Creating and Manipulating Models

Extracting and Modifying Subsystems

Subsystems relate subsets of the inputs and outputs of a system. The transfer matrix of a subsystem is a submatrix of the system transfer matrix. For example, if sys is a system with two inputs, three outputs, and I/O relation

then gives the relation between the first input and third output.

Accordingly, use matrix-like subindexing to extract this subsystem.

• SubSys = sys(3,1)
  

The resulting subsystem SubSys is an LTI model of the same type as sys, with its sample time, time delay, I/O name, and I/O group property values inherited from sys.

For example, if sys has an input group named controls consisting of channels one, two, and three, then SubSys also has an input group named controls with the first channel of SubSys assigned to it.

If sys is a state-space model with matrices a, b, c, d, the subsystem sys(3,1) is a state-space model with data a, b(:,1), c(3,:), d(3,1). Note the following rules when extracting subystems:

• In the expression sys(3,1), the first index selects the output channel while the second index selects the input channel.
• When extracting a subsystem from a given state-space model, the resulting state-space model may not be minimal. Use the command sminreal to eliminate unnecessary states in the subsystem.

You can use similar syntax to modify the LTI model sys. For example,

• sys(3,1) = NewSubSys
  

redefines the I/O relation between the first input and third output, provided NewSubSys is a SISO LTI model.

The following rules apply when modifying LTI models:

• sys, the LTI model that has had a portion reassigned, retains its original model type (TF, ZPK, SS, or FRD) regardless of the model type of NewSubSys.
• Subsystem assignment does not reassign any I/O names or I/O group names of NewSubSys that are already assigned to NewSubSys.
• Reassigning parts of a MIMO state-space model generally increases its order.
• If NewSubSys is an FRD model, then sys must also be an FRD model. Furthermore, their frequencies must match.

Other standard matrix subindexing extends to LTI objects as well. For example,

• sys(3,1:2)
  

extracts the subsystem mapping the first two inputs to the third output.

• sys(:,1)
  

selects the first input and all outputs, and

• sys([1 3],:)
  

extracts a subsystem with the same inputs, but only the first and third outputs.

For example, consider the two-input/two-output transfer function

.

To extract the transfer function from the first input to the first output, type

• T(1,1) 
  Transfer function:
     1
  -------
  s + 0.1
  

Next reassign to and modify the second input channel of T by typing

• T(1,1) = tf(1,[1 0.5]);
  T(:,2) = [ 1 ; tf(0.4,[1 0]) ]
  Transfer function from input 1 to output...
           1
   #1:  -------
        s + 0.5
   
            s - 1
   #2:  -------------
        s^2 + 2 s + 2
   
  Transfer function from input 2 to output...
   #1:  1
   
        0.4
   #2:  ---
         s
  

Precedence and Property Inheritance

Referencing FRD Models Through Frequencies