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The purpose of the next subsection is to show that the formulation of eqns (557-558) is equivalent to the formulation of eqns (561-562) Then Subsection 542 will derive two additional forms of the covariance update equation The various implementations are summarized in Table 55 Additional alternative implementations, including square-root implementations, are presented in [60] It is important that the reader remember that although the Kalman lter can be implemented by a variety of techniques, the techniques are equivalent theoretically However, some of the implementation techniques require less computation while others have better numeric properties It is up to the designer to select the most appropriate form for a given application For example, the form of the measurement update in eqns (557-558) is often not convenient for real-time implementations due to the required matrix inversion The alternative formulation of eqns (561-562) also involves a matrix inversion; however, the inverted matrix has dimensions corresponding to the number of measurements instead of the number of states Since the number of measurements is often signi cantly less than the number of states, the update of eqns (561-562) is often preferred In fact, when the problem can be formulated with only scalar measurements, as discussed in Section 561, then the matrix inversion in eqn (561) reduces to scalar division Such issues are discussed in Section 56 Table 55 summarizes the discrete-time Kalman lter equations The Kalman lter could be implemented by one of (at least) four techniques All the approaches would use the same time update (eqns (554-556)) For the measurement covariance update, four methods have been presented The user must select one approach In comparing the various covariance measurement update equations, eqn (562) is the simplest formula and requires the least computation Although eqn (557) results naturally from the derivation presented in Section 534, it is not as e cient to implement as the alternative, equivalent solutions Eqns (568) and (569) involve only symmetric operations Of the two, eqn (569) is more numerically stable Note that eqn (569) is the same as eqn (512) which is valid for.

L = {2, 3, 4, 5, 6} M = {6, 7, 8, 9, 10} Here is a pair of sets that overlap more: P = {21, 23, 25, 27, 29, 31, 33} Q = {25, 27, 29, 31, 33, 35, 37} Technically, these sets overlap too: R = {11, 12, 13, 14, 15, 16, 17, 18, 19} S = {12, 13, 14} Here, you can see that S is a subset of R In fact, S is a proper subset of R Now, let s look at a pair of infinite sets that overlap with four elements in common: W3 = {, 5, 4, 3, 2, 1, 0, 1, 2, 3} W0+ = {0, 1, 2, 3, 4, 5, } The notation W3 (read W sub three-minus ) means the set of all positive or negative whole numbers starting at 3 and decreasing, one by one, without end The notation W0+ (read W sub zero-plus ) means the set of whole numbers starting with 0 and increasing, one by one, without end That s the set of whole numbers as it is usually defined Figure 2-3 is a Venn diagram that shows two sets, T and U, with some elements in common, so they overlap You can imagine T as the set of all the points on or inside the circle, and

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4 Now that the top cover is removed, open the bottom cover Again, you may have to use a small screwdriver to get it open Locate the small potentiometer and pry back the plastic clips that hold it in place, as shown in Figure 49 You may have to actually break them off If you do break them off, make sure that they are removed from inside the servo and discarded If glue is holding the potentiometer in place, scrape it off with a small screwdriver

(choose one)

U as the set of all the points on or inside the oval When you have two overlapping sets, one of them can be a subset of the other, but this does not have to be the case It is clearly not true of the two sets T and U in Fig 2-3 Neither of these sets is a subset of the other, because both have some elements all their own

CHAPTER 5 OPTIMAL STATE ESTIMATION 0 x = E x0 P = var(x ) 0 0 Kk = P Hk Rk + Hk P Hk k k k k x+ = x + Kk ( k yk ) y P+ = [I Kk Hk ]P k k P+ = [I Kk Hk ] P [I Kk Hk ] + Kk RKk k k P+ = P Kk Rk + Hk P Hk Kk k k k (P+ ) 1 = (P ) 1 + Hk R 1 Hk k k k k+1 k x = k x+ + Gk uk P = k P+ k + Qdk k+1 k

The intersection of two sets is made up of all elements that belong to both of the sets When you have two sets, say V and W, their intersection is also a set, and it is written V W The upside-down U-like symbol is read intersect, so you would say V intersect W

20

Intersection of two congruent sets When two nonempty sets are congruent, their intersection is the set of all elements in either set You can write it like this, for any nonempty sets X and Y,

Table 55: Discrete-time Kalman lter equations Computation of the matrices k and Qdk is discussed in Section 471 any stabilizing state feedback gain vector L The approaches described in eqns (557), (562), and (568) apply only for the Kalman lter gain Kk

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