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All we have to do is find a numerical example where the two expressions don t agree There are plenty! Let x = 2, y = 9, and z = 1/2 Then xy = 29 = 512 If we raise 512 to the 1/2 power, we get 22627 , an endless, nonrepeating decimal Now consider y z That s 91/2, or 3 If we raise x, which is 2, to the power of 3, we get 8

Therefore, the scalar processing algorithm can be used with the measurement z having measurement noise with variance described by the diagonal matrix D, and related to the state by the observation matrix H

This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App A The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 Which of the following quantities can we reasonably suspect is irrational (a) 163/4 (b) (1/4)1/2 (c) ( 27) 1/3 (d) 271/2

1 2 3 4 8 7 6 5

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In many applications, a measurement expected at the k-th sample time may occasionally be missing Such instances have no e ect on the Kalman lter algorithm, as long as the missing measurement is ignored correctly When a measurement is missing, the corresponding measurement update (both state and covariance) must not occur Alternative, the missing measurement can be processed using an arbitrary value for yk as long as the Kalman gain vector is set to zero, which corresponds to Rk being in nite In addition, the state estimate and error covariance must be propagated through time to the time of the next expected measurement Even when a measurement is available, it is sometimes desirable to reject certain measurements as faulty or erroneous The Kalman lter provides a mechanism for detecting candidate measurements for exclusion Let the state error covariance be P before the incorporation of the scalar measurement y = Hx + Then the measurement residual r = y y is a scalar Gaussian random variable with zero mean and variance E rr = HP H + R where R = var( ) The designer can select a threshold such that P rob r2 > HP H + R =

2 Suppose we have an irrational number and we display the first few digits of its endless nonrepeating decimal expansion If we multiply this number by 10, is the result still irrational What if we multiply it by 100, or 1,000, or any natural-number power of 10 Will the results always be irrational 3 What is the cardinality of (a) The set of even natural numbers (b) The set of naturals divisible by 10 without a remainder (c) The set of naturals divisible by 100 without a remainder (d) The set of naturals divisible by any natural power of 10 without a remainder 4 Consider this equation in real variables x and y: 36x + 48y = 216 How can we simplify this so it has the minimum possible number of symbols (variables and digits) 5 The nonnegative square root of 18 can be simplified, or resolved, into a product of a natural number and an irrational number What are these numbers 6 The nonnegative square root of 83 cannot be resolved into a product of a natural number and an irrational number, other than the trivial case 1 831/2 How can we tell 7 The numbers 501/2 and 21/2 are both irrational But the ratio of 501/2 to 21/2 is a natural number What natural number 8 Using the sum of quotients rule, add the fractions 7/11 and 5/17, and then reduce this sum to lowest terms 9 Using the product of sums rule, multiply out the product (x + y)(x y) 10 Using the rules we have learned so far, derive a formula for multiplying out the following expression, where u, v, w, x, y, and z are real numbers: (u + v + w)(x + y + z)

where (0, 1) is usually quite small This probability can be evaluated using the probability density for chi-squared random variables which is de ned on p 164 When the condition r2 > (HP H + R) (581)

This is not a test! It s a review of important general concepts you learned in the previous nine chapters Read it though slowly and let it sink in If you re confused about anything here, or about anything in the section you ve just finished, go back and study that material some more

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