Optimal and Robust Control

Luigi Fortuna is a Full Professor of System Theory at the University of Catania, Italy. He was the coordinator of the courses in electronic engineering and Head of the Dipartimento di Ingegneria Elettrica Elettronica e dei Sistemi (DIEES). Since 2005, he has been the Dean of the Engineering Faculty. He currently teaches complex adaptive systems and robust control. He has published more than 450 technical papers and is the coauthor of ten scientific books. His scientific interests include robust control, nonlinear science and complexity, chaos, cellular neural networks, soft-computing strategies for control and robotics, micronanosensor and smart devices for control, and nanocellular neural networks modeling. Dr. Fortuna is an IEEE Fellow. Mattia Frasca received his PhD in Electronics and Automation Engineering in 2003, at the University of Catania, Italy. Currently, he is a research associate at the University of Catania, where he also teaches systems theory. His scientific interests include robust control, nonlinear systems and chaos, cellular neural networks, complex systems, and bio-inspired robotics. He is involved in many research projects and collaborations with industries and academic centers. He is also a referee for many international journals and conferences. Dr. Frasca was on the organizing committee of the 10th Experimental Chaos Conference and was co-chair of the 4th International Conference on Physics and Control. He is the coauthor of three research monographs, has published more than 150 papers in refereed international journals and international conference proceedings, and is the coauthor of two international patents. He is also a Senior Member of IEEE. For more information on Dr. Frasca’s work, see his web page at the University of Catania.

Modelling of uncertain systems and the robust control problem
Uncertainty and robust control
The essential chronology of major findings into robust control

Fundamentals of stability
Lyapunov criteria
Positive definite matrices
Lyapunov theory for linear time-invariant systems
Lyapunov equations
Stability with uncertainty
Exercises

Kalman canonical decomposition
Introduction
Controllability canonical partitioning
Observability canonical partitioning
General partitioning
Remarks on Kalman decomposition
Exercises

Singular value decomposition
Singular values of a matrix
Spectral norm and condition number of a matrix
Exercises

Open-loop balanced realization
Controllability and observability gramians
Principal component analysis
Principal component analysis applied to linear systems
State transformations of gramians
Singular values of linear time-invariant systems
Computing the open-loop balanced realization
Balanced realization for discrete-time linear systems
Exercises

Reduced order models
Reduced order models based on the open-loop balanced realization
Reduced order model exercises
Exercises

Symmetrical systems
Reduced order models for SISO systems
Properties of symmetrical systems
The cross-gramian matrix
Relations between W2c , W2o and Wco
Open-loop parameterization
Relation between the Cauchy index and the Hankel matrix
Singular values for a FIR filter
Singular values of all-pass systems
Exercises

Linear quadratic optimal control
LQR optimal control
Hamiltonian matrices
Resolving the Riccati equation by Hamiltonian matrix
The Control Algebraic Riccati Equation
Optimal control for SISO systems
Linear quadratic regulator with cross-weighted cost
Finite-horizon linear quadratic regulator
Optimal control for discrete-time linear systems
Exercises

Closed-loop balanced realization
Filtering Algebraic Riccati Equation
Computing the closed-loop balanced realization
Procedure for closed-loop balanced realization
Reduced order models based on closed-loop balanced realization
Closed-loop balanced realization for symmetrical systems
Exercises

Passive and bounded-real systems
Passive systems
Circuit implementation of positive-real systems
Bounded-real systems
Relationship between passive and bounded-real systems
Exercises

H∞ linear control
Introduction
Solution of the H∞ linear control problem
The H∞ linear control and the uncertainty problem
Exercises

Linear Matrix Inequalities for optimal and robust control
Definition and properties of LMI
LMI problems
Formulation of control problems in LMI terms
Solving a LMI problem
LMI problem for simultaneous stabilizability
Solving algebraic Riccati equations through LMI
Computation of gramians through LMI
Computation of the Hankel norm through LMI
H∞ control
Multiobjective control
Exercises

The class of stabilizing controllers
Parameterization of stabilizing controllers for stable processes
Parameterization of stabilizing controllers for unstable processes
Parameterization of stable controllers
Simultaneous stabilizability of two systems
Coprime factorizations for MIMO systems and unitary factorization
Parameterization in presence of uncertainty
Exercises

Recommended essential references

Appendix A. Norms

Appendix B. Algebraic Riccati Equations

Index