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A second story: At the end of my freshman year at St. Thomas, I made a dumb mistake. Caught up in the pressures and anxieties of final exams, I forgot to arrange for on-campus housing for the following fall. When I got back to school at the end of that summer for football practice, I discovered that I had no place to live. Well, no big deal. I got together with four of my teammates, and we rented the top floor of a house a few blocks from campus. To the extent that I thought about it at all, I guess I thought that I had come up with a good solution to a problem that should never have come up in the first place. I happened to mention the solution to my mother the next time I called home. Bright and early the next morning, my father called the dean of students. Send my son home, he told the dean in a non-nego-

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Figure 5.6: Standard deviation of the estimation errors of steady state lter (dashed-dotted) relative to time varying Kalman lter (solid). The error 2 2 variance of x1 is 1 . The error variance of x2 is 2 . navigation applications either the system dynamic equations or the measurement equations are not linear. For example, the GPS range measurement (5.92) r = (xs xr )2 + (ys yr )2 + (zs zr )2 is nonlinear in the receiver antenna position (xr , yr , zr ). It is therefore necessary to consider extensions of the linear optimal estimation equations to nonlinear systems. This is the objective of the present section. The section mainly considers continuous-time systems with discrete measurements, as they are most pertinent to navigation applications. The presentation herein is limited to two of the traditionally most common nonlinear ltering techniques referred to as the Linearized Kalman lter and the Extended Kalman lter. Other nonlinear optimal estimation approaches such as particle lters or unscented Kalman lters are discussed for example in [121]. The results of this section will be used extensively in the applications in Part II of this text. Consider the nonlinear system with dynamics and measurement described by x(t) = f (x(t), u(t), t) + (t) y(t) = h(x(t), t) + (t). (5.93) (5.94)

Recurrent generalized convulsions at a frequency that prevents regaining of consciousness in the interval between seizures (grand mal status) constitute the most serious therapeutic problem (an

5,720 0 0 0 1,716 0 9,152 0 9,152 1,976 3,120 48,256

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where p : the fluid's density {kg/m3t, V: its velocity (m/s), and & : dynamic viscosiry (N.sim2). In addition to appearing Eq. (6.l4), the Reynoldsnumber also serves in asthe criterion for whetherflow is turbulent(Re > 4000). In this casestudy,we will illustratehow the numericalmethodscoveredin this part of the book can be employedto determine/for air flow througha smooth,thin tube.Forthis case,theparameters p - 1.23kglm3,! : 1.79x I 0r N . slmz, : 0.005m, V = 40nr/s are D and e :0.0015 mm. Note that friction factorsrangefrom about0.008 to 0.08. In addition, an explicit formulationcalledthe Swamee-Juin equationprovidesan approximate estimate:

While statistics are Run to Daylight, p. 78. Keep in mind that From undated Vince Lombardi talk entitled Goal Posts, in Vince Lombardi, Jr. s collection. The ego destroys From a speech delivered by Vince Lombardi at the Fathers-Sons Communion Brunch at St. Bonaventure Prep, Sturtevant, Wisconsin, 4/16/67, but also in Vince Lombardi s 5/64 speech to the First Friday Club in Los Angeles. Whatever the reasons Winning is a Habit, p. 134. If I ever hear Maraniss, David. When Pride Still Mattered. Simon & Schuster, Inc. New York: 1999, p. 241. I ll never absolutely never Vince, p. 266. If you re black or white Vince, p. 265. Why, he asked his golfing buddies Winning is a Habit, p. 242. Every year I try Strive to Excel, p. 53; Vince, p. 262.

00,000,004,580,103.7864892022 Instead, it would be written as 4,580,103.7864892022 In a decimal numeral, commas are not customarily inserted to the right of the point, no matter how many digits there are.