A several complex variables approach to feedback stabilization of linear neutral delay-differential systems. (English) Zbl 0539.93064
The problem of uniform asymptotic stabilization of neutral delay differential systems is considered. The results are based on an algebraic model of neutral systems over a ring of rational functions \(R_{\delta}\) of the delay operators. In terms of the two matrices with the entries in \(R_{\delta}\) associated with a given system, sufficient conditions of stabilizability are formulated. The stabilizing feedback is constructed with the help of the solution of an algebraic Riccati equation with the coefficients analytically depending on a parameter in \({\mathbb{C}}^ n\). The Docquir-Grauert theorem and the multivariable argument principle of complex analysis are used as the important technical tools. The Docquir- Grauert theorem is applied to prove that every projective finitely- generated module over \(R_{\delta}\) is free. This result is then used to study the canonical Brunovsky forms of some systems over \(R_{\delta}\). The multivariable argument principle is applied to the analysis of the analytical properties of the solution of the algebraic Riccati equation, depending on a parameter. In this connection, a notion of solvability is introduced and then utilized in the formulation of the sufficient conditions of stabilizability. Numerous examples are considered. This paper gives an excellent example of the fruitful interaction between complex analysis and system theory.
Reviewer: L.E.Faibusovich
MSC:
| 93D15 | Stabilization of systems by feedback |
| 34K20 | Stability theory of functional-differential equations |
| 32A27 | Residues for several complex variables |
| 13C10 | Projective and free modules and ideals in commutative rings |
| 93B10 | Canonical structure |
| 15A24 | Matrix equations and identities |
Keywords:
uniform asymptotic stabilization; neutral delay differential systems; algebraic Riccati equation; Docquir-Grauert theorem; multivariable argument principle; canonical Brunovsky formsReferences:
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