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
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