SE201: Introduction to Systems Engineering Mathematical Modeling

Mathematical Models A mathematical model is a set of equations that relates different variables of the system Mathematical models are essential for the analysis and design of control systems

Falling Ball Example A ball falling from a height of 100 meters We need to determine a mathematical model that describes the behavior of the falling ball. Objectives of the model: answer these questions: 1. When does the ball reach ground? 2. What is the impact speed? Different assumptions results in different models

Falling Ball Example Can you list some of the assumptions?

Falling Ball Example Assumptions for Model 1 1. Initial position = 100 x(0) = Initial speed = 0 v(0) = 0 3. Location: near sea level 4. The only force acting on the ball is the gravitational force (no air resistance)

Falling Ball Example More models Other mathematical models are possible. One such model includes the effect of air resistance. Here the drag force is assumed to be proportional to the square of the velocity.

How far can this stunt driver jump? List some assumptions for solving this problem

Stunt driver Assumptions: Point mass Mass of car+driver =M Initial speed = v 0 Angle of inclination =a No drag force Model can be obtained to give the distance covered by the jump in terms of M,a, v 0,…

Classification of Models Static models: Models involve algebraic equations only Dynamic models involves differential equations Linear models: includes linear algebraic and/or differential equations Other types of Models (nonlinear, distributed, discrete-time, hybrid,…)

Transfer functions A transfer function is used to describe the relationship between the input and output of a system or a subsystem Transfer function is defined for systems described by linear algebraic and/or linear differential equations Transfer function G Input UOutput Y Y = G U

Static Models The relationship between steady state values of input and output of a linear system is described by a linear static model Transfer function G Input UOutput Y

Example Electric Motor Input (voltage supplied to motor) Volts Output (Motor speed ) revolution /min The transfer function represents the ratio of steady state value of output over steady state value of input Motor G U (volts)Y (rev/min)

Example Electric Motor steady state value of input 12 Volts transfer function G= 500 revolution /min/volt What is the steady state value of output? Motor G U (volts)Y (rev/min)

Block Diagrams Reductions

Objectives To understand the concept of Block diagrams and their assumptions To be able to reduce complex block diagrams to a simple (one block) diagram

Block diagrams A block diagram is used to represent linear relationship between the input and the output of a subsystem. Let the input and the output be U and Y, the relationship between U and Y is represented by a rectangular block having a transfer function G G UY = G U

Block diagrams A block diagram is pictorial representation of systems. It shows different components (or subsystems) that make a system and shows their interactions Each rectangular block represents a subsystem Inputs are represented by arrows entering the block Outputs are represented by arrows leaving the block G XY H G

Voltage Divider +e−+e− input output

Two-input-two-output systems inputs outputs

Block diagram manipulation H G H G H G H + G H G G. 1+H G _

Block diagram manipulation 2 G X Y G X Y G G X Y G X Y G V

Block Diagrams transformations Blocks in cascade G V X HG Y H X Y Two blocks in cascade are replaced by a single block whose transfer function is the product of transfer functions

Block Diagrams transformations Parallel Combination G+H Y G X Y H X Two blocks are in parallel they can be replaced by a single block whose transfer function is the sum of transfer functions

Block Diagrams transformations Moving a summing point G X Y G X Y G ─1 U U

Block Diagrams transformations Moving a summing point G X Y G X Y G U U

Block Diagrams transformations Moving a pickoff point (#1) G V Y G V Y G V V

Block Diagrams transformations Moving a pickoff point (#2) G X Y G X Y G ─1 X X

Block Diagrams transformations Moving a summing point G X Y G X Y G ─1 U U

Block Diagrams transformations Eliminating a Feedback Loop X G XY H Y

Exercise Find the transfer function relating X and Y G X Y H M K

BD transformations An example

Keywords Summing point Pickoff point Transfer function Cascade combination Parallel combination Feedback combination Block diagram Block diagram reduction

Example Find the transfer function of the following system GH K ?

Example Find the transfer function of the following system GH K KHG The transfer function of three blocks in series is the product of the three transfer functions U V WY Y = KW = K (HV) = K H (GU) = KHG U

Example Pressure Measurement System The pressure measurement system consists of A pressure sensor Signal conditioner Pressure Sensor Generate electric current proportional To the pressure G=0.1 mA/Pa Signal Conditioner (Amplifier) K=20 UY

Example Pressure Measurement System Pressure Sensor Generate electric current proportional To the pressure G=0.1 mA/Pa Signal Conditioner (Amplifier) K=20 PV T.F=20*0.1=2mA/Pa V P

Example Find the transfer function of the following system GH K ?

Example Find the transfer function of the following system GH K

Example Motor with Speed Control Find the transfer function of the following system Amplifier + relay + motor G=600 rev/min/volt Tachometer(speed sensor ) K=3mvolt/rev/min