Lecture 2 – Modeling, Laplace Transform & Partial Fractions

Modeling Dynamical Systems

→ Expression of system's behavior by some mathematical relations obtained by setting reasonable assumptions. Differential Eq, Integral Eq, Integro diff Eq

partial: number of independent variables is more than one continuous system

ordinary: Lumped/discretized

Lumped, discretizing (ayrık)

We have to talk about linear ODE's that change respect time

We use laplace to change differential equation to algebraic equation. ( s^2+s+a=r algebraic)

Differential theorem in laplace transform

output is y input is x

Most general ODE for a system.

Assuming that initial conditions are zero (to ignore transient respons)

Initial condition: n-1'e kadar

The max order of derivative define the output of the system

Impulse function in frequency domain

impulse response = T.F. (Transfer function)

zero initial condition

Find the D.E for the transfer function

Partial fraction expansion (PFE)

case 1: Poles are distinct and with multiplicity of unity!

tekrarlanmaz, the poles are unique, differents, not repeat

poles are = residues of the function

How to obtain the redisues.

Second method of residue calculation:

three poles of multiplicity one

case 2) Distinct poles of multiplicity k_i

Transfer function degree polynomial in the denominator: we expect n roots (poles)

make at least some potion of T.F. strictly proper (n>m)