intheCharacteristics tionsbasedon thesystem output are notincorrect be- the Characteristiccause
Yu‐Chi Ho · 1968
posed toasecond set. Thesecond problem isthedetection ofachange ofthecharacteristics fromthefirst setofvalues Byanextension tothetheory ofsequential detection withdependenttot scond asitchcuris.esh delwthe thseon of tothesecond asitoccurs. Weshall deal with thesecond of measurements, itispossible todevelop a sequential probability theseproblems. ratio test (SPRT) todetect changes inregime inaGauss-Markov Thedetection ofacharacteristic change isaproblem process rather than detecting which ofthetworegimes exists. Itis instatistical decision theory. Thebasic results usedhere shown howaposterior form ofthis extended SPRTmaybesimplified arethose ofsequential analysis, originated byWald. [1] The toreduce computational complexity. Thesimplified SPRT's arein analysis determines fromwhich oftwoprobability distrifact modifications oftheoriginal SPRTdetecting theregime andnotbutions a sequence ofuncorrelated samples comes, to thechange. Thetests areapplied totheproblem offault detection in withinseterrorbounds.Theprocedure istoderivefrom agyro navigational system; theresults ofadetailed computer simula- thesequence uptill atimek alikelihood function Ok. Its tion aregiven, value istested against twothresholds related totheset error bounds. Ifeither ofthethresholds isexceeded, then thetest isterminated bytaking theappropriate decision astowhichprobability distribution isshowntobecorrect. Ifneither threshold isexceeded, thena(k+l)th sample is taken, andthesameprocedure repeated fortheupdated sequence. A detailed andveryclear treatment ofthis Waldsequential probability ratio test(SPRT) isgiven in Hancock andWintz. [21 Thetheory wasextended tosequences ofcorrelated samples byBussgang andMiddleton,P3] andmorerecently ageneral extension hasbeenderived bymeansof thestate space approach bySchweppe.14' Inthepresent paper, weshall usethis approach exclusively. Section IIisdevoted toa fairly brief exposeof Schweppe's solution, because although Schweppe only deals withthefirst ofthetwoproblems previously stated, thesolution tothesecond problem isofaclosely similar form. InSection III, thecomplete solution tothesecond problem isgivenforthecasewherethecharacteristic change isachange ininput variance. Thissolution retains theadvantage oftheSPRTinthesolution ofthefirst problem, because ithastheproperty ofbeing thetest that, on average, requires theminimumnumberof