how to generate barcode using c#.net Discrete-time Equivalent Models in Software

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Discrete-time Equivalent Models
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When the dynamics of the system of interest evolve in continuous time, but analysis and implementation are more convenient in discrete-time, we will require a means for determining a discrete-time model in the form of eqn (465) which is equivalent to eqn (457) at the discrete-time instants tk = kT Speci cation of the equivalent discrete-time model requires computation of the discrete-time state transition matrix k for eqn (465) and the process noise covariance matrix Qd for eqn (467) These computations are discussed in the following two subsections for time invariant systems A method to compute and Qd over longer periods of time for which F or Q may not be constant is discussed in Section 7252
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Calculation of k from F(t)
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For equivalence at the sampling instants when F is a constant matrix, can be determined as in eqn (361): (t) = (t, 0) = eF t , ( , t) = eF( t) , and k = eFT
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(4103)
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where T = tk tk 1 is the sample period Methods for computing the matrix exponential are discussed in Section B12
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47 DISCRETE-TIME EQUIVALENT MODELS Example 420 Assume that for a system of interest, 0 F12 0 F = 0 0 F23 0 0 F33
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(4104)
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and the submatrices denoted by F12 , F23 , and F33 are constant over the interval t [t1 , t2 ] Then, (t2 , t1 ) = eFT where T = t2 t1 Expanding the Taylor series of 1 eFt = I + Ft + (Ft)2 2 is straightforward, but tedious The result is t t I F12 T2 F12 F23 t12 t1 eF33 s dsdt t (t2 , t1 ) = 0 (4105) I F23 t12 eF33 s ds F33 T2 0 0 e which is the closed form solution When F33 can be approximated as zero, the following reduction results 1 2 I F12 T2 2 F12 F23 T2 I F23 T2 (4106) (t2 , t1 ) = 0 0 0 I Eqn (4104) corresponds to the F matrix for certain INS error models after simpli cation If the state transition matrix is required for a time interval [Tm 1 , Tm ] of duration long enough that the F matrix cannot be considered constant, then it may be possible to proceed by subdividing the interval When the interval can be decomposed into subintervals Tm 1 < t1 < t2 < < Tm , where = max(tn tn 1 ) and the F matrix can be considered constant over intervals of duration less than , then by the properties of state transition matrices, (4107) (tn , Tm 1 ) = (tn , tn 1 ) (tn 1 , Tm 1 ) where (tn , tn 1 ) is de ned as in eqn (4103) with F considered as constant for t [tn , tn 1 ) The transition matrix (tn 1 , Tm 1 ) is de ned from previous iterations of eqn (4107) where the iteration is initialized at t = Tm 1 with (Tm 1 , Tm 1 ) = I The iteration continues for the interval of time propagation to yield (Tm , Tm 1 )
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Calculation of Qdk from Q(t)
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For equivalence at the sampling instants, the matrix Qdk must account for the integrated e ect of w(t) by the system dynamics over each sampling
CHAPTER 4 STOCHASTIC PROCESSES
period Therefore, by integration of eqn (457) and comparison with eqn (465), wk must satisfy x(tk+1 ) = eF(tk+1 tk ) x(tk ) +
tk+1 tk
eF(tk+1 ) G( )w( )d
(4108)
Comparison with eqn (465) leads to the de nition:
tk+1
wk =
eF(tk+1 ) G( )w( )d
(4109)
Then, with the assumption that w(t) is a white noise process, we can compute Qdk = cov(wk ) as follows: E =
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