By William G. Andrew, H.B. Williams

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Additional info for Applied Instrumentation in the Process Industries, Volume 3: Engineering Data and Resource Manual

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34) we have 38 Chapters. 31). 31), imply that £, sms e £00. 36) that from which we establish that ems e £2 and hence s e LI (due to m2s = 1 + n2}. 35) we have which together with j^ e £00 and \sms \ e £2 imply that # € £ 2 0 £00, and the proof of (i) is complete. The proof of parts (ii) and (iii) is longer and is presented in the web resource [94]. 2 The rate of convergence of 0 to 9* can be improved if we choose the design parameters so that 1 — y\ is as small as possible or, alternatively, y\ e (0, 1) is as close to 1 as possible.

In some cases, the unknown parameters cannot be expressed in the form of the linear in the parameters models. In such cases the PI algorithms based on such models cannot be shown to converge globally. A special case of nonlinear in the parameters models for which convergence results exist is when the unknown parameters appear in the special bilinear form or where z e 9t, 0 e 91", z\ € 91 are signals available for measurement at each time t, and p* e 91", 9* e 91" are the unknown parameters. The transfer function W(q) is a known stable transfer function.

53), we are not able to choose 9 — 0 using signals available for measurement to make V < 0. 52) cannot be generated since it depends on the unknown input O1$. 52) is used only for analysis. 54) are summarized by the following theorem. 1. 54) guarantees that exponentially fast. Proof. The proof for (i) is given above. The proof of (ii) is a long one and is given in [56 as well as in the web resource [94]. 54) is referred to as the adaptive law based on the SPR-Lyapunov synthesis approach.

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