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64 COVARIANCE DIVERGENCE
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10 P(t) for Q=00 Predicted Actual 5 K
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0 10 P(t) for Q=02
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Figure 66: Various forms of performance mismatch Top - Divergence Middle - Optimistic performance prediction Bottom - Pessimistic or conservative design
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Each row of plots in Figure 66 portrays the actual and predicted covari ance, and the Kalman gain for an assumed value of Q In the rst row, the analyst incorrectly models the state as a random constant (ie, Q = 0) Both the Kalman lter gain Kk and predicted covariance (dotted line) are approaching zero However, the actual error covariance (solid line) is diverging and will grow without bound This is due to the uncertainty in the actual state growing, while the (erroneously) computed variance of the estimated state approaches zero The second row corresponds to an optimistic design The design value of the driving noise spectral density is too small (ie, Q = 02), resulting in the Kalman lter predicting better performance than is actually achieved The third row corresponds to a pessimistic design in which the analyst has selected too large a value for the driving noise spectral density The Kalman lter gain is larger than the optimal gain, and the lter achieves better performance than it predicts When the lter design is known to not perfectly match the plant, the pessimistic approach corresponds to the preferred situation If the analyst designs a pessimistic lter and achieves the design speci cation, then the actual system will also achieve the design speci cation Figure 67 shows data from a single trial run corresponding to the lter design shown in the top row of Figure 66 (ie, Qh = 00) In this case, the measurement residual sequence is clearly not a white noise sequence with
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CHAPTER 6 PERFORMANCE ANALYSIS
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5 Residual
10 0 50 100 Time, t, s 150 200
Figure 67: Residual sequence from a single trial for the lter designed with Qh = 00 The solid line with x s is the residual The solid lines without x s is the lters predicted values for r where r = H P H + R
the variance predicted by P Analysis of residual sequences is one of the best methods to detect model de ciencies In doing so, the analyst should keep in mind that Hk Pk Hk + Rk is predicting the variance of the residual rk over an ensemble of experiments Therefore, for example, the residual sequence of Figure 67 does indicate a model de ciency, but by itself does not indicate that the residual is biased If the experiment was repeated N times, with the residual sequence from the i-th trial run denoted by rki , and all residual sequences tended to drift in the same direction, then the analyst might reconsider the system model to nd the cause of and model the bias AlterN natively, if the average of the residual sequences rk = i=1 rki is essentially zero mean, but the ensemble of residual sequences is not well-modeled by the P , then the analyst must consider possible alternative explanations To illustrate the nature of a residual sequence for a lter designed with a correct model, Figure 68 shows the residual for the lter designed with Qh = 05 The residual is essentially zero mean, has no noticeable trends, and has steady-state variance that closely matches the P
References and Further Reading
The main references for the presentation of this chapter were [58, 93]
66 EXERCISE
6 4 2 Residual 0 2 4 6 0 50 100 Time, t, s 150 200
Figure 68: Residual sequence from a single trial for the lter designed with Qh = 05 The solid line with x s is the residual The solid lines without x s is the lters predicted values for r where r = H P H + R
Exercise
Exercise 61 This example is intended to motivate the importance of having an accurate design model Let the actual system dynamics (ie, eqn (61)) be x y = = 0 2 n 1 0 1 0 x+ 0 1 nx
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