Video 1.4 - Cruise Controller (Example)

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Table of Contents:

• C) Cruise Controller - Example (🎦 video 1.4)
  • C.1) Defining the Model for a Car
  • C.2) Defining the Control Signal

C) Cruise Controller (Example)

Now that we have a way of describing "Dynamical systems" with "Differential Equations" in "Continuous Time", or "Difference Equations" in "Discrete Time”.

Let's do something interesting and build a "Cruise Controller" for a car.

The job of a "Cruise Controller" is to make a car drive at a desired speed "r"

C.1) Defining the Model for a Car

We can model the car using the Laws of Physics, and we can use Newton's 2nd Law to describe how a car (which has mass) behave.

Now, *what is the state of the system?
We need to somehow relate Newton's 2nd Law to the system and its state.

In this case, since we want the "Cruise Controller" to control the speed of the car, we can say that the state $x$ is the speed or velocity of the car.

But how does this relate with Newton's 2nd Law?

From the equation,

We know that the acceleration is the derivative of velocity (the derivative of our state "$x$" ).

The equation, $F=ma$ describes how the system (the car) behaves given a specific force "F".
So we can use "F" to control the speed of the car.

We can define F as,

with this equation we are mapping "stepping on the gas or the brake" onto the "force" that
is applied to the car. And this is done through some linear relationship.

As it was shown before "c" is the electro-mechanical transmission coefficient.

But…

• what happens if we don't know its value?
• can we still design a controller?
• or what happens if we design a controller for a specific car, will it work with another car with a different "c" value?

Remember, our control design cannot rely on us knowing "c", otherwise it won't be Robust. We can't have our controller be too highly dependent on what the particular parameters of a model are.

The same argument applies to the mass "m" of the car, the number of passengers can change, etc).

Let's now write the differential equation of our system.

We know that,

And if we equate the 2 equation of force that we have we get,

C.2) Defining the Control Signal

1. Assume we can measure the velocity of the car.

2. The control signal should be a function of the error signal "e"

3. Question: What properties should the control signal have for our application case?

• It should not overreact, in other words: small "e" gives small "u"
  

• "u" should not be "jerky", in other word,"u"should not vary too rapidly all the time.
  

• "u" should not depend on us knowing "c" and "m" exactly.
  

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Z) 🗃️ Glossary

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