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 linear the model. After you linearize your model, you can use the model for control design and linear analysis.
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
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.
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,
a linear ARX model using the same model orders
nk as the original model.
For Hammerstein-Wiener models,
a linear Output-Error (OE) model using the same model orders
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,
For more information, see the
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.
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:
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
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
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,
the initial states by transforming the data.