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Figure 47: Error standard deviation of 1-d inertial frame INS with position measurement aiding Either of the methods for determining Qd results in 833 10 4 125 10 3 167 10 7 Qd = 125 10 3 250 10 3 500 10 7 167 10 7 500 10 7 100 10 6 Figure 47 displays plots of the error standard deviation for each state as a function of time In this example, L = [0300, 0039, 0002] which produces eigenvalues for the discrete-time system near 090 007, and 085 Due to the integration of the noise processes va (t) and b (t), the variance grows between the time instants at which position measurements are available The position measurement occurs at one second intervals Typically, the 10Hz position measurement decreases the error variance of each state estimate, but this is not always the case Notice for example that at t = 10s the standard deviation of all three states increase Therefore, the gain vector L selected for this example is clearly not optimal At least at this time, the gain vector being zero would have been better Eventually, by t = 35s, an equilibrium condition has been reached where the growth in the error variance between measurements is equal to the decrease in error

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variance due to the measurement update Although each measurement has an error variance of 30m2 , the ltering of the estimates results in a steady state position error variance less that 10m2 In comparison with the results in the previous section, the position measurement results in bounded variance for the velocity and accelerometer bias estimation errors Notice that there is a distinct di erence in the state estimate error variance immediately prior and posterior to the measurement correction Performance analysis should clearly indicate which value is being presented The simulation is for an ideal situation where the initial error variance is identically zero In a typical situation, the initial error variance may be large, with the measurement updates decreasing the error variance until equilibrium is attained

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This week s survey saw bullish sentiment rise to 345%, below its long-term average of 389% Neutral sentiment rose to 331% below the long-term average of 311% And bearish sentiment fell to 324% above the long-term average of 300%

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The approach described in Section 494 and depicted in Figure 48 is an example of a feed-forward complementary lter implementation For this example, the kinematics are given by eqn (4141), u represents the acceler ation measurement in eqn (4142), the output prediction is y = p, and the error estimator is de ned in eqns (4145) and designed by the choice of L No + or superscript is indicated on the variable because the error x state estimator can compute either variable at any time needed to form the rightmost summation in the gure The error estimator can also propagate eqns (4125) and (4127) to maintain an estimate of the system accuracy The feed-forward complementary lter approach has the navigation sys tem integrating the variable u through the system kinematics to produce x and the error state estimator integrating x between aiding measurements to produce the corrected state estimate denoted as x+ in Figure 48 As an alternative approach, over the sampling interval t [tk 1 , tk ), starting from the initial condition x+ (tk 1 ) the navigation system could integrate u(t) through the system kinematics to produce x (tk ) At time

tk , the error state estimate + = Lk yk xk xk is added to the prior state estimate to produce the initial condition for the next interval of integration xk x+ (tk ) = x (tk ) + + Because the navigation system state now accounts for the estimated error, the expected value of the error in x+ (tk ) is zero and it is therefore proper to reset + = 0 xk This makes the time update portion of eqn (4145) trivial, so that it need not be implemented This feedback approach is theoretically equivalent to the feed-forward approach It is depicted in Figure 49 where the feedback of should be read as correcting the initial condition for each period of x integration

Source: American Association of Individual Investors (AAII)

This chapter has presented a very brief tour of random variables, stochastic processes, and linear systems with white Gaussian driving noise The main references for the writing of this chapter are [31, 58, 93, 107] Additional recent references on the topics of this chapter are [77, 84, 121] Prerequisites for the chapter were a basic knowledge of probability and statistics A few good references for this material are [31, 107] Examples of various probability density functions are graphed on p 78 in [107] The main reference for the approach of Section 4721 is [133] This chapter has modeled random processes by state space equations with white noise processes as inputs There are theoretical di culties with

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this approach, due to the fact that the white noise processes are not integrable in the conventional Riemann sense Fixing these theoretical issues requires fundamental theoretical changes to the concept of integration that are beyond the scope of this text Interested readers can study the topic of It calculus in, for example, [71, 72, 106] o Section 492 presented a very brief discussion of instrumentation speci cations and methods for their estimation Additional information on these topics is available in the various speci cation documents, eg, [9, 10, 11] The Allan variance method is discussed in [2]

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