# Examples of closed loop functions

• CraigH
In summary, closed loop functions are those where the line joins back to itself, and examples include the equation for a circle, y^2 + x^2 = r^2. They can also be represented in the form of x=x(t), y=y(t) with a pair of values t0 and t1, where x(t0)=x(t1) and y(t0)=y(t1). It is not possible to determine if a function is a closed loop without plotting it or knowing the pair of values. Closed loop functions can also be expressed as z=f(x,y) instead of y=f(x), such as z=x^2+y^2.
CraigH
Can someone please give me a list of examples of closed loop functions, the only one I know is the equation for a circle

$y^2 + x^2 = r^2$

Also are there any closed loop functions that aren't multi variable, i.e in the form y=f(x) and not z=f(x,y)

Is there a way to tell that a function is a closed loop without plotting it?

Thanks!

PS, I'm not asking about closed loop integrals here, just functions where the line joins back to itself.

CraigH said:
Also are there any closed loop functions that aren't multi variable, i.e in the form y=f(x) and not z=f(x,y)

Not in form of y=f(x), since will be multiple values.

More general is to present it by pair of functions x=x(t), y=y(t)

z=x^2+y^2 may be represented as x=sin(t), y=cos(t)

Is there a way to tell that a function is a closed loop without plotting it?

For any pair of functions x(t) and y(t), with pair of values t0, t1,
such that x(t0)=x(t1) and y(t0)=y(t1),

there is a loop of length t1 - t0

Ah okay thank you, this has cleared a few things up, but what do you mean by:
Graniar said:
For any pair of functions x(t) and y(t), with pair of values t0, t1,
such that x(t0)=x(t1) and y(t0)=y(t1),

there is a loop of length t1 - t0

This confuses me, do you mean y=f(t) and x=f(t)? And what do you mean by pair values?

For example, will take that z=x^2+y^2 <=> x=sin(t), y=cos(t)

Pair of functions: x=sin(t), y=cos(t)
Pair of values: t0=0, t1=2*pi

sin(t0) = sin(0) = 0 = sin(2*pi) = sin(t1)

cos(t0) = cos(0) = 1 = cos(2*pi) = cos(t1)

I can provide you with some examples of closed loop functions. Apart from the equation for a circle, some other examples include:

1. Ellipse: (x/a)^2 + (y/b)^2 = 1, where a and b are constants representing the major and minor axes.

2. Hyperbola: (x/a)^2 - (y/b)^2 = 1, where a and b are constants.

3. Lemniscate: (x^2 + y^2)^2 = a^2(x^2 - y^2), where a is a constant.

4. Cardioid: (x^2 + y^2) = a^2(1 + cosθ), where a is a constant and θ is the angle.

5. Torus: (a - √(x^2 + y^2))^2 + z^2 = b^2, where a and b are constants representing the radii of the inner and outer circles.

It is also possible to have closed loop functions in the form of y = f(x) that are not multi-variable. For example, the function y = sin(x) has a closed loop when plotted on a graph.

There are a few ways to tell if a function is a closed loop without plotting it. One way is to look at the equation and see if it contains terms that can create a closed loop, such as squares or trigonometric functions. Another way is to analyze the behavior of the function for different values of x. If the function repeats itself at certain intervals, it is likely to be a closed loop. Additionally, you can also try to manipulate the equation to see if it can be rearranged into a familiar closed loop form.

## 1. What is a closed loop function?

A closed loop function, also known as a feedback loop, is a control system in which the output of the system is fed back into the input to regulate and maintain a desired state or behavior.

## 2. What are some examples of closed loop functions in daily life?

Examples of closed loop functions in daily life include thermostats for regulating temperature, cruise control in cars, and homeostasis in living organisms.

## 3. How do closed loop functions differ from open loop functions?

Closed loop functions have a feedback mechanism, while open loop functions do not. This means that closed loop functions can adjust their output based on the input, while open loop functions have a predetermined output regardless of the input.

## 4. What are the advantages of using closed loop functions?

Closed loop functions allow for more precise control and regulation, as the system can adjust its output based on the input. They also tend to be more stable and resistant to external disturbances.

## 5. Can closed loop functions be found in other fields besides engineering?

Yes, closed loop functions can also be found in fields such as biology, psychology, and economics. They are used to explain and understand complex systems and behaviors in these fields.

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