Question

(40) 3. Make an approximation of the transfer function (s+10) G(s)-Y(s)/U(s) =__ (s+25) using Euler's approximation. Find the actual solution and the approximate solution for a sampling interval T-0.1 sec for t=0, 0.1, 0.2. u(t) = 10tfort O and u(t) =0 fort 30

Answer 1

Answer's o(s): Y(S)/vis) : ST10 S+25 By Tustin's appoonimation Z=3T S-41n (2) Part(1973: --- *} **] ) + 20 ( ] Thus, the appoonimated transfer function is G(2) - 615) 12-10() , 10 (**)+10 lo ]+25 102-16+ uz to 102-104 207 + 25 147 307 +15

Z ace) = consider the actual transfer function yes) UCS) = SHID 5725 4(3): Sino UCS) -O S+25 The input is ULE)-10 for to Apply Laplace transform U(S). Ź substitute UCS). Žin een yes) - Sites (3) = 25+20 5(+25) split into partial fractions Y(5) $ (5) + ($+25))

Apply inveose la place transform - y(t)= 4/4 + 6 eest for txo Now, the actual solution 1. for a sampling interval tro, 0.110.2 Y(o)+62560) 960) ang tama (1) 900.1) 4 +6826/01) cet 20.06208) 4+0.49248 4.492 -0.8984 (0-2)= “ +6 025(0-2) = a + (0-006737) • 4 +0.242532 4.242532 -0.8485 - 5

consider the appoonimated transfer function Gle): 462) Juce) - © The input is uch) = 0 tro Apply z transform U(z)= 2 2 2 substiture U(z) 2. in an o YCE 7037 ] LZ + Y(z) = 1 * * (2-1)(2+1) YE) - (2) : (2-1) (2+2)

Split into Partial fractions Yen (2) + (244) Y(2)=46) + Ślett) Apply inreose z toansform yen) sont (-a)" n70 now appoonimate solution for a sampling internal to for n=0,112 is Y(0) = + -tuº y to co 903 4+ (-) - 4600 -0.7 962).+) (-)? = “ + City) -0.8+120 = 120.8