# 0.9 Dynamical systems

 Page 1 / 1
Systems with memory

## "what is a dynamical system?"

When we talk about systems in the most general sense, we are talking about anything that takes in a certain number of inputsand produces a certain number of outputs based on those inputs.

In the figure above, the $u(t)$ inputs could be the jets on asatellite and the $y(t)$ outputs could be the gyros describing the"bearing" of the satellite.

There are two basic divisions of systems: static and dynamic. In a static system, the current outputs are based solely on the instantaneous values of the current inputs.An example of a static system is a resistor hooked up to a current source:

$V(t)=Ri(t)$

At any given moment, the voltage across the resistor (the output) depends only on the value of the current runningthrough it (the input). The current at any time $t$ is simply multiplied by the constant value describing the resistance $R$ to give the voltage $V$ . Now, let's see what happens if we replace the resistorwith a capacitor.

$I(t)=C\frac{d v(t)}{d t}}$

Solving for the voltage in the current voltage relationship above, we have:

$v(t)-v({t}_{0})=\frac{1}{C}\int_{{t}_{0}}^{t} i(t)\,d t$

So in the case of the capacitor, the output voltage depends on the history of the current flowing through it. In a sense, thissystem has memory. When a system depends on the present and past input, it is said to be a dynamical system.

## "describing dynamical systems"

As seen in voltage-current relationship of a capacitor, differential equations have memory and can thus be used todescribe dynamical systems. Take the following RLC circuit as an example:

In circuits (as well as in other applications), memory elements can be thought of as energy storage elements. In this circuitdiagram, there are two energy-storing components: the capacitor and the inductor. Since there are two memory elements, it makessense that the differential equation describing this system is second order.

$\frac{d^{2}y(t)}{dt^{2}}+\frac{7}{2}\frac{d^{1}y(t)}{dt^{1}}+9y(t)=6u(t)$

In the most general case of describing a system with differential equations, higher order derivatives of outputvariables can be described as functions of lower order derivatives of the output variables and some derivatives of theinput variables. Note that by saying "function" we make no assumptions about linearity or time-invariance.

By simply rearranging the equation for the RLC circuit above, we can show that that system is in fact covered by this general relationship.

Of course, dynamical systems are not limited to electrical circuits. Any system whose output depends on current and pastinputs is a valid dynamical system. Take for example, the following scenario of relating a satellite's position to itsinputs thrusters.

## "planar orbit satellite"

Using a simple model of a satellite, we can say that its position is controlled by a radial thruster ${u}_{r}$ , which contributes to its vertical motion, and a tangential thruster ${u}_{}$ which contributes to its motion tangential to its orbit. To simplify the analysis, let's assume that the satellite circles the earth in a planar orbit, and thatits position is described by the distance r from the satellite to the center of the Earth and theangleas shown in the figure.

Using the laws of motion, the following set of differential equations can be deduced:

$\frac{d^{2}r(t)}{dt^{2}}-\frac{d^{1}r(t)}{dt^{1}}^{2}={u}_{r}-\frac{k}{r^{2}}$
$2\frac{d^{1}r(t)}{dt^{1}}\frac{d^{1}(t)}{dt^{1}}+r\frac{d^{1}(t)}{dt^{1}}={u}_{}$

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
Do somebody tell me a best nano engineering book for beginners?
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
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!