Function Reference

inv

Invert LTI systems

Syntax

• isys = inv(sys)
  

Description

inv inverts the input/output relation

to produce the LTI system with the transfer matrix .

This operation is defined only for square systems (same number of inputs and outputs) with an invertible feedthrough matrix . inv handles both continuous- and discrete-time systems.

Example

Consider

At the MATLAB prompt, type

• H = [1 tf(1,[1 1]);0 1]
  Hi = inv(H)
  

to invert it. MATLAB returns

• Transfer function from input 1 to output...
   #1:  1
   
   #2:  0
   
  Transfer function from input 2 to output...
         -1
   #1:  -----
        s + 1
   
   #2:  1
  

You can verify that

• H * Hi
  

is the identity transfer function (static gain I).

Limitations

Do not use inv to model feedback connections such as

While it seems reasonable to evaluate the corresponding closed-loop transfer function as

• inv(1+g*h) * g
  

this typically leads to nonminimal closed-loop models. For example,

• g = zpk([],1,1)
  h = tf([2 1],[1 0])
  cloop = inv(1+g*h) * g
  

yields a third-order closed-loop model with an unstable pole-zero cancellation at s = 1.

• cloop
  
  Zero/pole/gain:
        s (s-1)
  -------------------
  (s-1) (s^2 + s + 1)
  

Use feedback to avoid such pitfalls.

• cloop = feedback(g,h)
  
  Zero/pole/gain:
        s
  -------------
  (s^2 + s + 1)
  

interp

iopzmap