Chapter 3 mathematical Modeling of Dynamic Systems

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Chapter 3 mathematical Modeling of Dynamic Systems
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Modeling
Mathematical models are developed from physical
laws chemical laws biological laws economic
laws etc Mathematical models take the form
of differential equations transfer
functions state equations block
diagrams signal flow graphs etc Mathematical
models are used to analyze dynamic
characteristics and to design control systems
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Models
Simplicity vs Accuracy lumped vs distributed
parameter models linear vs nonlinear
models time-invariant vs time varying models As
simple as possible with the required accuracy
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Transfer Function and Impulse-Response Function
Transfer Function Convolution Integral Impulse-Res
ponse Function
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Convolution Integral
ILT convolution
Multiplication in the frequency domain
corresponds to convolution in the time domain
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Convolution
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Impulse Response Function
The response of a differential equation to an
impulse (delta function) input is the Impulse
Response Function of that differential equation.
For an impulse input
The Laplace transform of the impulse response is
transfer function
The impulse response is the inverse Laplace
transform of the transfer function
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Impulse Response, Convolution, and solution to DEs
The output of a system due to the input r(t) is
the convolution of the input function and the
impulse response of the system. y(t)
conv(g(t),r(t)) There is an intimate relationship
between the impulse response of a system and the
response of the system to any other input.
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Automatic Control Systems
Block Diagrams Signals Blocks and Transfer
functions Summing point Branch point Block
diagram of a closed loop system Open-loop
transfer function and feedforward transfer
function Closed-loop transfer function Automatic
Controllers Industrial Controllers On-off,
P, I, PI, PD, PID Disturbances Block diagram
reduction
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Start by doing whats necessary, then whats
possible, and suddenly you are doing the
impossible. -St. Francis of Assisi
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Modeling review
Mechanical systems Electrical systems I hope to
come back to this topic later Skip 3-5, 3-9, 3-10
Mechanical examples? Electrical examples? Thermal
examples? Transfer function? Time constant
Mechanical examples? Electrical examples? Thermal
examples? Transfer function? Undamped natural
frequency damping ratio
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Transfer function
Input signal
Output signal
Transfer function
Block diagram
Equation
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Block Diagram reduction
Open loop TF? Feedforward TF? Closed loop TF?
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Steady-State Error
Steady-state means the output looks like the
input. Step inputs produce step (constant)
outputs Sinusoidal inputs produce sinusoidal
outputs Ramp inputs produce ramp outputs X
inputs produce X outputs Steady-state means that
the transients have died out. The output has (at
least) two terms, steady-state (mathematically
looks like the input) transient (decays to
zero) sometimes other terms (normally a bad
situation)
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Computing SS error

• To compute E(s)
• inverse Laplace transform to get e(t)
• take limit as t goes to infinity.
• Steady state error only exists if the limit
  exists.

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Chapter 3 Problems
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Assignments
Ungraded homework A1-5
Graded homework B1-7
Test 1 Solving DEs via Laplace transforms, Block
diagrams.
Laplace transforms definition 2 formulas for
functions derivatives Eulers identities ALGEBRA
CALCULUS Complex numbers/algebra