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Chapter 3 mathematical Modeling of Dynamic Systems

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

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

Transfer Function and Impulse-Response Function

Transfer Function Convolution Integral Impulse-Res

ponse Function

Convolution Integral

ILT convolution

Multiplication in the frequency domain

corresponds to convolution in the time domain

Convolution

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

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.

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

Start by doing whats necessary, then whats

possible, and suddenly you are doing the

impossible. -St. Francis of Assisi

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

Transfer function

Input signal

Output signal

Transfer function

Block diagram

Equation

Block Diagram reduction

Open loop TF? Feedforward TF? Closed loop TF?

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)

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.

Chapter 3 Problems

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

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