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Method of decomposing large-scale systems in a study of stability of motion. (English. Russian original) Zbl 0683.70023

Sov. Appl. Mech. 24, No. 8, 810-816 (1988); translation from Prikl. Mekh., Kiev 24, No. 8, 91-98 (1988).
F. Bailey [J. Soc. Industr. Appl. Math., Ser. A; Control 3(1965), 443-462 (1966; Zbl 0139.046)] proposed a method of using the Lyapunov vector function (LVF) to analyze the stability of complex (large-scale) systems (LSS). The method has since come into wide use. Investigators who have examined the problem of the stability of LSS’s have proceeded on the basis that the initial LSS (1.1) \(dx/dt=f(x)\), \(x\in R^ n\), f: \(R^ n\to R^ n\), \(f\in C^ 1(R^ n)\) or \(f\in C^ 1(D)\), \(D\subseteq R^ n\), can be decomposed into s interrelated subsystems (1.2) \(dx_ i/dt=f_ i(x_ i)+g_ i(x)\), \(i\in [1,s]\); \(x_ i\in R^{n_ i}\), \(R^ n=R^{n_ 1}\times...\times R^{n_ s}\), consisting of the independent subsystems (1.3) \(FS_ i:\) \(dx/dt=f_ i(x_ i)\), \(f_ t:R^{n_ i}\to R^{n_ i}\), \(f_ i\in C^ 1(R^{n_ i})\), \(f_ i(0)=0\), \(i\in [l,s]\) and subsystems linked together in the LSS by coupling functions \(IK_ i: g_ i=g_ i(x)\), \(g_ i: R^ n\to R^{n_ i}\). It is natural to refer to such a decomposition as a decomposition of level I.
We constructed the LVF for system (1.1). The components of the LVF are Lyapunov scalar functions (LSF) for independent subsystems (1.3). We then examined stability by two main approaches: we used the LVF and differential inequalities to construct a comparison system (CS) in the cosine \(K=\{u: u\in R^ s_+\}\) (s\(\leq n)\); we used the LVF to construct a scalar LF and a function in the cosine K which has its derivative as a majorant by virtue of system (1.1). Without dwelling on the relationship between these two approaches, we examine the second of them.

MSC:

70K20 Stability for nonlinear problems in mechanics
37-XX Dynamical systems and ergodic theory

Citations:

Zbl 0139.046
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References:

[1] L. T. Gruiich, A. A. Martynyuk, and M. Ribbens-Pavella, Stability of Large-Scale Systems with Structural and Singular Perturbations [in Russian], Naukova Dumka, Kiev (1984).
[2] A. A. Martynyuk, Stability of Motion of Complex Systems [in Russian], Naukova Dumka, Kiev (1975).
[3] A. A. Martynyuk and R., Gutovskii, Integral Equalities and Stability of Motion [in Russian], Naukova Dumka, Kiev (1979).
[4] A. A. Martynyuk and V. V. Shegai, ?Study of the stability of large-scale systems on the basis of Lyapunov matrix functions,? Prikl. Mekh.,22, No. 6, 106?113 (1986). · Zbl 0624.70019
[5] A. A. Voronov and V. M. Matrosov (eds.), The Method of Lyapunov Vector Functions in the Theory of Stability [in Russian], Nauka, Moscow (1987).
[6] R. A. Nelepin (ed.), Methods of Studying Nonlinear Automatic Control Systems [in Russian], Nauka, Moscow (1975).
[7] L. B. Rapoport, ?Lyapunov stability and fixing of the sign of the quadratic form in the cosine,? Prikl. Mat. Mekh.,50, No. 4, 674?679 (1986).
[8] M. Araki, ?Application of M-matrices to the stability problems of composite dynamical systems,? J. Math. Anal. Appl.,52, No. 2, 309?321 (1975). · Zbl 0324.34045 · doi:10.1016/0022-247X(75)90099-2
[9] F. N. Bailey, ?The application of Lyapunov’s second method to interconnected systems,? SIAM J. Contr.,3, 443?462 (1966).
[10] M. Z. Djordjevic, ?Stability analysis of large scale systems whose subsystems may be unstable,? Large Scale Systems,5, 255?262 (1983).
[11] M. Z. Grujic, ?Stability analysis of large-scale systems with stable and unstable subsystems? Int. J. Control.20, No. 3, 453?463 (1974). · Zbl 0291.93038 · doi:10.1080/00207177408932754
[12] L. Grujic, Large-Scale System Stability, Bac. Mech.-Engen, Belgrade (1974), pp. 1?167.
[13] L. T. Gruji?, A. A. Martynyuk, and M. Ribbens-Pavella, Large-Scale Systems Stability under Structural and Singular, Perturbations, Springer-Verlag (1987).
[14] A. N. Michel and R. K. Miller, Qualitative Analysis of Large-Scale Dynamical Systems, Academic Press, New York (1977). · Zbl 0494.93002
[15] D. D. Siljak, Large-Scale Dynamic Systems: Stability and Structure, North-Holland, New York (1978). · Zbl 0384.93002
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