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

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 Dynamical system  A dynamical system is a constantly changing system that connects outputs (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 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 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 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Everything is a dynamical system 5 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics 6 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Antique racecar Will Will https://flic.kr/p/7QJgoL 7 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Predator-prey system Picture M.dolly https://flic.kr/p/njqWCS 8 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics A mechanical system 9 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 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 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 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 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Slides courtesy of Prof. Rodolphe Sepulchre Feedback 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) Range localized O(1) Localized sensitivity 15 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Black principle: negative feedback ‘linearizes’ I V 1 ≡ - O(1) O(K) K V = sat(I - KV) ≡ V = sat (I) Sensitivity domain is spread by negative feedback (the essence of control theory) 16 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Black principle: positive feedback ‘quantizes’ (K large) I V 1 ≡ + O(1) O(K) K V = sat(I + KV) ≡ V = Sensitivity domain is spread by positive feedback Hysteretic behavior: memory, on-off devices (The essence of digital technology.) 17 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Black feedback principle + + - - • Negative feedback • Positive feedback linearizes quantizes • Continuous behavior • On-Off behavior • Analog technology • Digital technology • Exogenous • Endogenous (output primarily reflects (output primarily reflects the input) memory of the past) 18 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Balanced feedback ‘localizes’ K + K + + + I V I V I V - - K - K - ‘Linear’ ‘Localized’ ‘Memory’ |k| large |k| small |k| large O(K) k ≈ K + - K - 19 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Robust space + time localization by feedback I V Passive + RC-circuit High frequency behavior + Fast lag - Slow lag Low frequency behavior Necessary localization in same frequency range! 20 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 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics 22 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 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 Systems and Control Theory