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Figure 54: Kalman lter estimation performance for the simpli ed navigation system of Example 55 Left Standard deviation of estimation error Right Kalman lter gain sequence gains are plotted as one dot at each sampling instant to clearly indicate that they are only de ned at the sampling instants Due to the assumption of initial perfect knowledge (ie, P (0) = 0), the Kalman lter gains are initially small, but grow with time to optimally tradeo the noisy measurement with the increasingly noisy state estimates In comparison with Figure 47, each of the Kalman lter state error variance terms is smaller than the corresponding error variance terms of the pole placement observer, as expected of the optimal design

You are now invited to follow along with an extra-credit proof It will take some time and effort to understand it But why not try it It doesn t involve anything more complicated or sophisticated than facts of arithmetic you already know Let s prove that the value of 21/2 cannot be represented as a ratio of integers in lowest terms, and therefore that it s an irrational number We ll need two lemmas Let s accept them on faith Both of them can be proved, but that would be a distraction right now The square of an integer is always an integer Let s call this the integer-squared rule The square of an odd integer is always an odd integer Let s call this the odd-integer-squared rule

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Section 52 derived the Kalman lter in the stochastic process framework as an unbiased, minimum variance, linear stochastic estimator The results of that derivation approach can yield valuable insight into the Kalman lter and its performance Section 53 derived the Kalman lter as an extension of the weighted least squares approach from linear sets of algebraic equations to linear systems of ordinary di erence equations in the presence of additive process driving noise In this context, the Kalman lter is derived as an optimal combination (that turns out to be linear) of the time propagated estimate

Whenever we suspect that a proof might be involved, it s tempting to try reductio ad absurdum Let s use it now Table 9-2 does the job If you can t follow this proof as a whole, don t worry Try to understand each step, one at a time

The control/display circuit schematic is depicted in the schematic in Figure 20-4 It provides an additional gain of 200 over that of the sensor circuit The control/display circuit includes an adjustable low-pass notch filter, U2, a Linear Technology LTC 1062 By adjusting the clock frequency of the filter with resistor RI1 and the capacitors C4, C5 and C6, the filter cancels interference frequencies and noise in the 2 Hz to 10 kHz band The notch filter can also screen out 60 Hz noise Switch S2 inserts or removes the filter U3, an LM201A general purpose op-amp, filters out the clock noise generated within the filter chip The output of U3 is fed into the non-inverting input of U4-A, half of a dual 747 general purpose op-amp The overall gain of U4-A is adjusted by resistor R16 Trimmer potentiometer R15

.

Table 9-2

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