Question

3. Consider the Linear Time-Invariant (LTI) system decribed by the following differential equation: dy +504 + 4y = u(t) dt dt where y(t) is the output of the system and u(t) is the input. This is an Initial Value Problem (IVP) with initial conditions y(0) = 0, y = 0. Also by setting u(t) = (t) an input 8(t) is given to the system, where 8(t) is the unit impulse function. a. Write a function F(s) for a function f(t) that is known as the Laplace Transform and name one important use of this transform. Write the inverse of F(s) and state its use. [10 marks] b. Find the response of the system by solving for y(t) when u(t) = S(t) by using the Laplace Transform and the inverse laplace Transform. Use the following: 1. L[d” f(t)/dth] =s" f(s) – sn-1f(0) – sn-2 j (0) - ... - f(n)(0) 2. L[8(t)] = 1 3. L[e-at] = sta [15 marks]

Answer 1

Das The Laplace transform of a Signal (function) f(t) is the function F(s): dihet defined by Fes)= l fets estat + F is complex -vawed function of complex numbers = 's' is called the (complex) frequency variable, unit: seci ť is called time variable. Oniks: see. sist is unitless + The main use q Laplace Transform is used in system analysis, where complex differential equations are converted to simple algebraic equations of respective order, There by Simplifies Calculation complexity, without chaning any system parameters. - transform of Fl) is defined as The inverse Caploce L (EL)= HED Ctoo Ha a fisedt مهر The limits of the integration mean that the integral is carried out along a line parallel to the imaginary axis and at distance 'c' hom it,

ouse or inverse Caploce transform is to convert the Simplified output response of the System analysis in frequency domain to time domain. © Gruen dy + 5 dy +4y= ult) d 2 db ayo given 4(0)-0, 40-0 , ult- &16) by using the given formulas, applying Laplace transform to the above differential equation, weget sy(s) as'y(0) -5°ý(0+55465)+55910)+4 413) = 1 i s? Y43) +55 4499 +4903)=1 [*410-o, ycoso) 745) (5+554)=1 41523 _' 575564 4237 = 1 (5+1365+U2 applying partial factions we get you listu stu] now applying inverse coplace transform , wepot Lyst Let-e4t) / using formula & given) [*]L[54x)]