Question

The output y and input x of a device are related by y = x + 1.4x3 . (a) Find the values of the output for steady-state operation at the two operating points xo = 1 and xo = 2. (b) Obtain a linearized model for both operating points and compare them.

Answer 1

Answer:

a) $y(1) = 2.4$ , $y(2) = 13.2$, b) $y(x) = 2.4 + 5.2\cdot (x-1)$, $y(1) = 2.4$, $y(2) = 7.6$

Step-by-step explanation:

a) The values of the output for steady-state operation are:

$y(1) = 1 + 1.4\cdot (1)^{3}$

$y(1) = 2.4$

$y(2) = 2 + 1.4\cdot (2)^{3}$

$y(2) = 13.2$

b) The formula for linearization is:

$y(x) = y(1) +\frac{dy}{dx}|_{x = 1} \cdot (x-1)$

The first derivative of the formula evaluated at x = 1 is:

$\frac{dy}{dx}|_{x=1} = 1 + 4.2\cdot (1)^{2}$

$\frac{dy}{dx}|_{x=1} = 5.2$

The linearized model is:

$y(x) = 2.4 + 5.2\cdot (x-1)$

The output at x = 2 is presented below:

$y(2) = 2.4 + 5.2\cdot (2-1)$

$y(2) = 7.6$

Linearized model offers reasonable approximations for small intervals.