Fall 2001 16.31 16–17 Interpretations ? Withnoiseinthesystem,themodelisoftheform: ˙x = Ax+Bu+B w w, y= Cx+v – Andtheestimatorisoftheform: ˙ ?x = A?x+Bu+L(y??y) , ?y = C?x ? Analysis: inthiscase: ˙ ?x =˙x? ˙ ?x=[Ax+Bu+B w w]?[A?x+Bu+L(y??y)] = A(x??x)?L(Cx?C?x)+B w w?Lv = A?x?LC?x+B w w?Lv =(A?LC)?x+B w w?Lv ? Thisequationoftheestimationerrorexplicitlyshowsthecon?ict intheestimatordesignprocess. Mustbalancebetween: – Speedoftheestimatordecayrate,whichisgovernedby λ i (A? LC) – Impactofthesensingnoise v throughthegain L ? Fast state reconstruction requires rapid decay rate (typically re- quiresa large L), but thattends tomagnifythee?ect of v onthe estimationprocess. – Thee?ectoftheprocessnoiseisalwaysthere,butthechoiceof L willtendtomitigate/accentuatethee?ectof v on ?x(t). ? Kalman Filterprovidesanoptimalbalancebetweenthetwocon- ?ictingproblemsforagiven“size”oftheprocessandsensingnoises. Fall 2001 16.31 16–18 ? Filter Interpretation: Recallthat ˙ ?x =(A?LC)?x+Ly ? Consider a scalar system, and take the Laplace transform of both sidestoget: ? X(s) Y(s) = L sI ?(A?LC) ? Thisisthetransferfunctionfromthe“measurement”tothe“esti- matedstate” – Itlookslikealow-pass?lter. ? Clearly, by lowering r, and thus increasing L, we are pushing out thepole. – DCgainasymptotesto1/C as L→∞ 10 ?1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 ?2 10 ?1 10 0 Scalar TF from Y to \hat X for larger L Freq (rad/sec) |\hat X / Y| Increasing L Fall 2001 16.31 16–19 ? Second example: LightlyDampedHarmonicOscillator bracketleftbigg ˙x 1 ˙x 2 bracketrightbigg = bracketleftbigg 01 ?ω 2 0 0 bracketrightbiggbracketleftbigg x 1 x 2 bracketrightbigg + bracketleftbigg 0 1 bracketrightbigg w y = x 1 +v where R w =1andR v = r. ? Can sense the position state of the oscillator, but want to develop an estimator to reconstruct the velocity state. ? Findthelocationoftheoptimalpoles. G yw (s)= bracketleftbig 10 bracketrightbig bracketleftbigg s ?1 ω 2 0 s bracketrightbigg ?1 bracketleftbigg 0 1 bracketrightbigg = 1 s 2 +ω 2 0 = b(s) a(s) ? Sowemust?ndtheLHProotsof bracketleftbig s 2 +ω 2 0 bracketrightbigbracketleftbig (?s) 2 +ω 2 0 bracketrightbig + 1 r =(s 2 +ω 2 0 ) 2 + 1 r =0 ?1 ?0.8 ?0.6 ?0.4 ?0.2 0 0.2 0.4 0.6 0.8 1 ?1.5 ?1 ?0.5 0 0.5 1 1.5 Real Axis Imag Axis Symmetric root locus ? Notethatas r →0(cleansensor),theestimatorpolestendsto∞ alongthe±45degasymptotes,sothepolesareapproximately s ≈ ?1±j √ r ? Φ e (s)=s 2 + 2 √ r s+ 2 r =0 Fall 2001 16.31 16–20 ? Canusetheseestimatepolelocationsin acker,togetthat L = parenleftBigg bracketleftbigg 01 ?ω 2 0 0 bracketrightbigg 2 + 2 √ r bracketleftbigg 01 ?ω 2 0 0 bracketrightbigg + 2 r I parenrightBigg bracketleftbigg C CA bracketrightbigg ?1 bracketleftbigg 0 1 bracketrightbigg = bracketleftBigg 2 r ?ω 2 0 2 √ r ? 2 √ r ω 2 0 2 r ?ω 2 0 bracketrightBigg bracketleftbigg bracketleftbig 10 bracketrightbig bracketleftbig 01 bracketrightbig bracketrightbigg ?1 bracketleftbigg 0 1 bracketrightbigg = bracketleftBigg 2 √ r 2 r ?ω 2 0 bracketrightBigg ? Given L, A,andC,wecandeveloptheestimatortransferfunction fromthemeasurement y tothe ?x 2 ?x 2 y = bracketleftbig 01 bracketrightbig parenleftBigg sI ? bracketleftbigg 01 ?ω 2 0 0 bracketrightbigg + bracketleftBigg 2 √ r 2 r ?ω 2 0 bracketrightBigg bracketleftbig 10 bracketrightbig parenrightBigg ?1 bracketleftBigg 2 √ r 2 r ?ω 2 0 bracketrightBigg = bracketleftbig 01 bracketrightbig bracketleftBigg s+ 2 √ r ?1 2 r s bracketrightBigg ?1 bracketleftBigg 2 √ r 2 r ?ω 2 0 bracketrightBigg = bracketleftbig 01 bracketrightbig bracketleftBigg s 1 ?2 r s+ 2 √ r bracketrightBiggbracketleftBigg 2 √ r 2 r ?ω 2 0 bracketrightBigg 1 s 2 + 2 √ r s+ 2 r = ?2 r 2 √ r +(s+ 2 √ r )( 2 r ?ω 2 0 ) s 2 + 2 √ r s+ 2 r ≈ s? √ rω 2 0 s 2 + 2 √ r s+ 2 r ? Filterzeroasymptotesto s =0asr →0andthetwopoles→∞ ? Resultingestimatorlookslikea“band-limited”di?erentiator. – This was expected because we measure position and want to estimatevelocity. – Frequency band over which we are willing to perform the dif- ferentiationdeterminedbythe“relativecleanliness”ofthemea- surements. Fall 2001 16.31 16–21 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 10 3 10 ?4 10 ?2 10 0 10 2 10 4 Freq (rad/sec) Mag Vel sens to Pos state, sen noise r=0.01 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 10 3 0 50 100 150 200 Freq (rad/sec) Phase (deg) 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 10 3 10 ?4 10 ?2 10 0 10 2 10 4 Freq (rad/sec) Mag Vel sens to Pos state, sen noise r=0.0001 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 10 3 0 50 100 150 200 Freq (rad/sec) Phase (deg) 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 10 3 10 ?4 10 ?2 10 0 10 2 10 4 Freq (rad/sec) Mag Vel sens to Pos state, sen noise r=1e?006 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 10 3 0 50 100 150 200 Freq (rad/sec) Phase (deg) Vel sens to Pos state, sen noise r=1e?006 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 10 3 10 ?4 10 ?2 10 0 10 2 10 4 Freq (rad/sec) Mag Vel sens to Pos state, sen noise r=1e?008 10 ?3 10 ?2 10 ?1 10 0 10 1 10 2 10 3 0 50 100 150 200 Freq (rad/sec) Phase (deg)