Systems and control theory Lecture 1: Introduction Systems and - - PowerPoint PPT Presentation

▶
systems and control theory
SMART_READER_LITE
LIVE PREVIEW

Systems and control theory Lecture 1: Introduction Systems and - - PowerPoint PPT Presentation

Apr 10, 2024 294 likes •551 views

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Systems and control theory Lecture 1: Introduction Systems and Control Theory STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics

slide-1
SLIDE 1

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Systems and control theory

Lecture 1: Introduction

slide-2
SLIDE 2

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Dynamical system

  • A dynamical system is a constantly changing system that connects
  • utputs (denoted by y) and inputs (denoted by u).
  • The word dynamical refers to the fact that the system relates time-

changing signals.

  • ‘Everything is a dynamical system.
  • We will usually refer to dynamical systems with the word system.

2

slide-3
SLIDE 3

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

System theory

  • System theory occupies itself with the mathematical description and

study of systems.

  • The models describe the connections between the inputs, outputs and the

states.

  • Differential equations or difference equations are used.
  • It offers a set of tools that allow us to study almost every thinkable

system.

  • The first part of this course is about system theory.

3

slide-4
SLIDE 4

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

State and order

  • Next to inputs and outputs, states (denoted by a vector x) are a third

type of variable that is used to describe a system.

  • State variables represent the internal state of the system at a given

time.

  • The order of a system is the number of state-variables (the size of

the vector x).

4

slide-5
SLIDE 5

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Everything is a dynamical system

5

slide-6
SLIDE 6

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

6

slide-7
SLIDE 7

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Will Will https://flic.kr/p/7QJgoL

Antique racecar

7

slide-8
SLIDE 8

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Picture M.dolly https://flic.kr/p/njqWCS

Predator-prey system

8

slide-9
SLIDE 9

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

A mechanical system

9

slide-10
SLIDE 10

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Control Theory (Second part of the course)

  • In control theory we will apply our knowledge of system theory.
  • The goal is to find an input that results in the desired output:
  • Note that the input of the system that consists out of the original

system and the controller is the desired state.

10

slide-11
SLIDE 11

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Open loop

  • The system shown below is an open loop system.
  • The controller cannot see the effect of it actions.
  • This is hard to get the desired output.

11

slide-12
SLIDE 12

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Open loop

  • Take for example the following

system:

  • You are pouring a glass of water, but you

cannot look at the glass.

  • You know the desired output is a full

glass of water within reasonable time.

  • The input can have two values: on or off

(assume you are using a quite primitive tap).

  • You can imagine it will not be easy to do

this successfully.

12

slide-13
SLIDE 13

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Feedback

  • The solution is evident: look at the glass while pouring.
  • Then you get a feedback system and the input to the controller

becomes the error.

  • The input to the system as a whole still is the desired output.

13

slide-14
SLIDE 14

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Slides courtesy of

  • Prof. Rodolphe Sepulchre

Feedback

slide-15
SLIDE 15

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Range-localized sensitivity is a non linear behavior

V = sat(I)

O(1) O(1) Localized sensitivity Range localized

15

slide-16
SLIDE 16

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Black principle: negative feedback ‘linearizes’

K

  • I

V 1

≡

O(K) O(1)

V = sat(I - KV) ≡ V = sat (I)

Sensitivity domain is spread by negative feedback (the essence of control theory)

16

slide-17
SLIDE 17

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Black principle: positive feedback ‘quantizes’

K + I V 1

≡

O(K) O(1) (K large)

V = sat(I + KV) ≡ V =

Sensitivity domain is spread by positive feedback Hysteretic behavior: memory, on-off devices (The essence of digital technology.)

17

slide-18
SLIDE 18

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Black feedback principle

+

  • +
  • Negative feedback

linearizes

  • Continuous behavior
  • Analog technology
  • Exogenous

(output primarily reflects the input)

  • Positive feedback

quantizes

  • On-Off behavior
  • Digital technology
  • Endogenous

(output primarily reflects memory of the past)

18

slide-19
SLIDE 19

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Balanced feedback ‘localizes’

I V K- K+

  • +

I V K-

  • I

V K+ +

‘Linear’

|k| large

‘Memory’

|k| large

‘Localized’

|k| small O(K) k ≈ K+ - K-

19

slide-20
SLIDE 20

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Robust space + time localization by feedback

Passive RC-circuit I V Slow lag Fast lag + High frequency behavior +

  • Low frequency behavior

Necessary localization in same frequency range!

20

slide-21
SLIDE 21

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

What if… We are losing control?

  • https://www.youtube.com/watch?v=C221sI1W9Gk

21

slide-22
SLIDE 22

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

22

slide-23
SLIDE 23

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Millenium Bridge London

  • Resonance on the Millenium Bridge in London due to the rithm of walking people.
  • https://www.youtube.com/watch?v=eAXVa__XWZ8

23

slide-24
SLIDE 24

Systems and Control Theory

STADIUS - Center for Dynamical Systems,

Signal Processing and Data Analytics

Learning objectives

At the end of this lecture you should be able to:

  • provide your own example of a system,
  • recognize input/output and states of a system,
  • understand why feedback is essential in control.

24