The standard Kalman filter is a filter based on Bayesian filter (3), with two extra assumptions: 1) the process and measurement noises are white and Gaussian, and 2) the process and measurement models are linear. The document derives the prediction and update equations for the Kalman filter, which provide an optimal estimate of the state x given prior state estimates and current measurements. The prediction equations forecast the next state and its covariance. An update is then made using a proportional gain and current measurement to provide a corrected state estimate.

1. The standard Kalman filter is a filter based on Bayesian filter (3), with two extra assumptions:
1. The 3 functions:
2. All the noises (Process noise and measurement noise) are
rewrite the state equations previously declared in (1) and (2) as:We
where
𝑥(𝑘) ∈
𝐹 ∈ ℳ
𝑤(𝑘 −
𝐺 ∈ ℳ
𝑢[𝑘] ∈
𝑘 ∈ ℕ
As our goal is to calculate
assumption that
from the formulas we previously declared as (1, 2)
Therefore, the
This is because
linear, time
Same holds for the observation, it’s possible to estimate it back
The standard Kalman filter is a filter based on Bayesian filter (3), with two extra assumptions:
The 3 functions:
All the noises (Process noise and measurement noise) are
rewrite the state equations previously declared in (1) and (2) as:
∈ ℝ𝑛
is the state vector at time k.
ℳ𝑛×𝑛 is the state transition model.
− 1) ∈ ℝ𝑛
is the process noise.
ℳ𝑛×𝑛 is the control model.
∈ ℝ𝑛
is the control input vector.
ℕ is the time.
As our goal is to calculate
assumption that 𝑥𝑘
from the formulas we previously declared as (1, 2)
𝑥̂(𝑘
Therefore, the expected state transitionexpected state transitionexpected state transitionexpected state transition
This is because 𝑥̂
linear, time-invariant matrices (linear) and
Same holds for the observation, it’s possible to estimate it back
𝑧̅(𝑘)
Appendi
The standard Kalman filter is a filter based on Bayesian filter (3), with two extra assumptions:
The 3 functions: 𝑓𝑘, 𝑔𝑘 and
All the noises (Process noise and measurement noise) are
rewrite the state equations previously declared in (1) and (2) as:
𝑥(𝑘)
𝑧(𝑘)
is the state vector at time k.
is the state transition model.
is the process noise.
is the control model.
is the control input vector.
is the time.
As our goal is to calculate 𝑥
𝑘−1 is known to us, we can retrieve the formula needed to do that starting
from the formulas we previously declared as (1, 2)
𝑘|𝑘 − 1) = 𝐸
= 𝐹. 𝐸[
expected state transitionexpected state transitionexpected state transitionexpected state transition
𝑥̂(
𝑥̂(𝑘 − 1) = 𝐸
invariant matrices (linear) and
Same holds for the observation, it’s possible to estimate it back
) = 𝐸[𝑧(𝑘)]
Appendix E: Kalman demonstration
The standard Kalman filter is a filter based on Bayesian filter (3), with two extra assumptions:
and ℎ𝑘 are linear
All the noises (Process noise and measurement noise) are
rewrite the state equations previously declared in (1) and (2) as:
= 𝐹𝑥(𝑘 − 1)
= 𝐻𝑥(𝑘) + 𝑣
is the state vector at time k.
is the state transition model.
is the process noise.
is the control model.
is the control input vector.
𝑥̂𝑘, the expected value of
is known to us, we can retrieve the formula needed to do that starting
from the formulas we previously declared as (1, 2)
𝐸[𝑥(𝑘)] =
[𝑥(𝑘 − 1)] +
expected state transitionexpected state transitionexpected state transitionexpected state transition
̂(𝑘|𝑘 − 1) =
𝐸[𝑥(𝑘 − 1)
invariant matrices (linear) and
Same holds for the observation, it’s possible to estimate it back
= 𝐸[𝐻𝑥(𝑘)
E: Kalman demonstration
The standard Kalman filter is a filter based on Bayesian filter (3), with two extra assumptions:
linear.
All the noises (Process noise and measurement noise) are
rewrite the state equations previously declared in (1) and (2) as:
) + 𝐺𝑢[𝑘 − 1
𝑣(𝑘)
𝑛 ∈
𝐻 ∈
𝑧(𝑘
𝑣(𝑘
𝑚 ∈
, the expected value of
is known to us, we can retrieve the formula needed to do that starting
from the formulas we previously declared as (1, 2)
𝐸[𝐹𝑥(𝑘 − 1
+ 𝐺𝑢[𝑘 − 1
would be:
= 𝐹𝑥̂(𝑘 − 1)
)] and 𝐸[𝑤
invariant matrices (linear) and 𝑢[𝑘 − 1]
Same holds for the observation, it’s possible to estimate it back
) + 𝑣(𝑘)] =
E: Kalman demonstration
The standard Kalman filter is a filter based on Bayesian filter (3), with two extra assumptions:
All the noises (Process noise and measurement noise) are white
rewrite the state equations previously declared in (1) and (2) as:
1] + 𝑤(𝑘 − 1
∈ ℕ is the dimension of the state vector.
∈ ℳ𝑛×𝑚 is the measurement model.
𝑘) ∈ ℝ𝑚
is the measurement vector.
𝑘) ∈ ℝ𝑚
is the measurement noise.
∈ ℕ is the dim. of measurement vector.
, the expected value of 𝑥𝑘 for a given time k, using the
is known to us, we can retrieve the formula needed to do that starting
1) + 𝐺𝑢[𝑘 −
1] + 𝐸[𝑤(𝑘
) + 𝐺𝑢[𝑘 −
𝑤(𝑘 − 1)] =
is known control input
Same holds for the observation, it’s possible to estimate it back
𝐻. 𝐸[𝑥(𝑘)]
E: Kalman demonstration
The standard Kalman filter is a filter based on Bayesian filter (3), with two extra assumptions:
white and Gaussian
rewrite the state equations previously declared in (1) and (2) as:
1)
is the dimension of the state vector.
is the measurement model.
is the measurement vector.
is the measurement noise.
is the dim. of measurement vector.
for a given time k, using the
is known to us, we can retrieve the formula needed to do that starting
− 1] + 𝑤(𝑘 −
− 1)]
− 1]
0 (white noise)
is known control input
Same holds for the observation, it’s possible to estimate it back
= 𝐻𝑥̂(𝑘|𝑘 −
E: Kalman demonstration
The standard Kalman filter is a filter based on Bayesian filter (3), with two extra assumptions:
Gaussian.
is the dimension of the state vector.
is the measurement model.
is the measurement vector.
is the measurement noise.
is the dim. of measurement vector.
for a given time k, using the
is known to us, we can retrieve the formula needed to do that starting
− 1)]
(white noise), F and G are
is known control input
− 1)
The standard Kalman filter is a filter based on Bayesian filter (3), with two extra assumptions:
( 1 )
is the dimension of the state vector.
is the dim. of measurement vector.
for a given time k, using the
is known to us, we can retrieve the formula needed to do that starting
( 2 )
( 3 )
F and G are
( 4 )

2. since 𝐸
The resulting prediction equations are given
The reason
knowledge of the current observation but only previous data/model). The predicted state and
observation in equations (13) and (14) can be updated with the corresponding measurement
z(k) of the current time step k.
To correct the state estimate, we introduce a
from the current measurement
The main goal of the algorithm of Kalman,
way to calculate this gain
The error a priori covariance can be defined as follows:
The detailed calculation is completed in the following:
This is due to
(𝑥(𝑘) and
𝐸[𝑣(𝑘)] = 0
The resulting prediction equations are given
The reason 𝑥̂(𝑘|𝑘 −
knowledge of the current observation but only previous data/model). The predicted state and
observation in equations (13) and (14) can be updated with the corresponding measurement
f the current time step k.
To correct the state estimate, we introduce a
from the current measurement
The main goal of the algorithm of Kalman,
way to calculate this gain
The error a priori covariance can be defined as follows:
The detailed calculation is completed in the following:
𝑃𝑘|𝑘−1 = 𝐸
This is due to
and 𝑤(𝑘 − 1)
(white noise)
The resulting prediction equations are given
𝑥̂(𝑘|𝑘
− 1) is noted like that is that because it’s predicted a priori (without
knowledge of the current observation but only previous data/model). The predicted state and
observation in equations (13) and (14) can be updated with the corresponding measurement
f the current time step k.
To correct the state estimate, we introduce a
from the current measurement
𝑥̂(𝑘
The main goal of the algorithm of Kalman,
way to calculate this gain [5]. But first, we are going to prepare the stage for calculating it.
The error a priori covariance can be defined as follows:
𝑃𝑘|𝑘−1 = 𝐸
The detailed calculation is completed in the following:
𝐸[(𝐹(𝑥̂(𝑘 − 1
− 1))(𝐹
= 𝐹𝐸[
+ 𝐸[𝑤
𝐸
= 𝐸[ (𝑥̂(
) are uncorrelated and
(white noise).
The resulting prediction equations are given
𝑘 − 1) = 𝐹𝑥
𝑧̅(𝑘)
is noted like that is that because it’s predicted a priori (without
knowledge of the current observation but only previous data/model). The predicted state and
observation in equations (13) and (14) can be updated with the corresponding measurement
f the current time step k.
To correct the state estimate, we introduce a
from the current measurement 𝑧(𝑘)
𝑘|𝑘) = 𝑥̂(𝑘|
The main goal of the algorithm of Kalman,
. But first, we are going to prepare the stage for calculating it.
The error a priori covariance can be defined as follows:
𝐸 [(𝑥̂(𝑘|𝑘 −
The detailed calculation is completed in the following:
1|𝑘 − 1) − 𝑥(
𝐹(𝑥̂(𝑘 − 1|𝑘
[(𝑥̂(𝑘 − 1|𝑘 −
𝑤(𝑘 − 1)𝑤(𝑘 −
𝐸[ (𝑥̂(𝑘 − 1|𝑘
[ ̂(𝑘 − 1|𝑘 − 1
are uncorrelated and
The resulting prediction equations are given by:
𝑥̂(𝑘 − 1|𝑘 −
= 𝐻𝑥̂(𝑘|𝑘
is noted like that is that because it’s predicted a priori (without
knowledge of the current observation but only previous data/model). The predicted state and
observation in equations (13) and (14) can be updated with the corresponding measurement
To correct the state estimate, we introduce a proportional gainproportional gainproportional gainproportional gain
|𝑘 − 1) + 𝐾
The main goal of the algorithm of Kalman, and what is in the next lines is to get a proper
. But first, we are going to prepare the stage for calculating it.
The error a priori covariance can be defined as follows:
( 1) − 𝑥(𝑘))(
The detailed calculation is completed in the following:
(𝑘) + 𝑤(𝑘
𝑘 − 1) − 𝑥(𝑘)
− 1) − 𝑥(𝑘))
− 1)𝑇
]
𝑘 − 1) − 𝑥(𝑘
1) − 𝑥(𝑘))] 𝐸
are uncorrelated and 𝑤(𝑘 − 1) is a white noise with mean 0)
− 1) + 𝐺𝑢[𝑘
𝑘 − 1)
is noted like that is that because it’s predicted a priori (without
knowledge of the current observation but only previous data/model). The predicted state and
observation in equations (13) and (14) can be updated with the corresponding measurement
proportional gainproportional gainproportional gainproportional gain
𝐾(𝑧(𝑘) − 𝑧̅(
and what is in the next lines is to get a proper
. But first, we are going to prepare the stage for calculating it.
The error a priori covariance can be defined as follows:
)(𝑥̂(𝑘|𝑘 − 1)
The detailed calculation is completed in the following:
) + 𝑤(𝑘 − 1))
)(𝑥̂(𝑘 − 1|𝑘 −
𝑘))𝑤(𝑘 − 1)
) 𝐸[𝑤(𝑘 − 1)𝑇
is a white noise with mean 0)
𝑘 − 1]
is noted like that is that because it’s predicted a priori (without
knowledge of the current observation but only previous data/model). The predicted state and
observation in equations (13) and (14) can be updated with the corresponding measurement
proportional gainproportional gainproportional gainproportional gain using the feedback we get
̅(𝑘))
and what is in the next lines is to get a proper
. But first, we are going to prepare the stage for calculating it.
) − 𝑥(𝑘))
𝑇
]
)𝑇
]
− 1) − 𝑥(𝑘))
𝑇
]
𝑇
] = 0
is a white noise with mean 0)
is noted like that is that because it’s predicted a priori (without
knowledge of the current observation but only previous data/model). The predicted state and
observation in equations (13) and (14) can be updated with the corresponding measurement
using the feedback we get
)
and what is in the next lines is to get a proper
. But first, we are going to prepare the stage for calculating it.
) ]
𝑇
] 𝐹𝑇
is a white noise with mean 0)
( 5 )
( 6 )
is noted like that is that because it’s predicted a priori (without
knowledge of the current observation but only previous data/model). The predicted state and
observation in equations (13) and (14) can be updated with the corresponding measurement
using the feedback we get
( 7 )
and what is in the next lines is to get a proper
. But first, we are going to prepare the stage for calculating it.
( 8 )
( 9 )
( 10 )

3. The same holds for
𝐸[(𝑥̂(𝑘
𝑄𝑘−1 =
The residual covariance can
Which can be detailed as follows
Since 𝐸
𝑣(𝑘 − 1)
𝐸[(𝐻(𝑥
Besides,
( 14 )
We get
The error of a posteriori estimate is given by
To simplify the expression, we note
Its covariance matrix can be expressed using
The same holds for
( 𝑘 − 1|𝑘 − 1)
𝐸[𝑤(𝑘 − 1)
The residual covariance can
Which can be detailed as follows
𝑆(𝑘) = 𝐸 [
= 𝐻𝐸
𝐸[𝐻(𝑥(𝑘) −
) are uncorrelated and
(𝑥(𝑘) − 𝑥̂(𝑘|𝑘
Besides, 𝑃𝑘|𝑘−1 =
The error of a posteriori estimate is given by
=
To simplify the expression, we note
Its covariance matrix can be expressed using
The same holds for 𝐸[ (𝑥̂(
) − 𝑥(𝑘))(𝑥̂
)𝑤(𝑘 − 1)𝑇
]. We obtain the new representation of (17),
The residual covariance can be calculating using
𝑆(
Which can be detailed as follows
[(𝐻(𝑥(𝑘) −
[(𝑥(𝑘) − 𝑥̂(
( − 𝑥̂(𝑘|𝑘 − 1))
are uncorrelated and
𝑘 − 1)))𝑇
𝑣(𝑘
= 𝐸 [(𝑥(𝑘) −
The error of a posteriori estimate is given by
= 𝑥
= 𝑥̂(𝑘|𝑘 − 1
= (𝐼
To simplify the expression, we note
Its covariance matrix can be expressed using
[ ̂(𝑘 − 1|𝑘 −
)(𝑥̂(𝑘 − 1|𝑘 −
. We obtain the new representation of (17),
𝑃𝑘|𝑘−1 = 𝐹
be calculating using
(𝑘) = 𝐸[(𝑧(𝑘
Which can be detailed as follows:
( − 𝑥̂(𝑘|𝑘 − 1)
̂(𝑘|𝑘 − 1))(𝑥
)𝑣(𝑘)𝑇
𝐻𝑇
] =
are uncorrelated and 𝑣(𝑘 − 1) is a white noise with mean 0), the same holds for
) 𝑘)𝐻] = 0
( − 𝑥̂(𝑘|𝑘 − 1
𝑆(𝑘) =
The error of a posteriori estimate is given by
𝑘 =
𝑥̂(𝑘|𝑘 − 1) +
1) + 𝐾(𝐻(𝑥
𝐼 − 𝐾𝐻)(𝑥̂(
To simplify the expression, we note 𝑥̃𝑘|𝑘−1
Its covariance matrix can be expressed using
1) − 𝑥(𝑘))
− 1) − 𝑥(𝑘)
. We obtain the new representation of (17),
𝐹𝑃𝑘−1|𝑘−1𝐹
be calculating using
( 𝑘) − 𝑧̅(𝑘))(𝑧
)) + 𝑣(𝑘)) (
)(𝑥(𝑘) − 𝑥̂(𝑘|𝑘
] = 𝐻𝐸[𝑥(𝑘)
is a white noise with mean 0), the same holds for
1))(𝑥(𝑘) − 𝑥̂
= 𝐻𝑃𝑘|𝑘−1𝐻
The error of a posteriori estimate is given by
= 𝑥̂(𝑘|𝑘) − 𝑥
+ 𝐾(𝑧(𝑘) −
𝑥(𝑘) − 𝑥̂(𝑘|𝑘
( ̂(𝑘|𝑘 − 1) −
−1 = 𝑥̂(𝑘|𝑘
Its covariance matrix can be expressed using
)
𝑇
𝑤(𝑘 − 1)]
𝑇
], and by definition
. We obtain the new representation of (17),
𝐹𝑇
+ 𝑄𝑘−1
)(𝑧(𝑘) − 𝑧̅(𝑘))
) (𝐻(𝑥(𝑘) − 𝑥̂
𝑘 − 1))
𝑇
] 𝐻
− 𝑥̂(𝑘|𝑘 − 1
is a white noise with mean 0), the same holds for
( 𝑥̂(𝑘|𝑘 − 1))
𝑇
𝐻𝑇
+ 𝑅𝑘
𝑥(𝑘)
( − 𝑧̅(𝑘)) − 𝑥(
𝑘 − 1)) + 𝑣(
− 𝑥(𝑘)) + 𝐾𝑣
𝑘 − 1) − 𝑥(𝑘
] = 0, remarking
, and by definition
. We obtain the new representation of (17),
)
𝑇
]
( 𝑥̂(𝑘|𝑘 − 1))
) ] 𝐻𝑇
+ 𝐸[𝑣(𝑘)
1)]𝐸[𝑣(𝑘)𝑇
]𝐻
is a white noise with mean 0), the same holds for
)
𝑇
] and
) (𝑘)
(𝑘)) − 𝑥(𝑘
) 𝐾𝑣(𝑘)
𝑘)
emarking 𝑃𝑘
, and by definition
) + 𝑣(𝑘))
𝑇
]
)𝑣(𝑘)𝑇
]
]𝐻𝑇
= 0 (𝑥
is a white noise with mean 0), the same holds for
and 𝑅𝑘 = 𝐸[𝑣(
𝑘)
𝑘−1|𝑘−1 =
, and by definition
( 11 )
( 12 )
) ]
( 13 )
𝑥(𝑘) and
is a white noise with mean 0), the same holds for
(𝑘)𝑣(𝑘)𝑇
]
( 15 )
( 16 )

4. As we know that
the latter being a white noise with mean 0) and
Substituting
Using equation (23)
The idea of Kalman is to use Linear Quadratic Estimation of
minimizing
Jacobians)
Note 𝑃𝑘
𝑇
For the o
Therefore, the Kalman Gain, will have the following final expression
Method1
= 𝐸 [
=
As we know that 𝛼𝐸
the latter being a white noise with mean 0) and
Substituting 𝑃𝑘|𝑘−1
𝑃𝑘|𝑘
= (𝐼
= (
Using equation (23)
The idea of Kalman is to use Linear Quadratic Estimation of
minimizing 𝐽 = 𝐸[
Jacobians)
𝑘|𝑘−1
𝑇
= 𝑃𝑘|𝑘−
For the optimal 𝐾𝑜𝑝𝑡
Therefore, the Kalman Gain, will have the following final expression
Method1 Finally, replacing equation (29) in equation (
𝑃𝑘|𝑘 =
[((𝐼 − 𝐾𝐻)
= (𝐼 − 𝐾𝐻)𝐸
𝛼𝐸[𝑥̃𝑘|𝑘−1𝑣(
the latter being a white noise with mean 0) and
1 = 𝐸[𝑥̃𝑘|𝑘−1
= (𝐼 − 𝐾𝐻
𝐼 − 𝐾𝐻)𝑃𝑘|𝑘
(𝐼 − 𝐾𝐻)𝑃𝑘
Using equation (23), to ease writings, we note
𝑃𝑘|𝑘 = (
The idea of Kalman is to use Linear Quadratic Estimation of
[ 𝑘 𝑘
𝑇
] in regard to K, a first idea is to use
𝜕 𝑡𝑟(𝑃
𝜕𝐾
−1 (definite symmetric by definition (16) )
𝑜𝑝𝑡, we derive it from
Therefore, the Kalman Gain, will have the following final expression
Finally, replacing equation (29) in equation (
𝑃𝑘|𝑘 = (
= 𝑃𝑘|𝑘−1
= 𝑃𝑘|𝑘−1
= 𝐸 [(𝑥̂(𝑘|𝑘
)(𝑥̃𝑘|𝑘−1) + 𝐾𝑣
𝐸[𝑥̃𝑘|𝑘−1𝑥̃𝑘|𝑘
𝑇
𝑘)𝑇
] = 𝛼𝐸[
the latter being a white noise with mean 0) and
1𝑥̃𝑘|𝑘−1
𝑇
], 𝑅
𝐾𝐻)𝐸[𝑥̃𝑘|𝑘−1𝑥̃
𝑘−1 + 𝑃𝑘|𝑘−1
𝑃𝑘|𝑘−1 + 𝑃𝑘|𝑘−
, to ease writings, we note
(𝐼 − 𝐾𝐻)𝑃𝑘
The idea of Kalman is to use Linear Quadratic Estimation of
in regard to K, a first idea is to use
𝑃𝑘|𝑘)
𝐾
= −𝑃𝑘
𝑇
(definite symmetric by definition (16) )
, we derive it from
𝐾𝑜𝑝𝑡
Therefore, the Kalman Gain, will have the following final expression
𝐾𝑜𝑝𝑡
Finally, replacing equation (29) in equation (
(𝐼 − 𝐾𝐻)𝑃𝑘
1 − 𝐾𝐻𝑃𝑘|𝑘
1 − 𝐾(𝑃𝑘|𝑘−
( 𝑘) − 𝑥(𝑘))(𝑥
) 𝐾𝑣(𝑘)) ((𝐼
𝑘−1](𝐼 − 𝐾𝐻
[𝑥̃𝑘|𝑘−1]𝐸[𝑣(
the latter being a white noise with mean 0) and 𝛼 being a constant.
] = 𝐸[𝑣(𝑘)𝑣(
𝑥̃𝑘|𝑘−1
𝑇
](𝐼 − 𝐾𝐻
1𝐻𝑇
𝐾𝑇
− 𝐾𝐻
−1𝐻𝑇
𝐾𝑇
−
, to ease writings, we note 𝑆𝑘 instead of
𝑘|𝑘−1 + 𝑃𝑘|𝑘−
The idea of Kalman is to use Linear Quadratic Estimation of
in regard to K, a first idea is to use
𝑘|𝑘−1
𝑇
𝐻𝑇
− 𝑃
(definite symmetric by definition (16) )
, we derive it from
𝜕 𝑡𝑟(𝑃𝑘|𝑘)
𝜕𝐾
= 0
𝑜𝑝𝑡𝑆𝑘 = 𝑃𝑘|𝑘−
Therefore, the Kalman Gain, will have the following final expression
= 𝑃𝑘|𝑘−1𝐻
Finally, replacing equation (29) in equation (
𝑘|𝑘−1 + 𝑃𝑘|𝑘−
𝑘−1 + 𝐾𝑆𝑘𝐾
( −1𝐻𝑇
)𝑇
)(𝑥̂(𝑘|𝑘) − 𝑥(
) ( − 𝐾𝐻)(𝑥̃𝑘|
𝐾𝐻)𝑇
+ 𝐾𝐸[𝑣
(𝑘)𝑇
] = 0 (𝑥̃
being a constant.
(𝑘)𝑇
]
𝐾𝐻)𝑇
+ 𝐾𝐸
𝐾𝐻𝑃𝑘|𝑘−1𝐻𝑇
− 𝐾(𝐻𝑃𝑘|𝑘−1
instead of 𝑆(𝑘
−1𝐻𝑇
𝐾𝑇
−
The idea of Kalman is to use Linear Quadratic Estimation of
in regard to K, a first idea is to use 𝑃𝑘
𝑃𝑘|𝑘−1
𝑇
𝐻𝑇
+
(definite symmetric by definition (16) )
0,
−1𝐻𝑇
Therefore, the Kalman Gain, will have the following final expression
𝑇
𝑆𝑘
−1
Finally, replacing equation (29) in equation (27), we obtain
−1𝐻𝑇
𝐾𝑇
−
𝐾𝑇
− 𝐾𝑆𝑘𝐾
)
(𝑘))
𝑇
]
( |𝑘−1) + 𝐾𝑣(𝑘
[𝑣(𝑘)𝑣(𝑘)𝑇
]𝐾
𝑥̃𝑘|𝑘−1uncorrelated to
being a constant.
𝐸[𝑣(𝑘)𝑣(𝑘)𝑇
𝑇
𝐾𝑇
+ 𝐾𝑅
1𝐻𝑇
− 𝑅𝑘)𝐾
𝑘)
𝐾𝑆𝑘𝐾𝑇
The idea of Kalman is to use Linear Quadratic Estimation of 𝑥(𝑘), this could be done by
𝑘|𝑘’s trace (to avoid passing by
2𝐾𝑆𝑘
Therefore, the Kalman Gain, will have the following final expression
we obtain
𝐾𝑆𝑘𝐾𝑇
𝐾𝑇
)
𝑘))
𝑇
]
𝐾𝑇
uncorrelated to 𝑣
𝑇
]𝐾𝑇
𝑅𝑘𝐾𝑇
)𝐾𝑇
, this could be done by
’s trace (to avoid passing by
]
( 17 )
𝑣(𝑘) with
( 18 )
( 19 )
, this could be done by
’s trace (to avoid passing by
( 20 )
( 21 )
( 22 )
( 23 )

5. Again, we
in equation (20) ,
Method2
we pass th
Demonstrating both definitions of a posteriori estimate covariance matrix are compatible.
Another thing, we can assume that the noise covariance matrix
measurement error covariance matrix
calculations.
Another approach consists of estimating
constants, for instance
Least-Squares
Reference
Kallel Ahmed YahiaKallel Ahmed YahiaKallel Ahmed YahiaKallel Ahmed Yahia
Again, we should obtain
in equation (20) , 𝑆
Method2 In this second method, instead of passing through a residual error covariance matrix,
we pass through equation (29),
Demonstrating both definitions of a posteriori estimate covariance matrix are compatible.
Another thing, we can assume that the noise covariance matrix
measurement error covariance matrix
calculations.
Another approach consists of estimating
constants, for instance
Squares (ALS) algorithm.
Reference: MIT OpenCourseWare
Kallel Ahmed YahiaKallel Ahmed YahiaKallel Ahmed YahiaKallel Ahmed Yahia, Final Year project report’s Appendix
= 𝑃𝑘|𝑘−1
should obtain 𝑃𝑘|𝑘
𝑆𝑘 is symmetric and therefore
In this second method, instead of passing through a residual error covariance matrix,
rough equation (29),
𝑃𝑘|𝑘 = (
= (
Demonstrating both definitions of a posteriori estimate covariance matrix are compatible.
Another thing, we can assume that the noise covariance matrix
measurement error covariance matrix
Another approach consists of estimating
constants, for instance [6] describes a way to get both matrices using an
(ALS) algorithm.
: MIT OpenCourseWare
, Final Year project report’s Appendix
1 − 𝐾𝑆𝑘𝐾𝑇
= 𝑃𝑘|𝑘−1 −
is symmetric and therefore
In this second method, instead of passing through a residual error covariance matrix,
rough equation (29),
(𝐼 − 𝐾𝐻)𝑃𝑘
(𝐼 − 𝐾𝐻)𝑃𝑘
𝑃𝑘|𝑘 =
Demonstrating both definitions of a posteriori estimate covariance matrix are compatible.
Another thing, we can assume that the noise covariance matrix
measurement error covariance matrix 𝑄
Another approach consists of estimating
describes a way to get both matrices using an
(ALS) algorithm.
: MIT OpenCourseWare.
, Final Year project report’s Appendix
𝑇
𝐾𝑆𝑘
𝑇
𝐾𝑇
, but since by definition
is symmetric and therefore 𝑆𝑘
𝑇
In this second method, instead of passing through a residual error covariance matrix,
𝑘|𝑘−1 + 𝑃𝑘|𝑘−
𝑘|𝑘−1 + 𝐾𝑆𝑘
= (𝐼 − 𝐾𝐻)𝑃
Demonstrating both definitions of a posteriori estimate covariance matrix are compatible.
Another thing, we can assume that the noise covariance matrix
𝑄𝑘 = 𝑄 are both constant. This is to simplify the
Another approach consists of estimating 𝑄𝑘 and 𝑅
describes a way to get both matrices using an
, Final Year project report’s Appendix-E
, but since by definition
= 𝑆𝑘.
In this second method, instead of passing through a residual error covariance matrix,
−1𝐻𝑇
𝐾𝑇
−
𝑘𝐾𝑇
− 𝐾𝑆𝑘
𝑃𝑘|𝑘−1
Demonstrating both definitions of a posteriori estimate covariance matrix are compatible.
Another thing, we can assume that the noise covariance matrix
are both constant. This is to simplify the
𝑅𝑘 over time instead of setting them as
describes a way to get both matrices using an
, but since by definition
In this second method, instead of passing through a residual error covariance matrix,
𝐾𝑆𝑘𝐾𝑇
𝐾𝑇
Demonstrating both definitions of a posteriori estimate covariance matrix are compatible.
Another thing, we can assume that the noise covariance matrix
are both constant. This is to simplify the
over time instead of setting them as
describes a way to get both matrices using an
, but since by definition 𝑆𝑘 = 𝑆(𝑘)
In this second method, instead of passing through a residual error covariance matrix,
Demonstrating both definitions of a posteriori estimate covariance matrix are compatible.
Another thing, we can assume that the noise covariance matrix 𝑅𝑘 = 𝑅 and the
are both constant. This is to simplify the
over time instead of setting them as
describes a way to get both matrices using an Autocovariance
as seen
In this second method, instead of passing through a residual error covariance matrix,
( 24 )
Demonstrating both definitions of a posteriori estimate covariance matrix are compatible.
and the
are both constant. This is to simplify the
over time instead of setting them as
Autocovariance