]> LECTURE NOTES ON OPTIMAL CONTROL SYSTEMS

LECTURE NOTES ON OPTIMAL
CONTROL SYSTEMS

Piotr Grabowski

Originally published by Wydawnictwa AGH, Kraków, 1999
Revised internet edition
Copyright© 2002, by Piotr Grabowski
Last modification, February 8, 2006
1 PREFACE
2 PARAMETRIC OPTIMIZATION OF ABSTRACT SYSTEMS
 2.1 Introduction
 2.2 Formulation of the problem
 2.3 Semigroups and Riesz bases
 2.4 The Lyapunov operator equation
 2.5 Spectral properties of the system
 2.6 Semigroup generation
 2.7 Solution of the parametric optimization problem
 2.8 Solution of the Lyapunov equation
 2.9 Discussion of results
3 PARAMETRIC OPTIMIZATION OF TIME–DELAY SYSTEMS
 3.1 Introduction
  3.1.1 Example 1: Nuclear reactor temperature control I
  3.1.2 Example 2: Distortionless RLCG transmission line I
  3.1.3 Example 3: Automatic control system with PID controller
 3.2 Evaluation of the quadratic integrals
 3.3 Examples
  3.3.1 Example 1: Nuclear reactor temperature control II
  3.3.2 Example 2: Distortionless RLCG transmission line II
  3.3.3 Example 3: Automatic control system with PID controller II
4 THE BEST L2–APPROXIMATION WITH EXPONENTIAL SUMS
 4.1 Introduction
 4.2 Main results
 4.3 Computational aspects
  4.3.1 Example
 4.4 Discussion of results
5 APPROXIMATE PARAMETRIC OPTIMIZATION
 5.1 Introduction
 5.2 Representation of the orthoprojector
 5.3 Convergence of the approximation
 5.4 Application to parametric optimization
  5.4.1 Example 1: A simple time–delay control system
  5.4.2 Example 2: RC–transmission line
 5.5 Conclusions
6 THE LQ–CONTROLLER SYNTHESIS PROBLEM
 6.1 Introduction
 6.2 The lq controller problem
 6.3 Spectral factorization method the SISO case
 6.4 Examples of the synthesis of optimal controller
  6.4.1 Example 1
  6.4.2 Example 2
 6.5 Example
7 FOUNDATIONS OF BOUNDARY CONTROL
 7.1 Introduction
 7.2 The abstract model of boundary control I
  7.2.1 Admissible factor control vectors
  7.2.2 Duality theory
  7.2.3 The step response of an abstract system
 7.3 The transfer function
 7.4 The input–output map
 7.5 Example: Least–square identification
  7.5.1 Spectral properties of the system operator
  7.5.2 The step response of the system: the case of distributed
observation and control

  7.5.3 Least–square identification using the step response
  7.5.4 Simplification of the model
  7.5.5 Models with point observation/control
  7.5.6 The step response of the system: the case of point control
and observation

 7.6 The abstract model of boundary control II
8 THE LQ–CONTROLLER PROBLEM: BOUNDARY CONTROL CASE
 8.1 Introduction
 8.2 Statement of the lq problem
 8.3 Abstract dynamical model
 8.4 Spectral factorization
 8.5 Formal derivation of the optimal state feedback
 8.6 Physical realization of the optimal controller
 8.7 Formal derivation of the optimal cost
 8.8 Confirmation of the results by the Wiener – Hopf method
 8.9 Conclusions
Bibliography
Index