# Using FEEDBACK to Close Feedback Loops

This example shows why you should always use FEEDBACK to close feedback loops.

### Two Ways of Closing Feedback Loops

Consider the following feedback loop

where

K = 2; G = tf([1 2],[1 .5 3])

G = s + 2 --------------- s^2 + 0.5 s + 3 Continuous-time transfer function.

You can compute the closed-loop transfer function `H`

from r to y in at least two ways:

Using the

`feedback`

commandUsing the formula

$$H=\frac{G}{1+GK}$$

To compute `H`

using `feedback`

, type

H = feedback(G,K)

H = s + 2 --------------- s^2 + 2.5 s + 7 Continuous-time transfer function.

To compute `H`

from the formula, type

H2 = G/(1+G*K)

H2 = s^3 + 2.5 s^2 + 4 s + 6 ----------------------------------- s^4 + 3 s^3 + 11.25 s^2 + 11 s + 21 Continuous-time transfer function.

### Why Using FEEDBACK is Better

A major issue with computing `H`

from the formula is that it inflates the order of the closed-loop transfer function. In the example above, `H2`

has double the order of `H`

. This is because the expression `G/(1+G*K)`

is evaluated as a ratio of the two transfer functions `G`

and `1+G*K`

. If

$$G(s)=\frac{N(s)}{D(s)}$$

then `G/(1+G*K)`

is evaluated as:

$$\frac{N}{D}{\left(\frac{D+KN}{D}\right)}^{-1}=\frac{ND}{D(D+KN)}.$$

As a result, the poles of `G`

are added to both the numerator and denominator of `H`

. You can confirm this by looking at the ZPK representation:

zpk(H2)

ans = (s+2) (s^2 + 0.5s + 3) --------------------------------- (s^2 + 0.5s + 3) (s^2 + 2.5s + 7) Continuous-time zero/pole/gain model.

This excess of poles and zeros can negatively impact the accuracy of your results when dealing with high-order transfer functions, as shown in the next example. This example involves a 17th-order transfer function `G`

. As you did before, use both approaches to compute the closed-loop transfer function for `K=1`

:

load numdemo G H1 = feedback(G,1); % good H2 = G/(1+G); % bad

To have a point of reference, also compute an FRD model containing the frequency response of G and apply `feedback`

to the frequency response data directly:

w = logspace(2,5.1,100); H0 = feedback(frd(G,w),1);

Then compare the magnitudes of the closed-loop responses:

h = sigmaplot(H0,'b',H1,'g--',H2,'r'); legend('Reference H0','H1=feedback(G,1)','H2=G/(1+G)','location','southwest') setoptions(h,'YlimMode','manual','Ylim',{[-60 0]})

The frequency response of `H2`

is inaccurate for frequencies below 2e4 rad/s. This inaccuracy can be traced to the additional (cancelling) dynamics introduced near z=1. Specifically, `H2`

has about twice as many poles and zeros near z=1 as `H1`

. As a result, `H2(z)`

has much poorer accuracy near z=1, which distorts the response at low frequencies. See the example Using the Right Model Representation for more details.