PhD Thesis information
Title: Tools for Control System Design  Stratification of
Matrix Pairs and Periodic Riccati Differential Equation Solvers
Author: Ph Lic Stefan Johansson
Main supervisor: Prof. Bo Kågström
Assistant supervisor: Associate Prof. Erik Elmroth
The Thesis was publicly defended Friday 6th of March, 2009.
The summarizing chapters of the Thesis present a short
introduction to control theory and control system design. The
rest of the Thesis consists of the two major parts:
 Canonical structure information and stratification:
This part considers methods and algorithms for computing nearby
canonical structures
(see this
web page for more details). The qualitative information about
nearby structures is revealed by the theory of
stratification. A stratification reveals the closure
hierarchy of orbits and bundles of matrices, matrix pencils, and
system pencils. Paper I presents and review theory and
algorithms for computing the stratification for matrices, matrix
pencils, and matrix pairs. The explicit rules for the stratification
of matrix pairs are derived in Paper II.
Furthermore, in Paper I different canonical forms for matrices and
pencils are presented. These are the Jordan canonical form
for matrices, Kronecker canonical form for matrix pencils, and
(generalized) Brunovsky canonical form for system pencils.
 Periodic Riccati differential equations:
This part considers numerical solvers for the periodic Riccati
differential equation (PRDE). In Papers III and IV, methods
explicitly designed for
periodic systems are evaluated on both artificial systems
with known solutions and stabilization problems originating from
experimental control systems. The oneshot and multishot methods
have been implemented, where symplectic integrators are used to
solve the underlying Hamiltonian system.
This is a joint work with the
Control System group at
Department of Applied Physics and Electronics, Umeå University.
The Thesis consists of:
 Introduction and prelude
to the Thesis.
 Paper I: S. Johansson. Reviewing the closure hierarchy of
orbits and bundles of system pencils and their canonical
forms. Technical report, UMINF09.02, 2009.
 Paper II: E. Elmroth, S. Johansson, and B. Kågström.
Stratification of
Controllability and Observability Pairs  Theory and Use in
Applications. SIAM J. Matrix Analysis and Applications (accepted),
revised September 2008 (Also as Report UMINF 08.03).
 Paper III: S. Johansson, B. Kågström, A. Shiriaev, and
A. Varga. Comparing oneshot and
multishot methods for solving periodic Riccati differential
equations. In Proc. of the 3rd IFAC Workshop Periodic Control
Systems, PSYCO'07, St Petersburg, Russia, 2007.
 Paper IV: S. Gusev, S. Johansson, B. Kågström,
A. Shiriaev, and A. Varga.A
numerical evaluation of solvers for the Periodic Riccati
Differential Equation. Technical report, UMINF09.03, 2009.
