# retroCorrect

Correct filter with OOSM using retrodiction

## Description

The retroCorrect function corrects the state estimate and covariance using an out-of-sequence measurement (OOSM). To use this function, specify the MaxNumOOSMSteps property of the filter as a positive integer. Before using this function, you must use the retrodict function to successfully retrodict the current state to the time at which the OOSM was taken.

[retroCorrState,retroCorrCov] = retroCorrect(filter,z) corrects the filter with the OOSM measurement z and returns the corrected state and state covariance. The function changes the values of State and StateCovariance properties of the filter object to retroCorrState and retroCorrCov, respectively. If the filter is a trackingIMM object, the function also changes the ModelProbabilities property of the filter.

example

___ = retroCorrect(___,measparams) specifies the measurement parameters for the measurement z.

Caution

You can use this syntax only when the specified filter is a trackingEKF or trackingIMM object.

## Examples

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Generate a truth trajectory using the 3-D constant velocity model.

rng(2021) % For repeatable results
initialState = [1; 0.4; 2; 0.3; 1; -0.2]; % [x; vx; y; vy; z; vz]
dt = 1; % Time step
steps = 10;
sigmaQ = 0.2; % Standard deviation for process noise
states = NaN(6,steps);
states(:,1) = initialState;
for ii = 2:steps
w = sigmaQ*randn(3,1);
states(:,ii) = constvel(states(:,ii-1),w,dt);
end

Generate position measurements from the truths.

positionSelector = [1 0 0 0 0 0; 0 0 1 0 0 0; 0 0 0 0 1 0];
sigmaR = 0.2; % Standard deviation for measurement noise
positions = positionSelector*states;
measures = positions + sigmaR*randn(3,steps);

Show the truths and measurements in an x-y plot.

figure
plot(positions(1,:),positions(2,:),"ro","DisplayName","Truths");
hold on;
plot(measures(1,:),measures(2,:),"bx","DisplayName","Measures");
xlabel("x (m)")
ylabel("y (m)")
legend("Location","northwest")

Assume that, at the ninth step, the measurement is delayed and therefore unavailable.

delayedMeasure = measures(:,9);
measures(:,9) = NaN;

Construct an extended Kalman filter (EKF) based on the constant velocity model.

estimates = NaN(6,steps);
covariances = NaN(6,6,steps);

estimates(:,1) = positionSelector'*measures(:,1);
covariances(:,:,1) = 1*eye(6);
filter = trackingEKF(@constvel,@cvmeas, ...
"State",estimates(:,1),...
"StateCovariance",covariances(:,:,1), ...
"ProcessNoise",eye(3), ...
"MeasurementNoise",sigmaR^2*eye(3), ...
"MaxNumOOSMSteps",3);

Step through the EKF with the measurements.

for ii = 2:steps
predict(filter);
if ~any(isnan(measures(:,ii))) % Skip if unavailable
correct(filter,measures(:,ii));
end
estimates(:,ii) = filter.State;
covariances(:,:,ii) = filter.StateCovariance;
end

Show the estimated results.

plot(estimates(1,:),estimates(3,:),"gd","DisplayName","Estimates");

Retrodict to the ninth step, and correct the current estimates by using the out-of-sequence measurements at the ninth step.

[retroState,retroCov] = retrodict(filter,-1);
[retroCorrState,retroCorrCov] = retroCorrect(filter,delayedMeasure);

Plot the retrodicted state for the ninth step.

plot([retroState(1);retroCorrState(1)],...
[retroState(3),retroCorrState(3)],...
"kd","DisplayName","Retrodicted")

You can use the determinant of the final state covariance to see the improvements made by retrodiction. A smaller covariance determinant indicates improved state estimates.

detWithoutRetrodiciton = det(covariances(:,:,end))
detWithoutRetrodiciton = 8.5281e-06
detWithRetrodiciton = det(retroCorrCov)
detWithRetrodiciton = 7.9590e-06

Consider a target moving with a constant velocity model. The initial position is at [100; 0 ;1] in meters. The velocity is [1; 1; 0] in meters per second.

rng(2022) % For repeatable results
initialPosition = [100; 0; 1];
velocity = [1; 1; 0];

Assume the measurement noise covariance matrix is

measureCovaraince = diag([1; 1; 0.1]);

Generate a measurement every second for a duration of five seconds.

measurements = NaN(3,5);
dt = 1;
for i =1:5
measurements(:,i) = initialPosition + i*dt*velocity + sqrt(measureCovaraince)*randn(3,1);
end

Assume the measurement at the fourth second is out-of-sequence. It only becomes available after the fifth second.

oosm = measurements(:,4);
measurements(:,4) = NaN;

Create a trackingIMM filter with the true initial position using the initekfimm function. Set the maximum number of OOSM steps to five.

detection = objectDetection(0,initialPosition);
imm = initekfimm(detection);
imm.MaxNumOOSMSteps = 5;

Update the filter with the available measurements.

for i = 1:5
predict(imm,dt);
if ~isnan(measurements(:,i))
correct(imm,measurements(:,i));
end
end

Display the current state, diagonal of state covariance, and model probabilities.

disp("===============Before Retrodiction===============")
===============Before Retrodiction===============
disp("Current state:" + newline + num2str(imm.State'))
Current state:
106.7626       1.56623       6.15405      1.233862      1.000669    -0.1441939
disp("Diagonal elements of state covariance:" + newline + num2str(diag(imm.StateCovariance)'))
Diagonal elements of state covariance:
0.91884      1.1404     0.91861      1.2097     0.91569      1.1156
disp("Model probabities:" + newline + num2str(imm.ModelProbabilities'))
Model probabities:
0.51519   0.0016296     0.48318

Retrodict the filter and retrocorrect the filter with the OOSM.

[retroState, retroCov] = retrodict(imm,-1);
retroCorrect(imm,oosm);

Display the results after retrodiction. From the results, the magnitude of state covariance is reduced after the OOSM is applied, showing that retrodiction using OOSM can improve the estimates.

disp("===============After Retrodiction===============")
===============After Retrodiction===============
disp("Current state:" + newline + num2str(imm.State'))
Current state:
106.6937      1.621093      6.124384      1.261032      1.117407    -0.2363415
disp("Diagonal elements of state covariance:" + newline + num2str(diag(imm.StateCovariance)'))
Diagonal elements of state covariance:
0.80678      1.0429     0.81196      1.0962     0.80353      1.0231
disp("Model probabities:" + newline + num2str(imm.ModelProbabilities'))
Model probabities:
0.5191  0.00034574     0.48055

## Input Arguments

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Tracking filter object, specified as a trackingKF, trackingEKF, or trackingIMM object.

Out-of-sequence measurement, specified as a P-by-1 real-valued vector, where P is the size of the measurement.

Measurement parameters, specified as a structure or an array of structures. The structure is passed into the measurement function specified by the MeasurementFcn property of the tracking filter. The structure can optionally contain these fields:

 Field Description Frame Enumerated type indicating the frame used to report measurements. When detections are reported using a rectangular coordinate system, set Frame to 'rectangular'. When detections are reported in spherical coordinates, set Frame to 'spherical' for the first structure. OriginPosition Position offset of the origin of the child frame relative to the parent frame, represented as a 3-by-1 vector. OriginVelocity Velocity offset of the origin of the child frame relative to the parent frame, represented as a 3-by-1 vector. Orientation Frame orientation, specified as a 3-by-3 real-valued orthonormal frame rotation matrix. The direction of the rotation depends on the IsParentTochild field. IsParentToChild A logical scalar indicating whether Orientation performs a frame rotation from the parent coordinate frame to the child coordinate frame. If false, Orientation performs a frame rotation from the child coordinate frame to the parent coordinate frame instead. HasElevation A logical scalar indicating if the measurement includes elevation. For measurements reported in a rectangular frame, if HasElevation is false, measurement function reports all measurements with 0 degrees of elevation. HasAzimuth A logical scalar indicating if the measurement includes azimuth. HasRange A logical scalar indicating if the measurement includes range. HasVelocity A logical scalar indicating if the reported detections include velocity measurements. For measurements reported in a rectangular frame, if HasVelocity is false, the measurement function reports measurements as [x y z]. If HasVelocity is true, the measurement function reports measurements as [x y z vx vy vz].

## Output Arguments

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State corrected by retrodiction, returned as an M-by-1 real-valued vector, where M is the size of the filter state.

State covariance corrected by retrodiction, returned as an M-by-M real-valued positive-definite matrix.

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### Retrodiction and Retro-Correction

Assume the current time step of the filter is k. At time k, the posteriori state and state covariance of the filter are x(k|k) and P(k|k), respectively. An out-of-sequence measurement (OOSM) taken at time τ now arrives at time k. Find l such that τ is a time step between these two consecutive time steps:

$k-l\le \tau

where l is a positive integer and l < k.

Retrodiction

In the retrodiction step, the current state and state covariance at time k are predicted back to the time of the OOSM. You can obtain the retrodicted state by propagating the state transition function backward in time. For a linear state transition function, the retrodicted state is expressed as:

$x\left(\tau |k\right)=F\left(\tau ,k\right)x\left(k|k\right),$

where F(τ,k) is the backward state transition matrix from time step k to time step τ. The retrodicted covariance is obtained as:

$P\left(\tau |k\right)=F\left(\tau ,k\right)\left[P\left(k|k\right)+Q\left(k,\tau \right)-{P}_{xv}\left(\tau |k\right)-{P}_{xv}^{T}\left(\tau |k\right)\right]F{\left(\tau ,k\right)}^{T},$

where Q(k,τ) is the covariance matrix for the process noise and,

${P}_{xv}\left(\tau |k\right)=Q\left(k,\tau \right)-P\left(k|k-l\right){S}^{*}{\left(k\right)}^{-1}Q\left(k,\tau \right).$

Here, P(k|k-l) is the priori state covariance at time k, predicted from the covariance information at time k–l, and

${S}^{*}{\left(k\right)}^{-1}=P{\left(k|k-l\right)}^{-1}-P{\left(k|k-l\right)}^{-1}P\left(k|k\right)P{\left(k|k-l\right)}^{-1}.$

Retro-Correction

In the second step, retro-correction, the current state and state covariance are corrected using the OOSM. The corrected state is obtained as:

$x\left(k|\tau \right)=x\left(k|k\right)+W\left(k,\tau \right)\left[z\left(\tau \right)-z\left(\tau |k\right)\right],$

where z(τ) is the OOSM at time τ and W(k,τ), the filter gain, is expressed as:

$W\left(k,\tau \right)={P}_{xz}\left(\tau |k\right){\left[H\left(\tau \right)P\left(\tau |k\right){H}^{T}\left(\tau \right)+R\left(\tau \right)\right]}^{-1}.$

You can obtain the equivalent measurement at time τ based on the state estimate at the time k, z(τ|k), as

$z\left(\tau |k\right)=H\left(\tau \right)x\left(\tau |k\right).$

In these expressions, R(τ) is the measurement covariance matrix for the OOSM and:

${P}_{xz}\left(\tau |k\right)=\left[P\left(k|k\right)-{P}_{xv}\left(\tau |k\right)\right]F{\left(\tau ,k\right)}^{T}H{\left(\tau \right)}^{T},$

where H(τ) is the measurement Jacobian matrix.

The corrected covariance is obtained as:

$P\left(k|\tau \right)=P\left(k|k\right)-{P}_{xz}\left(\tau |k\right)S{\left(\tau \right)}^{-1}{P}_{xz}{\left(\tau |k\right)}^{T}.$

where

$S\left(\tau \right)=H\left(\tau \right)P\left(\tau |k\right)H{\left(\tau \right)}^{T}+R\left(\tau \right)$

IMM Retrodiction

For interactive multiple model (IMM) filter (trackingIMM), each member-filter is retrodicted to the time of OOSM in the same way as the process described above. Also, after obtaining the retrodicted state and measurement, each member-filter retro-corrects the current state of the filter as above.

Compared with a regular filter, an IMM filter needs to maintain the probability of each member-filter. In the retrodiction step, the probability of each model is first retrodicted using the probability transition matrix. Based on the OOSM, the filter can obtain the likelihood of each member-filter using the retrodicted state, filter probability, and measurement. Then, using the likelihood of each filter, the transition probability matrix, and the model probability at the current time, the filter obtains the updated model probability of each filter at the current time k. For more details, see [2].

## References

[1] Bar-Shalom, Y., Huimin Chen, and M. Mallick. “One-Step Solution for the Multistep out-of-Sequence-Measurement Problem in Tracking.” IEEE Transactions on Aerospace and Electronic Systems 40, no. 1 (January 2004): 27–37.

[2] Bar-shalom, Y. and Huimin Chen. “IMM Estimator with Out-of-Sequence Measurements.” IEEE Transactions on Aerospace and Electronic Systems, vol. 41, no. 1, Jan. 2005, pp. 90–98.

## Version History

Introduced in R2021b