trackingEKF

Extended Kalman filter for object tracking

Description

A trackingEKF object is a discrete-time extended Kalman filter used to track the positions and velocities of target platforms.

A Kalman filter is a recursive algorithm for estimating the evolving state of a process when measurements are made on the process. The extended Kalman filter can model the evolution of a state when the state follows a nonlinear motion model, when the measurements are nonlinear functions of the state, or when both conditions apply. The extended Kalman filter is based on the linearization of the nonlinear equations. This approach leads to a filter formulation similar to the linear Kalman filter, trackingKF.

The process and measurements can have Gaussian noise, which you can include in these ways:

Add noise to both the process and the measurements. In this case, the sizes of the process noise and measurement noise must match the sizes of the state vector and measurement vector, respectively.
Add noise in the state transition function, the measurement model function, or in both functions. In these cases, the corresponding noise sizes are not restricted.

Creation

Syntax

filter = trackingEKF

filter = trackingEKF(transitionfcn,measurementfcn,state)

filter = trackingEKF(___,Name,Value)

Description

filter = trackingEKF creates an extended Kalman filter object for a discrete-time system by using default values for the StateTransitionFcn, MeasurementFcn, and State properties. The process and measurement noises are assumed to be additive.

filter = trackingEKF(transitionfcn,measurementfcn,state) specifies the state transition function, transitionfcn, the measurement function, measurementfcn, and the initial state of the system, state.

example

filter = trackingEKF(___,Name,Value) configures the properties of the extended Kalman filter object by using one or more Name,Value pair arguments and any of the previous syntaxes. Any unspecified properties have default values.

Properties

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`State` — Kalman filter state
real-valued M-element vector

Kalman filter state, specified as a real-valued M-element vector, where M is the size of the filter state.

Example: [200; 0.2]

Data Types: double

`StateCovariance` — State estimation error covariance
positive-definite real-valued M-by-M matrix

State error covariance, specified as a positive-definite real-valued M-by-M matrix where M is the size of the filter state. The covariance matrix represents the uncertainty in the filter state.

Example: [20 0.1; 0.1 1]

`StateTransitionFcn` — State transition function
function handle

State transition function, specified as a function handle. This function calculates the state vector at time step k from the state vector at time step k – 1. The function can take additional input parameters, such as control inputs or time step size. The function can also include noise values.

The valid syntaxes for the state transition function depend on whether the filter has additive process noise. The table shows the valid syntaxes based on the value of the HasAdditiveProcessNoise property.

Valid Syntaxes (HasAdditiveProcessNoise = true) Valid Syntaxes (HasAdditiveProcessNoise = false)

Valid Syntaxes (`HasAdditiveProcessNoise = true`)	Valid Syntaxes (`HasAdditiveProcessNoise = false`)
x(k) = statetransitionfcn(x(k-1)) x(k) = statetransitionfcn(x(k-1),parameters) `x(k)` is the state at time `k`. `parameters` stands for all additional arguments required by the state transition function.	x(k) = statetransitionfcn(x(k-1),w(k-1)) x(k) = statetransitionfcn(x(k-1),w(k-1),dt) x(k) = statetransitionfcn(__,parameters) `x(k)` is the state at time `k`. `w(k)` is a value for the process noise at time `k`. `dt` is the time step of the `trackingEKF` filter, `filter`, specified in the most recent call to the `predict` function. The `dt` argument applies when you use the filter within a tracker and call the `predict` function with the filter to predict the state of the tracker at the next time step. For the nonadditive process noise case, the tracker assumes that you explicitly specify the time step by using this syntax: `predict(filter,dt)`. `parameters` stands for all additional arguments required by the state transition function.

x(k) = statetransitionfcn(x(k-1))
x(k) = statetransitionfcn(x(k-1),parameters)

x(k) is the state at time k.
parameters stands for all additional arguments required by the state transition function.

x(k) = statetransitionfcn(x(k-1),w(k-1))
x(k) = statetransitionfcn(x(k-1),w(k-1),dt)
x(k) = statetransitionfcn(__,parameters)

x(k) is the state at time k.
w(k) is a value for the process noise at time k.
dt is the time step of the trackingEKF filter, filter, specified in the most recent call to the predict function. The dt argument applies when you use the filter within a tracker and call the predict function with the filter to predict the state of the tracker at the next time step. For the nonadditive process noise case, the tracker assumes that you explicitly specify the time step by using this syntax: predict(filter,dt).
parameters stands for all additional arguments required by the state transition function.

Example: @constacc

Data Types: function_handle

`StateTransitionJacobianFcn` — Jacobian of state transition function
function handle

Jacobian of the state transition function, specified as a function handle. This function has the same input arguments as the state transition function.

The valid syntaxes for the Jacobian of the state transition function depend on whether the filter has additive process noise. The table shows the valid syntaxes based on the value of the HasAdditiveProcessNoise property.

Valid Syntaxes (HasAdditiveProcessNoise = true) Valid Syntaxes (HasAdditiveProcessNoise = false)

Valid Syntaxes (`HasAdditiveProcessNoise = true`)	Valid Syntaxes (`HasAdditiveProcessNoise = false`)
Jx(k) = statejacobianfcn(x(k)) Jx(k) = statejacobianfcn(x(k),parameters) `x(k)` is the state at time `k`. `Jx(k)` denotes the Jacobian of the predicted state with respect to the previous state. This Jacobian is an M-by-M matrix at time `k`. The Jacobian function can take additional input parameters, such as control inputs or time-step size. `parameters` stands for all additional arguments required by the Jacobian function, such as control inputs or time-step size.	[Jx(k),Jw(k)] = statejacobianfcn(x(k),w(k)) [Jx(k),Jw(k)] = statejacobianfcn(x(k),w(k),dt) [Jx(k),Jw(k)] = statejacobianfcn(__,parameters) `x(k)` is the state at time `k` `w(k)` is a sample Q-element vector of the process noise at time `k`. Q is the size of the process noise covariance. The process noise vector in the nonadditive case does not need to have the same dimensions as the state vector. `Jx(k)` denotes the Jacobian of the predicted state with respect to the previous state. This Jacobian is an M-by-M matrix at time k. The Jacobian function can take additional input parameters, such as control inputs or time-step size. `Jw(k)` denotes the M-by-Q Jacobian of the predicted state with respect to the process noise elements. `dt` is the time step of the `trackingEKF` filter, `filter`, specified in the most recent call to the `predict` function. The `dt` argument applies when you use the filter within a tracker and call the `predict` function with the filter to predict the state of the tracker at the next time step. For the nonadditive process noise case, the tracker assumes that you explicitly specify the time step by using this syntax: `predict(filter,dt)`. `parameters` stands for all additional arguments required by the Jacobian function, such as control inputs or time-step size.

Jx(k) = statejacobianfcn(x(k))
Jx(k) = statejacobianfcn(x(k),parameters)

x(k) is the state at time k.
Jx(k) denotes the Jacobian of the predicted state with respect to the previous state. This Jacobian is an M-by-M matrix at time k. The Jacobian function can take additional input parameters, such as control inputs or time-step size.
parameters stands for all additional arguments required by the Jacobian function, such as control inputs or time-step size.

[Jx(k),Jw(k)] = statejacobianfcn(x(k),w(k))
[Jx(k),Jw(k)] = statejacobianfcn(x(k),w(k),dt)
[Jx(k),Jw(k)] = statejacobianfcn(__,parameters)

x(k) is the state at time k
w(k) is a sample Q-element vector of the process noise at time k. Q is the size of the process noise covariance. The process noise vector in the nonadditive case does not need to have the same dimensions as the state vector.
Jx(k) denotes the Jacobian of the predicted state with respect to the previous state. This Jacobian is an M-by-M matrix at time k. The Jacobian function can take additional input parameters, such as control inputs or time-step size.
Jw(k) denotes the M-by-Q Jacobian of the predicted state with respect to the process noise elements.
dt is the time step of the trackingEKF filter, filter, specified in the most recent call to the predict function. The dt argument applies when you use the filter within a tracker and call the predict function with the filter to predict the state of the tracker at the next time step. For the nonadditive process noise case, the tracker assumes that you explicitly specify the time step by using this syntax: predict(filter,dt).
parameters stands for all additional arguments required by the Jacobian function, such as control inputs or time-step size.

If this property is not specified, the Jacobians are computed by numeric differencing at each call of the predict function. This computation can increase the processing time and numeric inaccuracy.

Example: @constaccjac

Data Types: function_handle

`ProcessNoise` — Process noise covariance
1 (default) | positive real scalar | positive-definite real-valued matrix

Process noise covariance, specified as a scalar or matrix.

When HasAdditiveProcessNoise is true, specify the process noise covariance as a positive real scalar or a positive-definite real-valued M-by-M matrix. M is the dimension of the state vector. When specified as a scalar, the matrix is a multiple of the M-by-M identity matrix.
When HasAdditiveProcessNoise is false, specify the process noise covariance as a Q-by-Q matrix. Q is the size of the process noise vector.
You must specify ProcessNoise before any call to the predict function. In later calls to predict, you can optionally specify the process noise as a scalar. In this case, the process noise matrix is a multiple of the Q-by-Q identity matrix.

Example: [1.0 0.05; 0.05 2]

`HasAdditiveProcessNoise` — Model additive process noise
`true` (default) | `false`

Option to model process noise as additive, specified as true or false. When this property is true, process noise is added to the state vector. Otherwise, noise is incorporated into the state transition function.

`MeasurementFcn` — Measurement model function
function handle

Measurement model function, specified as a function handle. This function can be a nonlinear function that models measurements from the predicted state. Input to the function is the M-element state vector. The output is the N-element measurement vector. The function can take additional input arguments, such as sensor position and orientation.

If HasAdditiveMeasurementNoise is true, specify the function using one of these syntaxes:
```
z(k) = measurementfcn(x(k))
```
```
z(k) = measurementfcn(x(k),parameters)
```
x(k) is the state at time k and z(k) is the predicted measurement at time k. The parameters argument stands for all additional arguments required by the measurement function.
If HasAdditiveMeasurementNoise is false, specify the function using one of these syntaxes:
```
z(k) = measurementfcn(x(k),v(k))
```
```
z(k) = measurementfcn(x(k),v(k),parameters)
```
x(k) is the state at time k and v(k) is the measurement noise at time k. The parameters argument stands for all additional arguments required by the measurement function.

Example: @cameas

Data Types: function_handle

`MeasurementJacobianFcn` — Jacobian of measurement function
function handle

Jacobian of the measurement function, specified as a function handle. The function has the same input arguments as the measurement function. The function can take additional input parameters, such sensor position and orientation.

If HasAdditiveMeasurmentNoise is true, specify the Jacobian function using one of these syntaxes:
```
Jmx(k) = measjacobianfcn(x(k))
```
```
Jmx(k) = measjacobianfcn(x(k),parameters)
```
x(k) is the state at time k. Jx(k) denotes the N-by-M Jacobian of the measurement function with respect to the state. The parameters argument stands for all arguments required by the measurement function.
If HasAdditiveMeasurmentNoise is false, specify the Jacobian function using one of these syntaxes:
```
[Jmx(k),Jmv(k)] = measjacobianfcn(x(k),v(k))
```
```
[Jmx(k),Jmv(k)] = measjacobianfcn(x(k),v(k),parameters)
```
x(k) is the state at time k and v(k) is an R-dimensional sample noise vector. Jmx(k) denotes the N-by-M Jacobian of the measurement function with respect to the state. Jmv(k) denotes the Jacobian of the N-by-R measurement function with respect to the measurement noise. The parameters argument stands for all arguments required by the measurement function.

If not specified, measurement Jacobians are computed using numerical differencing at each call to the correct function. This computation can increase processing time and numerical inaccuracy.

Example: @cameasjac

Data Types: function_handle

`MeasurementNoise` — Measurement noise covariance
1 (default) | positive scalar | positive-definite real-valued matrix

Measurement noise covariance, specified as a positive scalar or positive-definite real-valued matrix.

When HasAdditiveMeasurementNoise is true, specify the measurement noise covariance as a scalar or an N-by-N matrix. N is the size of the measurement vector. When specified as a scalar, the matrix is a multiple of the N-by-N identity matrix.
When HasAdditiveMeasurementNoise is false, specify the measurement noise covariance as an R-by-R matrix. R is the size of the measurement noise vector.
You must specify MeasurementNoise before any call to the correct function. After the first call to correct, you can optionally specify the measurement noise as a scalar. In this case, the measurement noise matrix is a multiple of the R-by-R identity matrix.

Example: 0.2

`HasAdditiveMeasurmentNoise` — Model additive measurement noise
`true` (default) | `false`

Option to enable additive measurement noise, specified as true or false. When this property is true, noise is added to the measurement. Otherwise, noise is incorporated into the measurement function.

Object Functions

`predict`	Predict state and state estimation error covariance of tracking filter
`correct`	Correct state and state estimation error covariance using tracking filter
`correctjpda`	Correct state and state estimation error covariance using tracking filter and JPDA
`distance`	Distances between current and predicted measurements of tracking filter
`likelihood`	Likelihood of measurement from tracking filter
`clone`	Create duplicate tracking filter
`residual`	Measurement residual and residual noise from tracking filter
`initialize`	Initialize state and covariance of tracking filter

Examples

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Constant-Velocity Extended Kalman Filter

Open Live Script

Create a two-dimensional trackingEKF object and use name-value pairs to define the StateTransitionJacobianFcn and MeasurementJacobianFcn properties. Use the predefined constant-velocity motion and measurement models and their Jacobians.

EKF = trackingEKF(@constvel,@cvmeas,[0;0;0;0], ...
    'StateTransitionJacobianFcn',@constveljac, ...
    'MeasurementJacobianFcn',@cvmeasjac);

Run the filter. Use the predict and correct functions to propagate the state. You may call predict and correct in any order and as many times you want. Specify the measurement in Cartesian coordinates.

measurement = [1;1;0];
[xpred, Ppred] = predict(EKF);
[xcorr, Pcorr] = correct(EKF,measurement);
[xpred, Ppred] = predict(EKF);
[xpred, Ppred] = predict(EKF)

xpred = 4×1

    1.2500
    0.2500
    1.2500
    0.2500

Ppred = 4×4

   11.7500    4.7500         0         0
    4.7500    3.7500         0         0
         0         0   11.7500    4.7500
         0         0    4.7500    3.7500

More About

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Filter Parameters

This table relates the filter model parameters to the object properties. M is the size of the state vector. N is the size of the measurement vector.

Filter Parameter	Description	Filter Property	Size
f	State transition function that specifies the equations of motion of the object. This function determines the state at time `k+1` as a function of the state and the controls at time `k`. The state transition function depends on the time-increment of the filter.	`StateTransitionFcn`	Function returns M-element vector
h	Measurement function that specifies how the measurements are functions of the state and measurement noise.	`MeasurementFcn`	Function returns N-element vector
x_k	Estimate of the object state.	`State`	M-element vector
P_k	State error covariance matrix representing the uncertainty in the values of the state.	`StateCovariance`	M-by-M matrix
Q_k	Estimate of the process noise covariance matrix at step `k`. Process noise is a measure of the uncertainty in the dynamic model. It is assumed to be zero-mean white Gaussian noise.	`ProcessNoise`	M-by-M matrix when `HasAdditiveProcessNoise` is `true`. Q-by-Q matrix when `HasAdditiveProcessNoise` is `false`
R_k	Estimate of the measurement noise covariance at step `k`. Measurement noise reflects the uncertainty of the measurement. It is assumed to be zero-mean white Gaussian noise.	`MeasurementNoise`	N-by-N matrix when `HasAdditiveMeasurementNoise` is `true`. R-by-R when `HasAdditiveMeasurementNoise` is `false`.
F	Function determining Jacobian of propagated state with respect to previous state.	`StateTransitionJacobianFcn`	M-by-M matrix
H	Function determining Jacobians of measurement with respect to the state and measurement noise.	`MeasurementJacobianFcn`	N-by-M for state vector Jacobian and N-by-R for measurement vector Jacobian

Algorithms

The extended Kalman filter estimates the state of a process governed by this nonlinear stochastic equation:

$x_{k + 1} = f (x_{k}, u_{k}, w_{k}, t)$

x_k is the state at step k. f() is the state transition function. Random noise perturbations, w_k, can affect the object motion. The filter also supports a simplified form,

$x_{k + 1} = f (x_{k}, u_{k}, t) + w_{k}$

To use the simplified form, set HasAdditiveProcessNoise to true.

In the extended Kalman filter, the measurements are also general functions of the state:

$z_{k} = h (x_{k}, v_{k}, t)$

h(x_k,v_k,t) is the measurement function that determines the measurements as functions of the state. Typical measurements are position and velocity or some function of position and velocity. The measurements can also include noise, represented by v_k. Again, the filter offers a simpler formulation.

$z_{k} = h (x_{k}, t) + v_{k}$

To use the simplified form, set HasAdditiveMeasurmentNoise to true.

These equations represent the actual motion and the actual measurements of the object. However, the noise contribution at each step is unknown and cannot be modeled deterministically. Only the statistical properties of the noise are known.

References

[1] Brown, R.G. and P.Y.C. Wang. Introduction to Random Signal Analysis and Applied Kalman Filtering. 3rd Edition. New York: John Wiley & Sons, 1997.

[2] Kalman, R. E. “A New Approach to Linear Filtering and Prediction Problems.” Transactions of the ASME–Journal of Basic Engineering. Vol. 82, Series D, March 1960, pp. 35–45.

[3] Blackman, Samuel and R. Popoli. Design and Analysis of Modern Tracking Systems. Artech House.1999.

[4] Blackman, Samuel. Multiple-Target Tracking with Radar Applications. Artech House. 1986.

Documentation

trackingEKF

Description

Creation

Syntax

Description

Properties

`State` — Kalman filter state
real-valued M-element vector

`StateCovariance` — State estimation error covariance
positive-definite real-valued M-by-M matrix

`StateTransitionFcn` — State transition function
function handle

`StateTransitionJacobianFcn` — Jacobian of state transition function
function handle

`ProcessNoise` — Process noise covariance
1 (default) | positive real scalar | positive-definite real-valued matrix

`HasAdditiveProcessNoise` — Model additive process noise
`true` (default) | `false`

`MeasurementFcn` — Measurement model function
function handle

`MeasurementJacobianFcn` — Jacobian of measurement function
function handle

`MeasurementNoise` — Measurement noise covariance
1 (default) | positive scalar | positive-definite real-valued matrix

`HasAdditiveMeasurmentNoise` — Model additive measurement noise
`true` (default) | `false`

Object Functions

Examples

Constant-Velocity Extended Kalman Filter

More About

Filter Parameters

Algorithms

References

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

See Also

Functions

Objects

Topics

Sensor Fusion and Tracking Toolbox Documentation

Support

Documentation

trackingEKF

Description

Creation

Syntax

Description

Properties

State — Kalman filter state real-valued M-element vector

StateCovariance — State estimation error covariance positive-definite real-valued M-by-M matrix

StateTransitionFcn — State transition function function handle

StateTransitionJacobianFcn — Jacobian of state transition function function handle

ProcessNoise — Process noise covariance 1 (default) | positive real scalar | positive-definite real-valued matrix

HasAdditiveProcessNoise — Model additive process noise true (default) | false

MeasurementFcn — Measurement model function function handle

MeasurementJacobianFcn — Jacobian of measurement function function handle

MeasurementNoise — Measurement noise covariance 1 (default) | positive scalar | positive-definite real-valued matrix

HasAdditiveMeasurmentNoise — Model additive measurement noise true (default) | false

Object Functions

Examples

Constant-Velocity Extended Kalman Filter

More About

Filter Parameters

Algorithms

References

Extended Capabilities

C/C++ Code Generation Generate C and C++ code using MATLAB® Coder™.

See Also

Functions

Objects

Topics

Sensor Fusion and Tracking Toolbox Documentation

Support

`State` — Kalman filter state
real-valued M-element vector

`StateCovariance` — State estimation error covariance
positive-definite real-valued M-by-M matrix

`StateTransitionFcn` — State transition function
function handle

`StateTransitionJacobianFcn` — Jacobian of state transition function
function handle

`ProcessNoise` — Process noise covariance
1 (default) | positive real scalar | positive-definite real-valued matrix

`HasAdditiveProcessNoise` — Model additive process noise
`true` (default) | `false`

`MeasurementFcn` — Measurement model function
function handle

`MeasurementJacobianFcn` — Jacobian of measurement function
function handle

`MeasurementNoise` — Measurement noise covariance
1 (default) | positive scalar | positive-definite real-valued matrix

`HasAdditiveMeasurmentNoise` — Model additive measurement noise
`true` (default) | `false`

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.