Unscented Kalman Filter

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

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Libraries:
Control System Toolbox / State Estimation
System Identification Toolbox / Estimators

Description

The Unscented Kalman Filter block estimates the states of a discrete-time nonlinear system using the discrete-time unscented 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 unscented Kalman filter algorithm, the block produces state estimates $$\widehat{x}$$ 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. For more information, see State Transition and Measurement Functions.

Examples

Estimate States of Nonlinear System with Multiple, Multirate Sensors

Use an Extended Kalman Filter block to estimate the states of a system with multiple sensors that are operating at different sampling rates.

• Open Live Script

Compute Residuals and State Estimation Errors

Estimate the states of a discrete-time Van der Pol oscillator and compute state estimation errors and residuals for validating the estimation. The residuals are the output estimation errors, that is, they are the difference between the measured and estimated outputs.

• Open Model

Ports

Input

y1,y2,y3,y4,y5 — Measured system outputs

vector

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 Apply, 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 Apply.

Data Types: single | double

StateTransitionFcnInputs — Additional optional input argument to state transition function

scalar | vector | matrix

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 Apply. 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 (Simulink) 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 (Simulink) blocks. No input ports are generated for the additional inputs in the Unscented Kalman Filter block. For an example, see Online State Estimation Using Identified Models - Nonlinear Models.

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

MeasurementFcn1Inputs,MeasurementFcn2Inputs,MeasurementFcn3Inputs,MeasurementFcn4Inputs,MeasurementFcn5Inputs — Additional optional input argument to each measurement function

scalar | vector | matrix

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],v[k],MeasurementFcn1Inputs)

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

y3[k] = h3(x[k],v[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 Apply. 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 (Simulink) 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 (Simulink) blocks. No input ports are generated for the additional inputs in the Unscented Kalman Filter block. For an example, see Online State Estimation Using Identified Models - Nonlinear Models.

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

Q — Time-varying process noise covariance

scalar | vector | matrix

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 Apply.

Data Types: single | double

R1,R2,R3,R4,R5 — Time-varying measurement noise covariance

matrix

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 Apply.

Data Types: single | double

Enable1,Enable2,Enable3,Enable4,Enable5 — Enable correction of estimated states when measured data is available

scalar

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 i^th 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 Apply.

Data Types: single | double | Boolean

Output

xhat — Estimated states

vector

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 (Simulink) block.

When the Use the current measurements to improve state estimates parameter is selected, the block outputs the corrected state estimate $$\widehat{x}[k|k]$$ at time step k, estimated using measured outputs until time k. If you clear this parameter, the block returns the predicted state estimate $$\widehat{x}[k|k-1]$$ 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

P — State estimation error covariance

matrix

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 (Simulink) block.

Dependencies

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

Data Types: single | double

Parameters

System Model Tab

State Transition

Function — State transition function name

myStateTransitionFcn (default) | function name

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)).

For more information, see State Transition and Measurement Functions.

You can create f using a Simulink Function (Simulink) 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 (Simulink) blocks and the additional inputs Us1,...,Usn using Inport (Simulink) blocks in the Simulink Function block. The Argument name parameter of the Argument Inport (Simulink) and Argument Outport (Simulink) blocks in the Simulink Function block must be different from each other. You do not provide Us1,...,Usn to the Unscented Kalman Filter block. For an example, see Online State Estimation Using Identified Models - Nonlinear Models.

Programmatic Use

Block Parameter: StateTransitionFcn

Type: character vector, string

Default: 'myStateTransitionFcn'

Process noise — Process noise characteristics

Additive (default) | Nonadditive

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

Block Parameter: HasAdditiveProcessNoise

Type: character vector

Values: 'Additive', 'Nonadditive'

Default: 'Additive'

Covariance — Time-invariant process noise covariance

1 (default) | scalar | vector | matrix

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'

Time-varying — Time-varying process noise covariance

'off' (default) | 'on'

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 — Initial state estimate

0 (default) | vector

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'

Initial covariance — State estimation error covariance

1 (default) | scalar | vector | matrix

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'

Unscented Transformation Parameters

Alpha — Spread of sigma points

1e-3 (default) | scalar value between 0 and 1

The unscented Kalman filter algorithm treats the state of the system as a random variable with a mean state value and variance. To compute the state and its statistical properties at the next time step, the algorithm first generates a set of state values distributed around the mean value by using the unscented transformation. These generated state values are called sigma points. The algorithm uses each of the sigma points as an input to the state transition and measurement functions to get a new set of transformed state points and measurements. The transformed points are used to compute the state and state estimation error covariance value at the next time step.

The spread of the sigma points around the mean state value is controlled by two parameters Alpha and Kappa. A third parameter, Beta, impacts the weights of the transformed points during state and measurement covariance calculations:

• Alpha — Determines the spread of the sigma points around the mean state value. Specify as a scalar value between 0 and 1 (0 < Alpha <= 1). It is usually a small positive value. The spread of sigma points is proportional to Alpha. Smaller values correspond to sigma points closer to the mean state.

• Kappa — A second scaling parameter that is typically set to 0. Smaller values correspond to sigma points closer to the mean state. The spread is proportional to the square-root of Kappa.

• Beta — Incorporates prior knowledge of the distribution of the state. For Gaussian distributions, Beta = 2 is optimal.

If you know the distribution of state and state covariance, you can adjust these parameters to capture the transformation of higher-order moments of the distribution. The algorithm can track only a single peak in the probability distribution of the state. If there are multiple peaks in the state distribution of your system, you can adjust these parameters so that the sigma points stay around a single peak. For example, choose a small Alpha to generate sigma points close to the mean state value.

For more information, see Unscented Kalman Filter Algorithm.

Programmatic Use

Block Parameter: Alpha

Type: character vector, string

Default: '1e-3'

Beta — Characterization of state distribution

2 (default) | scalar value greater than or equal to 0

Characterization of the state distribution that is used to adjust weights of transformed sigma points, specified as a scalar value greater than or equal to 0. For Gaussian distributions, Beta = 2 is the optimal choice.

For more information, see the description for Alpha.

Programmatic Use

Block Parameter: Beta

Type: character vector, string

Default: '2'

Kappa — Spread of sigma points

0 (default) | scalar value between 0 and 3

Spread of sigma points around mean state value, specified as a scalar value between 0 and 3 (0 <= Kappa <= 3). Kappa is typically specified as 0. Smaller values correspond to sigma points closer to the mean state. The spread is proportional to the square root of Kappa. For more information, see the description for Alpha.

Programmatic Use

Block Parameter: Kappa

Type: character vector, string

Default: '0'

Measurement

Function — Measurement function name

myMeasurementFcn (default) | function name

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.

For more information, see State Transition and Measurement Functions.

You can create h using a Simulink Function (Simulink) 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 MeasurementFcnInput to specify this argument.

• If you are using a Simulink Function block, specify x and v using Argument Inport (Simulink) blocks and the additional inputs Um1,...,Umn using Inport (Simulink) blocks in the Simulink Function block. The Argument name parameter of the Argument Inport (Simulink) and Argument Outport (Simulink) blocks in the Simulink Function block must be different from each other. You do not provide Um1,...,Umn to the Unscented Kalman Filter block. For an example, see Online State Estimation Using Identified Models - Nonlinear Models.

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'

Measurement noise — Measurement noise characteristics

Additive (default) | Nonadditive

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

Block Parameter: HasAdditiveMeasurementNoise1, HasAdditiveMeasurementNoise2, HasAdditiveMeasurementNoise3, HasAdditiveMeasurementNoise4, HasAdditiveMeasurementNoise5

Type: character vector

Values: 'Additive', 'Nonadditive'

Default: 'Additive'

Enable measurement wrapping — Enable measurement wrapping

off (default) | on

Select this parameter to enable measurement wrapping to estimate states when you have circular measurements that are independent of your model states. If you select this parameter, then the measurement function you specify must include the following two outputs:

1. The measurement, specified as a N-element output measurement vector of the nonlinear system at time step k, given the state vector at time step k. N is the number of measurements of the system.

2. The measurement wrapping bounds, specified as an N-by-2 matrix where, the first column provides the minimum measurement bound and the second column provides the maximum measurement bound.

Enabling the Enable measurement wrapping check box wraps the measurement residuals in a defined bound, which helps to prevent the filter from divergence due to incorrect measurement residual values. For an example, see State Estimation with Wrapped Measurements Using Extended Kalman Filter.

Programmatic Use

Block Parameter: EnableMeasurementWrapping1, EnableMeasurementWrapping2, EnableMeasurementWrapping3, EnableMeasurementWrapping4, EnableMeasurementWrapping5

Type: character vector

Values: 'off', 'on'

Default: 'off'

Covariance — Time-invariant measurement noise covariance

1 (default) | matrix

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 i^th 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'

Time-varying — Time-varying measurement noise covariance

off (default) | on

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 i^th 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'

Add Enable Port — Enable correction of estimated states only when measured data is available

off (default) | on

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 i^th 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

Use the current measurements to improve state estimates — Choose between corrected or predicted state estimate

on (default) | off

When this parameter is selected, the block outputs the corrected state estimate $$\widehat{x}[k|k]$$ at time step k, estimated using measured outputs until time k. If you clear this parameter, the block returns the predicted state estimate $$\widehat{x}[k|k-1]$$ 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'

Output state estimation error covariance — Output state estimation error covariance

off (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'

Data type — Data type for block parameters

double (default) | single

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'

Sample time — Block sample time

1 (default) | positive scalar

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

Enable multirate operation — Enable specification of different sample times for state transition and measurement functions

off (default) | on

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'

Sample times — State transition and measurement function sample times

positive scalar

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 i^th 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'

More About

State Transition and Measurement Functions

The algorithm computes the state estimates $$\widehat{x}$$ of the nonlinear system using state transition and measurement functions specified by you. You can specify up to five measurement functions, each corresponding to a sensor in the system. The software lets you specify the noise in these functions as additive or nonadditive.

• Additive Noise Terms — The state transition and measurements equations have the following form:

  $$\begin{array}{l}x[k+1]=f(x[k],u_s[k])+w[k]\\ y[k]=h(x[k],u_m[k])+v[k]\end{array}$$

  Here f is a nonlinear state transition function that describes the evolution of states x from one time step to the next. The nonlinear measurement function h relates x to the measurements y at time step k. w and v are the zero-mean, uncorrelated process and measurement noises, respectively. These functions can also have additional optional input arguments that are denoted by us and um in the equations. For example, the additional arguments could be time step k or the inputs u to the nonlinear system. There can be multiple such arguments.

  Note that the noise terms in both equations are additive. That is, x(k+1) is linearly related to the process noise w(k), and y(k) is linearly related to the measurement noise v(k). For additive noise terms, you do not need to specify the noise terms in the state transition and measurement functions. The software adds the terms to the output of the functions.

• Nonadditive Noise Terms — The software also supports more complex state transition and measurement functions where the state x[k] and measurement y[k] are nonlinear functions of the process noise and measurement noise, respectively. When the noise terms are nonadditive, the state transition and measurements equation have the following form:

  $$\begin{array}{l}x[k+1]=f(x[k],w[k],u_s[k])\\ y[k]=h(x[k],v[k],u_m[k])\end{array}$$

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using Simulink® Coder™.

The state transition and measurement functions that you specify must use only the MATLAB commands and Simulink blocks that support code generation. For a list of blocks that support code generation, see Blocks Supported for Code Generation (Simulink Coder). For a list of commands that support code generation, see Functions and Objects Supported for C/C++ Code Generation (MATLAB Coder).

Generated code uses an algorithm that is different from the algorithm that the Unscented Kalman Filter block itself uses. You might see some numerical differences in the results obtained using the two methods.

Version History

Introduced in R2017a

R2020b: Numerical Changes

Starting in R2020b, numerical improvements in the Unscented Kalman Filter algorithm might produce results that are different from the results you obtained in previous versions.