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CHAPTER 5 OPTIMAL STATE ESTIMATION
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Physical System + S x(t) 1 x(t) H y(t) G s T + F
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Figure 51: Continuous-time system with input u = 0 The topic of covariance analysis and its use in error budgeting is critical to making informed system level decisions at the design stage
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State Estimation: Review
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This section summarizes the linear stochastic state estimation results from the previous chapters For a linear continuous-time system such as that shown in Figure 51, to enable state estimation in discrete-time, we nd the discrete-time equivalent state-space model of the actual process: xk yk = = xk 1 + Guk 1 + k 1 Hxk + k (51) (52)
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where xk Rn , yk Rp , k Rp , and k Rn are the stochastic signals representing the state, output, measurement noise and driving noise, respectively; u Rm is a known input signal Assuming that the state is observable from the output signal, our objective is to provide an optimal estimate of the state vector The resulting algorithm is referred to as the Kalman lter The basic state estimation block diagram is shown in Figure 52 For simplicity of notation, the system is assumed to be time-invariant and the subscript k s on the system parameter matrices have been dropped (eg, k = ) The assumption of a time-invariant system is not a restriction on the approach The derivation does go through for and the Kalman lter is often applied to time-varying systems
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Discrete-time Equivalent model x wk + y x S k+1 1 k H S k S z + dy L + k nk F Computer
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^ xk+1 1 ^ k x z F
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^ yk
Figure 52: Discrete-time state estimator
51 STATE ESTIMATION: REVIEW
In this model, k and k are signals determined by the environment The designer does not know the actual values of these signals, but formulates the problem so that the following assumptions hold: E k = 0 E k = 0 var ( k , l ) = Rk kl var ( k , l ) = Qdk kl
and the cross-correlation between k and l is zero The symbol kl represents the Kronecker delta In addition, for all k 0, xk is uncorrelated with l for all l and xk is uncorrelated with l for l k The matrices Rk and Qdk are assumed to be positive de nite and the state is assumed to be controllable from k 0 For k > 0, assuming that x+ and positive de nite P+ are known, the 0 state estimate is computed using the algorithm described below The superscripts and + denote the estimates prior and posterior to incorporating k 1 k 1 the measurement, respectively At time k, given x+ , P+ and uk 1 , the prior estimate of the state and output are computed as k x yk = x+ + Guk 1 k 1 = H xk (53) (54)
When the k-th measurement yk becomes available, the measurement residual is computed as (55) yk = yk yk Given the state estimation gain vector Lk , the posterior state estimate is computed as k k (56) x+ = x + Lk yk Given the above state space models for the state and state estimate, with k k x = xk x and x+ = xk x+ , the previous chapters have derived the k k following equations for the prior state estimation error, the measurement residual, and the posterior state estimation error: x k+1
yk x+ k
= = =
x+ + k k H x k + k (I Lk Hk ) x Lk k k
(57) (58) (59)
With reference to eqn (4125), based on eqns (57 59), the covariance of the state estimation error prior to the measurement update is given by equation: (510) P = k P+ k + Qdk k+1 k As in eqn (4126), the covariance matrix for the predicted output error is P k = Hk P Hk + Rk y k (511)
CHAPTER 5 OPTIMAL STATE ESTIMATION
As in eqn (4127), the covariance matrix for the state estimation error posterior to the measurement correction is P+ = (I Lk Hk ) P (I Lk Hk ) + Lk Rk Lk k k (512)
Given that the state is controllable from k , it can be shown that P and k P+ are positive de nite matrices k Note that eqns (57 59) are not used in the state estimator implementation These equations are used for analysis to attain eqns (510 512) which serve as the basis for the derivation of the optimal state estimation gain in Section 52 The optimal state estimator will be implemented using eqns (53 56), eqns (510 512), and a formula for the optimal state estimation gain
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