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

V. ROOTS OF SECOND-ORDER AND THIRD-ORDER POLYNOMIALS

We consider conditions that second- and third-order polynomials have roots in the left half of the complex s-plane.

1/ Second-order polynomials

Second-order polynomials with real coefficients have either real or complex roots of the form

$\left(s+a\right)\left(s+b\right)=\text{0}\begin{array}{cc}& \end{array}\text{or}\begin{array}{cc}& \end{array}\left(s+a+\text{jc}\right)\left(s+\text{a - jc}\right)\text{= 0}$

where a>0 and b>0. The polynomials can be expressed as

${s}^{2}+\left(a+b\right)s+\text{ab}\begin{array}{cc}& \end{array}\text{or}\begin{array}{cc}& \end{array}{s}^{2}+2\text{as}+{a}^{2}+{c}^{2}=0$

Thus, both polynomials have the form

${s}^{2}+\mathrm{\alpha s}+\beta =0$

where α>0 and β>0. These conditions are both necessary and sufficient.

2/ Third-order polynomials

Third-order polynomials must have one real root and either a pair of real or complex roots of the form

$\left(s+a\right)\left(s+b\right)\left(s+c\right)=\text{0}\begin{array}{cc}& \end{array}\text{or}\begin{array}{cc}& \end{array}\left(s+b\right)\left(s+a+\text{jd}\right)\left(s+\text{a - jd}\right)\text{= 0}$

where a>0, b>0, and c>0. The polynomials can be expressed as

$\begin{array}{}{s}^{3}+\left(a+b+c\right){s}^{2}+\left(\text{ab}+\text{ac}+\text{bc}\right)s+\text{abc}=0\begin{array}{cc}& \end{array}\\ \text{or}\begin{array}{cc}& \end{array}{s}^{3}+\left(2a+b\right){s}^{2}+\left({a}^{2}+{d}^{2}+2\text{ab}\right)s+\left({a}^{2}+{d}^{2}\right)b=0\end{array}$

Thus, both can be put in the form

${s}^{3}+{\mathrm{\alpha s}}^{2}+\mathrm{\beta s}+\gamma =0$

Note that α>0, β>0, and γ>0. But in addition, β>γ/α. These conditions are both necessary and sufficient.

VI. ROOT LOCUS PLOTS FOR POSITION CONTROL SYSTEMS

1/ Proportional controller

$H\left(s\right)=\frac{K}{{s}^{2}+\text{101}s+\text{100}+K}$

Recall the step response of the position control system with proportional controller.

The poles of the closed-loop system function are at

${s}_{1,2}=-\frac{\text{101}}{2}±{\left[{\left[\frac{\text{101}}{2}\right]}^{2}-\text{100}-K\right]}^{1/2}$

The root locus plot is

2/ Integral controller

$H\left(s\right)=\frac{K}{{s}^{3}+\text{101}{s}^{2}+\text{100}s+K}$

Recall the step response of the position control system with integral controller.

The poles are the roots of the polynomial

${s}^{3}+\text{101}{s}^{2}+\text{100}s+K=0$

The root locus plot is

Note that all the poles are in the lhp for K>0 and 100>K/101 or 0<K<10100. For K>10100 two poles move into the rhp and the system is unstable.

VII. STABILIZATION OF UNSTABLE SYSTEMS

1/ Many common systems are unstable

Some common systems are annoyingly unstable (adapted from Figure 11.7 in Oppenheim&Willsky, 1983).

We can model the audio feedback system with SIMULINK as follows.

Another example of an unstable system is an inverted pendulum. For example, balancing a broomstick in your hand is an example of an inherently unstable system that is stabilized by your motor control system.

Figure adapted from Figure 11.2 in Oppenheim&Willsky, 1983.

2/ Inverted pendulum

We will analyze an inverted pendulum attached to a cart. A schematic diagram is shown on the left and a free-body diagram showing the forces on the pendulum is shown on the right.

The forces of attachment of the cart and pendulum on the pendulum are obtained from the equations of rectilinear motion.

The equation of rotational motion about the center of mass is

$J\frac{{d}^{2}\theta \left(t\right)}{{\text{dt}}^{2}}=\text{mgl}\text{sin}\theta \left(t\right)-\text{ml}\text{cos}\theta \left(t\right)\frac{{d}^{2}x\left(t\right)}{{\text{dt}}^{2}}$

where J is the moment of inertia of the mass about the mass, m is the mass, and g is acceleration of gravity. For small θ, this differential equation is linearized by noting that sinθ≈θ and cos θ ≈ 1. Therefore,

$J\frac{{d}^{2}\theta \left(t\right)}{{\text{dt}}^{2}}-\text{mgl}\text{sin}\theta \left(t\right)=-\text{ml}\frac{{d}^{2}x\left(t\right)}{{\text{dt}}^{2}}$

The system function is

$H\left(s\right)=\frac{\theta \left(s\right)}{X\left(s\right)}=\frac{-\left(\text{ml}/J\right){s}^{2}}{{s}^{2}-\left(\text{mgl}/J\right)}$

Therefore, the poles occur at

${s}_{1,2}=±{\left[\frac{\text{mgl}}{J}\right]}^{1/2}$

Hence, the system is inherently unstable since one of its poles (natural frequencies) is in the rhp.

3/ Stabilization of the inverted pendulum

To stabilize the inverted pendulum, a rotary potentiometer is used to measure θ(t). A current proportional to θ(t) − θo, where θo is the desired angle, drives a motor so as to increase x(t) (adapted from Figure 6.4-1, Siebert, 1986).

This inverted pendulum is connected in a feedback configuration as follows.

M(s) represents the system function of the motor dynamics, 1 + (a/s) is the system function of the proportional plus integral controller for θ(t), and c + bs is the system function of the proportional plus derivative controller for x(t).

Demo of stabilization of an inverted pendulum.

4/ Stabilize by pole cancellation?

Why not cascade an unstable system with another system that cancels the unstable pole?

• Perfect cancellation is very difficult.
• The unstable pole can be excited by other inputs.

VIII. CONCLUSIONS

• Interconnection of systems requires attention to their interactions.
• Feedback is a powerful method to improve the performance of systems. However, feedback systems have the capacity to become unstable.
• Instability can be determined by examining the poles of the closed-loop transfer function. It may also be important to examine the stability of components of the closed-loop system. Inherently unstable systems can be stabilized with feedback.

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
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Got questions? Join the online conversation and get instant answers!