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By The Analytic Sciences Corporation, Arthur Gelb

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ACTUAL DATA FITTED FIRST-OCtX k MOW FITTED SECOND -ORDER MODEL Finite data length implies an inability to determine the form of an autocorrelation function at large shifts, whereas finite data spacing implies an inability to determine the form at small shifts. Together, the two correspond to a limited quantity of data, thus resulting in classical statistical limitations in estimating autocorrelation fpnction parameters. These factors are treated analytically in Refs. 12 and 13; brief summaries are given here.

The homogeneous unforced matrix differential equation corresponding to Eq. 1-12) Suppose that at some time, to, all but one of the outputs of the system integrators are set to zero and no inputs are present. Also, assume that the nonzero integrator output isgiven a magnitude of one. 1-4 SingleAxis Schuler Loop Error Diagram No control p i s applied and, therefore, the rystem dynamics equation can be written by inspeetion as An equivalent form for the random forcing function is Now, due to the superposition principle, if integrators i and j both have nonzero outputs ci and cJ at time to, the system response is the sum of the individual response vectors, viz: But this can be written as a product of the matrix and the vector As previousl~noted the only requirement is that the product Gyy has the same dimension as the vector &.

Er. 111.. ",Pme. of the IEEE, Vol. 55, No. 10, pp. 1664-1674, October 1967. 5 Consider the single-rxls Error model fur an incrtial nrvigation systfln shown tn l'lg. 3-2. 7 For the constant discrete nthorder deterministic system driven by a scalar control Show that thecontrolhbility cnternon is that the rank of Q= I [ I hl9hl.. (e n - l ~ l be n where &,, &, and the u, are given. (Hint: Describe the vector & , ,u . -At- - - - 1-2-3-- - - - - - F i r e 3-2 , e-AtP, 4 In this simpliiicd model, gyro drift ratc errors are ignored.

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