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P 1 ( 2 x1 ) x 2 P 1 ( 1 x2 ) x 1
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2 Show that either equation in part 1 can be reduced to xc = P 1 + P 1 1 2
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P 1 x1 + P 1 x2 1 2
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512 EXERCISES
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3 In the special case that n = 1, show that either equation reduces to xc = P2 P1 x1 + x2 P1 + P2 P 1 + P2
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4 Discuss the physical intuition that supports this result Consider the limiting cases where P1 is near 0 or and P2 is a nite nonzero constant Exercise 55 For the system described as x = y = 0 1 0 0 1 0 x+ x + ny 0 1 nx
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with Gaussian white noise processes nx N (0, 001) and ny N (0, 1), complete the following 1 Calculate and Qd for the discrete-time equivalent model for sample time T 2 Assume that x(0) N (0, 0) and the measurement y is available at 10 Hz (a) Implement the Kalman lter gain and covariance propagation equations (b) Plot the Kalman gains and the diagonal of the covariance matrix versus time 3 Assume that x(0) N (0, 100I) Repeat the two steps of item 2 4 Compare the results of the two simulations for t near zero and for large t Discuss both the Kalman gain and the diagonal of the covariance matrix This problem is further discussed in Exercise 63 Exercise 56 This problem returns to the application considered in Section 111 As derived in Section 111, the model for the error state vector is p = v, v = b + 1 , b = b b + b (m/s2 ) 1 Assume that b = 600s and that the PSD of b is 001 Hz The vari300 m 2 ance of acceleration measurement is 005 s2 , The aiding sensor residual measurement model is y = p + p
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CHAPTER 5 OPTIMAL STATE ESTIMATION
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2 where p N (0, p ), p = 1m, and the measurement is available at 1Hz Assuming that the system is initially (nominally) stationary at an unknown location, design a Kalman lter by specifying the stochastic model parameters and initial conditions: x(0), P(0), F, Q, , H, , Qd, and R See also Exercise 75
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Exercise 57 In Example 42, the angle error model was (t) = bg (t) + (t) (5109)
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Assume that bg is a Gauss-Markov process with Rb ( )
2 = b e b | |
2 that N (0, g ) and that an angle measurement is available at 05Hz 2 that is modeled as = + where N (0, (10 ) ) Be careful with units Design a Kalman lter by specifying the stochastic model parameters and initial conditions: x(0), P(0), F, Q, , H, , Qd, and R See also Exercise 71
Exercise 58 Assume that y(t) = x(t) + b(t) + n(t)
2 where n(t) N (0, n ) is white and x and b are stationary, independent random processes with correlation functions
Rx ( ) = Rb ( ) =
2 x e x | | 2 b e b | |
If the state vector is z = [x, b] and measurements are available at 05 Hz, specify a set of state space model parameters (F, Q, , H, , Qd, and R) and initial conditions ( (0), P(0)) for the Kalman lter design x Exercise 59 For the system described in Exercise 37 on p 97, assume that the variance of the 01 Hz position measurement is 10m, and that a 2 N (0, a ) with a = 001 m/s and g N (0, g ) with g = 20 10 5 rad/s Hz Hz 1 Design a Kalman lter by specifying the discrete-time stochastic mod el parameters and initial conditions: x(0), P(0), F, Q, , H, , Qd, and R 2 Implement the Kalman lter (time and measurement) covariance propagation equations Plot the diagonal of the covariance matrix versus time
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