Video 1.5 - Control Design Basics

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Table of Contents:

• C.2) Control Design Basics - Comparing Control Designs (🎦 video 1.5)
  • C.2.1) Attempt 1 - Bang-Bang Controller
  • C.2.2) Attempt 2 - P-Regulator
    • C.2.2.1) P-Regulator: Stability but Not Tracking

C.2) Control Design Basics (Comparing Control Designs)

Let's continue with our "Cruise Controller" example and design a few controllers and compare them.

We have a "Car Model",

And we want the car to match the reference speed,

Note: In control design, we talk about "Asymptotic Properties", than means when $t\to \mathrm{\infty }$

which implies that,

We can also see it in the block diagram,

C.2.1) Attempt 1 - Bang-Bang Controller

Is this control strategy going to work?

Let's run a simulation with a reference speed r = 70

It works beautifully!
The system is stable, and we have reached the desired speed without any overshoot.

This type of control is known as "Bang-Bang Control" because we are switching between 2 extremes (full gas and full brake)

Are we done? Not really, let's see the control effort with our control signal

This is BAD!
First we accelerate until we reach the speed of 70 and then we start switching wildly between Umax and -Umax

Why? Because the system is to sensitive to any change in the measured speed and it will reach with full gas or full brake to compensate.

This will be a Bumpy ride and we will burn the actuators

C.2.2) Attempt 2 - P-Regulator

This control design solves our previous problem, because a small error yields a small control signal. It gives a nice and smooth behavior.

It is known as a "P-Regulator", where "P" stands for "proportional" because the input "u" is directly proportional to the error through a controlled gain "kp"

Is this control strategy going to work?

Let's run a simulation with a reference speed r = 70

It is nice, smooth and stable!

But there is a problem, the car never reaches a velocity of 70.
Even though the response is smooth we end up having Steady State Error, because we don't reach the reference.

Why did this happen?
Well, the problem is actually in our car model. We forgot an important force that affects cars: Wind Resistance

C.2.2.1) P-Regulator: Stability but Not Tracking

The "real" model is now augmented to include a wind-resistance term,

This is the model used in the previous simulations, and it is the reason why the P- Regulator failed with our Tracking Objective.

Let's analyse our model at "steady-state", in other words, when $x$ doesn't change any more,  $\dot{x}=0$

Remember that,

Therefore,

Now we can solve for $x$ and see why we have Steady State Error,

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Z) 🗃️ Glossary

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