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

**Library:**Control System Toolbox / State Estimation

System Identification Toolbox / Estimators

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.

`y1,y2,y3,y4,y5`

— Measured system outputsvector

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.

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 functionscalar | 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 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 Unscented Kalman Filter block.

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 functionscalar | 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 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 Unscented Kalman Filter block.

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 covariancescalar | 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.

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 covariancematrix

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.

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 availablescalar

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 **y** for
the

`i`

`i`

`0`

.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`

`xhat`

— Estimated statesvector

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 $$\widehat{x}[k|k]$$ 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 covariancematrix

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.

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

**Data Types: **`single`

| `double`

`Function`

— State transition function name`myStateTransitionFcn`

(default) | function nameThe 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 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 Unscented Kalman Filter block.

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))`

.

Block Parameter: `HasAdditiveProcessNoise` |

Type: character vector |

Values: `'Additive'` , `'Nonadditive'` |

Default: `'Additive'` |

`Covariance`

— Time-invariant process noise covariance`1`

(default) | scalar | vector | matrixTime-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.

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

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.

Block Parameter: `HasTimeVaryingProcessNoise` |

Type: character vector |

Values: `'off'` , `'on'` |

Default: `'off'` |

`Initial state`

— Initial state estimate`0`

(default) | vectorInitial 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.

Block Parameter: `InitialState` |

Type: character vector, string |

Default: `'0'` |

`Initial covariance`

— State estimation error covariance`1`

(default) | scalar | vector | matrixState 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**.

Block Parameter: `InitialStateCovariance` |

Type: character vector, string |

Default: `'1'` |

`Alpha`

— Spread of sigma points`1e-3`

(default) | scalar value between 0 and 1The 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.

Block Parameter:
`Alpha` |

Type: character vector,
string |

Default:
`'1e-3'` |

`Beta`

— Characterization of state distribution`2`

(default) | scalar value greater than or equal to 0Characterization 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**.

Block Parameter:
`Beta` |

Type: character vector,
string |

Default:
`'2'` |

`Kappa`

— Spread of sigma points`0`

(default) | scalar value between 0 and 3Spread 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**.

Block Parameter:
`Kappa` |

Type: character vector,
string |

Default:
`'0'` |

`Function`

— Measurement function name`myMeasurementFcn`

(default) | function nameThe 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 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 blocks and the additional inputs`Um1,...,Umn`

using Inport blocks in the Simulink Function block. You do not provide`Um1,...,Umn`

to the Unscented 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**.

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))`

.

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

Type: character vector |

Values: `'Additive'` , `'Nonadditive'` |

Default: `'Additive'` |

`Covariance`

— Time-invariant measurement noise covariance`1`

(default) | matrixTime-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 **R** to
specify the time-varying covariance for the

`i`

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

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

`i`

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 **y** for
the

`i`

Block Parameter: `HasMeasurementEnablePort1` , `HasMeasurementEnablePort2` , `HasMeasurementEnablePort3` , `HasMeasurementEnablePort4` , `HasMeasurementEnablePort5` |

Type: character vector |

Values: `'off'` , `'on'` |

Default: `'off'` |

`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.

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.

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.

Block Parameter: `DataType` |

Type: character vector |

Values: `'single'` , `'double'` |

Default: `'double'` |

`Sample time`

— Block sample time`1`

(default) | positive scalarBlock 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.

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

.

Block Parameter: `SampleTime` |

Type: character vector, string |

Default: `'1'` |

`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**.

Block Parameter: `EnableMultirate` |

Type: character vector |

Values: `'off'` , `'on'` |

Default: `'off'` |

`Sample time`

— State transition and measurement function sample timespositive 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*measurement function — Measured output^{th}**y**, additional input to measurement function`i`

**MeasurementFcn**, enable signal at portInputs`i`

**Enable**, and time-varying measurement noise covariance`i`

**R**. 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.`i`

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

.

Block Parameter: `StateTransitionFcnSampleTime` , `MeasurementFcn1SampleTime1` , `MeasurementFcn1SampleTime2` , `MeasurementFcn1SampleTime3` , `MeasurementFcn1SampleTime4` , `MeasurementFcn1SampleTime5` |

Type: character vector, string |

Default: `'1'` |

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`u`

and_{s}`u`

in the equations. For example, the additional arguments could be time step_{m}`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:$$\begin{array}{l}x[k+1]=f(x[k],w[k],{u}_{s}[k])\\ y[k]=h(x[k],v[k],{u}_{m}[k])\end{array}$$

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 Simulink Built-In Blocks That Support Code Generation (Simulink Coder). For a list of commands that support code generation, see Functions and Objects Supported for C/C++ Code Generation — Category List (MATLAB Coder).

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