Question

Problem 1:

You are given two tanks. Each with a separate inlet Fin and an outflow Fout. The relation of the level of water to the flow is given with the following differential equation:

A dh/dt = Fin − Fout .

To simplify the model of the system assume that the height of water in the tank vs the water flowing in can be linearized and represented by the following:

A dh/dt = Fin − αh;

Where alpha can be assumed to be a flow coefficient for the current flow operating range chosen.

The parameters for the 2 tanks are as such: alpha1 = 2, alpha2 = 10, Area1 = 400, Area2 = 1000.

1) Derive the transfer function for each system (where the level of water [h] is the output and the flow in [Fin] is the input.

2) What is the time constant and gain of each system

3) Assume that the Fout of tank 1 is going into tank 2. i.e. Fout,1 = Fin,2. What is the transfer function of the overall system? [where the level of water in tank 2 h2 is the output and the flow into tank 1 [Fin,1] is the input.

4) The transfer function in question 3 is second order. What is the gain, damping ratio, and natural frequency of this system?

5) Plot the step response of the overall system

6) Use the ‘stepinfo’ function (read the help in matlab) and determine the rise time (in seconds) and overshoot (in %) of the overall system to a step input.

Answer 1

A (dh/dt) = Fin - αh (given)

A (dh/dt) + αh = Fin

Taking laplace transform

A₁ s H'₁(s) + α H'₁(s) = F'_{1 in} (s)

(A₁ s + α₁) H'₁(s) = F'_{1 in} (s)

H'₁(s) / F'_{1 in} (s) = 1/(A₁s+α₁) ---------Transfer function for system 1

Similarly,

H'₂(s) / F'_{2 in} (s) = 1/(A₂s+α₂) -------------Transfer function for system 2

-------------------(solution for (1) ends here)----------------------------------------------

A₁ = 400; α₁ = 2 (given)

Substitute in Transfer function for system 1

we get

H'₁(s) / F'_{1 in} (s) = 1/(400s+2)

First order transfer function in the form 1/s+1

So, the above form will be divided by 2 in both numerator and denominator.

It becomes

H'₁(s) / F'_{1 in} (s) = 0.5/(200s+1)

From this,

₁ (time constant) = 200; K₁ (gain) = 0.5 (for system 1)

Similarly,

₂ (time constant) = 100; K₂(gain) = 0.1 (for system 2)

-------------------(solution for (2) ends here)----------------------------------------------

For two interacting system,

The general form is

H'2(s) / F'1in (s) = K₂ / ((₁s+1)*(₂s+1)

-------------------(solution for (3) ends here)----------------------------------------------

Substitute K₂, ₁, ₂ in above equation,

we get

H'2(s) / F'1in (s) = 0.1/(200s+1)*(100s+1)

H'2(s) / F'1in (s) = 0.1/ (20000s² + 300s + 1)

This is of the form K₁ / (²s²+2s +1)

² = 20000

= 141.42 (gain)

2 = 300

= 300 / 282.84

= 1.060 (damping ratio)

is larger than 1, So the natural frequency response of this system is less oscillatory

-------------------(solution for (4) ends here)----------------------------------------------