CHAPTER 5. FUNCTIONS IN THE TOOLBOX

5.2Multiple Model Systems

5.2.1 IMM Models

imm_filter

imm_filter

IMM filter prediction and update steps. Use this instead of separate prediction and update functions, if you don’t need the prediction estimates.

Syntax: [X_i,P_i,MU,X,P] = IMM_FILTER(X_ip,P_ip,MU_ip,p_ij, ind,dims,A,Q,Y,H,R)

AState transition matrices for each model as a cell array.

QProcess noise matrices for each model as a cell array.

YDx1 measurement vector.

HMeasurement matrices for each model as a cell array.

RMeasurement noise covariances for each model as a cell array.

X_p Updated state mean for each model as a cell array P_p Updated state covariance for each model as a cell array

Output: MU Model probabilities as vector

XCombined state mean estimate

PCombined state covariance estimate

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imm_predict

imm_predict

IMM filter prediction step.

Syntax: [X_p,P_p,c_j,X,P] =

IMM_PREDICT(X_ip,P_ip,MU_ip,p_ij,ind,dims,A,Q)

ˆ

X_ip Cell array containing Nj x 1 mean state estimate vector for each model j after update step of previous time step

ˆ ˆ

P_ip Cell array containing Nj x Nj state covariance matrix for

Input:

each model j after update step of previous time step MU_ip Vector containing the model probabilities at previous time

step

p_ij Model transition probability matrix

ind Indexes of state components for each model as a cell array dims Total number of different state components in the com-

bined system

AState transition matrices for each model as a cell array.

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imm_smooth

imm_smooth

Two filter fixed-interval IMM smoother.

Syntax: [X_S,P_S,X_IS,P_IS,MU_S] =

IMM_SMOOTH(MM,PP,MM_i,PP_i,

MU,p_ij,mu_0j,ind,dims,A,Q,R,H,Y)

MM NxK matrix containing the means of forward-time IMMfilter on each time step

PP NxNxK matrix containing the covariances of forwardtime IMM-filter on each time step

MM_i Model-conditional means of forward-time IMM-filter on each time step as a cell array

Input:

PP_i Model-conditional covariances of forward-time IMMfilter on each time step as a cell array

MU Model probabilities of forward-time IMM-filter on each time step

p_ij Model transition probability matrix mu_0j Prior model probabilities

ind Indices of state components for each model as a cell array dims Total number of different state components in the com-

bined system

AState transition matrices for each model as a cell array.

QProcess noise matrices for each model as a cell array.

RMeasurement noise matrices for each model as a cell array.

HMeasurement matrices for each model as a cell array

YMeasurement sequence

X_S Smoothed state means for each time step P_S Smoothed state covariances for each time step

Output: X_IS Model-conditioned smoothed state means for each time step

P_IS Model-conditioned smoothed state covariances for each time step

MU_S Smoothed model probabilities for each time step

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imm_update

imm_update

IMM filter measurement update step.

Syntax: [X_i,P_i,MU,X,P] =

IMM_UPDATE(X_p,P_p,c_j,ind,dims,Y,H,R)

ˆ

X_p Cell array containing Nj x 1 mean state estimate vector for each model j after prediction step

ˆ ˆ

P_p Cell array containing Nj x Nj state covariance matrix for

Input:

each model j after prediction step

c_j Normalizing factors for mixing probabilities

ind Indices of state components for each model as a cell array dims Total number of different state components in the com-

bined system

YDx1 measurement vector.

HMeasurement matrices for each model as a cell array.

RMeasurement noise covariances for each model as a cell array.

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