## Linear Approximation of Nonlinear Black-Box Models

### Why Compute a Linear Approximation of a Nonlinear Model?

Control design and linear analysis techniques using Control System Toolbox™ software require linear models. You can use your estimated nonlinear model in these applications after you linearize the model.

### Choosing Your Linear Approximation Approach

System Identification Toolbox™ software provides two approaches for computing a linear approximation of nonlinear ARX and Hammerstein-Wiener models.

To compute a linear approximation of a nonlinear model for a given input signal,
use the `linapp`

command. The resulting model is only valid for
the same input that you use to compute the linear approximation. For more
information, see Linear Approximation of Nonlinear Black-Box Models for a Given Input.

If you want a tangent approximation of the nonlinear dynamics that is accurate
near the system operating point, use the `linearize`

command. The
resulting model is a first-order Taylor series approximation for the system about
the operating point, which is defined by a constant input and model state values.
For more information, see Tangent Linearization of Nonlinear Black-Box Models.

### Linear Approximation of Nonlinear Black-Box Models for a Given Input

`linapp`

computes the best linear approximation, in a
mean-square-error sense, of a nonlinear ARX or Hammerstein-Wiener model for a given
input or a randomly generated input. The resulting linear model might only be valid
for the same input signal as you the one you used to generate the linear
approximation.

`linapp`

estimates the best linear model that is structurally
similar to the original nonlinear model and provides the best fit between a given
input and the corresponding simulated response of the nonlinear model.

To compute a linear approximation of a nonlinear black-box model for a given input, you must have these variables:

`linapp`

uses the specified input signal to compute a linear
approximation:

For nonlinear ARX models,

`linapp`

estimates a linear ARX model using the same model orders`na`

,`nb`

, and`nk`

as the original model.For Hammerstein-Wiener models,

`linapp`

estimates a linear Output-Error (OE) model using the same model orders`nb`

,`nf`

, and`nk`

.

To compute a linear approximation of a nonlinear black-box model for a randomly
generated input, you must specify the minimum and maximum input values for
generating white-noise input with a magnitude in this rectangular range,
`umin`

and `umax`

.

For more information, see the `linapp`

reference page.

### Tangent Linearization of Nonlinear Black-Box Models

`linearize`

computes a first-order Taylor series approximation
for nonlinear system dynamics about an *operating point*, which
is defined by a constant input and model state values. The resulting linear model is
accurate in the local neighborhood of this operating point.

To compute a tangent linear approximation of a nonlinear black-box model, you must have these variables:

To specify the operating point of your system, you must specify the constant input and the states. For more information about state definitions for each type of parametric model, see these reference pages:

If you do not know the operating point values for your system, see Computing Operating Points for Nonlinear Black-Box Models.

For more information, see the `idnlarx/linearize`

or `idnlhw/linearize`

reference page.

### Computing Operating Points for Nonlinear Black-Box Models

An *operating point* is defined by a constant input and model
state values.

If you do not know the operating conditions of your system for linearization, you
can use `findop`

to compute the operating point from
specifications.

#### Computing Operating Point from Steady-State Specifications

Use `findop`

to compute an operating point from
steady-state specifications:

Values of input and output signals.

If either the steady-state input or output value is unknown, you can specify it as`NaN`

to estimate this value. This is especially useful when modeling MIMO systems, where only a subset of the input and output steady-state values are known.More complex steady-state specifications.

Construct an object that stores specifications for computing the operating point, including input and output bounds, known values, and initial guesses. For more information, see

`idnlarx/operspec`

or`idnlhw/operspec`

.

For more information, see the `idnlarx/findop`

or `idnlhw/findop`

reference
page.

#### Computing Operating Points at a Simulation Snapshot

Compute an operating point at a specific time during model simulation
(snapshot) by specifying the snapshot time and the input value. To use this
method for computing the equilibrium operating point, choose an input that leads
to a steady-state output value. Use that input and the time value at which the
output reaches steady state (*snapshot* time) to compute the
operating point.

It is optional to specify the initial conditions for simulation when using this method because initial conditions often do not affect the steady-state values. By default, the initial conditions are zero.

However, for nonlinear ARX models, the steady-state output value might depend
on initial conditions. For these models, you should investigate the effect of
initial conditions on model response and use the values that produce the desired
output. You can use `data2state`

to map the input-output
signal values from before the simulation starts to the model's initial states.
Because the initial states are a function of the past history of the model's
input and output values, `data2state`

generates the initial
states by transforming the data.

## See Also

`idnlarx/linearize`

| `idnlhw/linearize`