Video 1.2 - What is Control Theory
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#Physics #Engineering #Control_Theory #RoboticsTable of Contents:
A) What is Control Theory anyway?
Control theory is an interdisciplinary field of engineering and mathematics, which has to do with the behavior of Dynamic Systems. The input of a system is called a reference. When one or more output variables of a system need to follow a certain reference over time, a controller manipulates the input to the system to obtain the desired effect on the output of the system feedback).
So remember,
- System: it is something that changes over time.
For example: Robots, Epidemics, stock markets, thermostats, circuits, Engines, Power grids, autopilots, etc.
In the study of dynamic systems, which are systems that change over time, there's a fascinating concept called the Lorenz attractor.
The Lorenz attractor is a set of chaotic solutions to the Lorenz oscillator equations (a set of equations by Edward Lorenz in 1963). When visualized, the Lorenz attractor looks like a butterfly or figure-eight. It's one of the most famous images associated with chaos theory.
This system shows that even simple equations can lead to unpredictable, chaotic behavior. It's a prime example of the "butterfly effect," where tiny changes can lead to vastly different outcomes!
- Control: it deals with how can we influence the change of a system.
The centrifugal governor, a device with spinning balls used to regulate engine speed, was a brilliant invention by James Watt for his steam engines. It's one of the earliest examples of a feedback control system!
When the engine runs too fast, the balls rise and reduce the steam input, slowing it down. If it's too slow, the balls drop, allowing more steam and speeding it up. This self-correcting mechanism laid the foundation for modern control systems, showcasing how systems can adjust themselves to maintain a desired state!
A.1) The Basic Building Blocks of a Control System
Let's start with trying to build up a "Control System" in terms of the basic needed building blocks.
A.1.1) The State of the System (x)
1st) We need some way of describing what the system is doing or where it is (in its behaviors. To do so, we define the "state" of the dynamic system.
The State is a representation of what the system is currently doing.
And we are going to use an "x" to describe what the state of the system is.
The State could be:
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The position and velocity of a robot
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The percentage of people infected by certain epidemic.
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etc
A.1.2) The Dynamics of the System
2nd) Now that we know what the state of a system is, the description of the change of the state as a function of time is known as the "dynamics" of the system.
It tells us what the system is actually doing.
The Dynamics is a description of how the state changes
A.1.3) The Reference Signal (r)
3th) Now that we have the building blocks of a dynamical system, we want some way of influencing its behavior so that it does what we want. We are going to introduce a "reference signal in order to tell the system what it is we want it to do.
The Reference is what we want the system to do.
And we are going to use an "r" for the reference
The reference could be:
- setting the cruis controller to 60 mph
- making a certain amount of money in the stock market
- make the temperature of a room 70° F
- etc
A.1.4) The Output of the System (y)
4th) How can I know if the behavior of my system is close to the reference signal? We need to measure the "output" of the system, now we can compare.
The Output is the measurement of (some aspects of) the system
And we are going to use a "y" for the output.
Note: Since we can't always measure the "state". The "outputs" are things that we are able to get out of the system, things we can measure.
A.1.5) The System Input (u)
5th) Now we need some way of mapping "reference signals" into actual "control signals" (also known as "inputs").
The Input is the control signal
And we are going to use "u" to for the input which is going to take the reference and produce a control signal that then hits the state of the system.
A.1.6) The Feedback and the Error Signal
6th) A good control design uses the measurements of the system in order to produce a proper control signal. So we need a final building block known as "feedback"
The Feedback is the mapping from outputs to inputs
We can take the difference between the reference signal and the output signal and produce an “ error signal", which can be translated into a control signal.
This is known as a "Closed loop system"
A.1.7) Summary - The Basic Building Blocks
Z) 🗃️ Glossary
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