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Feedback connection of two models

`sys = feedback(sys1,sys2) `

`sys = feedback(sys1,sys2) `

returns a model object `sys`

for the negative feedback
interconnection of model objects `sys1`

and `sys2`

.

The closed-loop model `sys`

has *u* as
input vector and *y* as output vector. The models `sys1`

and `sys2`

must
be both continuous or both discrete with identical sample times. Precedence
rules are used to determine the resulting model type (see Rules That Determine Model Type).

To apply positive feedback, use the syntax

sys = feedback(sys1,sys2,+1)

By default, `feedback(sys1,sys2)`

assumes negative
feedback and is equivalent to `feedback(sys1,sys2,-1)`

.

Finally,

sys = feedback(sys1,sys2,feedin,feedout)

computes a closed-loop model `sys`

for the
more general feedback loop.

The vector `feedin`

contains indices into the
input vector of `sys1`

and specifies which inputs *u* are
involved in the feedback loop. Similarly, `feedout`

specifies
which outputs *y* of `sys1`

are
used for feedback. The resulting model `sys`

has
the same inputs and outputs as `sys1`

(with their
order preserved). As before, negative feedback is applied by default
and you must use

sys = feedback(sys1,sys2,feedin,feedout,+1)

to apply positive feedback.

For more complicated feedback structures, use `append`

and `connect`

.

To connect the plant

$$G(s)=\frac{2{s}^{2}+5s+1}{{s}^{2}+2s+3}$$

with the controller

$$H(s)=\frac{5(s+2)}{s+10}$$

using negative feedback, type

G = tf([2 5 1],[1 2 3],'inputname','torque',... 'outputname','velocity'); H = zpk(-2,-10,5) Cloop = feedback(G,H)

These commands produce the following result.

Zero/pole/gain from input "torque" to output "velocity": 0.18182 (s+10) (s+2.281) (s+0.2192) ----------------------------------- (s+3.419) (s^2 + 1.763s + 1.064)

The result is a zero-pole-gain model as expected from the precedence
rules. Note that `Cloop`

inherited the input and
output names from `G`

.

Consider a state-space plant `P`

with five
inputs and four outputs and a state-space feedback controller `K`

with
three inputs and two outputs. To connect outputs 1, 3, and 4 of the
plant to the controller inputs, and the controller outputs to inputs
4 and 2 of the plant, use

feedin = [4 2]; feedout = [1 3 4]; Cloop = feedback(P,K,feedin,feedout)

You can form the following negative-feedback loops

by

Cloop = feedback(G,1) % left diagram Cloop = feedback(1,G) % right diagram

The feedback connection should be free of algebraic loop. If *D _{1}* and

`sys1`

and `sys2`

,
this condition is equivalent to:*I + D*nonsingular when using negative feedback_{1}D_{2}*I − D*nonsingular when using positive feedback._{1}D_{2}