Sensorless Torque Estimation using Adaptive Kalman Filter and ...

B. AKF based disturbance observer
In the extended system model (5-6), as the pair of the
system matrix and the output matrix is observable, full
order estimation is guaranteed by the DKF. Even the DKF
guarantees the full order estimation, in order to yield the best
result, the DKF needs an accurately selected measurement
covariance matrix in advance. However, obtaining the
accurate covariance matrix is not easy. Moreover, the
measurement covariance matrix can be changed with time.
To overcome the problems mentioned above, this paper
suggests the AKF based disturbance observer. For an LTI
system, the recursive algorithm of the AKF is described as
follows [14], [16]:
Time update
1 ) Project the state ahead
ˆx k|k+1 = F k−1ˆx k−1|k−1 + B k u k (7)
2 ) Project the error covariance ahead
Measurement update
1) Kalman gain
P k|k+1 = K k P k−1|k−1 F k T + Q ∗ k (8)
K k = P k|k−1 H T k (H k P k|k−1 H T k + ˆR ∗ k) −1 (9)
2) Update estimate with measurement Z k
ˆx k|k = ˆx k|k−1 + K k (z k − H k ˆx k|k−1 ) (10)
3) Update the error covariance
P k|k = (I − K k H k )P k|k−1 (11)
where F ∈ R n×n , B ∈ R n×n , K ∈ R n×m , and P ∈ R n×n
represent the system matrix, the input matrix,the kalman
gain, and the covariance state matrix, respectively. H ∈
R m×n , Q ∈ R n , and z k ∈ R m denote the output vector,
the covariance vector of system noise, and measurement,
respectively. The AKF based disturbance observer is different
to the DKF in terms of an adaptive covariance matrix
ˆR
k ∗ ∈ Rm×m in (9). The covariance matrix is updated by
a covariance uncertainties online estimator represented as
follows [16]:
ˆR ∗ (k i ) = 1 i∑
{[z j − H(k j )ˆx(k
N
j )][z j − H(k j )ˆx(k j )] T
j=i−N+1
+ H(k j )P (k j )H T (k j )}
(12)
The first term in the bracket represents a squared error
matrix between the measurements and the estimated states,
and diagonal elements of the second term are the covariances
of state variables. In the first term, the noisy(clear) measurement
makes the big(small) squared error term. Thus, ˆR∗ k
is
increased(decreased), and it decreases(increases) the Kalman
gain K k . Consequently, the dependence of the estimation
result between previously estimated states and current measurement
will be affected by the quality of measurement. As
we can see in (12), the covariance uncertainties estimator
needs N number of recent measurements. By an average of
the recent N number of terms in the bracket, the AKF based
disturbance observer updates the measurement covariance
matrix ˆR k ∗ at each processing time. In addition, large number
of the measurement will offer a more accurate measurement
covariance matrix( ˆR k ∗) estimation.
A. System model
IV. DISTURBANCE ESTIMATOR
This section proposes a signal processing based estimation
method. For the estimation, the DC motor model is used (see
Fig.2). In Fig.3, the upper part describes the system model,
and the middle and lower part represent the disturbance
estimator. In the system model, an input disturbance D input
and an output disturbance D output are described. The input
disturbance D input can be any kind of disturbance, e.g., load
force(torque), gravitational torque, parametric fluctuation or
any combinations of them. In order to validate the total
disturbance estimation, D output is injected. We assume the
disturbances are independent to the system and unbounded.
Subscript n denotes the nominal value of parameters.
B. Disturbance estimator
The disturbance estimator consists of an extraction part
and an estimation part. The two parts are cascade-connected.
As a first step, the extraction part calculates the system
output generated by the input and output disturbances. After
that, the estimation part generates the final estimation result.
Each process is described as follows:
1) Extraction part:
The composition of the extraction part is same to the
system model, but it does not have any disturbances. The
computation of the extraction part is described as follows:
(
K
θ D = −
s(Js + B) I + 1
s(Js + B) D input + 1 )
s D output
(
)
K n
+
s(J n s + B n ) I
(13)
The term in the upper bracket represents the actual
position result(θ I,D ) generated by the control input(I)
and both the input disturbance(D input ) and the output
disturbance(D output ). The term in the lower bracket
shows the position result((θ I ) generated by the control
input(I) only. Due to the subtraction, where the parameters
K, J, andB are same with K n , J n , andB n , the output of
the extraction part(θ D ) is the angular position generated
by the disturbances only. Otherwise, the output contains
contain the angular position generated by the parametric
fluctuation(uncertainty) also. It implies all the dynamics
not considered in extraction part are considered as the