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By Mark C. Draper, Xize Niu, Soongwon Cho, David I. James, Joshua B. Edel

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E. β(λ) = λr + βr−1 λr−1 + . . + β1 λ + β0 (54) Note that Sβ0 = In−1 0 and Sβn−1 = 0. Thus, if β(λ) is stable then there exists a functional observer of dimension r for the single-output system (44) with w(t) ˆ estimating w(t) = Kx(t) if and only if (52) is satisfied. To obtain the minimal order functional observer, one should starts with r = 1 and increase r until the condition (52) is satisfied. For multi-output case, one has several options. One option is to reduce the system to p coupled single-output systems and apply the same procedure.

Douglas Miller, 1993 (Last position: Unknown) MS thesis: “Applications of fuzzy logic to stock markets and financial planning” 12. Alex Martinez, 1993 (Last position: Unknown) MS thesis: “Fuzzy Control of Automotive Engines Idle Speed” 13. Steve Baugh, 1995 (Last position: Unknown) MS thesis: “A control approach for laser guidance systems” 14. ) MS thesis: “Intelligent navigation of mobile robots” Reflection on Four Decades of Contributions of My Graduate Students 7 15. Scott Beatty, 2005. (Last position: Lockheed Martin) MS thesis: “Simulation of a spacecraft electrical power distribution system using the simulink power system block set and soft computing techniques” 16.

Let tij ; j = 1, 2, . . , r be the j − th row of Ti ; i = 1, 2. Then using (40), one can expand (37) and (39) as follows ⎧ t21 = k2 ⎪ ⎪ ⎪ ⎪ ⎨t11 A12 + t21 A22 = t22 (41) ⎪... ⎪ ⎪ ⎪ ⎩ t1r A12 + t2r A22 = −αr t21 − αr−1 t22 − · · · − α1 t2r 20 B. Shafai and M. Saif These equations, except the last one, can express t2j ; j = 1, 2, . . , r in terms of t1j and substituting them in the last equation yields ⎡ ⎤ ⎤ I 0 ... 0 0 ⎡ A12 ⎢ α1 I ⎥ I . . 0 0 ⎢ ⎥ ⎢ A12 A22 ⎥ ⎥ ⎢ ⎥⎢ t1r t1,r−1 . . t12 t11 ⎢ α2 I α1 I .

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