Mathematical Modeling
of Systems
P.S. Gandhi
Mechanical Engineering
IIT Bombay
Recap
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Block diagram representation
Concepts of noise disturbances
represented on block diagram
Constraints on input
Regulation vs tracking problem
Today
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Modeling fundamentals
Need for modeling
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Assess system performance
Off-line Simulations to know the system
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Develop controllers
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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
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Friction model at: Low speeds vs. high speeds
Empirical models vs. models based on physics
Basis of Modeling
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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
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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
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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
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Properties of system : Linear vs Nonlinear, LTI
SISO, Multi-input systems