# Estimate Neural State-Space Model

## Description

The **Estimate Neural State-Space Model** task lets
you interactively estimate and validate a neural state-space model, using time-domain data.
You can define and vary the structure and the parameters of the networks and the solver. The
task automatically generates MATLAB^{®} code for your live script. For more information about Live Editor tasks, see
Add Interactive Tasks to a Live Script. For more information
about state-space estimation, see What Are State-Space Models?

The **Estimate Neural State-Space Model** task is independent
of the more general System Identification app. Use the System
Identification app when you want to compute and compare estimates for multiple model
structures.

To get started, load experiment data that contains input and output data into your MATLAB workspace and then import that data into the task. Then specify a model structure to estimate. The task gives you controls and plots that help you experiment with different model parameters and compare how well the output of each model fits the measurements.

### More

### Related Functions

The code that **Estimate Neural State-Space Model**
generates uses the following functions and objects.

The task estimates an `idNeuralStateSpace`

state-space model.

## Open the Task

To add the **Estimate Neural State-Space Model** task to a
live script in the MATLAB Editor:

On the

**Live Editor**tab, select**Task**>**Estimate Neural State-Space Model**.In a code block in your script, type a relevant keyword, such as

`neuralstatespace`

or`nlssest`

. Select`Estimate State-Space Model`

from the suggested command completions.

## Examples

### Estimate Neural State-Space Model with Live Editor Task

Use the **Estimate Neural State-Space Model** Live Editor Task to estimate a neural state-space model and compare the model output with the measurement data.

Open this example to see a preconfigured script containing the task.

**Generate Data**

For this example, generate data by simulating a first-order linear system. First, fix the random generator seed to guarantee reproducibility.

rng(0)

Create a first-order discrete dynamical system in `tf`

form with one input and one output, convert it to discrete time using a sample time of 0.1 sec, and use `ss`

to obtain a state-space realization.

Ts = 0.1; sys = ss(c2d(tf(1,[1 1]),Ts));

The identification of a neural state-space system requires you to have measurement of the system states. Therefore, transform the state-space coordinates so that the output is equal to the state. Alternatively you can augment the output equation to include the state among the measured signals.

sys.b = sys.b*sys.c; sys.c = 1;

In general, it is good practice to use multiple experiments, each containing a different trajectory, as doing so is more likely to yield a better coverage of the state-input space. Furthermore, using long trajectories tends to reduce both the accuracy and efficiency of the estimation. However, for this example, use a single trajectory for estimation.

Define a time vector and a random input sequence for estimation (training).

te = 0:Ts:10; ue = randn(length(te),size(sys.B,2));

Generate an output response to the random input sequence by simulating the system from a zero initial condition. The first (vertical) dimension in `ye`

must be time and the second (horizontal) dimension must be the specific output in the output vector signal.

ye = lsim(sys,ue,te,zeros(size(sys.B,1),1));

Define a shorter time vector and a random input sequence for validation.

tv = 0:Ts:1; uv = randn(length(tv),size(sys.B,2));

Generate an output response to the random input sequence by simulating the system, from a zero initial condition.

yv = lsim(sys,uv,tv,zeros(size(sys.B,1),1));

**Import Data into the Task**

In the **Select data** section, set **Data Type** to `Numeric`

, **Sample Time** to `0.01`

, **Estimation Data: Input (u)** to `ue`

, **Estimation Data: Output (y)** to `ye`

, **Validation Data: Input (u)** to `uv`

, and **Validation Data: Output (y)** to `yv`

.

**Specify Model Structure and State Network**

In the **Specify model structure** section, set the **Number of states** to 1 and select the discrete-time domain. In the **State network** section, set the **Number of states** to 1 and specify **Layer size** as 16. Leave the other options unchanged.

Note that since for this example the output is equal to the state, there is no **Output network** section.

**Examine Training and Display Options**

In the **Specify training options** section, the **Training algorithm** is set to `ADAM`

, with a **Learn rate** of `0.001`

. The number of epochs is set to `100`

, and so is the mini-batch size. For more information on these options, see `nssTrainingOptions`

.

In the **Display results** section, both the **Show fit to estimation data** and (since you have specified validation data) the **Show fit to validation data** are selected.

**Execute Live Task**

Execute the task from the **Live Editor** tab using **Run**. During training, a plot displays the training losses of the state and output networks.

`Generating estimation report...done.`

After training, two plots displays the model fit on the estimation and validation data.

**Generate Code**

To display the code that the task generates, click (Show code) at the bottom of the parameter section. The code that you see reflects the current parameter configuration of the task.

### Related Examples

## Parameters

**Select data**

`Data type`

— Data type for input and output data

`Numeric`

(default) | `Timetable`

| `iddata object`

The task accepts numeric measurement values that are uniformly sampled in time.
Input and output signals can contain multiple channels. Data can be packaged as numeric
arrays, in an `iddata`

object, or in a `timetable`

object. For multiexperiment data, numeric and timetable data can
be packaged as cell arrays. For cell arrays of timetables, all timetables must contain
the same variable names. Data objects handle multiexperiment data internally.

The data type you choose determines whether you must specify additional parameters.

`Numeric`

— Specify**Sample Time**and**Start Time**in the time unit that you select. Additionally, you need to specify different workspace variables containing the input and output signals to be used for estimation and (if available) validation.`Timetable`

— Specify no additional parameters because the timetable already contains the input and output signals and sampling information.`iddata object`

— Specify no additional parameters because the`iddata`

object already contains the input and output signals and sampling information.

`Sample time`

— Sample time

1 (default) | positive scalar

Sample time at which estimation and (if available) validation data are collected,
specified as a positive scalar, in the unit specified by the following time unit drop
down list. You can specify a sample time only when **Data Type** is
`Numeric`

.

`Time unit`

— Time unit

`seconds`

(default) | `minutes`

| `milliseconds`

| ...

Time unit for the **Sample time** and **Start
time** parameters. You can specify one of the following units:

`nanoseconds`

`microseconds`

`milliseconds`

`seconds`

`minutes`

`hours`

`days`

`weeks`

`months`

`years`

You can specify a sample time only when **Data Type** is
`Numeric`

.

`Start time`

— Start time

0 (default) | nonnegative scalar

Start time for the estimation and (if available) validation data, specified as a
nonnegative scalar, in the unit specified by the preceding time unit drop down list.
This value is relevant only if you deselect the **Time invariant**
checkbox. You can specify a start time only when **Data Type** is
`Numeric`

.

`Estimation data: Input (u)`

— Name of the input data variable used for estimation

valid variable name

Name of the input data variable used for estimation, selected from the MATLAB workspace choices. Use this parameter, along with **Estimation
data: Output (y)**, when **Data Type** is
`Numeric`

.

`Estimation data: Output (y)`

— Name of the output data variable used for estimation

valid variable name

Name of the output data variable used for estimation, selected from the MATLAB workspace choices. Use this parameter, along with **Estimation
data: Input (u)**, when **Data Type** is
`Numeric`

.

`Estimation data: Timetable`

— Variable name of timetable object containing input and output data for estimation

valid variable name

Select the timetable object variable name from the MATLAB workspace choices. If you use a use a cell arrays of timetables, all
timetables must contain the same variable names. Use this parameter when **Data
type** is `Timetable`

.

`Estimation data: Object`

— Variable name of data object containing input and output data for estimation

valid variable name

Select the `iddata`

object variable name from the MATLAB workspace choices. Use this parameter when **Data type**
is `iddata object`

.

`Validation data: Input (u)`

— Name of the input data variable used for validation

valid variable name

Name of the input data variable used for validation, selected from the MATLAB workspace choices. Use this parameter, along with **Validation
data: Output (y)**, when **Data Type** is
`Numeric`

.

`Validation data: Output (y)`

— Name of the output data variable used for validation

valid variable name

Name of the output data variable used for validation, selected from the MATLAB workspace choices. Use this parameter, along with **Validation
data: Input (y)**, when **Data Type** is
`Numeric`

.

`Validation data: Timetable`

— Variable name of timetable object containing input and output data for validation

valid variable name

Select the timetable object variable name from the MATLAB workspace choices. The timetables containing the validation data must have
the same variable names as the ones in the timetables selected for estimation in
**Estimation data: Timetable**. Use this parameter when
**Data type** is `Timetable`

. Specifying
validation data is optional but recommended.

`Validation data: Object`

— Variable name of data object containing input and output data for validation

valid variable name

Select the `iddata`

object variable name from the MATLAB workspace choices. Use this parameter when **Data type**
is `iddata object`

. Specifying validation data is optional but
recommended.

**Specify model structure**

`Number of states`

— Number of states

number of outputs (default) | positive integer

Number of states in the model to estimate. It must be less than or equal to the
number of outputs in the data. For more information, see `idNeuralStateSpace`

.

`Time invariant`

— Estimate a time invariant model

on (default) | off

Deselect this option to estimate a model in which the state equation explicitly
depends on time, other than states and inputs. When this option is left selected
(default), the state equation depends explicitly only on the current state and input
vectors. For more information, see the `isTimeInvariant`

property of
`idNeuralStateSpace`

.

`Time domain`

— Continuous or discrete time domain

`Continuous`

(default) | `Discrete`

Select a continuous-time or discrete-time model.

`Feedthrough`

— Estimate a model with direct feedthrough

off (default) | on

Select this option to estimate a model in which the output equation explicitly
depends on the input vector. When this option is left unselected (default), the output
equation does not depend explicitly on the input vector. For more information, see the
`HasFeedthrough`

property of `idNeuralStateSpace`

.

**State network**

`Activation function`

— Type of activation function for all hidden layers

`tanh`

(default) | `sigmoid`

You can specify one of the following:

`tanh`

— Use the hyperbolic tangent activation function for all hidden layers of the state network (default).`sigmoid`

—Use the sigmoid activation function for all hidden layers of the state network.

For more information, see `createMLPNetwork`

.

`Number of layers`

— Number of hidden layers

2 (default) | nonnegative integer

Number of hidden layers of the state network, specified as a nonnegative integer. It
must be equal to the number of elements of the vector you specify in **Layer
size** in the **State network** section. If you specify
`0`

, the state network has no hidden layer, and therefore expresses a
linear function.

For more information, see `createMLPNetwork`

.

`Layer size`

— Size of the hidden layers

`[64 64]`

(default) | vector of positive integers

Size of the hidden layers for the state network, specified as a vector of positive
integers. Each number specifies the number of neurons (network nodes) for each hidden
layer (each layer is fully-connected). For example, `[10 20 8]`

specifies a network with three hidden layers, the first (after the network input) having
`10`

neurons, the second having `20`

neurons, and
the last (before the network output), having `8`

neurons. Note that the
output layer is also a fully-connected, and you cannot change its size.

The number of elements in **Layer size** must be equal to the value
specified in **Number of layers** in the **State
network** section.

For more information, see `createMLPNetwork`

.

`Weights initializer`

— Weights initializer method

`glorot`

(default) | `he`

| `orthogonal`

| `narrow-normal`

| `zeros`

| `ones`

Weights initializer method for all the hidden layers of the state network. You can specify one of the following:

`glorot`

— Use the Glorot method (default).`he`

— Use the He method.`orthogonal`

— Use the orthogonal method.`narrow-normal`

— Use the narrow-normal method.`zeros`

— Initialize all weights to zero.`ones`

— Initialize all weights to one.

For more information, see `createMLPNetwork`

.

`Bias initializer`

— Bias initializer method

`zeros`

(default) | `ones`

| `narrow-normal`

Bias initializer method for all the hidden layers of the state network. You can specify one of the following:

`zeros`

— Initializes all biases to zero (default).`ones`

— Initializes all biases to one.`narrow-normal`

— uses the narrow-normal method.

For more information, see `createMLPNetwork`

.

**Output network**

`Activation function`

— Type of activation function for all hidden layers

`tanh`

(default) | `sigmoid`

You can specify one of the following:

`tanh`

— Use the hyperbolic tangent activation function for all hidden layers of the output network (default).`sigmoid`

—Use the sigmoid activation function for all hidden layers of the output network.

For more information, see `createMLPNetwork`

.

`Number of layers`

— Number of hidden layers

2 (default) | nonnegative integer

Number of hidden layers of the output network, specified as a nonnegative integer.
It must be equal to the number of elements of the vector you specify in **Layer
size** in the **Output network** section. If you specify
`0`

, the output network has no hidden layer, and therefore expresses
a linear function.

For more information, see `createMLPNetwork`

.

`Layer size`

— Size of the hidden layers

`[64 64]`

(default) | vector of positive integers

Size of the hidden layers for the output network, specified as a vector of positive
integers. Each number specifies the number of neurons (network nodes) for each hidden
layer (each layer is fully-connected). For example, `[10 20 8]`

specifies a network with three hidden layers, the first (after the network input) having
`10`

neurons, the second having `20`

neurons, and
the last (before the network output), having `8`

neurons. Note that the
output layer is also a fully-connected, and you cannot change its size.

The number of elements **Layer size** must be equal to the value
specified in **Number of layers** in the **Output
network** section.

For more information, see `createMLPNetwork`

.

`Weights initializer`

— Weights initializer method

`glorot`

(default) | `he`

| `orthogonal`

| `narrow-normal`

| `zeros`

| `ones`

Weights initializer method for all the hidden layers of the output network. You can specify one of the following:

`glorot`

— Use the Glorot method (default).`he`

— Use the He method.`orthogonal`

— Use the orthogonal method.`narrow-normal`

— Uses the narrow-normal method.`zeros`

— Initializes all weights to zero.`ones`

— Initializes all weights to one.

For more information, see `createMLPNetwork`

.

`Bias initializer`

— Bias initializer method

`zeros`

(default) | `ones`

| `narrow-normal`

Bias initializer method for all the hidden layers of the output network. You can specify one of the following:

`zeros`

— Initialize all biases to zero (default).`ones`

— Initialize all biases to one.`narrow-normal`

— Use the narrow-normal method.

For more information, see `createMLPNetwork`

.

**ODE Solver options**

`Input step size`

— Initial step size

`Auto`

(default) | positive scalar

Initial step size used to simulate the model (when continuous-time). It is specified
as either `Auto`

or a positive scalar. If you specify
`Auto`

, then the solver bases the initial step size on the slope of
the solution at the initial time point.

For more information, see `odeset`

, `nssTrainingADAM`

and `nssTrainingSGDM`

.

`Maximum step size`

— Maximum step size

`Auto`

(default) | positive scalar

Maximum step size used to simulate the model (when continuous-time). It is an upper
bound on the size of any step taken by the solver, and it is specified as either
`Auto`

or a positive scalar. If you specify `Auto`

,
then the value used is one-tenth of the difference between final and initial
time.

For more information, see `odeset`

, `nssTrainingADAM`

and `nssTrainingSGDM`

.

`Absolute tolerance`

— Absolute tolerance

`0.01`

(default) | positive scalar

Absolute tolerance used to simulate continuous time models, specified as a positive
scalar. It is the largest allowable absolute error. That is, when the solution
approaches 0, `AbsoluteTolerance`

is the threshold below which you do
not worry about the accuracy of the solution since it is effectively 0.

For more information, see `odeset`

, `nssTrainingADAM`

and `nssTrainingSGDM`

.

`Relative tolerance`

— Relative tolerance

`0.01`

(default) | positive scalar

Relative tolerance used to simulate the continuous time models, specified as a positive scalar. This tolerance measures the error relative to the magnitude of each solution component. That is, it controls the number of significant digits in a solution (except when is smaller than the absolute tolerance).

For more information, see `odeset`

, `nssTrainingADAM`

and `nssTrainingSGDM`

.

**Specify training options**

`Training algorithm`

— Training algorithm used to train the networks

`ADAM`

(default) | `Mean of square error`

You can specify one of the following:

`ADAM`

— use the Adam (name derived from adaptive moment estimation) algorithm.`SGDM`

—use the SGDM (stochastic gradient descent with momentum) algorithm.

For more information, see `nssTrainingOptions`

.

`Gradient decay factor`

— Decay rate of gradient moving average

`0.9`

(default) | nonnegative scalar less than `1`

Decay rate of gradient moving average for the Adam solver, specified as a
nonnegative scalar less than `1`

. The default value works well for most
tasks. You can specify a **Gradient decay factor** only when
**Training algorithm** is `ADAM`

.

For more information, see `nssTrainingADAM`

and `TrainingOptionsADAM`

(Deep Learning Toolbox).

`Squared gradient decay factor`

— Decay rate of squared gradient moving average

`0.999`

(default) | nonnegative scalar less than `1`

Decay rate of squared gradient moving average for the Adam solver, specified as a
nonnegative scalar less than `1`

.

Typical values of the decay rate are `0.9`

,
`0.99`

, and `0.999`

, corresponding to averaging
lengths of `10`

, `100`

, and `1000`

parameter updates, respectively. You can specify a **Gradient decay
factor** only when **Training algorithm** is
`ADAM`

.

For more information, see `nssTrainingADAM`

and `TrainingOptionsADAM`

(Deep Learning Toolbox).

`Momentum`

— Contribution of previous step

`0.95`

(default) | nonnegative scalar less than `1`

Contribution of the parameter update step of the previous iteration to the current
iteration of the SGDM algorithm, specified as a scalar from `0`

to
`1`

.

A value of `0`

means no contribution from the previous step,
whereas a value of `1`

means maximal contribution from the previous
step. The default value works well for most tasks. You can specify a **Gradient
decay factor** only when **Training algorithm** is
`SGDM`

.

For more information, see `nssTrainingSGDM`

and `TrainingOptionsSGDM`

(Deep Learning Toolbox).

**Example: **`0.9`

`Loss function`

— Type of function used to calculate loss

`Mean of absolute error`

(default) | `Mean of square error`

You can specify one of the following:

`Mean of absolute error`

— Use the mean value of the absolute error.`Mean of square error`

— Use the mean value of the squared error.

For more information, see `nssTrainingADAM`

and `nssTrainingSGDM`

.

`Learn Rate`

— Learning rate used for training

`0.001`

(default) | positive scalar

Learning rate used for training, specified as a positive scalar. If the learning rate is too small, then training can take a long time. If the learning rate is too large, then training might reach a suboptimal result or diverge.

For more information, see `nssTrainingADAM`

and `nssTrainingSGDM`

.

`Maximum number of epochs`

— Maximum number of epochs

100 (default) | positive integer

Maximum number of epochs to use for training, specified as a positive integer. An epoch is the full pass of the training algorithm over the entire training set.

For more information, see `nssTrainingADAM`

and `nssTrainingSGDM`

.

`Size of mini-batch`

— Validation period

100 (default) | positive integer

Size of the mini-batch to use for each training iteration, specified as a positive integer. A mini-batch is a subset of the training set that is used to evaluate the gradient of the loss function and update the weights.

If the mini-batch size does not evenly divide the number of training samples, then the estimation process discards the training data that does not fit into the final complete mini-batch of each epoch.

For more information, see `nssTrainingADAM`

and `nssTrainingSGDM`

.

`Input intersample`

— Input interpolation method

`Spline`

(default) | `zoh`

| `foh`

| `cubic`

| `makima`

| `makima`

You can select one of the following options:

`zoh`

— Zero-order hold interpolation method`foh`

— First-order hold interpolation method`cubic`

— Cubic interpolation method`makima`

— Modified Akima interpolation method`pchip`

— Shape-preserving piecewise cubic interpolation method`spline`

— Spline interpolation method (default)

This is the interpolation method used to interpolate the input when integrating
continuous-time neural state-space models. For more information, see interpolation
methods in `interp1`

.

**Example: **`"foh"`

`Show fit to validation data during training`

— Enable plot showing comparison of predicted and measured estimation outputs

on (default) | off

Enable displaying a validation plot periodically during training. The validation plot shows a comparison between the predicted output response to measured validation inputs and the measured validation outputs. The plot also displays the model fit percentage.

`Validation data fit frequency`

— Validation period

20 (default) | positive integer

This is the number of epochs after which the validation plot is updated with a new
comparison (new predicted output against measured outputs). For example, if
**Validation data fit frequency** is 10, the validation plot is
updated every 10 epochs. For more information, see `nlssest`

.

`Show training loss plot`

— Show training loss plot

on (default) | off

Enable displaying a training plot during training (estimation). The training plot shows how the state and output network loss values evolve after each training epoch.

**Display results**

`Show fit to estimation data`

— Enable plot showing comparison of predicted and measured estimation outputs

on (default) | off

After estimation (training), plot a comparison between the predicted output response to measured estimation inputs and the measured estimation outputs. Selecting this parameter also displays the model fit percentage.

`Show fit to validation data`

— Enable plot showing comparison of predicted and measured validation outputs

on (default) | off

After estimation (training), plot a comparison between the predicted output response
to measured validation inputs and the measured validation outputs. Selecting this
parameter also displays the model fit percentage. This parameter is available only if
you select validation data in the **Select Data** section.

## Version History

**Introduced in R2023b**

## See Also

### Objects

### Functions

`createMLPNetwork`

|`nssTrainingOptions`

|`nlssest`

|`generateMATLABFunction`

|`idNeuralStateSpace/evaluate`

|`idNeuralStateSpace/linearize`

|`sim`

### Blocks

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