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Minimum Variance Gain Derivation
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Eqn (512) is useful for the selection of the estimator gain Lk , because it allows us to evaluate the covariance matrices P+ that would result from alk ternative choices of Lk In fact, the diagonal of P+ contains the variance of k each element of the state vector The scalar function T r(P+ ) = trace(P+ ) k k is the sum of the variances of the individual states (ie, the mean-squarederror); therefore, for the purposes of optimization it is reasonable to minimize the cost function J(Lk ) = T r(P+ ) k
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Kalman Gain Derivation
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(513)
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Dropping the subscripts and multiplying out expression (512) yields P+ = P LHP P H L + L HP H + R L
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This expression is a second order matrix polynomial equation in the variable L Therefore, the mean-squared posterior state estimation error, which can be computed conveniently as T r(P+ ), is also a function of L Using the properties of the T r function, which are reviewed in Section B2, we can reduce the expression for T r(P+ ) as follows: T r(P+ ) = T r P LHP P H L + L HP H + R L = T r[P ] 2T r[LHP ] + T r[L HP H + R L ] Selection of L to minimize T r(P+ ) requires di erentiation of the scalar function T r(P+ ) with respect to L Using eqns (B55 B56) for the derivative of T r with respect to a matrix, results in d T r(P+ ) = 2 HP dL + 2L HP H + R
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52 MINIMUM VARIANCE GAIN DERIVATION
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Setting this expression equal to the zero vector and solving for L yields the formula for the Kalman gain vector, which will be denoted by K instead of L: 1 (514) Kk = P Hk Hk P Hk + Rk k k where we have used the fact that P is symmetric Since P and R are positive de nite, the matrix Hk P Hk + Rk is also positive de nite, which k ensures that the inverse in the above formula exist Since Hk P Hk + Rk k is also the second derivative of T r(P+ ) with respect to L, the Kalman gain Kk yields a minimum value for the mean-squared cost
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Kalman Gain: Posterior Covariance
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When the Kalman gain is used, a simpli ed equation can be derived for the posterior error covariance matrix P+ The derivation uses the fact that Kk Hk P Hk + Rk = P Hk k k (515)
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which is easily derived from eqn (514) The derivation proceeds from eqn (513) as follows: P+ = P KHP + KRK + KHP H K P H K = [I KH] P + K R + HP H K P H K = [I KH] P + P H K P H K = [I KH] P (516)
Note that the nal expression in eqn (516) is only valid for the Kalman gain K, while (512) is valid for any estimator gain vector L
Summary
This section has presented a derivation of the Kalman gain formula shown in eqn (514) and has derived a simpli ed formula for the posterior error covariance matrix that is shown in eqn (516) The Kalman gain is the time-varying gain sequence that minimizes the mean-square state estimation error at each measurement instant Computation of the Kalman gain sequence requires computation of the prior and posterior error covariance matrices The prior covariance P is used directly in eqn (514) The k posterior covariance P+ is used to compute the P for use with the next k k+1 measurement The derivation of this section is short and clearly shows the meansquared optimality The following section presents an alternative derivation that obtains the same results The alternative derivation is considerably longer, but also provides additional insight into the operation of the Kalman lter
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