inv (Control System Toolbox)

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

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 or star to avoid such pitfalls.

cloop = feedback(g,h)

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

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