Engineering HELP!
Got a resit on wednesday and really struggling with the past papers (again). Totally do not understand this one module even though on a 2.1 overall. Could anybody help me with this question?
(A) state three properties of the system transfer function
(B) Assuming zero condition show that the unit step response of the 1st order model:
T(dx/dt) + x = Ku(t) is: x(t) = K(1-e^-t/T)
Note: the laplace transforms of a unit step and first-order differential are as follows:
L{u(t)} = 1/s
L{(dx/dt)} = sX(s) - x(0)
(C) Consider the differential equation model:
d2x/dt2 - dx/dt +0.5x = du(t)/dt + 2u(t)
Note: L{d2x/dt2} = s^2 X(s) - sx(0) - dx(0)/dt
(i) Find the transfer function, X(s)
(ii) Calculate the steady state gain for a given step input
(iii) State the characteristic equation
(iv) Find the poles and zeros for the above system
(v) hence determine the stability of the system
(vi) sketch the poles and zeros on a complex s-plane
(vii) sketch the form of the likely time response mode
(viii) How do the locations of the zeros influence the response?
(D) Briefly explain the necessary conditions for stability. Determine the range of values of K for which the system having the following characteristic equation is stable.
s^3 + s^2 + (5+K)s + 3K = 0
Note: given s^3 + a1s^2 + a2s + a3 = 0 then the Hurwitz determinant will be :
H3 = a1 a3 0
1 a2 0
0 a1 a3
(A) state three properties of the system transfer function
(B) Assuming zero condition show that the unit step response of the 1st order model:
T(dx/dt) + x = Ku(t) is: x(t) = K(1-e^-t/T)
Note: the laplace transforms of a unit step and first-order differential are as follows:
L{u(t)} = 1/s
L{(dx/dt)} = sX(s) - x(0)
(C) Consider the differential equation model:
d2x/dt2 - dx/dt +0.5x = du(t)/dt + 2u(t)
Note: L{d2x/dt2} = s^2 X(s) - sx(0) - dx(0)/dt
(i) Find the transfer function, X(s)
(ii) Calculate the steady state gain for a given step input
(iii) State the characteristic equation
(iv) Find the poles and zeros for the above system
(v) hence determine the stability of the system
(vi) sketch the poles and zeros on a complex s-plane
(vii) sketch the form of the likely time response mode
(viii) How do the locations of the zeros influence the response?
(D) Briefly explain the necessary conditions for stability. Determine the range of values of K for which the system having the following characteristic equation is stable.
s^3 + s^2 + (5+K)s + 3K = 0
Note: given s^3 + a1s^2 + a2s + a3 = 0 then the Hurwitz determinant will be :
H3 = a1 a3 0
1 a2 0
0 a1 a3
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