The Stability and Control of Discrete Processes

1. Front Matter

   Pages i-vii

   PDF

2. Introduction

   • J. P. LaSalle

   Pages 1-7

3. Liapunov’s Direct Method

   • J. P. LaSalle

   Pages 7-12

4. Linear systems x’ = Ax.

   • J. P. LaSalle

   Pages 13-19

5. An algorithm for computing An

   • J. P. LaSalle

   Pages 19-24

6. A characterization of stable matrices. Computational criteria

   • J. P. LaSalle

   Pages 24-32

7. Liapunov’s characterization of stable matrices. A Liapunov function for x’ = Ax

   • J. P. LaSalle

   Pages 32-37

8. Stability by the linear approximation

   • J. P. LaSalle

   Pages 38-40

9. The general solution of x’ = Ax. The Jordan Canonical Form.

   • J. P. LaSalle

   Pages 40-45

10. Higher order equations. The general solution of ψ(z)y = 0.

    • J. P. LaSalle

    Pages 45-51

11. Companion matrices. The equivalence of x’ = Ax and ψ(z)y = 0.

    • J. P. LaSalle

    Pages 51-65

12. Another algorithm for computing An.

    • J. P. LaSalle

    Pages 66-70

13. Nonhomogeneous linear systems x’ = Ax + f(n). Variation of parameters and undetermined coefficients.

    • J. P. LaSalle

    Pages 70-83

14. Forced oscillations.

    • J. P. LaSalle

    Pages 84-87

15. Systems of higher order equations P(z)y = 0. The equivalence of polynomial matrices.

    • J. P. LaSalle

    Pages 87-93

16. The control of linear systems. Controllability.

    • J. P. LaSalle

    Pages 94-107

17. Stabilization by linear feedback. Pole assignment.

    • J. P. LaSalle

    Pages 107-116

18. Minimum energy control. Minimum time-energy feedback controls.

    • J. P. LaSalle

    Pages 117-125

19. Observability. Observers. State Estimation. Stabilization by dynamic feedback.

    • J. P. LaSalle

    Pages 125-146

20. Back Matter

    Pages 147-150

    PDF

Professor J. P. LaSalle died on July 7, 1983 at the age of 67. The present book is being published posthumously with the careful assistance of Kenneth Meyer, one of the students of Professor LaSalle. It is appropriate that the last publi­ cation of Professor LaSalle should be on a subject which con­ tains many interesting ideas, is very useful in applications and can be understood at an undergraduate level. In addition to making many significant contributions at the research level to differential equations and control theory, he was an excel­ lent teacher and had the ability to make sophisticated con­ cepts appear to be very elementary. Two examples of this are his books with N. Hasser and J. Sullivan on analysis published by Ginn and Co. , 1949 and 1964, and the book with S. Lefschetz on stability by Liapunov's second method published by Academic Press, 1961. Thus, it is very fitting that the present volume could be completed. Jack K. Hale Kenneth R. Meyer TABLE OF CONTENTS page 1. Introduction 1 2. Liapunov's direct method 7 3. Linear systems Xl = Ax. 13 4. An algorithm for computing An. 19 5. Acharacterization of stable matrices. Computational criteria. 24 6. Liapunovls characterization of stable matrices. A Liapunov function for Xl = Ax. 32 7. Stability by the linear approximation. 38 8. The general solution of Xl = Ax. The Jordan Canonical Form. 40 9. Higher order equations. The general solution of ~(z)y = O.

• Approximation
• Parameter
• Variation
• algorithms
• matrices

Policies and ethics