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Additional info for Absolute Stability of Nonlinear Control Systems

Sample text

1. 1) is globally stable. The zero solution of (2. 1. •AI' the zero solution of (2. 1. 1) is globally stable. e (or ahsolutely stahle in [O,kJ)? Lurie didn't find these necessary and sufficient conditions. When studying the absolute stability of (2. 1. 1), we usually make the following classification. We say that (2. 1. 1) is a direct control system. or the basic case (principal case) if Re A(A) < O. that is. A is sta bie. In other words. the eigenvalues of A have negative real parts only. 1. 1) i8 a indirect control system.

Proo/. taking /(a) = 1) f(1 Suppose there exists an eigenvalue Ao such that Re Ao (A) >0. By (O < E« 1). 1) becomes + dxT)x. 2) « 1. the coefficient matrix of (2. 1. 2) must have an eigenvalue lo such that Re lo > o. This is Since eigenvalues depend on coefficients continuously. for O < E inconsistent with the fact that the zero solution of (2. 1. 1) is absolutely stable. 29 2. 2. Neceseary and Sufficient Conditiona for AbeoIute Stabi6ty . 1. 2) is expressed as If cTb = . -b/ ;-1 + E~c;b, < O. ;-1 > O.

5) implies that S(x,tI) ~O. )) ~ - a/(O')tI is negative definite for n. Therefore, from Theorem 2. 2. 1 it follows that the conclusion is true. • 44 2. AUTONOMOUSOONTROLSYSTEMS Sup~ k = Corollary 2. 3. 4. + 00 anii 0111! 01 the lolWwing condttions hold: 1) 2) (,B~Tc + Pb) T R- 1( ,B~Tc + Pb) + ,BcTb < (~~TC + Pb) T R-I C< O [( ,B~Tc + Pb) T R-1( ,B~Tc + Pb) + {JcTb J~ O, (,B~Tc + Pb) T R-I C ~ O [( anii ,B~Tc + PbrR-lcr - (cTR-1c)[( 3) O, ,B~Tc + PbrR-lc r - (cTR-I C )[( ,B~Tc Then there exists a > anii + Pbr R-1( ,B~Tc + Pb) + {JcTb]> o.

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