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 noisesFall 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 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 ˜x(t). • Kalman Filter provides an optimal balance between the two con- flicting problems for a given “size” of the process and sensing noises