Neural state-space model with identifiable network weights
Since R2022b
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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,
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.
For more information on neural state-space models, see What are Neural State-Space Models?.
Creation
Syntax
nss = idNeuralStateSpace(nx)
nss = idNeuralStateSpace(___,Name=Value)
Description
creates an autonomous (no-input) time-invariant continuous-time neural state-space object with nss
= idNeuralStateSpace(nx)nx
state variables and output identical to state.
example
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.nss
= idNeuralStateSpace(___,Name=Value)
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).
example
Input Arguments
expand all
nx
— Number of state variables
positive integer
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
NumInputs
— Number of input variables
0 (default) | nonnegative integer
Number of input variables, specified as a nonnegative integer.
Example: NumInputs=2
NumOutputs
— Number of output variables
nx
(default) | nonnegative integer
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
HasFeedthrough
— Option to set direct feedthrough
false
(default) | true
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 ifNumInputs
is greater than zero. This is the default case, and y2(t) = H(t,x).
This argument sets the value of the read-only property FeedthroughInOutputNetwork
of nss
.
Example: false
Properties
expand all
StateNetwork
— State function network
dlnetwork
object
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 64 nodes.
Two activation layers: each featuring a 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")
You can also directly assign a custom dlnetwork
object as the state function network when the sizes of the input layers are distinguishable. For example:
nss.StateNetwork = dlnet;
where dlnet
is a custom dlnetwork
object with the configuration as mentioned under the dlnet
argument in setNetwork.
For more information on custom networks and how to assign them to an idNeuralStateSpace
object when the input layer sizes are not distinguishable, see setNetwork.
To train both state and output networks, use nlssest. For example:
options1 = nssTrainingOptions("adam"); 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 the network, you can either manually assign the learnables of the
dlnetwork
object or initialize it.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.
OutputNetwork
— Output function networks
dlnetwork
object
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 and HasFeedthrough
is true
), 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 64 nodes.
See AlsoTrain Robust Deep Learning Network with Jacobian Regularization - MATLAB & Simulink - MathWorks 日本Quantify Image Quality Using Neural Image Assessment - MATLAB & Simulink - MathWorks 日本createMLPNetwork - Create and initialize a Multi-Layer Perceptron (MLP) network to be used within a neural state-space system - MATLAB - MathWorks 日本dlnetwork - 深層学習ニューラル ネットワーク - MATLAB - MathWorks 日本Two activation layers: each featuring a 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")
You can also directly assign a custom dlnetwork
object as the output function network when the sizes of the input layers are distinguishable. For example:
nss.OutputNetwork = dlnet;
where dlnet
is a custom dlnetwork
object with the configuration as mentioned under the dlnet
argument in setNetwork.
For more information on custom networks and how to assign them to an idNeuralStateSpace
object when the input layer sizes are not distinguishable, see setNetwork.
To train both state and output networks, use nlssest. For example:
options1 = nssTrainingOptions("adam")options2 = nssTrainingOptions("sgdm")nss = nlssest(U, Y, nss, [options1; options2])
Encoder
— Encoder function network
[]
(default) | dlnetwork
object
Encoder function network, specified as a dlnetwork (Deep Learning Toolbox) object. The encoder maps the state to a latent state (usually, of a lower dimension), which is the input to the state function network. The dimension of this latent state is specified by LatentDim.
The default value is []
, which means that you are not using the encoder. You add an encoder to your model by changing the default value of LatentDim
. This encoder network, by default contains two layers of size 64 with tanh
as the activation function.
To change the default encoder network configuration, use createMLPNetwork. For example:
nss.Encoder = createMLPNetwork(nss,"encoder",... LayerSizes=[4 4],... Activations="sigmoid")
You can also directly assign a custom dlnetwork
object as the encoder function network. For example:
nss.Encoder = dlnet;
where dlnet
is a custom dlnetwork
object with one input representing the original state and one output representing the latent state.
For more information on autoencoders, see What are Neural State-Space Models?
Decoder
— Decoder function network
[]
(default) | dlnetwork
object
Decoder function network, specified as a dlnetwork (Deep Learning Toolbox) object. The output of the state function network is the input of the decoder. The decoder maps the latent state back to the original state. The dimension of the latent state is specified by LatentDim.
The default value is []
, which means that you are not using the decoder. You add a decoder to your model by changing the default value of LatentDim
. This decoder network, by default, contains two layers of size 64 with tanh
as the activation function.
To change the default decoder network configuration, use createMLPNetwork. For example:
nss.Decoder = createMLPNetwork(nss,"decoder",... LayerSizes=[4 4],... Activations="sigmoid")
You can also directly assign a custom dlnetwork
object as the decoder function network. For example:
nss.Decoder = dlnet;
where dlnet
is a custom dlnetwork
object with one input representing the latent state and one output representing the original state.
For more information on autoencoders, see What are Neural State-Space Models?
LatentDim
— Dimension of internal state
NaN
(default) | finite positive integer
Dimension of the internal (latent) state, specified as NaN
or a positive integer. To add an encoder or decoder to your model, specify LatentDim
as a positive integer.
LatentDim Value | Model Framework |
---|---|
NaN | |
A finite positive scalar | |
Example: 2
IsTimeInvariant
— Flag indicating time invariance
true
(default) | false
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)
FeedthroughInOutputNetwork
— Flag indicating direct feedthrough
false
(default) | true
| array of logical
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]
StateName
— State names
{'x1','x2',...}
(default) | character vector | cell array of character vectors
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'}
StateUnit
— State units
{''}
(default) | character vector | cell array of character vectors or stings | string | string array
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'
Example: {'rpm','rad/s'}
TimeVariable
— Independent variable name
"t"
(default) | string | char vector
Independent variable name, specified as a string or character vector, for the state, input and output functions.
Example: "t"
NoiseVariance
— Innovation covariance matrix
matrix
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)
InputName
— Names of input channels
{''}
(default) | character vector | cell array of character vectors | string | string array
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
.
InputUnit
— Units of input signals
{''}
(default) | character vector | cell array of character vectors | string | string array
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'}
InputGroup
— Input channel groups
structure with no fields (default) | structure
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])
OutputName
— Names of output channels
{''}
(default) | character vector | cell array of character vectors or strings | string | string array
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
.
OutputUnit
— Units of output signals
{''}
(default) | character vector | cell array of character vectors | string | string array
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'}
OutputGroup
— Output channel groups
structure with no fields (default) | structure
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])
Notes
— Text notes about model
[0×1 string]
(default) | string | character vector | cell array of character vectors or strings | string array
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.Notessys2.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.
UserData
— Data associated with model
[]
(default) | any data type
Data of any kind that you want to associate and store with the model, specified as any MATLAB® data type.
Ts
— Sample time
nonnegative scalar
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
TimeUnit
— Model time units
'seconds'
(default) | 'minutes'
| 'milliseconds'
| ...
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.
Report
— Summary report
report field values
This property is read-only.
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 |
setNetwork | Assign dlnetwork object as the state or output function of a neural state-space model |
generateMATLABFunction | Generate MATLAB functions that evaluate the state and output functions, and their Jacobians, of a nonlinear grey-box or neural state-space model |
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
collapse all
Create Continuous-Time Neural State-Space Object
Open Live Script
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 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 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.
Create Discrete-Time Neural State-Space Object
Open Live Script
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: tanhg(.) network: Deep network with 2 fully connected, hidden layers Activation function: tanh Inputs: u1, u2Outputs: y1, y2, y3, y4States: x1, x2, x3Sample 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(:) 5 'y[k]' Addition Element-wise addition of 2 inputs
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 9 'y[k]' Addition Element-wise addition of 2 inputs
nss.UserData = ['Created on ' char(datetime)];nss.UserData
ans = 'Created on 20-Jul-2024 13:09:43'
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
See Also
Objects
- nssTrainingADAM | nssTrainingSGDM | nssTrainingRMSProp | nssTrainingLBFGS | idss | idnlgrey
Functions
- createMLPNetwork | setNetwork | nssTrainingOptions | nlssest | generateMATLABFunction | idNeuralStateSpace/evaluate | idNeuralStateSpace/linearize | sim
Blocks
- Neural State-Space Model
Live Editor Tasks
- Estimate Neural State-Space Model
Topics
- What are Neural State-Space Models?
- Estimate Neural State-Space System
- Estimate Nonlinear Autonomous Neural State-Space System
- Neural State-Space Model of Simple Pendulum System
- Reduced Order Modeling of a Nonlinear Dynamical System using Neural State-Space Model with Autoencoder
- Augment Known Linear Model with Flexible Nonlinear Functions
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