# 0.11 Lecture 12:interconnected systems and feedback  (Page 2/5)

III. SIMPLE LINEAR FEEDBACK SYSTEM

1/ Black’s formula

K(s) is called the open loop system function, and H(s) = Y (s)/X(s) is called the closed-loop system function. Note, when β(s) = 0, H(s) = K(s)

We can find H(s) by combining

$E\left(s\right)=X\left(s\right)-\beta \left(s\right)Y\left(s\right)\begin{array}{cc}& \end{array}\text{and}\begin{array}{cc}& \end{array}Y\left(s\right)=K\left(s\right)E\left(s\right)$

to obtain

$Y\left(s\right)=K\left(s\right)X\left(s\right)-\beta \left(s\right)Y\left(s\right)$

which can be solved to obtain Black’s formula,

$H\left(s\right)=\frac{Y\left(s\right)}{X\left(s\right)}=\frac{K\left(s\right)}{1+\beta \left(s\right)K\left(s\right)}=\frac{\text{forward}\begin{array}{cc}& \end{array}\text{transmission}}{1\begin{array}{cc}& \end{array}-\begin{array}{cc}& \end{array}\text{loop}\begin{array}{cc}& \end{array}\text{gain}}$

Two-minute miniquiz problem

Problem 8-1 Simple position control system

The objective of the position control system shown below is for the output position Y (s) to track the input signal X(s).

a) Determine the closed-loop system function H(s) = Y (s)/X(s).

b) For x(t) = u(t), a unit step, determine the steady state value of y(t).

Solution

1. We can use Black’s formula to find H(s) as follows

$H\left(s\right)=\frac{\frac{K}{\left(s+1\right)\left(s+\text{100}\right)}}{1+\frac{K}{\left(s+1\right)\left(s+\text{100}\right)}}=\frac{K}{\left(s+1\right)\left(s+\text{100}\right)+K}$

b) The steady-state response to a unit step is simply the response to the complex exponential $x\left(t\right)=\text{1}\text{.}{\text{e}}^{0\text{.}\text{t}}=\text{1}$ which is $y\left(t\right)=\text{1}\text{.}\text{H}\left(0\right)\text{.}{e}^{0\text{.}\text{t}}=\text{K/}\left(\text{100}+\text{K}\right)$ . The position error $\epsilon =K/\left(\text{100}+K\right)-1\text{=}-\text{100/}\left(\text{100 +}Κ\right)$ Hence, this position controller (with proportional feedback) has an error that diminishes as the gain increases. However, the error is never zero no matter how large the gain.

Effect of feedback on system performance

Feedback is used to enhance system performance.

• Stabilize gain
• Reduce the effect of an output disturbance
• Improve dynamic characteristics — increase bandwidth, improve response time
• Reduce noise
• Reduce nonlinear distortion

Properties of feedback

• Increase input impedance
• Decrease output impedance

2/ Stabilize gain

$\text{overall}\begin{array}{cc}& \end{array}\text{gain}=\text{10}\begin{array}{cc}& \end{array}\text{Overall}\begin{array}{cc}& \end{array}\text{gain}=\frac{\text{100}×\text{10}}{1+0\text{.}\text{099}×\text{100}×\text{10}}=\text{10}$

Note that both the open-loop and the same gain which equals 10.

But now suppose that the gain of the power amplifier is reduced to 5.

$\text{overall}\begin{array}{cc}& \end{array}\text{gain}=\text{10}\begin{array}{cc}& \end{array}\text{Overall}\begin{array}{cc}& \end{array}\text{gain}=\frac{\text{100}×5}{1+0\text{.}\text{099}×\text{100}×5}=9\text{.}9$

Note that a change in gain of the power amplifier of 50% leads to a change in gain of the feedback system of only 1%.

The stabilization of the gain resulting from feedback can be appreciated more generally from Black’s formula.

$H\left(s\right)=\frac{K\left(s\right)}{1+\beta \left(s\right)K\left(s\right)}$

If K(s) is large so that |β(s)K(s)|>>1 then

$H\left(s\right)\approx \frac{1}{\beta \left(s\right)}$

So if H(s) has a gain then β(s) must have an attenuation. If the attenuation β(s) is determined precisely but the gain K(s) varies (with time, temperature, etc.), then we can make an amplifier whose gain is independent of K(s) and determined almost entirely by β(s).

But how can we make β(s) precise?

Consider the non-inverting amplifier with a non-ideal (finite gain) op-amp — network (left), block diagram (right).

For an op-amp (model 741) the gain is typically $K\approx {\text{10}}^{7}$ . Hence, provided ${\text{KR}}_{2}/\left({R}_{1}+{R}_{2}\right)\text{>>1}$ , we have

$\frac{{V}_{o}}{{V}_{i}}=\frac{{\text{10}}^{7}}{1+{\text{10}}^{7}\frac{{R}_{2}}{{R}_{1}+{R}_{2}}}\approx \frac{{R}_{1}+{R}_{2}}{{R}_{2}}$

Conclusion — the gain of the feedback amplifier depends primarily on the values of the resistors and not on the gain of the op-amp which depends on parameters of transistors which change with time, temperature, etc.

3/ Reduce the effect of an output disturbance

The transfer functions for the input and the disturbance are

$\frac{Y\left(s\right)}{X\left(s\right)}=\frac{K\left(s\right)}{1+\beta \left(s\right)K\left(s\right)}\begin{array}{cc}& \end{array}\text{and}\begin{array}{cc}& \end{array}\frac{Y\left(s\right)}{D\left(s\right)}=\frac{1}{1+\beta \left(s\right)K\left(s\right)}$

Therefore, if β(s)K(s) is made arbitrarily large then

I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!