Beschreibung
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).
Autorenporträt
Inhaltsangabe1. 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-Szegö 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 d
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