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<br />- <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br /> <br />(13) <br /> <br /> <br />which can be translated into the state equation <br /> <br />[:/t1 =[ 0 1] [X<t)] + [0] <br /> <br />X2(t)J -aO-a1 X21t) 1 <br /> <br />w(t) <br /> <br />By comparing equations 11 and 13, it can be seen that n equals 2, <br /> <br />? =I 0 1] <br />tao-al <br /> <br />, <br /> <br />G =[~] <br /> <br />and a single, random-forcing function, w(t), drives the system. It should <br /> <br />also be noted that x2(t) equals qT(t) minus qR(t). <br /> <br /> <br />Identification of the structure of w(t) must be accomplished through <br /> <br />analysis of the measurements qM(t), because w(t) itself cannot be measured <br /> <br />directly. For example, if measurements indicate that x2(t) is a first-order <br /> <br />~mrkovian process, a state equation for xZ(t) is <br /> <br />X2(t) = -SxZ(tJ + w(t) <br /> <br />(14) <br /> <br />where S is the inverse of the correlation time constant of the Markovian <br /> <br />proces~. By substituting equation 14 into equation 13, the state equation <br /> <br />for a Markovian XZ(t) can be written <br /> <br />[:1 (tJ] <br />x2 (t) <br /> <br />= <br /> <br />[0 1] [x/tJJ- <br />o -S x2(tJ <br /> <br />+ <br /> <br />GJ w(t) <br /> <br />(15) <br /> <br />An error-free discharge measurement at time t would yield a single <br /> <br />value of x2(t) for any discharge rating. Discharge measurements are not <br /> <br />error-free',but Kalman filters deal with measurement errors by means of <br /> <br />the general measurement equation <br /> <br />;:Jt) = iI ~JtJ + !:'.(tJ (16) <br /> <br /> <br />where ~(t) is the vector of measurements at time t~ n is a matrL~ of <br /> <br />coefficients and v(t) is a vector of measurement errors. In the discharge- <br /> <br />22 <br /> <br />. <br />