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It is important to emphasize that the Kalman lter is the optimal lter for the modeled process If the model is an inaccurate representation of the physical system, the optimal lter for the model cannot be expected to be optimal for the actual system Covariance analysis provides a method for analyzing the performance of a lter design relative to a truth model Even when these analysis techniques predict that the system will achieve a desired level of performance, the actual implementation may not achieve that level of performance Assuming that the algorithm is implemented correctly, the divergence of the actual performance from the predicted performance indicates that some important aspect of the system model has been neglected Typical causes of divergence are neglecting unstable or marginally stable states, or the process noise being too small Both of these e ects result in the calculated error variance being too small Therefore, the calculated Kalman gain is too small and the measurements are not weighted enough Analysis of the e ects of and sensitivity to modeling errors is one of the main objectives of covariance analysis Covariance analysis for which the system state and parameters { , Qd, H, R, P(0)} are identical to those of the design model are useful only for demonstrating the optimal performance that can be achieved under perfect modeling conditions A good analyst will follow this optimal analysis with a study to determine a set of lter parameters that results in near optimal performance while being robust to expected ranges of model uncertainty State estimators designed to be robust to model error and noise parameter uncertainty in the system model are called robust state estimators Such methods are reviewed in, for example, [121] Example 63 To illustrate various ways that the actual and predicted lter performance might not match, consider the example of estimating the state of the random walk plus random constant system described by x(t) = (t) (634).

with P (0) = var(x(0)) = 10 and Q(t) = var( (t)) = 2 from discrete measurements occurring at two second intervals: y(tk ) = x(tk ) + (tk ) where tk = 20k seconds and R = var( ) = 50 Therefore, the Kalman lter is designed using H = 1, = 1, P (0) = 10 and R = 5 Assume for 2 the purpose of this example that the value of is not accurately known by the analyst at the design stage For the purposes of the example, the actual value will be taken as Q = 05

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Pot Core Even more inductance per turn can be obtained with a pot core This is a shell of ferromagnetic material that is wrapped around a loop-shaped coil The core is manufactured in two halves (Fig 18-7) You wind the coil inside one of the halves, and then bolt the two together The final core completely surrounds the loop, and the magnetic flux is confined to the core material

Eqn (847) contains three unknowns which are the components of p Therefore, at least three equations will be required to determine a solution However, m must still be greater or equal to 4 because the m-th pseudorange is subtracted from the rst (m 1) pseudoranges For a set of m 4 satellites, with the m-th measurement subtracted from each of the others, (m 1) equations result This set of (m 1) equations can be solved as in Sections 822 and B142, using pk+1 = pk + H H where H=

= h

Like the toroid, the pot core is self-shielding There is essentially no coupling to external components A pot core can be used to wind a single, high-inductance coil Inductance values of more than 1 H are possible with a reasonable number of wire turns In a pot-core transformer, the primary and secondary must be wound next to each other This is unavoidable because of the geometry Therefore, the interwinding capacitance of a pot-core transformer is high Pot cores are useful at AF and the lowest-frequency parts of the RF spectrum They are rarely employed at high radio frequencies

(848)

Autotransformer In some situations, there is no need to provide dc isolation between the primary and secondary windings of a transformer In a case of this sort, an autotransformer can be used It has a single, tapped winding Figure 18-8 shows three autotransformer configurations The unit shown at A has an air core, and is a step-down type The unit at B has a laminated iron core, and is a step-up type The unit at C has a powdered iron core, and is a step-up type You ll sometimes see autotransformers in radio receivers or transmitters Autotransformers work well in impedance-matching applications, and also perform well as solenoidal loopstick antennas Autotransformers are occasionally, but not often, used in AF applications and in 60-Hz utility wiring In utility circuits, autotransformers can step the voltage down by a large factor, but they aren t used to step voltages up by more than a few percent

is represented as a row vector Alternatively, to prop-

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