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Introduction to Automatic Control

# Introduction to Automatic Control - Dr Atilla Dogan MAE...

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Dr. Atilla Dogan MAE 4310 Automatic Control – Fall 2006 Homework–1 (Due date: Tuesday, Sep 12, 2006) Textbook Readings: Sections 1–1, 1–2, 1–3, 2–1, 2–2, 2–3, 2–4, 2–5, 2–6,2–7, 3–3, 3–7 1. For each of the following systems; find the transfer function from u to y , draw the pole–zero map, characterize the stability (internal and BIBO) and the nature (exponential, oscillatory, growth, decay) of the response to non–zero initial conditions. ... y ( t ) + 4¨ y ( t ) + ˙ y ( t ) + 4 y ( t ) = u ( t ) ¨ y ( t ) + 4 ˙ y ( t ) + 4 y ( t ) = 3 ˙ u ( t ) + 2 u ( t ) ... y ( t ) + ¨ y ( t ) + ˙ y ( t ) - 3 y ( t ) = u ( t ) ¨ y ( t ) + ˙ y ( t ) = 2 ˙ u ( t ) .... y ( t ) + 2¨ y ( t ) + y ( t ) = u ( t ) 2. Consider the differential equation ¨ y ( t ) + 4 ˙ y ( t ) + 3 y ( t ) = 3 ˙ u ( t ) + 2 u ( t ) (i) Obtain the transfer function G(s) of the system from u to y . (ii) Let u ( t ) = 1(t) . Find U ( s ) and use (i) to write out an expression for Y ( s ). (iii) Find the partial fraction expansion of Y ( s ). Verify your result using MATLAB. To do this, type >> num = numerator of Y(s) >> den = denominator of Y(s) >> [r, p, k]=residue(num,den) The elements of the vector r are the residues of the partial fraction expansion corre- sponding to the elements of the vector p which contains the poles of the system. (iv) Use the partial fraction expansion of Y ( s ) to obtain the forced response of the system to u ( t ) = 1. Mechanical and Aerospace Engineering - UTA 1 of 5

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Dr. Atilla Dogan MAE 4310 Automatic Control – Fall 2006 3a. (MATLAB) Plot and print the function f ( t ) = 5 sin t . At the MATLAB prompt ( >> ), type the following commands: >> t = 0:0.1:10; >> f = 5*sin(t); >> plot(t,y) 3b. (MATLAB) Calculate the roots of the polynomial 0 . 0346 λ 3 + 0 . 0535 λ 2 + 0 . 5297 λ + 0 . 6711 Type >> poly = [0.0346 0.0535 0.5297 0.6711]; >> rts = roots(poly); The first statement creates a coefficient vector named ” poly ” and the second statement cal- culates its roots.
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