Question

3. A math model of a system as in Eg.2 with initial condition y(0) 1 is expressed as 2y+5y 10 (a) Using Laplace, determine the complete solution y(t) (show all working and steps, do not just write the solution). (b) Determine the system time constant (c) Determine y(oo)and y(T). (d) Sketch the dynamic output response y(t) of system showing all relevant details on your sketch.

Please provide a step by step solution (System Dynamics)

Answer 1

(a):

The given equation is:

Let the laplacet tranform of y(t) be represented by Y(s). Then, the laplace transform of the above equation is given by:

$2(sY(s)-y(0))+5Y(s)=\frac{10}{s}$

$=>2sY(s)-2+5Y(s)=\frac{10}{s}$

$=>(2s+5)Y(s)=\frac{10}{s}+2$

$=>Y(s)=\frac{10+2s}{s(2s+5)}$

$=>Y(s)=\frac{s+5}{s^2+2.5s}=\frac{2}{s}-\frac{1}{s+2.5}$

Taking the inverse laplace of the above equation gives:

(b):

The time constant of the first order system can be found by observing the term with the exponent part. In this case it is: $e^{-2.5t}$

The time constant $\tau$ is given by:

$\tau = \frac{1}{2.5 } = 0.4secs$

(c):

$y(\infty ) = 2 - e^{-2.5\infty } = 2-0 = 2$

$y(\tau ) = 2 - e^{-2.5\tau } = 2-e^{-1 } = 1.63212$

(d):

The matlab code to plot the above function is given below:

t = 0:0.01:10;
y = 2-exp(-2.5*t);
plot(t,y); xlabel('Time'); ylabel('y(t)')
tau = 0.4;
y_tau = 2-exp(-2.5*tau);
hold on
plot(tau,y_tau,'x','MarkerSize',10,'MarkerEdgeColor','r');
tau = 0.4;
text(tau,y_tau,'\leftarrow y(\tau)')

The graph is shown below: