In this lecture we're going to be a little bit more precise about what we mean when

we say control theory, and I've been throwing around this term controls quite a

bit so far, even though we've only seen one lecture. and what I want to do now is

make this a little bit more precise so that we get a slightly better

understanding ofwhat it is that we're going to be doing in the class. And,

ultimately controls, it deal with dynamical systems. These are abstractions

that describe something that changes over time. This could be a car that's moving.

This could be the price of a particular stock. But ultimately, controls is dealing

with how can we best or how can we influence this change of the system. And

some examples, and we're going to see all of these examples in the class of systems

that one can control, would be robots, epidemics, you could come up with

vaccination strategies, stock markets, thermostat's for keeping the temperature

pleasant indoors, electrical circuits, dc engines, power grids, or autopilots on a

aircraft and all of these will show up and see which of these are easy to control and

which of these are actually very hard to control. And we're also going to

understand what makes certain systems harder to control than others. I put

robots in red though because this at the end of the day is the main focus of the

course from the point of view. So let's start with trying to build up a

control system in terms of the basic needed building blocks. And the first

thing you need is some way of describing what the system is doing or, more

importantly, where it is. If I want to control a car, let's say I want to build a

self-driving car that drives from my home to my office, well I need to know where

the car is, and central to this is the notion of a state. The state represents

what the system is currently doing, what state that it's in, and we're going to use

x to describe what the state of the system is. This could be the position or the

velocity of a, of a, robot, this could be the p ercentage of people that are

infected by a certain, epidemic, this could be a number of different things, but

ultimately the state is the key thing that describes what the system is up to. And

what it is actually doing is the dynamics. And the dynamics is the description of the

change of the state as a function of time. Now, this is all good. But, we want some

way of influencing this. So, we're going to have a reference signal that is going

to. We're going to use it as a way of telling the system what it is that we want

it to do. So the reference could be set the cruise controller to 60 miles per

hour, or make me a certain amount of money on the stock market, or make the

temperature in the room 70 degrees. What do I know? now we can do that all we want,

but let's say that we want a cruise controller to go at 60 miles per hour,

that's not going to work until we can actually measure how fast the system is

going. So we also need an output. So we're going to use r for the reference, y for

the output. And out-, the outputs are the things that we're able to get out of the

system. This is telling us what the system is doing. So we can't always measure the

state y is the output. Now this picture is actually a blatant lie, because if I tell

the stock market to go make me a certain amount of money there's no way it's going

to do that. If I just you know yell 65 at my car, it's not going to go 65 miles per

hour so we need some way of mapping reference signals into actual control

signals, the inputs. So u is going to be the thing that takes the reference and

produces a control signal that then hits the state of the system. Now this is all

fine and good but this control design is not particularly good because the control

signal has nothing to do with the measurements. So we need the final

building block which is the feedback. This is a mapping from outputs to inputs in the

sense that what we're doing is we're taking Y here. Oh, sorry.

We're taking y here, let's say y, and then we're taking the refer ence and out here

comes the reference minus y, which is going to be the error in terms of how the

system is performance. And this error is translating into some control signal

that's then hitting the system. So this feedback mapping is really the key to

doing any kind of controls in an effective way. Good.

So, now that we know a little bit about what are the basic building blocks. I want

to talk about examples just so we can try to understand what these building blocks

actually represent. So, the state of a robot is typically, if it's a mobile robot

on the ground, it's the position, maybe the orientation of the robot, maybe the

velocity of the robot. The state is ultimately what we need to describe what

it's doing. If it's a manipulator robot, it would be the angles, the joint angles

of the various, segments of the, of the robot. so that would be the state. The

dynamic simply says how the robot is moving. The control signal, or the control

input, would be things like velocities or torques or accelerations. Some ways of

moving the robots around, and the measurements would be things like, where

is the robot? What is it seeing in the environment? The epidemics, like I said a

state would be maybe what percentage of the population is infected? The thing we

can control, so the input would be vaccination strategies. In this case, the

output is tricky. It;s very hard to know what percentage of a population is

infected for instance with, with an infection? So, there, the output would be

some kind of measurements at hospitals of how many people show up with this

particular disease. stockmarket state could be, in fact no one really knows what

the state is of this stock market which is why impart so hard to make money. the

output would be, well, the prices of the stock and the input is simply buy or sell.

Thermostats, temperature, outputs temperature, the thing we control is

turning things on and off. Circuits typically the state would be currents

voltages and, and currents, and the inputs are typically voltages that you're

applying and the outputs are measured currents of voltages in the circuits.

Engines you can apply, typically voltages and out comes a torque. the power grid,

well the state there somehow is power consumption across the different modules.

The things you can control are putting loads and generators onto the grid, maybe

same thing with autopilots on aircraft, the states would be positions and

velocities, the things you can control are accelerations or change the, the control

surfaces of the aircraft, and the measurements are a whole bunch of things,

how fast is it going, where is it going, what's the altitude? What's the wind

resistance and things like that. So, what we need to do is come up with a rather

general way of describing all of these systems, in a way that makes sense, that

we can use. And, that more importantly, can use to control mobile robots.

What's Control Theory, Anyway?

Control of Mobile Robots

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