Fall 2001 16.3116-17 Interpretations With noise in the system, the model is of the form =AC+ Bu+ Buw, y= Ca +U And the estimator is of the form =Ai+ Bu+L(y-9,y=Ci e Analysis: in this case: C-I=[AT+ Bu+Buw-[Ac+ Bu+L(y-gI A(-)-L(CI-Ca)+B A元-LC+B10-Lu (A-LC).C+Buw-Lv This equation of the estimation error explicitly shows the conflict in the estimator design process. Must balance between Speed of the estimator decay rate, which is governed by Ai(A LC) Impact of the sensing noise v through the gain L Fast state reconstruction requires rapid decay rate(typically re- quires a large L), but that tends to magnify the effect of v on the estimation process The effect of the process noise is always there, but the choice of L will tend to mitigate/ accentuate the effect of v on c(t) Kalman Filter provides an optimal balance between the two con- Hicting problems for a given"size"of the process and sensing noises
Fall 2001 16.31 16–17 Interpretations • With noise in the system, the model is of the form: x˙ = Ax + Bu + Bww, y = Cx + v – And the estimator is of the form: ˙ xˆ = Axˆ + Bu + L(y − yˆ) , yˆ = Cxˆ • Analysis: in this case: ˙ x˜ = ˙x − ˙ xˆ = [Ax + Bu + Bww] − [Axˆ + Bu + L(y − yˆ)] = A(x − xˆ) − L(Cx − Cxˆ) + Bww − Lv = Ax˜ − LCx˜ + Bww − Lv = (A − LC)˜x + Bww − Lv • This equation of the estimation error explicitly shows the conflict in the estimator design process. Must balance between: – Speed of the estimator decay rate, which is governed by λi(A − LC) – Impact of the sensing noise v through the gain L • Fast state reconstruction requires rapid decay rate (typically requires a large L), but that tends to magnify the effect of v on the estimation process. – The effect of the process noise is always there, but the choice of L will tend to mitigate/accentuate the effect of v on ˜x(t). • Kalman Filter provides an optimal balance between the two con- flicting problems for a given “size” of the process and sensing noises
Fall 2001 16.3116-18 Filter Interpretation: Recall that A=(A-LC)i+ Ly e Consider a scalar system. and take the Laplace transform of both sides to get L Y) sI-(A- LC ● This is the transfer function from the“ measurement” to the“esti- mated state It looks like a low-pass filter Clearly, by lowering r, and thus increasing L, we are pushing out the pole DC gain asymptotes to1/CasL→∞ Scalar TF from Y to hat x for larger L
Fall 2001 16.31 16–18 • Filter Interpretation: Recall that ˙ xˆ = (A − LC)ˆx + Ly • Consider a scalar system, and take the Laplace transform of both sides to get: Xˆ (s) Y (s) = L sI − (A − LC) • This is the transfer function from the “measurement” to the “estimated state” – It looks like a low-pass filter. • Clearly, by lowering r, and thus increasing L, we are pushing out the pole. – DC gain asymptotes to 1/C as L → ∞ 10−1 100 101 102 103 104 105 106 10−2 10−1 100 Scalar TF from Y to \hat X for larger L Freq (rad/sec) |\hat X / Y| Increasing L
Fall 2001 16.3116-19 Second example: Lightly Damped Harmonic Oscillator 0 1+0 where Ru= 1 and Ro Can sense the position state of the oscillator, but want to develop an estimator to reconstruct the velocity state Find the location of the optimal poles S-1 b(s) e So we must find the lhp roots of 2+[(-s)2+4]+=(52+6)2+=0 Note that as r-0(clean sensor), the estimator poles tends to oo along the +45 deg asymptotes, so the poles are approximately 1±j 0 /r →重(s)=s2+
Fall 2001 16.31 16–19 • Second example: Lightly Damped Harmonic Oscillator x˙ 1 x˙ 2 = 0 1 −ω2 0 0 x1 x2 + 0 1 w y = x1 + v where Rw = 1 and Rv = r. • Can sense the position state of the oscillator, but want to develop an estimator to reconstruct the velocity state. • Find the location of the optimal poles. Gyw(s) = 1 0 s −1 ω2 0 s −1 0 1 = 1 s2 + ω2 0 = b(s) a(s) • So we must find the LHP roots of s2 + ω2 0 (−s) 2 + ω2 0 + 1 r = (s2 + ω2 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 • Note that as r → 0 (clean sensor), the estimator poles tends to ∞ along the ±45 deg asymptotes, so the poles are approximately s ≈ −1 ± j √r ⇒ Φe(s) = s2 + 2 √r s + 2 r = 0
Fall 2001 16.3116-20 Can use these estimate pole locations in acker, to get that 01 2「0112 0 L 0 2001 0 CA 1 Given L, A, and C, we can develop the estimator transfer function from the measurement y to the i2 01 01 01 S 0 s+22 22 +(S+ 0 S S2+-=s+ S+ ● Filter zero asymptotes to s=0asr→0 and the two poles→ . Resulting estimator looks like a" band-limited"differentiator This was expected because we measure position and want to estimate velocity. Frequency band over which we are willing to perform the dif- ferentiation determined by the relative cleanliness" of the mea- surements
Fall 2001 16.31 16–20 • Can use these estimate pole locations in acker, to get that L = 0 1 −ω2 0 0 2 + 2 √r 0 1 −ω2 0 0 + 2 r I C CA −1 0 1 = 2 r − ω2 0 √ 2 r −√ 2 r ω2 0 2 r − ω2 0 1 0 0 1 −1 0 1 = √ 2 r 2 r − ω2 0 • Given L, A, and C, we can develop the estimator transfer function from the measurement y to the ˆx2 xˆ2 y = 0 1 sI − 0 1 −ω2 0 0 + √ 2 r 2 r − ω2 0 1 0 −1 √ 2 r 2 r − ω2 0 = 0 1 s + √ 2 r −1 2 r s −1 √ 2 r 2 r − ω2 0 = 0 1 s 1 −2 r s + √ 2 r √ 2 r 2 r − ω2 0 1 s2 + √ 2 r s + 2 r = −2 r √ 2 r + (s + √ 2 r )(2 r − ω2 0) s2 + √ 2 r s + 2 r ≈ s − √rω2 0 s2 + √ 2 r s + 2 r • Filter zero asymptotes to s = 0 as r → 0 and the two poles → ∞ • Resulting estimator looks like a “band-limited” differentiator. – This was expected because we measure position and want to estimate velocity. – Frequency band over which we are willing to perform the differentiation determined by the “relative cleanliness” of the measurements.
Fall 2001 16.3116-21 Vel sens to pos state sen no se f=D. 01 Vel sens to pos state. sen noise fafe-006 Vel sens to pos state. sen noise fale-008
Fall 2001 16.31 16–21 10−3 10−2 10−1 100 101 102 103 10−4 10−2 100 102 104 Freq (rad/sec) Mag Vel sens to Pos state, sen noise r=0.01 10−3 10−2 10−1 100 101 102 103 0 50 100 150 200 Freq (rad/sec) Phase (deg) 10−3 10−2 10−1 100 101 102 103 10−4 10−2 100 102 104 Freq (rad/sec) Mag Vel sens to Pos state, sen noise r=0.0001 10−3 10−2 10−1 100 101 102 103 0 50 100 150 200 Freq (rad/sec) Phase (deg) 10−3 10−2 10−1 100 101 102 103 10−4 10−2 100 102 104 Freq (rad/sec) Mag Vel sens to Pos state, sen noise r=1e−006 10−3 10−2 10−1 100 101 102 103 0 50 100 150 200 Freq (rad/sec) Phase (deg) Vel sens to Pos state, sen noise r=1e−006 10−3 10−2 10−1 100 101 102 103 10−4 10−2 100 102 104 Freq (rad/sec) Mag Vel sens to Pos state, sen noise r=1e−008 10−3 10−2 10−1 100 101 102 103 0 50 100 150 200 Freq (rad/sec) Phase (deg)
