Hyperstability of control systems
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Bibliographic Information
Hyperstability of control systems
(Die Grundlehren der mathematischen Wissenschaften, Bd. 204)
Springer , Editura Academiei, 1973
[Rev. ed]
- : gw
- : us
- : gw : pbk
- Other Title
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Hiperstabilitatea sistemelor automate
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Note
"This is the revised edition of 'Hiperstabilitatea sistemelor automate'"--T.p. verso
Originally published as Hiperstabilitatea sistemelor automate: Bucureşti : Editura Academiei, 1966
Bibliography: p. [395]-400
Description and Table of Contents
- Volume
-
: gw ISBN 9783540063735
Description
Table of Contents
- 1. Introduction.- 1. Stability as a property of a family of systems.- 2. The families of systems considered in the problem of absolute stability.- 3. Selecting the most natural families of systems.- 4. Introducing new families of systems.- 5. The concept of hyperstability.- 6. Indications on the use of the monograph.- 2. Classes of Equivalent Systems.- 1. Equivalence classes for quadratic forms with relations between the variables.- 1. Transformations of quadratic forms with relations between the variables.- 2. Successive transformations.- 3. More about the group G.- 4. Partitioning of the set E into classes.- 5. Other equivalence classes.- 2. Classes of single-input systems.- 1. The system.- 2. Transformations.- 3. Some particular transformations.- 4. The polarized system and its properties.- 3. The characteristic polynomial of single-input systems.- 1. The characteristic function of single-input systems.- 2. The characteristic polynomial and its properties.- 3. Relations between the characteristic functions of systems belonging to the same class.- 4. Invariance of the characteristic polynomial under the transformations introduced in 2.- 4. Conditions under which all systems with the same characteristic polynomial belong to the same class.- 1. Some supplementary assumptions.- 2. A. one-to-one correspondence between the characteristic polynomials and certain particular systems.- 3. A property of "completely controllable systems".- 4. Methods for bringing completely controllable systems to special forms.- 5. Properties of systems with the same ? (??, ?).- 5. Equivalence classes for multi-input systems.- 1. Definition and properties of the classes of multi-input systems.- 2. The characteristic function.- 3. Properties of the determinants of H(?, ?) and C(?).- 4. The characteristic polynomial and its invariance.- 5. Systems with a fixed differential equation.- 6. Equivalence classes for discrete systems.- 1. Definition of the classes of discrete systems.- 2. The characteristic function and the characteristic polynomial.- 3. Relations between discrete systems with the same characteristic function.- 7. Equivalence classes for systems with time dependent coefficients.- 3. Positive Systems.- 8. Single-input positive systems.- 1. Definition of single-input positive systems.- 2. Theorem of positiveness for single-input systems.- 3. Remarks on the theorem of positiveness.- 4. Proof of the theorem of positiveness.- 5. The Yakubovich-Kalman lemma.- 6. Special forms for completely controllable single-input positive systems.- 9. Multi-input positive systems.- 1. The theorem of positiveness for multi-input systems.- 2. Proof of the theorem.- 3. Generalization of the Yakubovich-Kalman lemma.- 4. Special forms for multi-input positive systems.- 10. Discrete positive systems.- 1. The theorem of positiveness for discrete systems.- 2. Proof of the theorem.- 3. Generalization of the Kalman-Szego lemma.- 11. Positive systems with time-dependent coefficients.- 12. Nonlinear positive systems.- 4. Hyperstable Systems and Blocks.- 13. General properties of the hyperstable systems.- 1. Linear systems of class H.- 2. Hypotheses concerning the systems of class H.- 3. Other properties of the systems belonging to class H.- 4. Definition of the property of hyperstability.- 5. A consequence of property Hs.- 6. A sufficient condition of hyperstability.- 7. Hyperstability of systems which contain "memoryless elements".- 8. The "sum" of two hyperstable systems.- 9. Hyperstable blocks and their principal properties.- 14. Single-input hyperstable systems.- 15. Simple hyperstable blocks.- 16. Multi-input hyperstable systems.- 17. Multi-input hyperstable blocks.- 18. Discrete hyperstable systems and blocks.- 19. Hyperstability of more general systems.- 20. Integral hyperstable blocks.- 1. Description of completely controllable integral blocks.- 2. Definition of the hyperstable integral blocks.- 3. A method of obtaining the desired inequalities.- 4. Hyperstability theorem for integral blocks.- 5. Multi-input integral blocks.- 21. Lemma of I. Barb?lat and its use in the study of asymptotic stability.- 22. Other methods for studying asymptotic stability.- 23. Conditions of asymptotic stability of single-input and multi-input systems with constant coefficients.- 24. Characterization of the hyperstability property by the stability of systems with negative feedback.- 5. Applications.- 25. Inclusion of the problem of absolute stability in a problem of hyperstability.- 1. The absolute stability problem for systems with one nonlinearity.- 2. Definition of an auxiliary problem of hyperstability.- 3. A frequency criterion.- 4. Discussion of the condition of minimal stability.- 5. Sufficient conditions for absolute stability.- 6. Sufficient conditions for asymptotic stability.- 7. Simplifying the frequency criterion.- 8. Using hyperstable blocks to treat the problem of absolute stability.- 9. Determining the largest sector of absolute stability.- 10. Other generalizations of the problem of absolute stability.- 26. Determination of some Liapunov functions.- 1. Necessary conditions for the existence of Liapunov functions of the Lur'e-Postnikov type.- 2. Functions of the Liapunov type for systems with a single non-linearity.- 27. Stability in finite domains of the state space.- 1. An auxiliary lemma.- 2. Stability in the first approximation.- 28. Stability of systems containing nuclear reactors.- 29. Stability of some systems with non-linearities of a particular form.- 1. Systems with monotone non-linear characteristics.- 2. Stability of a system with a non-linearity depending on two variables.- 30. Optimization of control systems for integral performance indices.- Appendix A. Controllability
- Observability
- Nondegeneration.- 31. Controllability of single-input systems.- 1. Definition of the complete controllability of single-input systems.- 2. Theorem of complete controllability of single-input systems.- 3. Discussion.- 4. Proof of the theorem.- 5. Relations between single-input completely controllable systems.- 32. Single-output completely observable systems.- 33. Nondegenerate systems.- 1. Definition of the property of non degeneration and statement of the theorem of non degeneration.- 2. Remarks on the theorem of nondegeneration.- 3. Proof of the theorem of non degeneration.- 4. Bringing nondegenerate systems into the Jordan-Lur'e-Lefschetz form.- 34. Controllability of multi-input systems.- 1. Definition and theorem of the complete controllability of multi-input systems.- 2. Proof of Theorem 1.- 3. Other properties of completely controllable multi-input systems.- 35. Completely observable multi-output systems.- 36. Special forms for multi-input blocks.- Appendix B. Factorization of Polynomial Matrices.- 37. Auxiliary proposition.- 38. Theorem of factorization on the unit circle.- 1. Statement of the theorem.- 2. Preliminary remarks.- 3. Some additional assumptions.- 5. A family of factorization relations.- 6. A special way of writing polynomial matrices.- 7. A nonsingular factorization.- 8. Properties of the nonsingular factorizations.- 9. Bringing the nonsingular factorization to the form required in Theorem 1.- 10. More about Assumption (e).- 11. Eliminating restrictions (C) and (e).- 39. The theorem of factorization on the imaginary axis.- 1. Statement of the theorem.- 2. Definition of a matrix factorizable on the unit circle.- 3. Relations between ?(?) and ?(?).- 4. Factorization of the imaginary axis.- Appendix C. Positive Real Functions.- Appendix D. The Principal Hyperstable Blocks.- Appendix E. Notations.- Appendix F. Bibliography.
- Volume
-
: gw : pbk ISBN 9783642656569
Description
Table of Contents
- 1. Introduction.- 1. Stability as a property of a family of systems.- 2. The families of systems considered in the problem of absolute stability.- 3. Selecting the most natural families of systems.- 4. Introducing new families of systems.- 5. The concept of hyperstability.- 6. Indications on the use of the monograph.- 2. Classes of Equivalent Systems.- 1. Equivalence classes for quadratic forms with relations between the variables.- 1. Transformations of quadratic forms with relations between the variables.- 2. Successive transformations.- 3. More about the group G.- 4. Partitioning of the set E into classes.- 5. Other equivalence classes.- 2. Classes of single-input systems.- 1. The system.- 2. Transformations.- 3. Some particular transformations.- 4. The polarized system and its properties.- 3. The characteristic polynomial of single-input systems.- 1. The characteristic function of single-input systems.- 2. The characteristic polynomial and its properties.- 3. Relations between the characteristic functions of systems belonging to the same class.- 4. Invariance of the characteristic polynomial under the transformations introduced in 2.- 4. Conditions under which all systems with the same characteristic polynomial belong to the same class.- 1. Some supplementary assumptions.- 2. A. one-to-one correspondence between the characteristic polynomials and certain particular systems.- 3. A property of "completely controllable systems".- 4. Methods for bringing completely controllable systems to special forms.- 5. Properties of systems with the same ? (??, ?).- 5. Equivalence classes for multi-input systems.- 1. Definition and properties of the classes of multi-input systems.- 2. The characteristic function.- 3. Properties of the determinants of H(?, ?) and C(?).- 4. The characteristic polynomial and its invariance.- 5. Systems with a fixed differential equation.- 6. Equivalence classes for discrete systems.- 1. Definition of the classes of discrete systems.- 2. The characteristic function and the characteristic polynomial.- 3. Relations between discrete systems with the same characteristic function.- 7. Equivalence classes for systems with time dependent coefficients.- 3. Positive Systems.- 8. Single-input positive systems.- 1. Definition of single-input positive systems.- 2. Theorem of positiveness for single-input systems.- 3. Remarks on the theorem of positiveness.- 4. Proof of the theorem of positiveness.- 5. The Yakubovich-Kalman lemma.- 6. Special forms for completely controllable single-input positive systems.- 9. Multi-input positive systems.- 1. The theorem of positiveness for multi-input systems.- 2. Proof of the theorem.- 3. Generalization of the Yakubovich-Kalman lemma.- 4. Special forms for multi-input positive systems.- 10. Discrete positive systems.- 1. The theorem of positiveness for discrete systems.- 2. Proof of the theorem.- 3. Generalization of the Kalman-Szegoe lemma.- 11. Positive systems with time-dependent coefficients.- 12. Nonlinear positive systems.- 4. Hyperstable Systems and Blocks.- 13. General properties of the hyperstable systems.- 1. Linear systems of class H.- 2. Hypotheses concerning the systems of class H.- 3. Other properties of the systems belonging to class H.- 4. Definition of the property of hyperstability.- 5. A consequence of property Hs.- 6. A sufficient condition of hyperstability.- 7. Hyperstability of systems which contain "memoryless elements".- 8. The "sum" of two hyperstable systems.- 9. Hyperstable blocks and their principal properties.- 14. Single-input hyperstable systems.- 15. Simple hyperstable blocks.- 16. Multi-input hyperstable systems.- 17. Multi-input hyperstable blocks.- 18. Discrete hyperstable systems and blocks.- 19. Hyperstability of more general systems.- 20. Integral hyperstable blocks.- 1. Description of completely controllable integral blocks.- 2. Definition of the hyperstable integral blocks.- 3. A method of obtaining the desired inequalities.- 4. Hyperstability theorem for integral blocks.- 5. Multi-input integral blocks.- 21. Lemma of I. Barb?lat and its use in the study of asymptotic stability.- 22. Other methods for studying asymptotic stability.- 23. Conditions of asymptotic stability of single-input and multi-input systems with constant coefficients.- 24. Characterization of the hyperstability property by the stability of systems with negative feedback.- 5. Applications.- 25. Inclusion of the problem of absolute stability in a problem of hyperstability.- 1. The absolute stability problem for systems with one nonlinearity.- 2. Definition of an auxiliary problem of hyperstability.- 3. A frequency criterion.- 4. Discussion of the condition of minimal stability.- 5. Sufficient conditions for absolute stability.- 6. Sufficient conditions for asymptotic stability.- 7. Simplifying the frequency criterion.- 8. Using hyperstable blocks to treat the problem of absolute stability.- 9. Determining the largest sector of absolute stability.- 10. Other generalizations of the problem of absolute stability.- 26. Determination of some Liapunov functions.- 1. Necessary conditions for the existence of Liapunov functions of the Lur'e-Postnikov type.- 2. Functions of the Liapunov type for systems with a single non-linearity.- 27. Stability in finite domains of the state space.- 1. An auxiliary lemma.- 2. Stability in the first approximation.- 28. Stability of systems containing nuclear reactors.- 29. Stability of some systems with non-linearities of a particular form.- 1. Systems with monotone non-linear characteristics.- 2. Stability of a system with a non-linearity depending on two variables.- 30. Optimization of control systems for integral performance indices.- Appendix A. Controllability
- Observability
- Nondegeneration.- 31. Controllability of single-input systems.- 1. Definition of the complete controllability of single-input systems.- 2. Theorem of complete controllability of single-input systems.- 3. Discussion.- 4. Proof of the theorem.- 5. Relations between single-input completely controllable systems.- 32. Single-output completely observable systems.- 33. Nondegenerate systems.- 1. Definition of the property of non degeneration and statement of the theorem of non degeneration.- 2. Remarks on the theorem of nondegeneration.- 3. Proof of the theorem of non degeneration.- 4. Bringing nondegenerate systems into the Jordan-Lur'e-Lefschetz form.- 34. Controllability of multi-input systems.- 1. Definition and theorem of the complete controllability of multi-input systems.- 2. Proof of Theorem 1.- 3. Other properties of completely controllable multi-input systems.- 35. Completely observable multi-output systems.- 36. Special forms for multi-input blocks.- Appendix B. Factorization of Polynomial Matrices.- 37. Auxiliary proposition.- 38. Theorem of factorization on the unit circle.- 1. Statement of the theorem.- 2. Preliminary remarks.- 3. Some additional assumptions.- 5. A family of factorization relations.- 6. A special way of writing polynomial matrices.- 7. A nonsingular factorization.- 8. Properties of the nonsingular factorizations.- 9. Bringing the nonsingular factorization to the form required in Theorem 1.- 10. More about Assumption (e).- 11. Eliminating restrictions (C) and (e).- 39. The theorem of factorization on the imaginary axis.- 1. Statement of the theorem.- 2. Definition of a matrix factorizable on the unit circle.- 3. Relations between ?(?) and ?(?).- 4. Factorization of the imaginary axis.- Appendix C. Positive Real Functions.- Appendix D. The Principal Hyperstable Blocks.- Appendix E. Notations.- Appendix F. Bibliography.
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