The Simulink^{®} product
provides the ` `

, `linmod`

`linmod2`

,
and `dlinmod`

functions to extract linear models
in the form of the state-space matrices *A*, *B*, *C*,
and *D*. State-space matrices describe the linear
input-output relationship as

$$\begin{array}{l}\dot{x}=Ax+Bu\\ y=Cx+Du,\end{array}$$

where *x*, *u*, and *y* are
state, input, and output vectors, respectively. For example, the following
model is called `lmod`

.

To extract the linear model of this system, enter this command.

[A,B,C,D] = linmod('lmod') A = -2 -1 -1 1 0 0 0 1 -1 B = 1 0 0 C = 0 1 0 0 0 -1 D = 0 1

Inputs and outputs must be defined using Inport and Outport blocks from the Ports & Subsystems library. Source and sink blocks do not act as inputs and outputs. Inport blocks can be used in conjunction with source blocks, using a Sum block. Once the data is in the state-space form or converted to an LTI object, you can apply functions in the Control System Toolbox™ product for further analysis:

Conversion to an LTI object

sys = ss(A,B,C,D);

Bode phase and magnitude frequency plot

bode(A,B,C,D) or bode(sys)

Linearized time response

step(A,B,C,D) or step(sys) impulse(A,B,C,D) or impulse(sys) lsim(A,B,C,D,u,t) or lsim(sys,u,t)

You can use other functions in the Control System Toolbox and the Robust Control Toolbox™ products for linear control system design.

When the model is nonlinear, an operating point can be chosen
at which to extract the linearized model. Extra arguments to `linmod`

specify
the operating point.

[A,B,C,D] = linmod('sys', x, u)

For discrete systems or mixed continuous and discrete systems,
use the function `dlinmod`

for
linearization. This function has the same calling syntax as `linmod`

except that the second right-hand
argument must contain a sample time at which to perform the linearization.

You can use `linmod`

to extract a linear model
from a Simulink environment that contains Model blocks.

In Normal mode, the `linmod`

command applies
the block-by-block linearization algorithm on blocks inside the referenced
model. If the Model block is in Accelerator mode, the `linmod`

command
uses numerical perturbation to linearize the referenced model. Due
to limitations on linearizing multirate Model blocks in Accelerator
mode, you should use Normal mode simulation for all models referenced
by Model blocks when linearizing with referenced models. For an explanation
of the block-by-block linearization algorithm, see the Simulink
Control Design™ documentation.

For example, consider the `f14`

model `mdlref_f14`

.
The Aircraft Dynamics Model block refers to the model `mdlref_dynamics`

.

To linearize the `mdlref_f14`

model, call the `linmod`

command
on the top `mdlref_f14`

model as follows.

[A,B,C,D] = linmod('mdlref_f14')

The resulting state-space model corresponds to the complete `f14`

model,
including the referenced model.

You can call `linmod`

with a state and input
operating point for models that contain Model blocks. When using operating
points, the state vector `x`

refers to the total
state vector for the top model and any referenced models. You must
enter the state vector using the structure format. To get the complete
state vector, call

x = Simulink.BlockDiagram.getInitialState(topModelName)

Calling the `linmod`

command
with the `'v5'`

argument invokes the perturbation
algorithm created prior to MATLAB^{®} software version 5.3. This
algorithm also allows you to specify the perturbation values used
to perform the perturbation of all the states and inputs of the model.

[A,B,C,D]=linmod('sys',x,u,para,xpert,upert,'v5')

`linmod`

with
the `'v5'`

option to linearize a model that contains
Derivative or Transport Delay blocks can be troublesome. Before linearizing,
replace these blocks with specially designed blocks that avoid the
problems. These blocks are in the Simulink Extras library in
the Linearization sublibrary.You access the Extras library by opening the Blocksets & Toolboxes icon:

For the Derivative block, use the Switched derivative for linearization.

When using a Derivative block, you can also try to incorporate the derivative term in other blocks. For example, if you have a Derivative block in series with a Transfer Fcn block, it is better implemented (although this is not always possible) with a single Transfer Fcn block of the form

$$\frac{s}{s+a}.$$

In this example, the blocks on the left of this figure can be replaced by the block on the right.

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