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Estimate states of discrete-time nonlinear system using particle filter

**Library:**System Identification Toolbox / Estimators

The Particle Filter block estimates the states of a discrete-time nonlinear system using the discrete-time particle filter algorithm.

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

The algorithm computes the state estimates $$\widehat{x}$$ of the nonlinear system using the state transition and measurement likelihood functions you specify.

You create the nonlinear state transition function and measurement likelihood 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 likelihood functions, each corresponding to a sensor in the system.

`y1,y2,y3,y4,y5`

— Measured system outputsvector

Measured system outputs corresponding to each measurement likelihood
function that you specify in the block. The number of ports equals the
number of measurement likelihood functions in your system. You can
specify up to five measurement likelihood functions. For example, if
your system has two sensors, you specify two measurement likelihood
functions in the block. The first port **y1** is
available by default. Click **Add Measurement**, to
generate port **y2** corresponding to the second
measurement likelihood 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 two-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**.

`StateTransitionFcnInputs`

— Optional input argument to state transition functionscalar | vector | matrix

Optional input argument to the state transition function
`f`

other than the state
`x`

.

If you create `f`

using a MATLAB^{®} function (`.m`

file), the software
generates the port **StateTransitionFcnInputs** when
you enter the name of your function, and click
**Apply**.

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 Particle 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 particles.

`MeasurementLikelihoodFcn1Inputs,...,MeasurementLikelihoodFcn5Inputs`

— Optional input argument to each measurement likelihood functionscalar | vector | matrix

Optional inputs to the measurement likelihood functions other than the
state `x`

and measurement `y`

.

**MeasurementLikelihoodFcn1Inputs** corresponds to
the first measurement likelihood function that you specify, and so
on.

If you specify two measurement inputs using MATLAB functions (`.m`

files) in
**Function**, the software generates ports
**MeasurementLikelihoodFcn1Inputs** and
**MeasurementLikelihoodFcn2Inputs** when you click
**Apply**. You can specify the inputs to these
ports as scalars, vectors, or matrices.

If your measurement likelihood 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 Particle Filter block.

A port corresponding to a measurement likelihood function
`h`

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

You specify measurement input

`h`

in**Function**using a MATLAB function, and`h`

is on the MATLAB path.`h`

requires only one additional input argument apart from particles and measurement.

`Enable1,Enable2,Enable3,Enable4,Enable5`

— Enable correction of estimated states when measured data is availablescalar

Enable correction of estimated states when measured data is available.

For example, consider that measured output data is not available at
all time points at the port **y1** that corresponds to
the first measurement likelihood function. Then, 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`

.If you select **Add Enable port** for a
measurement likelihood function, a port corresponding to that
measurement likelihood function is generated. The port appears when
you click **Apply**.

`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.

`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.

You can output the error covariance only if you select
**Output state estimation error covariance** in the
**Block outputs, Multirate** tab, and click
**Apply**.

This parameter is available if in the **Block outputs,
Multirate** tab, the **State estimation
method** parameter is set to
`'Mean'`

.

`Particles`

— Particle values used for state estimationarray

Particle values used for state estimation, returned as an
*Ns*-by-*Np* or
*Np*-by-*Ns* array.
*Ns* is the number of states of the system, and
*Np* is the number of particles.

If the

`StateOrientation`

parameter is specified as`'column'`

, then**Particles**is returned as an*Ns*-by-*Np*array.If the

`StateOrientation`

parameter is specified as`'row'`

, then**Particles**is returned as an*Np*-by-*Ns*array.

This port is generated if you select **Output all
particles** in the **Block outputs,
Multirate** tab, and click
**Apply**.

`Weights`

— Particle weights used for state estimationvector

Particle weights used for state estimation, returned as a
1-by-*Np* or *Np*-by-1 vector,
where *Np* is the number of particles used for state
estimation.

If the

`StateOrientation`

parameter is specified as`'column'`

, then**Weights**is returned as a 1-by-*Np*vector, where each weight is associated with the particle in the same column in the`Particles`

array.If the

`StateOrientation`

parameter is specified as`'row'`

, then**Weights**is returned as a*Np*-by-1 vector, where each weight is associated with the particle in the same row in the`Particles`

array.

This port is generated if you select **Output
weights** in the **Block outputs,
Multirate** tab, and click
**Apply**.

`Function`

— State transition function name`'vdpParticleFilterStateFcn'`

(default) | function nameThe particle filter state transition function calculates the particles
at time step *k+1*, given particles at time step
*k* per the dynamics of your system and process
noise. This function has the syntax:

particlesNext = f(particles, param1, param2, ...)

where, *particles* and
*particlesNext* have dimensions
*Ns*-by-*Np* if **State
Orientation** is specified as `'column'`

,
or *Np*-by-*Ns* if **State
Orientation** is specified as `'row'`

.
Also, `param_i`

represents optional input arguments you
may specify. For more information on optional input arguments, see StateTransitionFcnInputs.

You create the state transition function and specify the function name
in **Function**. For example, if
`vdpParticleFilterStateFcn.m`

is the state
transition function that you created and saved, specify
**Function** as
`'vdpParticleFilterStateFcn'`

.

You can create **Function** using a Simulink Function block or
as a MATLAB function (`.m`

file).

Block Parameter:
`StateTransitionFcn` |

Type: character vector,
string |

Default:
`'vdpParticleFilterStateFcn'` |

`Number of Particles`

— Number of particles used in the filter1000 (default) | positive scalar integer

Number of particles used in the filter, specified as a positive scalar integer. Each particle represents a state hypothesis in the system. A higher number of particles increases the state estimation accuracy, but also increases the computational effort required to run the filter.

Block Parameter:
`NumberOfParticles` |

Type: positive scalar
integer |

Default:
`1000` |

`Distribution`

— Initial distribution of particles`'Gaussian'`

(default) | `'Uniform'`

| `'Custom'`

Initial distribution of particles, specified as
`'Gaussian'`

, `'Uniform'`

, or
`'Custom'`

.

If you choose `'Gaussian'`

, the initial set of
particles or state hypotheses are distributed per the multivariate
Gaussian distribution, where you specify the **Mean**
and **Covariance**. The initial weight of all particles
is assumed to be equal.

If you choose `'Uniform'`

, the initial set of
particles are distributed per the uniform distribution, where you
specify the upper and lower **State bounds**. The
initial weight of all particles is assumed to be equal.

`'Custom'`

allows you to specify your own set of
initial particles and their weights. You can use arbitrary probability
distributions for **Particles** and
**Weights** to initialize the filter.

Block Parameter:
`InitialDistribution` |

Type: character
vector |

Values:
`'Gaussian'` ,
`'Uniform'` ,
`'Custom'` |

Default:
`'Gaussian'` |

`Mean`

— Initial mean value of particles[0,0] (default) | vector

Initial mean value of particles, specified as a vector. The number of states to be estimated defines the length of the vector.

This parameter is available if in the **System
model** tab, the **Distribution**
parameter is set to `Gaussian`

.

Block Parameter:
`InitialMean` |

Type: array |

Default:
`[0,0]` |

`Covariance`

— Initial covariance of particles1 (default) | scalar | vector | matrix

Initial covariance of particles, specified as a scalar, vector, or matrix.

If **Covariance** is specified as:

A scalar, then it must be positive. The covariance is assumed to be a [

*Ns**Ns*] matrix with this scalar on the diagonals. Here,*Ns*is the number of states.A vector, then each element must be positive. The covariance is assumed to be a [

*Ns**Ns*] matrix with the elements of the vector on the diagonals.A matrix, then it must be positive semidefinite.

This parameter is available if in the **System
model** tab, the **Distribution**
parameter is set to `Gaussian`

.

Block Parameter:
`InitialCovariance` |

Type: scalar, vector, or
matrix |

Default:
`1` |

`Circular Variables`

— Circular variables used for state estimation0 (default) | scalar | vector

Circular variables used for state estimation, specified as a scalar,
or *Ns*-element vector, where *Ns* is
the number of states.

If **Circular Variables** is specified as a scalar,
the software extends it to a vector where each element is equal to this
scalar. Circular (or angular) distributions use a probability density
function with a range of [`-π`

`π`

]. Use circular variables if some of the states in
your system represent angular quantities like the orientation of an
object.

Block Parameter:
`CircularVariables` |

Type: scalar,
vector |

Default:
`0` |

`State Orientation`

— Orientation of input system states`'column'`

(default) | `'row'`

Orientation of system states, specified as `'column'`

or `'row'`

.

If **State Orientation** is specified as:

`'column'`

, then the first input argument to the state transition and measurement likelihood function is [*Ns**Np*]. In this case,*i*column of this matrix is the^{th}*i*particle (state hypothesis). Also, the states estimates^{th}**xhat**is output as a [*Ns*1] vector. Here,*Ns*is the number of states, and*Np*is the number of particles.`'row'`

, then the first input argument to the state transition and measurement likelihood function is [*Np**Ns*], and each row of this matrix contains a particle. Also, the states estimates**xhat**is output as a [1*Ns*] vector.

Block Parameter:
`StateOrientation` |

Type: character
vector |

Values:
`'column'` ,
`'row'` |

Default:
`'column'` |

`State bounds`

— Initial bounds on system states[-3 3;-3 3] (default) | array

Initial bounds on system states, specified as an
*Ns*-by-2 array, where *Ns* is the
number of states.

The *i ^{th}* row lists the
lower and upper bound of the uniform distribution for the initial
distribution of particles of the

This parameter is available if in the **System
model** tab, the **Distribution**
parameter is set to `Uniform`

.

Block Parameter:
`InitialStateBounds` |

Type: array |

Default: ```
[-3 3;-3
3]
``` |

`Particles`

— Custom particle distribution for state estimation[] (default) | array

Custom particle distribution for state estimation, specified as an
*Ns*-by-*Np* or
*Np*-by-*Ns* array.
*Ns* is the number of states of the system, and
*Np* is the number of particles.

If the

`StateOrientation`

parameter is specified as`'column'`

, then**Particles**is an*Ns*-by-*Np*array.If the

`StateOrientation`

parameter is specified as`'row'`

, then**Particles**is an*Np*-by-*Ns*array.

This parameter is available if in the **System
model** tab, the **Distribution**
parameter is set to `Custom`

.

Block Parameter:
`InitialParticles` |

Type: array |

Default:
`[]` |

`Weights`

— Custom particle weight values for state estimation[] (default) | positive vector

Custom particle weight values for state estimation, specified as a
1-by-*Np* or *Np*-by-1 positive
vector, where *Np* is the number of particles used for
state estimation.

If the

`StateOrientation`

parameter is specified as`'column'`

, then**Weights**is a 1-by-*Np*vector. Each weight in the vector is associated with the particle in the same column in the`Particles`

array.If the

`StateOrientation`

parameter is specified as`'row'`

, then**Weights**is a*Np*-by-1 vector. Each weight in the vector is associated with the particle in the same row in the`Particles`

array.

This parameter is available if in the **System
model** tab, the **Distribution**
parameter is set to `Custom`

.

Block Parameter:
`InitialWeights` |

Type: positive
vector |

Default:
`[]` |

`Function`

— Measurement likelihood function name`'vdpMeasurementLikelihoodFcn'`

(default) | function nameThe measurement likelihood function calculates the likelihood of
particles (state hypotheses) using the sensor measurements. For each
state hypothesis (particle), the function first calculates an
*Nm*-element measurement hypothesis vector. Then
the likelihood of each measurement hypothesis is calculated based on the
sensor measurement and the measurement noise probability distribution.
this function has the
syntax:

likelihood = h(particles, measurement, param1, param2, ...)

`'column'`

,
or `'row'`

.
You create the measurement likelihood function and specify the
function name in **Function**. For example, if
`vdpMeasurementLikelihoodFcn.m`

is the measurement
likelihood function that you created and saved, specify
**Function** as
`'vdpMeasurementLikelihoodFcn'`

.

You can create **Function** using a Simulink Function block or
as a MATLAB function (`.m`

file).

You can use a MATLAB function only if

*h*has zero or one additional input argument`param_i`

other than**Particles**and**Measurement**.The software generates an additional input port

**MeasurementLikelihoodFcn**to specify this argument for theInputs`i`

*i*measurement likelihood function, and click^{th}**Apply**.If you are using a Simulink Function block, specify

`x`

and`y`

using Argument Inport blocks and the additional inputs`param_i`

using Inport blocks in the Simulink Function block. You do not provide`param_i`

to the Particle Filter block.

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

Block Parameter:
`MeasurementLikelihoodFcn1` ,
`MeasurementLikelihoodFcn2` ,
`MeasurementLikelihoodFcn3` ,
`MeasurementLikelihoodFcn4` ,
`MeasurementLikelihoodFcn5` |

Type: character vector,
string |

Default:
`'vdpMeasurementLikelihoodFcn'` |

`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 likelihood function. To generate an input port
**Enable1**, select **Add Enable
port**. 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'` |

`Resampling method`

— Method used for particle resampling`'Multinomial'`

(default) | `'Systemic'`

| `'Stratified'`

Method used for particle resampling, specified as one of the following:

`'Multinomial'`

`'Systematic'`

`'Stratified'`

Block Parameter:
`ResamplingMethod` |

Type: character
vector |

Values:
`'Multinomial'` ,
`'Systemic'` ,
`'Stratified'` |

Default:
`'Multinomial'` |

`Trigger method`

— Method to determine when resampling occurs`'Ratio'`

(default) | `'Interval'`

Method to determine when resampling occurs, specified as either
`'Ratio'`

or `'Interval'`

. The
`'Ratio'`

value triggers resampling based on the
ratio of effective total particles. The `'Interval'`

value triggers resampling at regular time steps of the particle filter
operation.

Block Parameter:
`TriggerMethod` |

Type: character
vector |

Values:
`'Ratio'` ,
`'Interval'` |

Default:
`'Ratio'` |

`Minimum effective particle ratio`

— Minimum desired ratio of the effective number of particles to the total number of particles 0.5 (default) | positive scalar

Minimum desired ratio of the effective number of particles to the total number of particles, specified as a positive scalar. The effective number of particles is a measure of how well the current set of particles approximates the posterior distribution. A lower effective particle ratio implies that a lower number of particles are contributing to the estimation and resampling is required.

If the ratio of the effective number of particles to the total number of particles falls below the minimum effective particle ratio, a resampling step is triggered.

Specify minimum effective particle ratio as any value from 0 through 1.

This parameter is available if in the **System
model** tab, the **Trigger method**
parameter is set to `Ratio`

.

Block Parameter:
`MinEffectiveParticleRatio` |

Type: scalar |

Values: Range
`[0,1]` |

Default:
`0.5` |

`Sampling Interval`

— Fixed interval between resampling1 (default) | positive scalar integer

Fixed interval between resampling, specified as a positive scalar
integer. The sampling interval determines during which correction steps
the resampling is executed. For example, a value of two means the
resampling is executed every second correction step. A value of
`inf`

means that resampling is never
executed.

This parameter is available if in the **System
model** tab, the **Trigger method**
parameter is set to `Interval`

.

Block Parameter:
`SamplingInterval` |

Type: positive scalar
integer |

Default:
`1` |

`Randomness`

— Whether the random numbers are repeatable`'Repeatable'`

(default) | `'Not repeatable'`

Whether the random numbers are repeatable, specified as either
`'Repeatable'`

or ```
'Not
repeatable'
```

. If you want to be able to produce the same
result more than once, set **Randomness** to
`'Repeatable'`

, and specify the same random number
generator seed value in **Seed**.

Block Parameter:
`Randomness` |

Type: character
vector |

Values:
`'Repeatable'` , ```
'Not
repeatable'
``` |

Default:
`'Repeatable'` |

`Seed`

— Seed value for repeatable random numbers0 (default) | scalar

Seed value for repeatable random numbers, specified as a scalar.

This parameter is available if in the **System
model** tab, the **Randomness**
parameter is set to `'Repeatable'`

.

Block Parameter:
`Seed` |

Type: scalar |

Default:
`0` |

`Sample time`

— Block sample time`1`

(default) | positive scalarBlock sample time, specified as a positive scalar.

Use the **Sample time** parameter if your state
transition and all measurement likelihood functions have the same sample
time. Otherwise, select the **Enable multirate
operation** option in the **Multirate**
tab, and specify sample times in the same tab.

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

.

Block Parameter:
`SampleTime` |

Type: character vector,
string |

Default:
`'1'` |

`State Estimation Method`

— Method used for extracting a state estimate from particles`'Mean'`

(default) | `'MaxWeight'`

| `'None'`

Method used for extracting a state estimate from particles, specified as one of the following:

`'Mean'`

— The Particle Filter block outputs the weighted mean of the particles, depending on the parameters**Weights**and**Particles**, as the state estimate.`'Maxweight'`

— The Particle Filter block outputs the particle with the highest weight as the state estimate.`'None'`

— Use this option to implement a custom state estimation method by accessing all particles using the**Output all particles**parameter from the**Block outputs, Multirate**tab.

Block Parameter:
`StateEstimationMethod` |

Type: character vector,
string |

Values:
`'Mean'` , `'MaxWeight'` ,
`'None'` |

Default:
`'Mean'` |

`Output all particles`

— Output all particles`'off'`

(default) | `'on'`

If you select this parameter, an output port for particles used in the
estimation, **Particles** is generated in the
block.

If the

`StateOrientation`

parameter is specified as`'column'`

, then the particles are output as an*Ns*-by-*Np*array.*Ns*is the number of states of the system, and*Np*is the number of particles.If the

`StateOrientation`

parameter is specified as`'row'`

, then the particles are output as an*Np*-by-*Ns*array.

Block Parameter:
`OutputParticles` |

Type: character
vector |

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

Default:
`'off'` |

`Output weights`

— Output particle weights`'off'`

(default) | `'on'`

If you select this parameter, an output port for particle weights used
in the estimation, **Weights** is generated in the
block.

If the

`StateOrientation`

parameter is specified as`'column'`

, then the particle weights are output as a 1-by-*Np*vector. Here, where each weight is associated with the particle in the same column in the`Particles`

array.*Np*is the number of particles used for state estimation.If the

`StateOrientation`

parameter is specified as`'row'`

, then the particle weights are output as a*Np*-by-1 vector.

Block Parameter:
`OutputWeights` |

Type: character
vector |

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

Default:
`'off'` |

`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.

This parameter is available if in the **Block outputs,
Multirate** tab, the **State estimation
method** parameter is set to
`'Mean'`

.

Block Parameter:
`OutputStateCovariance` |

Type: character
vector |

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

Default:
`'off'` |

`Use the current measurements to improve state estimates`

— Option to use current measurements for state estimation`'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:
`'on'` ,
`'off'` |

Default:
`'on'` |

`Enable multirate operation`

— Enable specification of different sample times for state transition and measurement likelihood functions`'off'`

(default) | `'on'`

Select this parameter if the sample times of the state transition or
any of the measurement likelihood functions differ from the rest. 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 likelihood function sample timespositive scalar

If the sample times for state transition and measurement likelihood
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**. 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 likelihood functions.Ports corresponding to

*i*measurement likelihood function — Measured output^{th}**y**, additional input to measurement likelihood function`i`

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

**Enable**. The sample times of these ports for the same measurement likelihood function must always be the same, but can differ from the sample time for the state transition function and other measurement likelihood functions.`i`

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

.

Block Parameter:
`StateTransitionFcnSampleTime` ,
`MeasurementLikelihoodFcn1SampleTime1` ,
`MeasurementLikelihoodFcn1SampleTime2` ,
`MeasurementLikelihoodFcn1SampleTime3` ,
`MeasurementLikelihoodFcn1SampleTime4` ,
`MeasurementLikelihoodFcn1SampleTime5` |

Type: character vector,
string |

Default:
`'1'` |

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

The state transition and measurement likelihood 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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