Biomedical Engineering Reference
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xk þ 1
ð
Þ ¼ F i ðÞ x ðÞþ G i ðÞ u ðÞþ v i ðÞ;
z ðÞ¼ H i ðÞ x ðÞþ w i ðÞ
ð 3 : 1 Þ
where x ðÞ is the n-dimensional state vector and u ðÞ is an n-dimensional known
vector (which is not used in our application). The subscript i denotes quantities
attributed to model M i : v ðÞ and w ðÞ are process noise and measurement noise
with
the
property
of
the
zero-mean
white
Gaussian
noise
with
covariance,
Ev ðÞ v ðÞ T ¼ Q ðÞ and Ew ðÞ w ðÞ T
¼ R ðÞ , respectively. The matrices F,
G, H, Q, and R are assumed known and possibly time-varying. That means that the
system can be time-varying and the noise non-stationary.
The Kalman filter estimates a process by using a form of feedback control. So the
equations for the Kalman filter divide into two groups: time update equations and
measurement update equations. The estimation algorithm starts with the initial
estimate x ðÞ of x ðÞ and associated initial covariance P ðÞ . The problem for-
mulation of the predicted state and the state prediction covariance can be written as:
^xk þ 1
ð
Þ ¼ F ðÞ ^x ðÞþ G ðÞ u ðÞ;
ð 3 : 2 Þ
Þ ¼ F ðÞ P ðÞ F ðÞ T þ Q ðÞ:
Pk þ 1
ð
For the proposed MC-IMME, we use Eqs. ( 3.2 ) and ( 3.3 ) with a different model
of filters, i.e., a constant velocity model and a constant acceleration model.
3.2.2 Interacting Multiple Model Framework
Multiple model algorithms can be divided into three generations: autonomous
multiple models (AMM), cooperating multiple models (CMM), and variable
structure multi-models (VSMM) [ 14 , 16 ]. The AMM algorithm uses a fixed
number of motion models operating autonomously. The AMM output estimate is
typically computed as a weighted average of the filter estimates. The CMM
algorithm improves on AMM by allowing the individual filters to cooperate. The
VSMM algorithm has a variable group of models cooperating with each other. The
VSMM algorithm can add or delete models based on performance, eliminating
poorly performing ones and adding candidates for improved estimation. The well-
known IMME algorithm is part of the CMM generation [ 16 ].
The main feature of the interacting multiple model (IMM) is the ability to
estimate the state of a dynamic system with several behavior models. For the IMM
algorithm, we have implemented two different models based on Kalman filter
(KF): (1) a constant velocity (CV) filter in which we use the direct discrete-time
kinematic models, and (2) a constant acceleration (CA) filter in which the third-
order state equation is used [ 24 - 26 , 43 , 45 - 47 ]. The IMME is separated into four
distinct steps: interaction, filtering, mode probability update, and combination
[ 43 ]. Figure 3.2 depicts a two-filter IMM estimator, where x is the system state,
P is the filter estimate probability, z is the measurement data, and l are mixing
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