Video 1.6 - Performance Objectives

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#Physics #Engineering #Control_Theory #Robotics

Table of Contents:

C.2.2.2) Control Theory: Recall the "Performance Objectives" (🎦 video 1.6)
- C.2.3) Attempt 3 - Forcing a Tracking Controller
- C.2.4) Attempt 4 - PI-Regulator

C.2.2.2) Control Theory: Recall the "Performance Objectives"

Let's recall the first three "Performance Objectives" we have in control and see how we can improve our "P-Regulator"

A controller should provide:

Stability to the system - also known as BIBO Stability

In other words, if the input is reasonable, our system doesn't blow up.

Our P-Regulator can do this, so that is a win!

Tracking - means we should get to the reference value that we want (or at least close enough, within a 2% error).

Our "P-Regulator" fails at this because the error at steady state is too much.

The reference is at 70 and we can only achieve 58 due to wind-resistance.
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Robustness - means we shouldn't have to know much about parameters that we really have no way of knowing. And preferably we should be able to fight noise as well.

So, let's try to achieve "Tracking" with a new controller.

C.2.3) Attempt 3 - Forcing a Tracking Controller

What is this new term doing? Let's analyse at steady state, $\dot{x} = 0$ ,

Remember that,

Therefore, if we substitute for $u$ ,

We have lost "Robustness" with this control approach. Just take a look at our controller "u"

All of a sudden we have to know all these physical parameters that we don't know, so this is not a robust control design.

Let's go back to our "P-Regulator" and see what we can learn.

What is actually happening is that the proportional error is doing a fine job pushing the system up to close to where it should be, but then it can't push hard enough to overcome the effect of wind-resistance.

The error starts to accumulate over time. If we could "collect" all these errors over time and use them in our controller, we would eventually be able to speed up the car.

Spoiler Alert

We can use an "integral" and integrate $e (t)$ in an interval of time, let's say from $τ = 0$ (when we turn on the controller) to $τ = t$ (the current time)

C.2.4) Attempt 4 - PI-Regulator

Note

The PI-Regulator is one of the most common regulators found anywhere in the world. In fact, it's almost 2/3 of all commercial grade cruise controllers

Let's see its time response,

It works perfectly!

The system behaves quickly, nice and smoothly.

The car reaches the reference of 70 mph.

It is stable, it can track the reference and it is robust since we don't need to know the parameters of the car.

Note

PI regulators can induce unwanted oscillations in some systems, that is why we also have:
PID - Regulators

This is an extremly useful controller that shows up a lot. It is a great control structure and we are going to get quite good at designing PID regulators.

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