Volterra series representation of nonlinear systems

Flake, Robert H.

doi:10.1109/tai.1963.6371765

Cited by 38 publications

(8 citation statements)

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“…Local Inverse Theorem: Let U and V be normed spaces of which a t least one is a Banach space. Suppose that a polynomial operator P = 1 Hk E @(U,V), where n > 2 and H,, # 0, satisfies the following condit,ions: 2 ) Hz E E z ( U , V ) , . .…”

Section: Introductionmentioning

confidence: 99%

Generalized polynomial operators for nonlinear systems analysis

Halme¹,

Orava²

1972

IEEE Trans. Automat. Contr.

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226.-IEEE TRANSACTIONS ON AUTOXATIC CONTROL, URIL 1972 was established as a set of sufficient conditions for the ESL of system (9) (by means of the vector Lyapunov approach). Since cll 5 c12 and e l _< & it follows that (27) yields less conservative results t.han (28). VI. CONCLUSIONSNew stability results for composite systems with linear and nonlinear interconnections were established by utilizing scalar Lyapunov functions. It was shown that for large classes of problems, the use of scalar Lyapunov functions leads t.0 results which are in general less conservat.ive than those obtained from t,he vector Lyapunov function approach. ACKNOWLEDQMENTThe authors wish to thank one of t.he reviewers for his const,ructive suggestions. REFERENCES netted systems," SIA% J . Contr.. ser. A, vo?; 3, pp. 113-462. 1966. F. X. Bailey "The a plication of Lyapunov's second method to intercon-A . A. Piontkovskn and L. D. Rutkovskaya. Investigation of certain stability-theory problems by the vector Lyapunov function method," Automat. Remote Contr., v01. 28, pp.,l422-1429, 0,ct. 1965. H. K. Ramaqriyan and N. Bailey, A decomposition method for the sta-Space Congr., Oct. 1970. bility analysls of a spacecraft attitude control system." in Proc. 1st West. H. K. Ramapriyan. Study of coupling in interconnected systems," Ph.D. dissertation, Univ. hfj,nnesota, Minneapolis. BIar. 1970.Abstract-The concept of the so-called generalized polynomial operators is considered and applied especially to systems described by certain types of nonlinear dzerential equations. A theorem concerning local invertibility of polynomial operators is given. By an example it is shown how this theorem can be used to prove the existence of solutions, to construct these solutions, and to find a region of BIB0 stability of the aforementioned systems. The treatment is quite general, being based on functional analysis. In particular, it can be applied to the systems analyzed by using functional series of Volterra type.

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Section: Introductionmentioning

confidence: 99%

Generalized polynomial operators for nonlinear systems analysis

Halme¹,

Orava²

1972

IEEE Trans. Automat. Contr.

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“…Wiener [8] und Brilliant [9] verwendeten sie zur Analyse von nichtlinearen Systemen und zeigten, daß für analytische Systeme durch die Funktionalpotenzreihe eine geschlossene Ein-Ausgangsdarstellung möglich ist. In der Zwischenzeit hat sich diese Beschreibungsform als besonders vorteilhaft zur Behandlung konkreter technisch-physikalischer Probleme erwiesen [10], wobei im besonderen die Volterra-Reihe [11] zur Analyse von realen nichtlinearen Systemen in der Regelungstechnik verwendet wird.…”

Section: Prozeßmodelle Für Nichtlineare Systemeunclassified

“…(11). Durch diese Differenzbildung entfällt der Gleichwert r x in den Vektoren zur Bestimmung der Prozeßparameter.…”

Section: Prozeß-und Störmodell (Bild 3) Mit Der Differenzengleichungunclassified

Ein parameteradaptiver Regler für nichtlineare Prozesse / A parameter-adaptive controller for nonlinear processes

Lachmann¹,

Goedecke²

1982

at - Automatisierungstechnik

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“…The inverse of the original function (5) will in general be a multivalued relation between y and x. The series (6) is single valued, therefore it only specifies one part of the inverse relation. This is the part which is zero when y is zero.…”

Section: Inversion and Other Operationsmentioning

confidence: 99%