# idNeuralStateSpace

Neural state-space model with identifiable network weights

Since R2022b

## Description

Use idNeuralStateSpace to create a black-box continuous-time or discrete-time neural state-space model with identifiable (estimable) network weights and bias. You can use the trained black-box model for control, estimation, optimization, and reduced order modeling.

Continuous-time neural state space models have the following general form,

$\begin{array}{l}\stackrel{˙}{x}\left(t\right)=F\left(t,x\left(t\right),u\left(t\right)\right)\\ y\left(t\right)=\left[\begin{array}{c}{y}_{1}\left(t\right)\\ {y}_{2}\left(t\right)\end{array}\right]=\left[\begin{array}{c}x\left(t\right)+{e}_{1}\left(t\right)\\ H\left(t,x\left(t\right),u\left(t\right)\right)+{e}_{2}\left(t\right)\end{array}\right]\end{array}$

where the state function F and the nontrivial output function H are approximated by neural networks. Because you need to measure all the states to properly train the state function, the states measurements are considered to be part of the output function. Here, e1 and e2 are measurement noises in the data sets which are minimized by the network training algorithm.

For discrete-time state-space systems, the state and output functions have this form.

$\begin{array}{l}x\left(t+1\right)=F\left(t,x\left(t\right),u\left(t\right)\right)\\ y\left(t\right)=\left[\begin{array}{c}{y}_{1}\left(t\right)\\ {y}_{2}\left(t\right)\end{array}\right]=\left[\begin{array}{c}x\left(t\right)+{e}_{1}\left(t\right)\\ H\left(t,x\left(t\right),u\left(t\right)\right)+{e}_{2}\left(t\right)\end{array}\right]\end{array}$

Note

Defining and estimating a neural state space system requires that:

1. You know what the states of the systems are (to your best knowledge).

2. The states are measured, and thus, their measurements are part of experiment data set.

## Creation

### Description

example

nss = idNeuralStateSpace(nx) creates an autonomous (no-input) time-invariant continuous-time neural state-space object with nx state variables and output identical to state.

example

nss = idNeuralStateSpace(___,Name=Value) specifies name-value pair arguments after any of the input argument in the previous syntax. You can use name-value pair arguments to set the number of inputs and outputs and other system configurations such as time domain, whether the system is time invariant and whether the system output has feed-through.

For example, nss = idNeuralStateSpace(3,NumInputs=2,NumOutputs=4,Ts=0.1) creates a time-invariant discrete-time neural state-space object with 3 states, 2 inputs, four outputs (the first three are state measurements), and sample time 0.1. The system is also time invariant (both state and output functions do not explicitly depend on time) and does not have direct feed-through (the input does not have immediate impact on output).

### Input Arguments

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Number of state variables, specified as a positive integer.

Example: 2

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Use name-value pair arguments to specify NumInputs, NumOutputs and the Ts, IsTimeInvariant, and HasFeedthrough properties of nss.

Example: Ts=0.1

Number of input variables, specified as a nonnegative integer.

Example: NumInputs=2

Number of output variables, specified as a positive integer greater than or equal to nx. The value must be greater than nx because all the states are measured.

For example, if nx is 2, NumOutputs=4 means that the state space system has four outputs, with the first two outputs being state measurements, and the last two are outputs from the output function H.

Example: NumOutputs=4

Option to set direct feedthrough, specified as one of the following:

• true — the nontrivial output measurement y2 is an explicit function of the input, that is y2(t) = H(t,x,u).

• false — the nontrivial output measurement y2 is not an explicit function of the input, even if NumInputs is greater than zero. This is the default case, and y2(t) = H(x,u).

This argument sets the value of the read-only property FeedthroughInOutputNetwork of nss.

Example: false

## Properties

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State function network, specified as a dlnetwork (Deep Learning Toolbox) object. This network approximates the state function of the state-space system (F). For continuous state-space systems the state function returns the system state derivative with respect to time, while for discrete-time state-space systems it returns the next state. The inputs of the function are time (if IsTimeInvariant is false), the current state, and the current input (if NumInputs is positive), in that order.

When an idNeuralStateSpace model is constructed, a default state network is created. It is a multi-layer perceptron (MLP) network with the following features:

• Two hidden layers: each is a fully-connected layer with 128 nodes.

• Two activation layers: each featuring an hyperbolic tangent (tanh) function.

• One output layer: a fully-connected layer with nx nodes.

To change the default network configuration, use createMLPNetwork. For example:

nss.StateNetwork = createMLPNetwork(nss, 'state', ...
LayerSizes=[64 64 64], ...
Activations="sigmoid")
To train both state and output networks, use nlssest. For example:
nss = nlssest(U, Y, nss, options1);

Note

• To train the network, use nlssest which updates the weights and biases of the network. After training completes, the network weights and biases are said to be "trained".

• A new training starts with the previously trained network. To reset weights and bias, use createMLPNetwork to create a new network.

• Multi-layer perceptron (MLP) networks with at least one hidden layer featuring squashing functions (such as hyperbolic tangent or sigmoid) are universal approximators, that is, are theoretically capable of approximating any function to any desired degree of accuracy provided that sufficiently many hidden units are available.

• Deeper networks (networks with more hidden layers) can approximate compositional functions as well as shallow networks but with exponentially lower number of training parameters and sample complexity.

Output function networks, specified as a 2-by-1 array of dlnetwork (Deep Learning Toolbox) objects. The first network represents the identity relation between y1 and x, since all the states are measured. This network has no learnable parameters, is fixed and cannot be changed or trained.

The second network approximates the output function H of the state-space system, which is a function of time (if IsTimeInvariant is false), the current state, and the current input (if NumInputs is positive), in that order.

When you create an idNeuralStateSpace model, the default network created to approximate H is a multi-layer perceptron (MLP) network with the following features:

• Two hidden layers: each is a fully-connected layer with 128 nodes.

• Two activation layers: each featuring an hyperbolic tangent (tanh) function.

• One output layer: a fully-connected layer with NumOutputs - nx nodes.

To change the default network configuration, use createMLPNetwork. For example:

nss.OutputNetwork = createMLPNetwork(nss, 'output', ...
LayerSizes=[64 64 64], ...
Activations="sigmoid")
To train both state and output networks, use nlssest. For example:
options2 = nssTrainingOptions('sgdm')
nss = nlssest(U, Y, nss, [options1; options2])

Flag indicating time invariance, returned as one of the following:

• true — (default), the system is time invariant, neither the state function F of the output function H depend explicitly on time.

• false — the system is time varying, both the state of the output function depend explicitly on time.

This property is read-only and cannot be set using dot notation. You can only specify this properly when you create nss. To do so, use the corresponding name-value pair argument in idNeuralStateSpace. For example:

nss = idNeuralStateSpace(3,NumInputs=2,IsTimeInvariant=false)

Example: true

Flag indicating direct feedthrough in the output networks, returned as false or as an array logical values.

If NumOutputs = nx, FeedthroughInOutputNetwork is false, because the only output is the measured state, and there is no contribution from any input.

If NumOutputs > nx, FeedthroughInOutputNetwork is a 1-by-2 logical array in which the elements are as follows.

• The first logical value corresponds to y1 and is always false.

• The second value corresponds to y2 and is the same value that you specify with the name-value pair argument HasFeedThrough when you create the object. When this value is true, then y2 is an explicit function of the input, otherwise, as default, there is no explicit contribution from the input to y2.

Note

This property is read-only and you can change it only when you create nss, using the HasFeedThrough argument in idNeuralStateSpace.

Example: [false, false]

State names, specified as one of these values:

• Character vector — For first-order models

• Cell array of character vectors — For models with two or more states

• '' — For unnamed states

You can specify StateName using a string, such as "velocity", but the state name is stored as a character vector, 'velocity'.

Example: {'velocity','distance'}

State units, specified as:

• A character vector or string — For first-order models

• A cell array of character vectors or string array — For models with two or more states

• '' — For states without specified units

Use StateUnit to keep track of the units each state is expressed in. StateUnit has no effect on system behavior.

If you specify StateUnit using a string, such as "mph", the state units are stored as a character vector, 'mph'.

Example: 'mph'

Independent variable name, specified as a string or character vector, for the state, input and output functions.

Example: "t"

Innovation covariance matrix, specified as an NumOutputs-by-NumOutputs positive semi-definite matrix. Typically this property is automatically set by the estimation algorithm.

Example: 1e-3*eye(2)

Names of input channels, specified as:

• A character vector or string — For single-input models

• A cell array of character vectors or a string array — For models with two or more inputs

• '' — For inputs without specified names

You can use automatic vector expansion to assign input names for multi-input models. For example, if sys is a two-input model, you can specify InputName as follows.

sys.InputName = 'controls';

The input names automatically expand to {'controls(1)';'controls(2)'}.

You can use the shorthand notation u to refer to the InputName property. For example, sys.u is equivalent to sys.InputName.

Input channel names have several uses, including:

• Identifying channels on model display and plots

• Extracting subsystems of MIMO systems

• Specifying connection points when interconnecting models

If you specify InputName using a string or string array, such as "voltage", the input name is stored as a character vector, 'voltage'.

When you estimate a model using an iddata object, data, the software automatically sets InputName to data.InputName.

Units of input signals, specified as:

• A character vector or string — For single-input models

• A cell array of character vectors or string array — For models with two or more inputs

• '' — For inputs without specified units

Use InputUnit to keep track of the units each input signal is expressed in. InputUnit has no effect on system behavior.

If you specify InputUnit using a string, such as "voltage", the input units are stored as a character vector, 'voltage'.

Example: 'voltage'

Example: {'voltage','rpm'}

Input channel groups, specified as a structure where the fields are the group names and the values are the indices of the input channels belonging to the corresponding group. When you use InputGroup to assign the input channels of MIMO systems to groups, you can refer to each group by name when you need to access it. For example, suppose you have a five-input model sys, where the first three inputs are control inputs and the remaining two inputs represent noise. Assign the control and noise inputs of sys to separate groups.

sys.InputGroup.controls = [1:3];
sys.InputGroup.noise = [4 5];

Use the group name to extract the subsystem from the control inputs to all outputs.

sys(:,'controls')

Example: struct('controls',[1:3],'noise',[4 5])

Names of output channels, specified as:

• A character vector or string— For single-output models

• A cell array of character vectors or string array — For models with two or more outputs

• '' — For outputs without specified names

You can use automatic vector expansion to assign output names for multi-output models. For example, if sys is a two-output model, you can specify OutputName as follows.

sys.OutputName = 'measurements';

The output names automatically expand to {'measurements(1)';'measurements(2)'}.

You can use the shorthand notation y to refer to the OutputName property. For example, sys.y is equivalent to sys.OutputName.

Output channel names have several uses, including:

• Identifying channels on model display and plots

• Extracting subsystems of MIMO systems

• Specifying connection points when interconnecting models

If you specify OutputName using a string, such as "rpm", the output name is stored as a character vector, 'rpm'.

When you estimate a model using an iddata object, data, the software automatically sets OutputName to data.OutputName.

Units of output signals, specified as:

• A character vector or string — For single-output models

• A cell array of character vectors or string array — For models with two or more outputs

• '' — For outputs without specified units

Use OutputUnit to keep track of the units each output signal is expressed in. OutputUnit has no effect on system behavior.

If you specify OutputUnit using a string, such as "voltage", the output units are stored as a character vector, 'voltage'.

Example: 'voltage'

Example: {'voltage','rpm'}

Output channel groups, specified as a structure where the fields are the group names and the values are the indices of the output channels belonging to the corresponding group. When you use OutputGroup to assign the output channels of MIMO systems to groups, you can refer to each group by name when you need to access it. For example, suppose you have a four-output model sys, where the second output is a temperature, and the rest are state measurements. Assign these outputs to separate groups.

sys.OutputGroup.temperature = [2];
sys.OutputGroup.measurements = [1 3 4];

Use the group name to extract the subsystem from all inputs to the measurement outputs.

sys('measurements',:)

Example: struct('temperature',[2],'measurement',[1 3 4])

Text notes about the model, specified as a string or character vector. The property stores whichever of these two data types you provide. For instance, suppose that sys1 and sys2 are dynamic system models. You can set their Notes properties to a string and a character vector, respectively.

sys1.Notes = "sys1 has a string.";
sys2.Notes = 'sys2 has a character vector.';
sys1.Notes
sys2.Notes
ans =

"sys1 has a string."

ans =

'sys2 has a character vector.'

You can also specify Notes as string array or a cell array of character vectors or strings.

Data of any kind that you want to associate and store with the model, specified as any MATLAB® data type.

Sample time, specified as a nonnegative scalar, in units specified by the TimeUnit property. For a continuous time model, Ts is equal to 0 (default). Changing the value of Ts has no impact on the system data and does not discretize or resample the model.

Note

If you change Ts to a different value after networks are trained, you need to train the networks again because the original trained networks are no longer valid.

Example: 0.1

Model time units, specified as:

• 'nanoseconds'

• 'microseconds'

• 'milliseconds'

• 'seconds'

• 'minutes'

• 'hours'

• 'days'

• 'weeks'

• 'months'

• 'years'

If you specify TimeUnit using a string, such as "hours", the time units are stored as a character vector, 'hours'.

Model properties such as sample time Ts, InputDelay, OutputDelay, and other time delays are expressed in the units specified by TimeUnit. Changing this property has no effect on other properties, and therefore changes the overall system behavior. Use chgTimeUnit to convert between time units without modifying system behavior.

Summary report that contains information about the estimation options and results for a state-space model obtained using estimation commands. Use Report to find estimation information for the identified model, including the:

• Status (estimated or constructed)

• Estimation method

• Estimation options

• Search termination conditions

• Estimation data fit and other quality metrics

For more information on this property and how to use it, see the Output Arguments section of the corresponding estimation command reference page and Estimation Report.

## Object Functions

 createMLPNetwork Create and initialize a Multi-Layer Perceptron (MLP) network to be used within a neural state-space system generateMATLABFunction Generate MATLAB functions that evaluate the state and output functions of a neural state-space object, and their Jacobians sim Simulate response of identified model idNeuralStateSpace/evaluate Evaluate a neural state-space system for a given set of state and input values and return state derivative (or next state) and output values idNeuralStateSpace/linearize Linearize a neural state-space model around an operating point

## Examples

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Use idNeuralStateSpace to create a continuous-time neural state-space object with two states, no inputs, and outputs identical to states.

nss = idNeuralStateSpace(2)
nss =

Continuous-time Neural ODE in 2 variables
dx/dt = f(x(t))
y(t) = x(t) + e(t)

f(.) network:
Deep network with 2 fully connected, hidden layers
Activation function: Tanh
g(.) network:
Deep network with 0 fully connected, hidden layers
Activation function:

Variables: x1, x2

Status:
Created by direct construction or transformation. Not estimated.

Use dot notation to access the object properties.

nss.StateNetwork
ans =
dlnetwork with properties:

Layers: [6x1 nnet.cnn.layer.Layer]
Connections: [5x2 table]
Learnables: [6x3 table]
State: [0x3 table]
InputNames: {'x'}
OutputNames: {'dxdt'}
Initialized: 1

View summary with summary.

nss.Name = "myNssObject";
nss.UserData = ['Created on ' char(datetime)]
nss =

Continuous-time Neural ODE in 2 variables
dx/dt = f(x(t))
y(t) = x(t) + e(t)

f(.) network:
Deep network with 2 fully connected, hidden layers
Activation function: Tanh
g(.) network:
Deep network with 0 fully connected, hidden layers
Activation function:

Variables: x1, x2

Status:
Created by direct construction or transformation. Not estimated.

You can now re-configure the state network using createMLPNetwork, if needed, and then use time-domain data to perform estimation and validation.

Use idNeuralStateSpace to create a discrete-time neural state-space object with three states, two inputs, four outputs, and sample time 0.1.

nss = idNeuralStateSpace(3,NumInputs=2,NumOutputs=4,Ts=0.1)
nss =

Discrete-time Neural State-Space Model with 4 outputs, 3 states, and 2 inputs
x(t+1) = f(x(t),u(t))
y_1(t) = x(t) + e_1(t)
y_2(t) = g(x(t),u(t)) + e_2(t)
y(t) = [y_1(t); y_2(t)]

f(.) network:
Deep network with 2 fully connected, hidden layers
Activation function: Tanh
g(.) network:
Deep network with 2 fully connected, hidden layers
Activation function: Tanh

Inputs: u1, u2
Outputs: y1, y2, y3, y4
States: x1, x2, x3
Sample time: 0.1 seconds

Status:
Created by direct construction or transformation. Not estimated.

Use dot notation to access the object properties.

nss.OutputNetwork.Layers
ans =
5x1 Layer array with layers:

1   'x[k]'   Feature Input   3 features
2   'u[k]'   Feature Input   2 features
3   'yx'     Function        @(x)x(:)
4   'yu'     Function        @(u)zeros(nx,nu)*u(:)

ans =
9x1 Layer array with layers:

1   'x[k]'   Feature Input     3 features
2   'fc1'    Fully Connected   64 fully connected layer
3   'act1'   Tanh              Hyperbolic tangent
4   'fc2'    Fully Connected   64 fully connected layer
5   'act2'   Tanh              Hyperbolic tangent
6   'yx'     Fully Connected   1 fully connected layer
7   'u[k]'   Feature Input     2 features
8   'yu'     Function          @(u)zeros(ny,nu)*u(:)

nss.UserData = ['Created on ' char(datetime)];
nss.UserData
ans =
'Created on 12-Feb-2024 23:31:13'

Note that by default the output does not explicitly depend on the input.

nss.FeedthroughInOutputNetwork
ans = 1x2 logical array

0   0

You can now re-configure the state and output networks using createMLPNetwork, if needed, and then use time-domain data to perform estimation and validation.

## References

[1] Chen, Ricky T. Q., Yulia Rubanova, Jesse Bettencourt, and David Duvenaud. “Neural Ordinary Differential Equations.” arXiv, December 13, 2019. http://arxiv.org/abs/1806.07366.

## Version History

Introduced in R2022b