title	Control Theory
textbook	#control-theory

control theory

Design of Autonomous Systems

csci 6907/4907-Section 86

Prof. Sibin Mohan

consider a simple problem → how do you balance a ball?

Note:

discuss pen balance
show example in class
what is the "input"?
what is the "output"?
what is the "desired state"?

consider a simple problem → how do you balance a ball?

ok, that's a bit hard!

let's simplify → in a one-dimensional plane?

balance a ball → in the middle of a table

pretty good attempt but unstable!

goals

the ball remains stable and
it is in the middle of the table

options

tilt the table down on the left (anti-clockwise)
title the table down on the right (clockwise)

options

tilt the table down on the left (anti-clockwise)
title the table down on the right (clockwise)

options

tilt the table down on the left (anti-clockwise)
title the table down on the right (clockwise)
control the speed at which the table tilts

parameters for the problem

type	options
inputs	speed (clockwise, anticlockwise)

parameters for the problem

type	options
inputs	speed (clockwise, anticlockwise)
output	ball velocity, acceleration

we need to control outputs → modifying inputs to system

control theory

multidisciplinary field → applied mathematics+engineering

control theory

multidisciplinary field → applied mathematics+engineering
wide use → e.g., mechanical, aerospace, electrical, chemical, etc.

control theory

multidisciplinary field → applied mathematics+engineering
wide use → e.g., mechanical, aerospace, electrical, chemical, etc.
even biological sciences, finance, you name it!

anything that you,

want to control and

anything that you,

want to control and
can develop a model

anything that you,

want to control and
can develop a model

develop a controller → using control theory

lots of everyday applications as well

adaptive cruise control, thermostats, ovens, lawn sprinkler systems, etc.

control theory | basic idea

understand a process or a system → develop a model

relationships between → inputs and outputs

control theory | model

adjust the inputs :model → to get desired outputs

relationship between inputs and outputs → empirical analysis

make changes to the input

relationship between inputs and outputs → empirical analysis

make changes to the input
wait for the system to respond

relationship between inputs and outputs → empirical analysis

make changes to the input
wait for the system to respond
observe changes in the output

Note:

Even if the model is based on an equation from physics, the parameters within the model are still identified experimentally or through computer simulations.

this is not what we really want!

so, what is is that we want?

what is the objective?

e.g., balance the ball

what is the objective?

e.g., balance the ball → control the output

control the output → tune the input!

simple example | lightbulb

assume we don't know relationship between bulb and switch

conduct a few experiments → capture the relationship

results of empirical analysis

switch state (input)	bulb state (output)
off	off
on	on

results of empirical analysis

switch state (input)	bulb state (output)
off	off
on	on

"model" of input (switch state) → output (ligthbulb state)

results of empirical analysis

switch state (input)	bulb state (output)
off	off
on	on

"model" of input (switch state) → output (ligthbulb state)

not the control model

why?

we care about → if bulb is on (or off)

not if switch is on (or off)

so to develop a control model → invert above relationship

control model | lightbulb

desired output lightbulb state	corresponding input switch state

control model | lightbulb

desired output lightbulb state	corresponding input switch state
on	on
off	off

let's formalize things a little

consider following mathematical model

Note:

The model says that if we change the input u the output y will change to be twice the value of the input u.

consider following mathematical model

remember → we want to get to a specific output, say $y^*$

manipulate model → to get control model

$$u = \frac{y^*}{2}$$

for any desired value of $y^*$ → can identify input u

$$u = \frac{y^*}{2}$$

for any desired value of $y^*$ → can identify input u

we have jus designed our first controller!

desired value, $y^*$ → setpoint

course map

Open-Loop vs Closed-Loop Control

if our model is accurate+no disturbances

$$ y = y^*$$

if our model is accurate+no disturbances

$$ y = y^*$$

nothing guarantees this

open loop controller

desire certain outcome → hope controller actually gets there

open loop controller

what we really want → ensure controllers gets to setpoint

Note:

problem is that while the input drives the output, there is no way to guarantee that the controller will get to the set point

the solution → feedback

from output → input

closed loop controller

adjust u

closed loop controller

adjust u
ensure that we get to $y^*$

closed loop controller

adjust u
ensure that we get to $y^*$

(or, at least as close to it as possible)

closed loop controller

feedback from → output of controller model we created

closed loop controller

feedback from → output of controller model we created

closed loop controller

feedback from → output of controller model we created

closed loop controller

feedback from → output of controller model we created

feedback can be positive or negative

how to ensure car remains in center of lane?

apply a correction to direction of motion

how do we apply the corrections?
how much?
when do we stop?

Feedback Control

compare system state to the desired state

Feedback Control

compare system state to the desired state
apply a change to system inputs → counteract deviations

Feedback Control

compare system state to the desired state
apply a change to system inputs → counteract deviations
repeat until desired outcome → setpoint

e.g., temperature control of a room

Note:

thermostat needs to control/provide inputs to a furnace/AC,

e.g., temperature control of a room

Note:

which then affects the temperature in the room:

e.g., temperature control of a room

so we're...done?

real world has disturbances

disturbances → heat loss, bad insulation, physical problems, etc.

input not sufficient to achieve set point

provide "feedback" to controller

Note:

Essentially the temperature reading of the room, after the thermostat and furnace/AC have completed their operations based on the original inputs (desired temperature).

some technical terms

Note: -The "controller" is based on the "control model" that we developed earlier. It sends commands ("actuation signals") to an actuator and then affects the process under control. Finally, the process variable (the "output" from the earlier discussions) is what we want to drive towards the set point.

another example → cruise control

note → how the feedback reaches controller

closed-loop feedback control system

some of these inputs/edges have specific names

main goal → error is minimized (ideally 0)

quantity	definition
$r(t)$	reference/set point
$e(t)$	error
$u(t)$	control signal/"input"
$y(t)$	(expected/final) output
$\overline{y(t)}$	"feedback"/estimate

Feedback Control for Lane Following

we want to keep car → center of its lane

question → how do you find the center of lane?

consider road with lane markings on either side,

assume → can track white lines

need to find the center of the lane, as marked in the figure:

$$x_{center} = \frac{x_{left-end}+x_{right-end}}{2}$$

car need not be in actual center of the lane,

assuming camera is mounted → center of car,

car's position: $x_{car} = \frac{width}{2}$

we calculate the cross-track error (CTE),

$$CTE = x_{car} - x_{center}$$

what happens when,

$CTE > 0$
$CTE < 0$?

keeping the car in center of lane,

CTE → as small as possible and
applying corrections

$64,000 question → how?

feedback control

problem statement:

given the CTE, how do we compute the control signal
so the car stays in the middle of the lane?

final "corrections", when applied, may look something like:

start with one goal → lateral position control


process variable	$\textbf{y(t)}$	$y$ position at time, $t$

start with one goal → lateral position control


process variable	$\textbf{y(t)}$	$y$ position at time, $t$
goal	$y = 0$	keep the car at position 0

start with one goal → lateral position control


process variable	$\textbf{y(t)}$	$y$ position at time, $t$
goal	$y = 0$	keep the car at position 0
control signal	$u(t)$	steering

say we have the car's start and end positions,

we know relationship between → $u(t)$ and $y(t)$

we want $u(t)$ to be negative → $y$ tends towards eventual goal, $y = 0$.

so, what should be our control input?

$$e(t) = ?$$

input → decreasing value of the feedback

$$e(t) = -y(t)$$

Proportional (P) Control

correction is proportional to size of error, i.e.,

Proportional (P) Control

correction is proportional to size of error, i.e.,

apply proportional control → lateral control

following choices:

$K_p > 0$
$K_p < 0$

apply proportional control → lateral control

following choices:

$K_p > 0$
$K_p < 0$

following from $e(t) = -y(t)$,

$$ K_p e(t) = - K_p y(t)$$

car moves towards reference/goal, $y=0$

let's consider a few situations:

what if $K_p$ → too small (small "gain")?

what if $K_p$ → too small (small "gain")?

what if $K_p$ → too small (small "gain")?

response is too slow/gradual → may never reach goal!

what if $K_p$ → too large (large "gain")?

what if $K_p$ → too large (large "gain")?

what if $K_p$ → too large (large "gain")?

response is too sudden → system may overshoot the goal!

so, can the car be stabilized at $y=0$?

unlikely using only proportional control method since,

gain	effect
small	stead-state error
large	oscillations

how can we reduce oscillations

Note: can we get to the following situation (smoother, actual approach to the goal)?

Derivative (D) Control

improves the dynamic response of system by,

Derivative (D) Control

improves the dynamic response of system by,

studying the rate of change of the error and
decreasing oscillation

Derivative (D) Control

improves the dynamic response of system

Derivative (D) Control

improves the dynamic response of system

measuring and trying to control → the rate of change

$$y(t-1) \rightarrow y(t)$$

car actually get close to reference/goal, $y=0$

proportional/derivative control used together

counteract each others' influences

proportional+derivative controller

two options for derivative "gain":

$\frac{dy(t)}{dt} < 0$
$\frac{dy(t)}{dt} > 0$

derivative controller's job:

steer away from the reference line
act as a counter to proportional controller

$$\frac{dy(t)}{dt} < 0$$

derivative controller acts like a brake and counteracts correctional force

Note:

derivative controller acts like a brake and counteracts the correctional force. It reduces overshoot by slowing the correctional factor → as the reference goal approaches.
We see numerous uses of the combination, known as the P-D controllers in every day life, from the smallest to the largest, even rocket ships!

Tuning P-D Controllers

need to pick the right values for the gains, $K_p$ and $K_d$

Tuning P-D Controllers

consider following values for two coefficients, $K_p$ and $K_d$:

Tuning P-D Controllers

consider following values for two coefficients, $K_p$ and $K_d$:

tuning $K_d$ has a significant impact!

Note:

As we see, tuning $K_d$ has a significant impact! The oscillations pretty much go away and we quickly get to the reference line with very little oscillation. Of course, the car overshoots a little but the combination of P-D brings it back soon enough.

Tuning P-D Controllers

what if we increase $K_d$ → making it very large?

Tuning P-D Controllers

what if we increase $K_d$ → making it very large?

can improve or make things worse → drive car away from goal!

Note:

While we see from the first graph ($K_p = 0.2$, $K_d = 4.0$) that the oscillations have gone away, increasing $K_d$ further make

fix → tuning paramters, e.g., K_p = 3.0

before	after

a quick, "smooth" path to the reference!

Note:

In fact, a lot of the design of control systems involves the tuning of such parameters, depending on the system, to get things "just right".

are we done?

let's take a closer look at the results:

Note:

As we see from this image, even though we reached the reference, the behavior is not smooth!

many reasons, e.g., steering drift

unmodeled disturbances and systemic errors ("bias")

actuators and processes → not ideal

unmodeled disturbances and systemic errors ("bias")

actuators and processes → not ideal
friction, steering, drift, changing workloads, misalignments, etc.

signal may never reach set points!

may end up "settling" near reference, which is not ideal

Integral (I) Control

define steady state error (SSE):

difference between the reference and the steady-state process variable

Integral (I) Control

when time goes to infinity,

$$SSE = \lim_{x\to\infty} [r(t) - y(t)]$$

Integral (I) Control

applied correction must be,

proportional to error and duration of error
essentially it sums the error over time

Integral (I) Control

unless the error is zero, this term will grow!


error adds up correction must also increase drives steady state error to zero

unless the error is zero, this term will grow!


error adds up correction must also increase drives steady state error to zero

I control used in conjunction with P/D controllers

PID Control

examples of tuning PID parameters

Let's increase $K_p$ now and see the effect:

system stabilized → around reference point

(if we zoom in, we see fewer disturbances)

what if we keep increasing $K_p$?

wait, the signal oscillates?

what if we keep increasing $K_p$?

the I term → not zero when crossing the reference, $y(t) = 0$

takes a little while → wash out the cumulative error

summary

P is required