Control Systems question?

Let x(t) be the input and y(t) be the output of the system and X(s) and Y(s) be the corresponding Laplace transforms.

The transforms are related by:

Y(s) = X(s) · G(s)

To find y(t) transform x(t) insert the expression to this equation and obtain y(t) from the inverse transform of Y(s)

x(t) = 2 · cos(2t - 45°)

= 2·cos(2t)·cos(- 45°) - 2·sin(2t)·sin(- 45°)

= 2·cos(45°) · cos(2t) + 2·sin(45°) · sin(2t)

= √2 · cos(2t) + √2 ·sin(2t)

From the transformation table you get the transform:

X(s) = √2 · s/(s²+4) + √2 ·2/(s²+4)

= √2 · (s + 2)/(s²+4)

The Laplace transform of the answer of the system is:

Y(s) = X(s) · G(s)

= 10 · √2 · (s + 2) / [(s²+4)·(s+11)]

expand to partial fractions

= A/(s+11) + B·s/(s²+4) + C·2/(s²+4)

with

A = -18·√2 / 25

B = 18·√2 / 25

C = 26·√2 / 25

Take the inverse transform from the table:

y(t) = A·e^(-11t) + B·cos(2t) + C·sin(2t)

The first term vanishes for large t, hence the steady state solution is

ys(t) = B·cos(2t) + C·sin(2t)

You can rearrange this to

ys(t) = D·cos(2t + φ)

by

D·cos(2t + φ ) = D·cos(φ )·cos(2t) - D·sin(φ)·sin(2t)

Compare the coefficients

B = D·cos(φ)

C = -D·sin(φ)

Hence

D = √(B² + C²) = √[(18² + 26²) ·2 / 252]

= √(16/5) = 1.78885

φ = arctan(-C/B) = - arctan(C/B)

= -arctan(13/9)

= -55.3°

=>

ys(t) = 1.78885 · cos(2t - 55.3°)

Note:

The argument of cosine in the input function need to be in degrees, because if you take the argument in radians, you get a wrong result for φ.

Edit:

Control engineers would take a different way to the solution.

The Laplace transform is not only a nifty mathemathecal tool, the transfer function describes the properties of the system in the frequency domain. The transfer function of the complex argument j·ω gives you the response to a sinussidoil input of frequency ω.

By evaluating the magnitude of transfer function, you get the ratio of the output amplitude to the input amplitude. The angle of the value for the transfer function in the complex plane is the phase shift between inpout and output.

For this problem:

G(j·ω) = 10 / (11 + j·ω) = 10· (11 - j·ω) / (11² - (j·ω)²)

= (11 - j·ω) ·10/(11² + ω²)

The magnitude is the square root of the squares of real part and of imaginary part::

|G(j·ω)| = √(11² + ω²) ·10/(11² + ω²) = 10/√(11² + ω²)

The angle in the complex plane is the inversetangent of the ratio imaginary part to real part:

φ = arctan(-ω/11) = -arctan(ω/11)

For an input function with ω = 2 and amplitude x_max=2

=> x = 2sin(2t)

The amplitude ratio is

y_max / x_max = |G(j·2)| = 10/√(11² + 2²) = 10/√(11² + 2²)

= 10 / √125

<=>

y_max = 20/√125 = 4/√5 = 1.78885

The phase shift is:

φ = - arctan(2/11) = -10.3°

For an input function with a certain phase shift:

like x = 2sin(2t + φ_out), you get the out put phase shift from

φ_out = φ_in + φ

A cosine input function is nothing else that a phase shifted sine function. Hence it will behave similar.

For

x(t) = 2cos(2t - 45°)

=>

y_max = 4/√5

φ_out = -45° - 10.3° = -55.3°

and

ys = (4/√5) · cos(2t - 55.3°)

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