# 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:

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

what is the meaning of money and inflation
what is a bar chart
what's economic
what is the meaning of imperfect market structure
the theroy of demand
Tinuke
what is market economy
Market Economy:Is a system where the laws of supply and those demand direct the production of goods and services.
Iyabo
what is the meaning of money and inflation
Tinuke
what's demand in economics?
Demand in economic is the good a consumer is willing or able to purchase at a particular time
Hoyindamolah
explanation of graph,bar chart,pie chart with examples
Abdulmojeed
4. Assume, after completing your economics class, you explain your friend that about 65% of GDP is spending on consumption. Your friend tells you that people are greedy and it is better for GDP if they spend on services or experiences. What would be your answer to your friend?
What is power
how can I choose a model for analyzing primary data?
economics is science because it used sciencetific ways in solving it problem
Mhizz
Is economics a science, more expanciate
yh because it adopts scientific methods in research and analysis of economic issues
maame
Analyse Eskom in terms of the characteristics of a monopoly
microeconomics deals with individuals and firms while macroeconomics deal with the activities of the society in large
Morning guys I need questions under any topic in economics
C-Stixxs
Shalom
maame
why Economics is science
Ok
Olamilekan
what is economic
is a science which studies human behavior as a relationship with ends n scarce which have alternative uses
C-Stixxs
Explain four reason why scale of preference is necessery
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Got questions? Join the online conversation and get instant answers!