Linearizing Models

The Simulink^® product provides the linmod, linmod2, and dlinmod functions to extract linear state-space models from Simulink models. However, these functions provide only basic linearization functionality. For full linearization functionality, use Simulink Control Design™ software. For more information, see Choose Linearization Tools (Simulink Control Design).

The linmod, linmod2, and dlinmod functions 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.