Mathematical Modeling
of Systems
P.S. Gandhi
Mechanical Engineering
IIT Bombay
Recap
Block diagram representation
Concepts of noise disturbances
represented on block diagram
Constraints on input
Regulation vs tracking problem
Today
Modeling fundamentals
Need for modeling
Assess system performance
Off-line Simulations to know the system
Develop controllers
Behavior
Response to disturbances
Response to various inputs
Test performance of a controller
Get control input history
Substantial saving in time and cost of
control system design
Criteria for Modeling
Simplicity Vs Accuracy
Simple model amenable to development of
control
Accuracy of representation of dynamics of
interest
Example
Friction model at: Low speeds vs. high speeds
Empirical models vs. models based on physics
Basis of Modeling
Based on physics of phenomenon
Example: Newton’s Law: for dynamical systems
Kirchoff’s Law: for electrical circuits
Heat conduction, fluid dynamics
Based on input output experimental
observations: why this is needed???
Example: Some chemical processes
Some fluid dynamics models
Example 1
F
m
F
x
1/m
F = mx
x
∫
x
∫
x
Integrator
Block Diagram Representation of differential equation
Example: 2
mx + cx + kx = F
F
x
1/m
+
+
+
∫
-
x
c/m
k/m
∫
x
Example 3
dh(t )
A
= Qi (t ) − Qo (t )
dt
Qo (t ) = k h(t )
Example 4: DC Motor
Simple model of a DC motor
Load torque
Input Voltage
Plant
θ(t)
Motor Dynamics
Armature Free Body Diagram
Electrical circuit Diagram
Steps in Modeling
Identify the most important physical phenomenon
Assess their complexity to be represented using
equations governing physics
Take decision regarding I/O model vs. physics
Apply corresponding laws to get the final equations
Verify models by matching model simulation and
experimental results
Next Class
Properties of system : Linear vs Nonlinear, LTI
SISO, Multi-input systems
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