23 EMT4801/101/0/2012Assignment 03Unique number: 858321Recommended closing date: 31 August 2012Based...

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23 EMT4801/101/0/2012Assignment 03Unique number: 858321Recommended closing date: 31 August 2012Based on all Units in Module 3 & 4Question 1Use Laplace transforms to solve the following system of differential equations;x′ + y′ + 5x+ 3y = e−t2x′ + y′ + x+ y = 3y (0) = 1, x (0) = 0.[Hint: First solve for y. Then determine x using the above equations.] [10]Question 2Suppose we are given a system which is initially at rest with input u (t) and output x (t) ,described by the differential equationx′′ − 6x′ + 13x = 2u′ + u2.1 Write down the transfer function G (p) for the system and say with adequate justi-fication if this system is stable or not. (5)2.2 Write down a state–space model for the system(yielding the same transfer function). (3)2.3 Use the initial value theorem to find g (0+) where g (t) = L−1 [G (p)] . (3)2.4 Can we in this case trust the final value theorem to give limt→∞g (t) where g is asefore? Justify your answer.(3)2.5 Determine the response of the system if a forcing function of u (t) = e−12t is appliedmomentarily at time t = 0. (5)[Hint: p2 − 6p + 13 = (p− XXXXXXXXXX.] [19]24Question 3Consider a system which is initially in a quiescent state and is described by the differentialequationx(4) + 2x′′ − x′ + 2x = 2u′ + 3uWrite down the transfer function of this system and use the Routh–Hurwitz criterion totest the stability of the system. [5]Question 4Consider the periodic function f (t) with graph as shown below, and let f1 be the functionwhich agrees with f on [0, 4] , and is zero elsewhere:2 4 6 824.1 Determine f1 in terms of Heaviside unit step functions. (4)4.2 Find the Laplace transform F1 (p) = L {f1 (t)}. (2)4.3 Use the answer to 4.2 to compute F (p) = L {f (t)}. (1)[7]Question 5In the circuit below, vc = x3 =1C∫ t0 i1 is the voltage drop across the capacitor C.Obtain the state-space model characterising the given network in the form ẋ= Ax+Bu; y =Cx by applying Kirchoff’s laws to the left and right loops.Note: ẋ =ẋ1ẋ2ẋ3; x =x1x2x3; y =y1y2y3[12]25 EMT4801/101/0/2012Question 6A system is characterised by the following equation:ẋ1ẋ2 =−70 010 −40x1x2+10 400.and the initial conditions are x1 (0) = 0 and x2 (0) = 0. Take Laplace transforms of thestate equation andsolve. [10]Question 7Use the method of convolution to find the inverse Laplace transformL−1[p2+q(p2−32)2](Hint: You can simplify this one by setting p2+9(p2−32)2= 1(p2−32) +18(p2−32)2.)[8]26Question 8Suppose we are given a system initially in a quiescent state (so all initial conditions are0) described by the difference equation6yk+2 − yk+1 − yk = 6uk+1 + 12uk.8.1 Find the transfer function of the system, and say with justification if the system isstable, marginally stable, or unstable. (4)8.2 Write down a state space representation of this system. (4)[8]Question 9Solve the difference equationyn+2 + 4yn+1 + 3yn = un+1 − 2unif the initial conditions are given as y0 = 0, y1 = 0, and the input is uk = 5 for all k. [9]Question 10Solve the following state–space equation by taking a Z–transform and using an inversematrix, given that x1 (0) = x2 (0) = 0 and uk = 2.x1 (k + 1)x2 (k + 1) =0 1−18 −34x1 (k)x2 (k)+10ukyk = [1 − 2]x1 (k)x2 (k)[12]TOTAL: [100]

David · Accepted Answer

Solution1:
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Solution 2:
2.1 
Since it is initially at rest therefore 
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Characteristic equation of the transfer function is: 
  
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Using routh Hurwitz criteria 
p^2 1 14 
p -4   0 
1 (-4*14-0)/(-4) 
Since not all terms of column 1 are positive hence system is not stable 
2.2 
Forming state space equation: 
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2.3 
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For initial value theorem: 
   
   
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2.4 
No we cannot put final value in this as at t→ ∞ exponential value is ∞ 
2.

23 EMT4801/101/0/2012 Assignment 03 Unique number: 858321 Recommended closing date: 31 August 2012 Based on all Units in Module 3 & 4 Question 1 Use Laplace transforms to solve the following system of...

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