Hyperstability of Control Systems: (204 Grundlehren der Mathematischen Wissenschaften)

15 Liapunov's "second method" eliminates this drawback and leads to accurate conclusions regarding the stability of well-defined families of systems. This method made it natural to introduce the new notion of "absolute stability" whose origin can be traced to a work of . A. I. Lur'e and V. N. Post- nikov [1]. The results of the many investigations effected to date in the field of absolute stability have been presented in a number of monographs, from which we mention (in chrono- logical order) . A. I. Lur'e [1], . A. M. Letov [1], . A. Halanay Fig. 1. 2 [1], M . . A . . Aizerman and F. R. Gantmacher [1] and S. Lef- schetz [1]. Without going into a detailed exposition of these results (see the final chapter of this book), we shall discuss here only the manner in which one defines the families of systems that are studied. These systems are characterized by the fact that in Relation (4) - which describes the non-linear block B2 (Fig. 1. 1) - the function cp is continuous, vanishes for v = 0 and satisfies the inequality cp( v)v > 0 for every v =/= o. (6) In other words, the graph of function cp is entirely contained in the quadrants I and III; it may have, for instance, a shape similar to that shown in Fig. 1. 2. The object of the study of absolute stability consists in finding a criterion which secures simultaneously the stability of all the systems characterized by Condition (6) .

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.