYSTEMS AND H2 AND H∞ SPACES
3. INTERNAL STABILITY
   3.1 Feedback structure
   3.2 Internal stability
   3.3 Coprime factorization
4. PERFORMANCE SPECIFICATIONS AND LIMITATIONS
   4.1 Feedback properties
   4.2 Weighting functions and performance
   4.3 Bode formulas
   4.4 Analyticity constraints
5. UNCERTAINTY AND ROBUSTNESS
   5.1 Model uncertainty
   5.2 Small Gain Theorem
   5.3 Robust stability
   5.4 Robust Performance
6. LINEAR FRACTIONAL TRANSFORMATION
   6.1 Basic principle
   6.2 Redheffer Star Products
7. CONTROLLER PARAMETERIZATION
   7.1 Existence and parameterization of stabilizing controllers
   7.2 Coprime factorization approach
8. ALGEBRAIC RICCATI EQUATIONS
   8.1 Stabilizing solution and Riccati operator
   8.2 Inner functions
9. H2 OPTIMAL CONTROL
   9.1 Introduction to the Regulator Problem
   9.2 Standard and Extended LQR Problems
   9.3 Guaranteed stability margins
   9.4 Standard 2 H Problem
10. H∞ CONTROL
    10.1 Problem formulation
    10.2 General H∞ solutions
    10.3 H2 and H∞ Integral Control
    10.4 H∞ Filtering

• Assessment:

Attendance at lectures (participation) and a final exam (problems).

• Number of credits:

5 ECTS – Second semester

• Faculty:

Robert Griñó Cubero (coordinator)

• Selected references:

1. Zhou K., Doyle J.C. (1998), "Essentials of Robust Control", Prentice-Hall
2. Skogestad S., Postlethwaite I. (1996), "Multivariate Feedback Control. Analysis and Design", John Wiley & Sons.
3. Zhou, K., Doyle, J.C., Glover, K. (1995), “Robust and Optimal Control”, Prentice-Hall.
4. Maciejowski, J.M. (1994), Multivariable Feedback Design, Addison-Wesley
5. Doyle J.C., Francis B., Tannenbaum A. (1992) “Feedback Control Theory”, Macmillan Pub.