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Linearize nonlinear ARX model

`SYS = linearize(NLSYS,U0,X0)`

`SYS = linearize(NLSYS,U0,X0)`

linearizes
a nonlinear ARX model about the specified operating point `U0`

and `X0`

.
The linearization is based on tangent linearization. For more information
about the definition of states for `idnlarx`

models,
see Definition of idnlarx States.

`NLSYS`

:`idnlarx`

model.`U0`

: Matrix containing the constant input values for the model.`X0`

: Model state values. The states of a nonlinear ARX model are defined by the time-delayed samples of input and output variables. For more information about the states of nonlinear ARX models, see the`getDelayInfo`

reference page.

**Note**

To estimate `U0`

and `X0`

from
operating point specifications, use the `findop`

command.

`SYS`

is an`idss`

model.When the Control System Toolbox™ product is installed,

`SYS`

is an LTI object.

The following equations govern the dynamics of an `idnlarx`

model:

$$\begin{array}{l}X(t+1)=AX(t)+B\tilde{u}(t)\\ y(t)=f(X,u)\end{array}$$

where *X*(*t*)
is a state vector, *u*(*t*) is the
input, and *y*(*t*) is the output. *A* and *B* are
constant matrices. $$\tilde{u}(t)$$ is [*y*(*t*), *u*(*t*)]* ^{T}*.

The output at the operating point is given by

*y** = *f*(*X**, *u**)

where *X** and *u** are the
state vector and input at the operating point.

The linear approximation of the model response is as follows:

$$\begin{array}{l}\Delta X(t+1)=(A+{B}_{1}{f}_{X})\Delta X(t)+({B}_{1}{f}_{u}+{B}_{2})\Delta u(t)\\ \Delta y(t)={f}_{X}\Delta X(t)+{f}_{u}\Delta u(t)\end{array}$$

where

$$\Delta X(t)=X(t)-{X}^{*}(t)$$

$$\Delta u(t)=u(t)-{u}^{*}(t)$$

$$\Delta y(t)=y(t)-{y}^{*}(t)$$

$$B\tilde{U}=[{B}_{1},{B}_{2}]\left[\begin{array}{c}Y\\ U\end{array}\right]={B}_{1}Y+{B}_{2}U$$

$${{f}_{X}=\frac{\partial}{\partial X}f(X,U)|}_{X*,U*}$$

$${{f}_{U}=\frac{\partial}{\partial U}f(X,U)|}_{X*,U*}$$

**Note**

For linear approximations over larger input ranges, use `linapp`

.