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56 IMPLEMENTATION ISSUES
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8 Given that the state is controllable from the driving noise and is observable from the measurements, the Kalman lter estimation error dynamics are asymptotically stable Therefore, for an observable system, the e ects of the initial conditions (x and P ) decay away and 0 0 do not a ect the solution as k Since the above properties rely on three assumptions, it is natural to consider the reasonableness of these assumptions Although most physical systems are in fact non-linear, the linearity assumption can be locally applied when the system non-linearities are linearizable and the distance from the linearizing trajectory is small These conditions are usually valid in navigation problems, especially when aiding information such as GPS is available The white noise assumption is also valid, since colored driving noise can be modeled by augmenting a linear system with white driving noise to the system model The Gaussian assumption is valid for most driving noise sources as expected based on the Central Limit Theorem [107] Even when an application involves non-Gaussian noise, it is typical to proceed as if the noise source were Gaussian with appropriately de ned rst and second moments In such cases, although a better nonlinear estimator may exist, the Kalman lter will provide the minimum variance, linear, unbiased state estimate
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This section discusses several issues related to the real-time implementation of Kalman lters
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Scalar Measurement Processing
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The Kalman lter algorithm as presented in Table 55 is formulated to process a vector of m simultaneous measurements The portion of the measurement update that requires the most computing operations (ie, FLOP s) is the covariance update and gain vector calculation For example, the standard algorithm K x+ P+ = P H R + HP H = x + K( H ) y x = (I KH) P
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(572) (573) (574)
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can be programmed to require 3 n2 m + 3 nm2 + nm + m3 + 1 m2 + 1 m 2 2 2 2 FLOP s Alternatively, when R is a diagonal matrix, the measurements can be equivalently treated as m sequential measurements with a zero-width time-interval between measurements, which results in signi cant computational savings
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CHAPTER 5 OPTIMAL STATE ESTIMATION At time t = kT , de ne P1 = (k) P H1 H= Hm x1 = (k) x R1 R= 0 0
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(575)
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Rm
Then, the equivalent scalar measurement processing algorithm is, for i = 1 to m, Ki xi+1 Pi+1 Pi Hi Ri + Hi Pi Hi = xi + Ki ( i Hi xi ) y = = (I Ki Hi ) Pi , (576) (577) (578)
with the state and error covariance matrix posterior to the set of measurements de ned by x+ (k) P+ (k) = xm+1 = Pm+1 (579) (580)
The total number of computations for the m scalar measurement updates is m 3 n2 + 5 n plus m scalar divisions Thus it can be seen that m scalar 2 2 updates are computationally cheaper for all m At the completion of the m scalar measurements x+ (k) and P+ (k) will be identical to the values that would have been computed by the corresponding vector measurement update The state Kalman gain vectors Ki corresponding to the scalar updates are not equal to the columns of the state feedback gain matrix that would result from the corresponding vector update This is due to the di erent ordering of the updates a ecting the error covariance matrix Pi at the intermediate steps during the scalar updates Example 56 To illustrate the di erences between the vector and scalar measurement updates, consider one measurement update for a second-order system having P (k) = H= 100 0 0 , x (k) = 0 100 0 10 00 1024 2120 00 10 , y= 07 03 1391 05 05 1484 , , and R = I4
56 IMPLEMENTATION ISSUES For the vector update, by eqn (572), the Kalman gain is K(k) = 06276 02139 02139 03752 02069 08136 00944 02999
which results by eqn (573-574) in the state estimate x+ (k) = 101823 208216 with covariance P+ (k) = 06276 02139 02139 08136
For the set of m = 4 scalar updates, by eqn (576) the gain sequence is m Ki 1 09901 00000 2 00000 09901 3 04403 01887 4 02069 02999
The sequence of state estimates from eqn (577) is m xi 1 0 0 2 101386 00000 3 101386 209901 4 103658 210874 5 101823 208216
By eqn (578) the second, fourth, and nal error covariance matrices are P2 = 09901 00000 00000 1000000 , P4 = 06850 01308 01308 09341 , and
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