The document discusses different types of discrete Kalman filters, including the standard discrete Kalman filter, one with a fixed forgetting factor λ, and one with a varying forgetting factor λ. It provides the mathematical equations for each type of filter. An example application to a time series problem is shown. The results demonstrate that the discrete Kalman filter with a varying forgetting factor λ produces the minimum mean and standard deviation of the estimation error compared to the other methods. MATLAB code examples for each filter type are also provided.

1. Contents
1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Discrete Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Discrete Kalman Filter with Forgetting Factor λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Discrete Kalman Filter with Varying Forgetting Factor λ . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.7 MATLAB Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.9 Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1

2. 1.1. DEFINITIONS Discrete Kalman Filter
1.1 Definitions
• x : process model states
• φ : state-transition matrix
• H : measurement matrix
• z : measurement matrix
• ˆx−
0 : initial conditions of states
• P−
o : initial error covariance matrix
• R : variance of measurement error matrix
• Q : variance of process model noise
• λ : forgetting factor 0 < λ < 2
• λ∗ : initial forgetting factor
1.2 Notations
•ˆ: estimate
• ˆx−
k : a prior estimate of xk
• ˆx+
k : a posterior estimate of xk
1.3 Discrete Kalman Filter
1. Guess initial values of P−
0 and ˆx−
0
2. Calculate the gain :
• Kk = P−
k HT HP−
k HT + R
−1
3. Update estimate :
• ˆx+
k = ˆx−
k + Kk zk − Hˆx−
k
4. Update error :
• P+
k = (I − KkH) P−
k
5. Project ahead :
• ˆx−
k+1 = φˆx+
k
• P−
k+1 = φP+
k φT + Q
• P−
k+1 =
P−
k+1+P−T
k+1
2
1.4 Discrete Kalman Filter with Forgetting Factor λ
1. Guess initial values of P−
0 and ˆx−
0
2. Calculate the gain :
• Kk = P−
k HT HP−
k HT + Rλ
−1
3. Update estimate :
• ˆx+
k = ˆx−
k + Kk zk − Hˆx−
k
4. Update error :
• P+
k = (I − KkH)
P−
k
λ
5. Project ahead :
• ˆx−
k+1 = φˆx+
k
• P−
k+1 = φP+
k φT + Q
• P−
k+1 =
P−
k+1+P−T
k+1
2
Mohamed Mohamed El-Sayed Atyya Page 2 of 12

3. 1.5. DISCRETE KALMAN FILTER WITH VARYING FORGETTING FACTOR λ Discrete Kalman Filter
1.5 Discrete Kalman Filter with Varying Forgetting Factor λ
1. Guess initial values of P−
0 , ˆx−
0 and λ∗
2. Calculate the gain :
• Kk = P−
k HT HP−
k HT + Rλk
−1
3. Update estimate :
• ˆx+
k = ˆx−
k + Kk zk − Hˆx−
k
4. Update error :
• P+
k = (I − KkH)
P−
k
λk
5. Project ahead :
• ˆx−
k+1 = φˆx+
k
• P−
k+1 = φP+
k φT + Q
• P−
k+1 =
P−
k+1+P−T
k+1
2
6. Forgetting factor λ:
• = zk − Hˆx+
k
• λk+1 = 1 −
1−ˆx+T
k Kk
2
σ2( )µ( )
• λk+1 =



> 0.95 λk+1 = 0.95
< 0.3 λk+1 = 0.3
else λk+1 = λk+1
1.6 Example
• φ = 1
• H = 1
• R = 100
• Q = 1
• ˆx−
0 = 1
• P−
0 = 0
• ∆t = 1.0256
• x(1) = 1
• x(k + 1) = φx(k) + ∆t + normrnd(0,
√
Q)
• z(k) = Hx(k) + normrnd(0,
√
R)
• λ = 0.9
• λ∗ = 0.9
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4. 1.6. EXAMPLE Discrete Kalman Filter
Results
Discrete Kalman Filter
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5. 1.6. EXAMPLE Discrete Kalman Filter
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6. 1.6. EXAMPLE Discrete Kalman Filter
Discrete Kalman Filter with Forgetting Factor λ
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7. 1.6. EXAMPLE Discrete Kalman Filter
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8. 1.6. EXAMPLE Discrete Kalman Filter
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9. 1.6. EXAMPLE Discrete Kalman Filter
Discrete Kalman Filter with Varying Forgetting Factor λ
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10. 1.6. EXAMPLE Discrete Kalman Filter
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11. 1.6. EXAMPLE Discrete Kalman Filter
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12. 1.7. MATLAB CODES Discrete Kalman Filter
Comments
From results we see that the error% have the data,
DKF DKF with λ DKF with varying λ
µ 38.3429 26.5472 22.7235
σ 20.224 21.2737 17.5789
It’s seems that DKF with varying λ has the min. µ and σ
1.7 MATLAB Codes
1.3 http://goo.gl/s8KB0e
1.4 http://goo.gl/NiVVE0
1.5 http://goo.gl/KCCsGF
1.8 References
1. Robert Grover Brown and Patrick Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering
1.9 Contacts
mohamed.atyya94@eng-st.cu.edu.eg
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