System modeling Lesson 3 Systems and Control Theory STADIUS - - PowerPoint PPT Presentation

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics System modeling Lesson 3 Systems and Control Theory

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Introduction Modeling of dynamical systems We can derive the mathematical model of a system in two ways mainly:  Physical Modeling It consists of applying various laws of physics, chemistry, thermodynamics, etc., to derive ODE or PDE models. It is modeling from “First Principles” . Mass-spring system Inverted pendulum Tubular chemical reactor T T T J1 J2 J3 reactant reactant A A   A B product C , T C T , in in B   E C C     RT v k Ce   0 t z   E T T       v G Ce RT H T ( T )   r r w t z 2 Systems and Control Theory 2

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Introduction  System identification or Empirical Modeling It consists of developing models from observed or collected data. Dynamical system u t 1 ( ) y t 1 ( ) u t 2 ( ) y t 2 ( ) y ( ) t u t ( ) m n T T s s Identification Tuning parameters Algorithm For example: Mathematical   Model     y ( ) t f y ( t 1), ( y t 2), , ( ), ( u t u t 1), 3 Systems and Control Theory 3

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

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: Mass Spring System  Dynamical system Systems and Control Theory 5

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Mass Spring Damper System Systems and Control Theory 6

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Mass Spring Damper System https://www.youtube.com/watch?v=8DuJEpy-ODo Systems and Control Theory 7

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: Pendulum Systems and Control Theory 8

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Inverted pendulum Systems and Control Theory 9

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Flying Inverted Pendulum https://www.youtube.com/watch?v=15DIidigArA Systems and Control Theory 10

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: LC circuit Systems and Control Theory 11

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: RLC circuit  Equations for each component  Let V2 and i be the states. (They are already in the derivative).  Model output as function of states and inputs Systems and Control Theory 12

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: linear circuit  Equations from previous slide  Writing the equations as matrices results in state space representation Systems and Control Theory 13

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Force-Voltage Analogy Systems and Control Theory 14

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Force-Voltage Analogy  Force F  Voltage e  Mass m  Inductance L  Viscous-friction coefficient b  Resistance R  Spring constant k  Reciprocal of capacitance 1/C  Displacement  Charge q  Velocity  Current i Systems and Control Theory 15

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: Hoover dam  :inflow of water in  :current volume of water in  :outflow to river in  :the current water level (height) in Systems and Control Theory 16

STADIUS - Center for Dynamical Systems, Signal Processing and Data Analytics Example: Hoover dam  What happens when we open the gate?  Outflow (like a brick of milk)  We assume (falsely) that depends linearly on . Systems and Control Theory 17