Algebraic Methods in Statistical Mechanics and Quantum Field by Gerard G. Emch

By Gerard G. Emch

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Extra resources for Algebraic Methods in Statistical Mechanics and Quantum Field Theory (Physics & Astronomical Monograph)

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48 IDENTIFICA TION OF THE OPTIMAL L Y APUNOV FUNCTION FOR SYSTEMS WITH STRUCTURED UNCERTAINTIES Andrew alas and Ela alas Olas & Associates 1135 NW Circle Corvallis, OR 97330 ABSTRACT The problem of determination of the optimal Lyapunov function is modified and formulated as the problem of an identification of such a function. The objects considered are systems which are nominally linear but subject to nonlinear, time-varying structured uncertain ties. In the paper the identification problem of the optimal Lyapunov function from the class of quadratic forms is solved.

Contrary to the previous method, the controller obtained guarantees robust stability. The drawback of this method is that the Lyapunov function determined at the procedure is not an optimal one. Thus the controller designed may be a conservative one. For some applications, numerical results show that the design is still conservative, see Olas and Ahmadkhanlou [10]. The rationale for a present paper is as follows. e. the number l. such that the inequality V s l. V is fulfilled. If l. < 0 the Lyapunov function time constant T = -1/ l.

Or, equivalently, iff H(P;R) ~ H(P;Q) + H(Q;R) (4) for every PEE. Now let us state: Theorem 3. Let E and Ek, k = 1,2, ... ). ), that the Ek's, k = I, 2, .. and E are convex and variation closed, and that (5) Ek::::> Ek+l, k = 1,2, ... , n;=1 Ek=E. (6) Then (i) There is a unique Csiszar's projection of R on each set Ek, k = 1,2, ... and on E, namely Qk, k == 1,2, ... and Q*, respectively. (ii) H(Qk;R) ~ H(Qoo;R), and H( Qoo;Qk) --+ 0 as k --+ 00 • (iii) Qk converges in variation to Qoo, Qoo = Q* ,as k .....

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