System Identification Toolbox™

idnlgrey - Class representing nonlinear ODE models

Syntax

m = idnlgrey('filename',Order,Parameters) m = idnlgrey('filename',Order,Parameters,InitialStates) m = idnlgrey('filename',Order,Parameters,InitialStates,Ts) m = idnlgrey('filename',Order,Parameters,InitialStates,Ts,P1,V1,...,PN,VN)

Description

idnlgrey is an object that represents the nonlinear grey-box model.

For information about the nonlinear grey-box models, see Estimating Nonlinear Grey-Box Models.

The information in these reference pages summarizes the idnlgrey model constructor and properties. It discusses the following topics:

idnlgrey Constructor

Use the following syntax to define the idnlgrey model object:

m = idnlgrey('filename',Order,Parameters)

m = idnlgrey('filename',Order,Parameters,InitialStates)

m = idnlgrey('filename',Order,Parameters,InitialStates,Ts)

m = idnlgrey('filename',Order,Parameters,InitialStates,Ts,P1,V1,...,PN,VN)

The idnlgrey arguments are defined as follows:

'filename' — Name of the M-file or MEX-file storing the model structure (ODE file).
Order — Vector with three entries [Ny Nu Nx], specifying the number of model outputs Ny, the number of inputs Nu, and the number of states Nx.
Parameters — Parameters, specified as struct arrays, cell arrays, or double arrays.
InitialStates — Specified in a same way as parameters. Must be fourth input to the idnlgrey constructor.
The command
```
m = idnlgrey('filename',Order,Parameters,...
              InitialStates,Ts,P1,V1,...,PN,VN)
```
specifies idnlgrey property-value pairs. See information on properties of idnlgrey objects below.

Estimate the parameters of this object using pem.

idnlgrey Properties

You can include property-value pairs in the model estimator or constructor to specify the model structure and estimation algorithm properties.

After creating the object, you can use get or dot notation to access the object property values. For example:

% Get the model time unit
get(m,'TimeUnit')
m.TimeUnit

The following table summarizes idnlgrey model properties. The general idnlmodel properties also apply to this nonlinear model object (see the corresponding reference pages).

Property Name	Description
`Algorithm`	A structure that specifies the estimation algorithm options, as described in idnlgrey Algorithm Properties.
`CovarianceMatrix`	Covariance matrix of the estimated `Parameters`. Assignable values: `'None'` to omit computing uncertainties and save time during parameter estimation. `'Estimate'` to estimation covariance. Symmetric and positive N_p-by-N_p matrix (or `[]`) where N_p is the number of free model parameters.
`EstimationInfo`	A read-only structure that stores estimation settings and results, as described in idnlgrey EstimationInfo Properties.
`FileArgument`	Contains auxiliary variables passed to the ODE file (M-file or MEX-file) specified in `FileName`. These variables may be used as extra inputs for specifying the state and/or output equations.`FileArgument` should be specified as a cell array. Default: `{}`.
`FileName`	File name string (without extension) or a function handle for computing the states and the outputs. If `'FileName'` is a string, then it must point to an M-file or MEX-file. For more information about the file variables, see Specifying the Nonlinear Grey-Box Model Structure.
`InitialStates`	An `Nx`-by-1 structure array with fields as follows. Here, `Nx` is the number of states of the model. `Name`: Name of the state (a string). Default value is `'x#i'`, where `#i` is an integer in `[1, Nx]`. `Unit`: Unit of the state (a string). Default value is `''`. `Value`: Initial value of the initial state(s). Assignable values are: A finite real scalar A finite real 1-by-Ne vector, where `Ne` is the number of experiments in the data set to be used for estimation `Minimum`: Minimum value of the initial state(s). Must be a real scalar/1-by-`Ne` vector of the same size as Value and such that Minimum <= Value for all components. Default value: -`Inf(size(Value))`. `Maximum`: Maximum value of the initial state(s). Must be a real scalar/1-by-`Ne` vector of the same size as `Value` and such that `Value` <= `Maximum` for all components. Default value: `Inf(size(Value))`. `Fixed`: Specifies which component(s) of the initial state(s) are fixed to their known values. Must be a Boolean scalar/1-by-`Ne` vector of the same size as `Value`. Default value: `true(size(Value))` (that is, do not estimate the initial states). For an `idnlgrey` model `M`, the `i`:th initial state is accessed through `M.InitialStates(i)` and its sub-fields as `M.InitialStates(i).FIELDNAME`.
`Order`	Structure with following fields: `ny` — Number of outputs of the model structure. `nu` — Number of inputs of the model structure. `nx` — Number of states of the model structure. For time series, `nu` is 0. For static model structures, `nx` is 0.
`Parameters`	`Np`-by-1 structure array with information about the model parameters containing the following fields: `Name`: Name of the parameter (a string). Default value is `'p#i'`, where `#i` is an integer in `[1, Np]`. `Unit`: Unit of the parameter (a string). Default value is `''`. `Value`: Initial value of the parameter(s). Assignable values are: A finite real scalar A finite real column vector A 2–dimensional real matrix `Minimum`: Minimum value of the parameter(s). Must be a real scalar/column vector/matrix of the same size as `Value` and such that `Minimum` <= `Value` for all components. Default value: -`Inf(size(Value))`. `Maximum`: Maximum value of the parameter(s). Must be a real scalar/column vector/matrix of the same size as `Value` and such that `Value` <= `Maximum` for all components. Default value: `Inf(size(Value))`. `Fixed`: Specifies which component(s) of the parameter(s) are fixed to their known values. Must be a boolean scalar/column vector/matrix of the same size as `Value`. Default value: `false(size(Value))`, (estimate all parameter components). For an `idnlgrey` model `M`, the `i`:th parameter is accessed through `M.Parameters(i)` and its sub-fields as `M.Parameters(i).FIELDNAME`.

idnlgrey Algorithm Properties

The following table summarizes the fields of the Algorithm idnlgrey model properties. Algorithm is a structure that specifies the estimation-algorithm options.

Property Name	Description
`Advanced`	A structure that specifies additional estimation algorithm options, as described in idnlgrey Advanced Algorithm Properties.
`Criterion`	Specifies criterion used during minimization. `Criterion` can have the following values: `'Det'`: Minimize `det(E'E)` where `E` represents the prediction error. This is the optimal choice in a statistical sense and leads to the maximum likelihood estimates in case nothing is known about the variance of the noise. It uses the inverse of the estimated noise variance as the weighting function. This is the default criterion used for all models, except `idnlgrey` which uses `'Trace'` by default. `'Trace'`: Minimize the trace of the weighted prediction error matrix `trace(E'EW)`, where `E` is the matrix of prediction errors, with one column for each output, and `W` is a positive semi-definite symmetric matrix of size equal to the number of outputs. By default, `W` is an identity matrix of size equal to the number of model outputs (so the minimization criterion becomes `trace(E'E)`, or the traditional least-sum-of-squared-errors criterion. You can specify the relative weighting of prediction errors for each output using the `Weighting` field of the `Algorithm` property.
`LimitError`	Robustification criterion that limits the influence of large residuals, specified as a positive real value. Residual values that are larger than `'LimitError'` times the estimated residual standard deviation have a linear cost instead of the usual quadratic cost. Default: `0` (no robustification).
`MaxIter`	Maximum number of iterations for the estimation algorithm, specified as a positive integer. Default: `20`.
`SearchMethod`	Method used by the iterative search algorithm. Assignable values: `'Auto'` — Automatically chooses from the following methods. `'gn'` — Gauss-Newton method. `'gna'` — Adaptive Gauss-Newton method. `'grad'` — A gradient method. `'lm'` — Levenberg-Marquardt method. `'lsqnonlin'` — Nonlinear least-squares method (requires the Optimization Toolbox™ product). This method handles only the `'Trace'` criterion.
`Tolerance`	Specifies to terminate the iterative search when the expected improvement of the parameter values is less than `Tolerance`, specified as a positive real value in %. Default: `0.01`.
`GradientOptions`	A structure that specifies the options related to calculation of gradient of the cost, idnlgrey Gradient Options.
`SimulationOptions`	A structure that specifies the simulation method and related options, as described in idnlgrey Simulation Options.
`Trace`	Toggles displaying or hiding estimation progress information in the MATLAB^® Command Window. Default: `'Off'`. Assignable values: `'Off'` — Hide estimation information. `'On'` — Display estimation information.
`Weighting`	Positive semi-definite matrix `W` used for weighted trace minimization. When `Criterion` = `'Trace'`, `trace(E'EW)` is minimized. `Weighting` can be used to specify relative importance of outputs in multiple-input multiple-output models (or reliability of corresponding data) when `W` is a diagonal matrix of nonnegative values. `Weighting` is not useful in single-output models. By default, `Weighting` is an identity matrix of size equal to the number of outputs.

Note The Criterion property setting is meaningful in multiple-output cases only. In single-output models, the two criteria are equivalent. Both the Det and Trace criteria are derived from a general requirement of minimizing a weighted sum of least squares of prediction errors. The Det criterion can be interpreted as estimating the covariance matrix of the noise source and using the inverse of that matrix as the weighting. You should specify the weighting when using the Trace criterion.

If you want to achieve better accuracy for a particular channel in multiple-input multiple-output models, you should use Trace with weighting that favors that channel. Otherwise it is natural to use Det. When using Det you can check cond(model.NoiseVariance) after estimation. If the matrix is ill-conditioned, it may be more robust to use the Trace criterion. You can also use compare on validation data to check whether the relative error for different channels corresponds you your needs or expectations. Use the Trace criterion if you need to modify the relative errors, and check model.NoiseVariance to determine what weighting modifications to specify.

The search method of lsqnonlin supports the Trace criterion only.

idnlgrey Advanced Algorithm Properties

The following table summarizes the fields of the Algorithm.Advanced model properties. The fields in the Algorithm.Advanced structure specify additional estimation-algorithm options.

Property Name Description

GnPinvConst

When the search direction is computed, the algorithm discards the singular values of the Jacobian that are smaller than GnPinvConst*max(size(J))*norm(J)*eps. Singular values that are closer to 0 are included when GnPinvConst is decreased.
Default: 1e4.
Assign a positive, real value.

LMStartValue

(For Levenberg-Marquardt search algorithm) The starting level of regularization when using the Levenberg-Marquardt search method (Algorithm.SearchMethod='lm').
Default: 0.001.
Assign a positive real value.

LMStep

(For Levenberg-Marquardt search algorithm) Try this next level of regularization to get a lower value of the criterion function. The level of regularization is LMStep times the previous level. At the start of a new iteration, the level of regularization is computed as1/LMStep times the value from the previous iteration.
Default: 10.
Assign a real value >1.

MaxBisections

Maximum number of bisections performed by the line search algorithm along the search direction (number of rotations of search vector for 'lm'). Used by 'gn', 'lm', 'gna' and 'grad' search methods (Algorithm.SearchMethod property)
Default: 10.
Assign a positive integer value.

MaxFunEvals

The iterations are stopped if the number of calls to the model file exceeds this value.
Default: Inf.
Assign a positive integer value.

MinParChange

The smallest parameter update allowed per iteration.
Default: 1e-16.
Assign a positive, real value.

RelImprovement

The iterations are stopped if the relative improvement of the criterion function is less than RelImprovement.
Default: 0.
Assign a positive real value.

Note Does not apply to Algorithm.SearchMethod='lsqnonlin'

StepReduction

(For line search algorithm) The suggested parameter update is reduced by the factor 'StepReduction' after each try until either 'MaxBisections' tries are completed or a lower value of the criterion function is obtained.
Default: 2.
Assign a positive, real value >1.

Note Does not apply to Algorithm.SearchMethod='lsqnonlin'

idnlgrey Simulation Options

The following table summarizes the fields of Algorithm.SimulationOptions model properties.

Property Name	Description
`AbsTol`	Absolute error tolerance. This scalar applies to all components of the state vector. `AbsTol` applies only to the variable step solvers. Default: `1e-6`. Assignable value: A positive real value.
`FixedStep`	(For fixed-step time-continuous solvers) Step size used by the solver. Default: `'Auto'`. Assignable values: `'Auto'` — Automatically chooses the initial step. A real value such that `0<FixedStep<=1`.
`InitialStep`	(For variable-step time-continuous solvers) Specifies the initial step at which the ODE solver starts. Default: `'Auto'`. Assignable values: `'Auto'` — Automatically chooses the initial step. A positive real value such that `MinStep<=InitialStep<=MaxStep`.
`MaxOrder`	(For ode15s) Specifies the order of the Numerical Differentiation Formulas (NDF). Default: `5`. Assignable values: 1, 2, 3, 4 or 5.
`MaxStep`	(For variable-step time-continuous solvers) Specifies the largest time step of the ODE solver. Default: `'Auto'` — 1/15 of the simulation interval. Assignable values: `'Auto'` — Automatically chooses the time step. A positive real value > `MinStep`.
`MinStep`	(For variable-step time-continuous solvers) Specifies the smallest time step of the ODE solver. Default: `'Auto'`. Assignable values: `'Auto'` — Automatically chooses the time step. A positive real value < `MaxStep`.
`RelTol`	(For variable-step time-continuous solvers) Relative error tolerance that applies to all components of the state vector. The estimated error in each integration step satisfies `\|e(i)\| <= max(RelTol*abs(x(i)), AbsTol(i))`. Default: `1e-3` (0.1% accuracy). Assignable value: A positive real value.
`Solver`	ODE (Ordinary Differential/Difference Equation) solver for solving state space equations. A. Variable-step solvers for time-continuous `idnlgrey` models: `'ode45'` — Runge-Kutta (4,5) solver for nonstiff problems. `'ode23'` — Runge-Kutta (2,3) solver for nonstiff problems. `'ode113'` — Adams-Bashforth-Moulton solver for nonstiff problems. `'ode15s'` — Numerical Differential Formula solver for stiff problems. `'ode23s'` — Modified Rosenbrock solver for stiff problems. `'ode23t'` — Trapezoidal solver for moderately stiff problems. `'ode23tb'` — Implicit Runge-Kutta solver for stiff problems. B. Fixed-step solvers for time-continuous `idnlgrey` models: `'ode5'` — Dormand-Prince solver. `'ode4'` — Fourth-order Runge-Kutta solver. `'ode3'` — Bogacki-Shampine solver. `'ode2'` — Heun or improved Euler solver. `'ode1'` — Euler solver. C. Fixed-step solvers for time-discrete `idnlgrey` models: `'FixedStepDiscrete'` D. General: `'Auto'` — Automatically chooses one of the previous solvers (default).

idnlgrey Gradient Options

The following table summarizes the fields of the Algorithm.GradientOptions model properties. Algorithm is a structure that specifies the estimation-algorithm options.

Property Name	Description
`DiffMaxChange`	Largest allowed parameter perturbation when computing numerical derivatives. Default: `Inf`. Assignable value: A positive real value `>'DiffMinChange'`.
`DiffMinChange`	Smallest allowed parameter perturbation when computing numerical derivatives. Default: `0.01*sqrt(eps)`. Assignable value: A positive real value `<'DiffMaxChange'`.
`DiffScheme`	Method for computing numerical derivatives with respect to the components of the parameters and/or the initial state(s) to form the Jacobian. Default: `'Auto'` Assignable values: `'Auto'` - Automatically chooses from the following methods. `'Central approximation'` `'Forward approximation'` `'Backward approximation'`
`GradientType`	Method used when computing derivatives (Jacobian) of the parameters or the initial states to be estimated. Default: `'Auto'`. Assignable values: `'Auto'` — Automatically chooses from the following methods. `'Basic'` — Individually computes all numerical derivatives required to form each column of the Jacobian. `'Refined'` — Simultaneously computes all numerical derivatives required to form each column of the Jacobian.

idnlgrey EstimationInfo Properties

The following table summarizes the fields of the EstimationInfo model properties. The read-only fields of the EstimationInfo structure store estimation settings and results.

Property Name	Description
`Status`	Shows whether the model parameters were estimated.
`Method`	Names of the solver and the optimizer used during estimation.
`LossFcn`	Value of the loss function, equal to `det(E'*E/N)`, where `E` is the residual error matrix (one column for each output) and `N` is the total number of samples. Provides a quantitative description of the model quality.
`FPE`	Value of Akaike's Final Prediction Error (see `fpe`).
`DataName`	Name of the data from which the model is estimated.
`DataLength`	Length of the estimation data.
`DataTs`	Sampling interval of the estimation data.
`DataDomain`	`'Time'` means time domain data. `'Frequency'` is not supported.
`DataInterSample`	Intersample behavior of the input estimation data used for interpolation: `'zoh'` means zero-order-hold, or piecewise constant. `'foh'` means first-order-hold, or piecewise linear.
`WhyStop`	Reason for terminating parameter estimation iterations.
`UpdateNorm`	Norm of the search vector (Gauss-Newton vector) at the last iteration. Empty when `'lsqnonlin'` is the search method.
`LastImprovement`	Criterion improvement in the last iteration, shown in %. Empty when `SearchMethod='lsqnonlin'` is the search method.
`Iterations`	Number of iterations performed by the estimation algorithm.
`InitialGuess`	Structure with the fields `InitialStates` and `Parameters`, specifying the values of these quantities before the last estimation.
`Warning`	Any warnings encountered during parameter estimation.

Definition of idnlgrey States

The states of an idnlgrey model are defined explicitly by the user in the M-file or MEX-file (as specified in the FileName property of the model) storing the model structure . The concept of states is useful for functions such as sim, predict, compare, and findstates.

Note The initial values of the states are configured by the InitialStates property of the idnlgrey model.

`pem`
`getinit`
`setinit`
`getpar`
`setpar`