Control Systems/Examples/Second Order Systems

Second Order Systems: ExamplesEdit

Example 1Edit

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, Kd are known constants and given as:

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TasksEdit

  1. 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?
  2. Find the corresponding time-domain response of the system.
  3. 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 1Edit

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 ωn 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 2Edit

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

   
   
   

We can plot the output of this system:

Task 3Edit

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