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# Extended Kalman Filter

Estimate states of discrete-time nonlinear system using extended Kalman filter

• Library:
• Control System Toolbox / State Estimation

System Identification Toolbox / Estimators

## Description

The Extended Kalman Filter block estimates the states of a discrete-time nonlinear system using the first-order discrete-time extended Kalman filter algorithm.

Consider a plant with states x, input u, output y, process noise w, and measurement noise v. Assume that you can represent the plant as a nonlinear system.

Using the state transition and measurement functions of the system and the extended Kalman filter algorithm, the block produces state estimates $\stackrel{^}{x}$ for the current time step. For information about the algorithm, see Extended and Unscented Kalman Filter Algorithms for Online State Estimation.

You create the nonlinear state transition function and measurement functions for the system and specify these functions in the block. The block supports state estimation of a system with multiple sensors that are operating at different sampling rates. You can specify up to five measurement functions, each corresponding to a sensor in the system. You can also specify the Jacobians of the state transition and measurement functions. If you do not specify them, the software numerically computes the Jacobians. For more information, see State Transition and Measurement Functions.

## Ports

### Input

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Measured system outputs corresponding to each measurement function that you specify in the block. The number of ports equals the number of measurement functions in your system. You can specify up to five measurement functions. For example, if your system has two sensors, you specify two measurement functions in the block. The first port y1 is available by default. When you click , the software generates port y2 corresponding to the second measurement function.

Specify the ports as N-dimensional vectors, where N is the number of quantities measured by the corresponding sensor. For example, if your system has one sensor that measures the position and velocity of an object, then there is only one port y1. The port is specified as a 2-dimensional vector with values corresponding to position and velocity.

#### Dependencies

The first port y1 is available by default. Ports y2 to y5 are generated when you click Add Measurement, and click .

Data Types: single | double

Additional optional input argument to the state transition function f other than the state x and process noise w. For information about state transition functions see, State Transition and Measurement Functions.

Suppose that your system has nonadditive process noise, and the state transition function f has the following form:

x(k+1) = f(x(k),w(k),StateTransitionFcnInputs)

Here k is the time step, and StateTransitionFcnInputs is an additional input argument other than x and w.

If you create f using a MATLAB® function (.m file), the software generates the port StateTransitionFcnInputs when you click . You can specify the inputs to this port as a scalar, vector, or matrix.

If your state transition function has more than one additional input, use a Simulink Function block to specify the function. When you use a Simulink Function block, you provide the additional inputs directly to the Simulink Function block using Inport blocks. No input ports are generated for the additional inputs in the Extended Kalman Filter block.

#### Dependencies

This port is generated only if both of the following conditions are satisfied:

• You specify f in Function using a MATLAB function, and f is on the MATLAB path.

• f requires only one additional input argument apart from x and w.

Data Types: single | double

Additional optional inputs to the measurement functions other than the state x and measurement noise v. For information about measurement functions see, State Transition and Measurement Functions.

MeasurementFcn1Inputs corresponds to the first measurement function that you specify, and so on. For example, suppose that your system has three sensors and nonadditive measurement noise, and the three measurement functions h1, h2, and h3 have the following form:

y1[k] = h1(x[k],v1[k],MeasurementFcn1Inputs)

y2[k] = h2(x[k],v2[k],MeasurementFcn2Inputs)

y3[k] = h3(x[k],v3[k])

Here k is the time step, and MeasurementFcn1Inputs and MeasurementFcn2Inputs are the additional input arguments to h1 and h2.

If you specify h1, h2, and h3 using MATLAB functions (.m files) in Function, the software generates ports MeasurementFcn1Inputs and MeasurementFcn2Inputs when you click . You can specify the inputs to these ports as scalars, vectors, or matrices.

If your measurement functions have more than one additional input, use Simulink Function blocks to specify the functions. When you use a Simulink Function block, you provide the additional inputs directly to the Simulink Function block using Inport blocks. No input ports are generated for the additional inputs in the Extended Kalman Filter block.

#### Dependencies

A port corresponding to a measurement function h is generated only if both of the following conditions are satisfied:

• You specify h in Function using a MATLAB function, and h is on the MATLAB path.

• h requires only one additional input argument apart from x and v.

Data Types: single | double

Time-varying process noise covariance, specified as a scalar, vector, or matrix depending on the value of the Process noise parameter:

• Process noise is Additive — Specify the covariance as a scalar, an Ns-element vector, or an Ns-by-Ns matrix, where Ns is the number of states of the system. Specify a scalar if there is no cross-correlation between process noise terms, and all the terms have the same variance. Specify a vector of length Ns, if there is no cross-correlation between process noise terms, but all the terms have different variances.

• Process noise is Nonadditive — Specify the covariance as a W-by-W matrix, where W is the number of process noise terms in the state transition function.

#### Dependencies

This port is generated if you specify the process noise covariance as Time-Varying. The port appears when you click .

Data Types: single | double

Time-varying measurement noise covariances for up to five measurement functions of the system, specified as matrices. The sizes of the matrices depend on the value of the Measurement noise parameter for the corresponding measurement function:

• Measurement noise is Additive — Specify the covariance as an N-by-N matrix, where N is the number of measurements of the system.

• Measurement noise is Nonadditive — Specify the covariance as a V-by-V matrix, where V is the number of measurement noise terms in the corresponding measurement function.

#### Dependencies

A port is generated if you specify the measurement noise covariance as Time-Varying for the corresponding measurement function. The port appears when you click .

Data Types: single | double

Suppose that measured output data is not available at all time points at the port y1 that corresponds to the first measurement function. Use a signal value other than 0 at the Enable1 port to enable the correction of estimated states when measured data is available. Specify the port value as 0 when measured data is not available. Similarly, if measured output data is not available at all time points at the port yi for the ith measurement function, specify the corresponding port Enablei as a value other than 0.

#### Dependencies

A port corresponding to a measurement function is generated if you select Add Enable port for that measurement function. The port appears when you click .

Data Types: single | double | Boolean

### Output

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Estimated states, returned as a vector of size Ns, where Ns is the number of states of the system. To access the individual states, use the Selector block.

When the Use the current measurements to improve state estimates parameter is selected, the block outputs the corrected state estimate $\stackrel{^}{x}\left[k|k\right]$ at time step k, estimated using measured outputs until time k. If you clear this parameter, the block returns the predicted state estimate $\stackrel{^}{x}\left[k|k-1\right]$ for time k, estimated using measured output until a previous time k-1. Clear this parameter if your filter is in a feedback loop and there is an algebraic loop in your Simulink® model.

Data Types: single | double

State estimation error covariance, returned as an Ns-by-Ns matrix, where Ns is the number of states of the system. To access the individual covariances, use the Selector block.

#### Dependencies

This port is generated if you select Output state estimation error covariance in the System Model tab, and click .

Data Types: single | double

## Parameters

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### System Model Tab

#### State Transition

The state transition function calculates the Ns-element state vector of the system at time step k+1, given the state vector at time step k. Ns is the number of states of the nonlinear system. You create the state transition function and specify the function name in Function. For example, if vdpStateFcn.m is the state transition function that you created and saved, specify Function as vdpStateFcn.

The inputs to the function you create depend on whether you specify the process noise as additive or nonadditive in Process noise.

• Process noise is Additive — The state transition function f specifies how the states evolve as a function of state values at previous time step:

x(k+1) = f(x(k),Us1(k),...,Usn(k)),

where x(k) is the estimated state at time k, and Us1,...,Usn are any additional input arguments required by your state transition function, such as system inputs or the sample time. To see an example of a state transition function with additive process noise, type edit vdpStateFcn at the command line.

• Process noise is Nonadditive — The state transition function also specifies how the states evolve as a function of the process noise w:

x(k+1) = f(x(k),w(k),Us1(k),...,Usn(k)).

You can create f using a Simulink Function block or as a MATLAB function (.m file).

• You can use a MATLAB function only if f has one additional input argument Us1 other than x and w.

x(k+1) = f(x(k),w(k),Us1(k))

The software generates an additional input port StateTransitionFcnInputs to specify this argument.

• If you are using a Simulink Function block, specify x and w using Argument Inport blocks and the additional inputs Us1,...,Usn using Inport blocks in the Simulink Function block. You do not provide Us1,...,Usn to the Extended Kalman Filter block.

#### Programmatic Use

 Block Parameter: StateTransitionFcn Type: character vector, string Default: 'myStateTransitionFcn'

Jacobian of state transition function f, specified as one of the following:

• off — The software computes the Jacobian numerically. This computation may increase processing time and numerical inaccuracy of the state estimation.

• on — You create a function to compute the Jacobian, and specify the name of the function in Jacobian. For example, if vdpStateJacobianFcn.m is the Jacobian function, specify Jacobian as vdpStateJacobianFcn. If you create the state transition function f using a Simulink Function block, then create the Jacobian using a Simulink Function block. If you create f using a MATLAB function (.m file), then create the Jacobian using a MATLAB function.

The function calculates the partial derivatives of the state transition function with respect to the states and process noise. The number of inputs to the Jacobian function must equal the number of inputs of the state transition function and must be specified in the same order in both functions. The number of outputs of the Jacobian function depends on the Process noise parameter:

• Process noise is Additive — The function calculates the partial derivative of the state transition function f with respect to the states ($\partial f/\partial x$). The output is an Ns-by-Ns Jacobian matrix, where Ns is the number of states.

To see an example of a Jacobian function for additive process noise, type edit vdpStateJacobianFcn at the command line.

• Process noise is Nonadditive — The function must also return a second output that is the partial derivative of the state transition function f with respect to the process noise terms ($\partial f/\partial w$). The second output is returned as an Ns-by-W matrix, where W is the number of process noise terms in the state transition function.

#### Programmatic Use

 Block Parameter: HasStateTransitionJacobianFcn Type: character vector Values: 'off','on' Default: 'off'
 Block Parameter: StateTransitionJacobianFcn Type: character vector, string Default: ''

Process noise characteristics, specified as one of the following values:

• Additive — Process noise w is additive, and the state transition function f that you specify in Function has the following form:

x(k+1) = f(x(k),Us1(k),...,Usn(k)),

where x(k) is the estimated state at time k, and Us1,...,Usn are any additional input arguments required by your state transition function.

• Nonadditive — Process noise is nonadditive, and the state transition function specifies how the states evolve as a function of the state and process noise at the previous time step:

x(k+1) = f(x(k),w(k),Us1(k),...,Usn(k)).

#### Programmatic Use

Time-invariant process noise covariance, specified as a scalar, vector, or matrix depending on the value of the Process noise parameter:

• Process noise is Additive — Specify the covariance as a scalar, an Ns-element vector, or an Ns-by-Ns matrix, where Ns is the number of states of the system. Specify a scalar if there is no cross-correlation between process noise terms and all the terms have the same variance. Specify a vector of length Ns, if there is no cross-correlation between process noise terms but all the terms have different variances.

• Process noise is Nonadditive — Specify the covariance as a W-by-W matrix, where W is the number of process noise terms.

If the process noise covariance is time-varying, select Time-varying. The block generates input port Q to specify the time-varying covariance.

#### Dependencies

This parameter is enabled if you do not specify the process noise as Time-Varying.

#### Programmatic Use

 Block Parameter: ProcessNoise Type: character vector, string Default: '1'

If you select this parameter, the block includes an additional input port Q to specify the time-varying process noise covariance.

#### Programmatic Use

 Block Parameter: HasTimeVaryingProcessNoise Type: character vector Values: 'off', 'on' Default: 'off'

#### Initialization

Initial state estimate value, specified as an Ns-element vector, where Ns is the number of states in the system. Specify the initial state values based on your knowledge of the system.

#### Programmatic Use

 Block Parameter: InitialState Type: character vector, string Default: '0'

State estimation error covariance, specified as a scalar, an Ns-element vector, or an Ns-by-Ns matrix, where Ns is the number of states of the system. If you specify a scalar or vector, the software creates an Ns-by-Ns diagonal matrix with the scalar or vector elements on the diagonal.

Specify a high value for the covariance when you do not have confidence in the initial state values that you specify in Initial state.

#### Programmatic Use

 Block Parameter: InitialStateCovariance Type: character vector, string Default: '1'

#### Measurement

The measurement function calculates the N-element output measurement vector of the nonlinear system at time step k, given the state vector at time step k. You create the measurement function and specify the function name in Function. For example, if vdpMeasurementFcn.m is the measurement function that you created and saved, specify Function as vdpMeasurementFcn.

The inputs to the function you create depend on whether you specify the measurement noise as additive or nonadditive in Measurement noise.

• Measurement noise is Additive — The measurement function h specifies how the measurements evolve as a function of state Values:

y(k) = h(x(k),Um1(k),...,Umn(k)),

where y(k) and x(k) are the estimated output and estimated state at time k, and Um1,...,Umn are any optional input arguments required by your measurement function. For example, if you are using a sensor for tracking an object, an additional input could be the sensor position.

To see an example of a measurement function with additive process noise, type edit vdpMeasurementFcn at the command line.

• Measurement noise is Nonadditive— The measurement function also specifies how the output measurement evolves as a function of the measurement noise v:

y(k) = h(x(k),v(k),Um1(k),...,Umn(k)).

To see an example of a measurement function with nonadditive process noise, type edit vdpMeasurementNonAdditiveNoiseFcn.

You can create h using a Simulink Function block or as a MATLAB function (.m file).

• You can use a MATLAB function only if h has one additional input argument Um1 other than x and v.

y[k] = h(x[k],v[k],Um1(k))

The software generates an additional input port MeasurementFcniInputs to specify this argument for the ith measurement function.

• If you are using a Simulink Function block, specify x and v using Argument Inport blocks and the additional inputs Um1,...,Umn using Inport blocks in the Simulink Function block. You do not provide Um1,...,Umn to the Extended Kalman Filter block.

If you have multiple sensors in your system, you can specify multiple measurement functions. You can specify up to five measurement functions using the Add Measurement button. To remove measurement functions, use Remove Measurement.

#### Programmatic Use

 Block Parameter: MeasurementFcn1, MeasurementFcn2, MeasurementFcn3, MeasurementFcn4, MeasurementFcn5 Type: character vector, string Default: 'myMeasurementFcn'

Jacobian of measurement function h, specified as one of the following:

• off — The software computes the Jacobian numerically. This computation may increase processing time and numerical inaccuracy of the state estimation.

• on — You create a function to compute the Jacobian of the measurement function h, and specify the name of the function in Jacobian. For example, if vdpMeasurementJacobianFcn.m is the Jacobian function, specify MeasurementJacobianFcn as vdpMeasurementJacobianFcn. If you create h using a Simulink Function block, then create the Jacobian using a Simulink Function block. If you create h using a MATLAB function (.m file), then create the Jacobian using a MATLAB function.

The function calculates the partial derivatives of the measurement function h with respect to the states and measurement noise. The number of inputs to the Jacobian function must equal the number of inputs to the measurement function and must be specified in the same order in both functions. The number of outputs of the Jacobian function depends on the Measurement noise parameter:

• Measurement noise is Additive — The function calculates the partial derivatives of the measurement function with respect to the states ($\partial h/\partial x$). The output is as an N-by-Ns Jacobian matrix, where N is the number of measurements of the system and Ns is the number of states.

To see an example of a Jacobian function for additive measurement noise, type edit vdpMeasurementJacobianFcn at the command line.

• Measurement noise is Nonadditive — The function also returns a second output that is the partial derivative of the measurement function with respect to the measurement noise terms ($\partial h/\partial v$). The second output is returned as an N-by-V Jacobian matrix, where V is the number of measurement noise terms.

#### Programmatic Use

 Block Parameter: HasMeasurementJacobianFcn1, HasMeasurementJacobianFcn2, HasMeasurementJacobianFcn3, HasMeasurementJacobianFcn4,HasMeasurementJacobianFcn5 Type: character vector Values: 'off','on' Default: 'off'
 Block Parameter: MeasurementJacobianFcn1, MeasurementJacobianFcn2, MeasurementJacobianFcn3, MeasurementJacobianFcn4, MeasurementJacobianFcn5 Type: character vector Default: ''

Measurement noise characteristics, specified as one of the following values:

• Additive — Measurement noise v is additive, and the measurement function h that you specify in Function has the following form:

y(k) = h(x(k),Um1(k),...,Umn(k)),

where y(k) and x(k) are the estimated output and estimated state at time k, and Um1,...,Umn are any optional input arguments required by your measurement function.

• Nonadditive — Measurement noise is nonadditive, and the measurement function specifies how the output measurement evolves as a function of the state and measurement noise:

y(k) = h(x(k),v(k),Um1(k),...,Umn(k)).

#### Programmatic Use

Time-invariant measurement noise covariance, specified as a matrix. The size of the matrix depends on the value of the Measurement noise parameter:

• Measurement noise is Additive — Specify the covariance as an N-by-N matrix, where N is the number of measurements of the system.

• Measurement noise is Nonadditive — Specify the covariance as a V-by-V matrix, where V is the number of measurement noise terms.

If the measurement noise covariance is time-varying, select Time-varying. The block generates input port Ri to specify the time-varying covariance for the ith measurement function.

#### Dependencies

This parameter is enabled if you do not specify the process noise as Time-Varying.

#### Programmatic Use

 Block Parameter: MeasurementNoise1, MeasurementNoise2, MeasurementNoise3, MeasurementNoise4, MeasurementNoise5 Type: character vector, string Default: '1'

If you select this parameter for the measurement noise covariance of the first measurement function, the block includes an additional input port R1. You specify the time-varying measurement noise covariance in R1. Similarly, if you select Time-varying for the ith measurement function, the block includes an additional input port Ri to specify the time-varying measurement noise covariance for that function.

#### Programmatic Use

 Block Parameter: HasTimeVaryingMeasurementNoise1, HasTimeVaryingMeasurementNoise2, HasTimeVaryingMeasurementNoise3, HasTimeVaryingMeasurementNoise4, HasTimeVaryingMeasurementNoise5 Type: character vector Values: 'off', 'on' Default: 'off'

Suppose that measured output data is not available at all time points at the port y1 that corresponds to the first measurement function. Select Add Enable port to generate an input port Enable1. Use a signal at this port to enable the correction of estimated states only when measured data is available. Similarly, if measured output data is not available at all time points at the port yi for the ith measurement function, select the corresponding Add Enable port.

#### Programmatic Use

 Block Parameter: HasMeasurementEnablePort1, HasMeasurementEnablePort2, HasMeasurementEnablePort3, HasMeasurementEnablePort4, HasMeasurementEnablePort5 Type: character vector Values: 'off', 'on' Default: 'off'

#### Settings

When this parameter is selected, the block outputs the corrected state estimate $\stackrel{^}{x}\left[k|k\right]$ at time step k, estimated using measured outputs until time k. If you clear this parameter, the block returns the predicted state estimate $\stackrel{^}{x}\left[k|k-1\right]$ for time k, estimated using measured output until a previous time k-1. Clear this parameter if your filter is in a feedback loop and there is an algebraic loop in your Simulink model.

#### Programmatic Use

 Block Parameter: UseCurrentEstimator Type: character vector Values: 'off', 'on' Default: 'on'

If you select this parameter, a state estimation error covariance output port P is generated in the block.

#### Programmatic Use

 Block Parameter: OutputStateCovariance Type: character vector Values: 'off','on' Default: 'off'

Use this parameter to specify the data type for all block parameters.

#### Programmatic Use

 Block Parameter: DataType Type: character vector Values: 'single', 'double' Default: 'double'

Block sample time, specified as a positive scalar. If the sample times of your state transition and measurement functions are different, select Enable multirate operation in the Multirate tab, and specify the sample times in the Multirate tab instead.

#### Dependencies

This parameter is available if in the Multirate tab, the Enable multirate operation parameter is off.

#### Programmatic Use

 Block Parameter: SampleTime Type: character vector, string Default: '1'

### Multirate Tab

Select this parameter if the sample times of the state transition and measurement functions are different. You specify the sample times in the Multirate tab, in Sample time.

#### Programmatic Use

 Block Parameter: EnableMultirate Type: character vector Values: 'off', 'on' Default: 'off'

If the sample times for state transition and measurement functions are different, specify Sample time. Specify the sample times for the measurement functions as positive integer multiples of the state transition sample time. The sample times you specify correspond to the following input ports:

• Ports corresponding to state transition function — Additional input to state transition function StateTransitionFcnInputs and time-varying process noise covariance Q. The sample times of these ports must always equal the state transition function sample time, but can differ from the sample time of the measurement functions.

• Ports corresponding to ith measurement function — Measured output yi, additional input to measurement function MeasurementFcniInputs, enable signal at port Enablei, and time-varying measurement noise covariance Ri. The sample times of these ports for the same measurement function must always be the same, but can differ from the sample time for the state transition function and other measurement functions.

#### Dependencies

This parameter is available if in the Multirate tab, the Enable multirate operation parameter is on.

#### Programmatic Use

 Block Parameter: StateTransitionFcnSampleTime, MeasurementFcn1SampleTime1, MeasurementFcn1SampleTime2, MeasurementFcn1SampleTime3, MeasurementFcn1SampleTime4, MeasurementFcn1SampleTime5 Type: character vector, string Default: '1'

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