# Control Systems/Examples/Second Order Systems

## Second Order Systems: Examples[edit | edit source]

### Example 1[edit | edit source]

A damped control system for aiming a hydrophonic array on a minesweeper vessel has the following open-loop transfer function from the driveshaft to the array.

The gain parameter *K* can be varied. The moment of inertia, *J*, of the array and the force due to viscous drag of the water, *K _{d}* are known constants and given as:

#### Tasks[edit | edit source]

- The system is arranged as a closed loop system with unity feedback. Find the value of
*K*such that, when the input is a unit step, the closed loop response has at most a 50% overshoot (approximately). You may use standard response curves. Should*K*be greater or less than this value for less overshoot? - Find the corresponding time-domain response of the system.
- The system is now given an input of constant angular velocity,
*V*. For the limiting value of*K*found above, calculate the maximum value of*V*such that the array follows the input with at most 5° error.

#### Task 1[edit | edit source]

First, let us draw the block diagram of the system. We know the open-loop transfer function, and that there is unit feedback. Therefore, we have:

The closed-loop gain is given by:

We now need to express the closed-loop transfer function in the standard second order form.

We can now express the natural frequency *ω _{n}* and damping ratio,

*ζ*:

We now look at the standard response curves for second order systems.

We see that for 50% overshoot, we need *ζ*=0.2 or more.

This is the maximum permissible value, thus *K* should be less than this value for less overshoot. We can now evaluate the natural frequency fully:

#### Task 2[edit | edit source]

The output of the second order system is given by the following equation:

We can plot the output of this system:

#### Task 3[edit | edit source]

The tracking error signal, *E(s)*, is equal to the output's deviation from the input.

Now, we can find the gain from the reference input, *R(s)* to the error tracking signal:

The gain from the input to the error tracking signal of a unity feedback system like this is simply .

Now, *R(s)* is given by the Laplace transform of a ramp of slope *V*:

We now use the final value theorem to find the value of *E(s)* in the steady state:

We require this to be less than