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.
Note
In Normal mode, the |
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.
For the Transport Delay block, use the Switched transport delay for linearization. (Using this block requires that you have the Control System Toolbox product.)
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.