Design Case Studies

Model Discretization

The task here is to design a digital controller that provides accurate positioning of the read/write head. The design is performed in the digital domain. First, discretize the continuous plant. Because our plant will be equipped with a digital-to-analog converter (with a zero-order hold) connected to its input, use c2d with the 'zoh' discretization method. Type

• Ts = 0.005;     % sampling period = 0.005 second
  Hd = c2d(H,Ts,'zoh')
  Transfer function:
  6.233e-05 z + 6.229e-05
  -----------------------
   z^2 - 1.973 z + 0.998
   
  Sampling time: 0.005
  

You can compare the Bode plots of the continuous and discretized models with

• bode(H,'-',Hd,'--')
  
  
  
  

To analyze the discrete system, plot its step response, type

• step(Hd)
  
  
  
  

The system oscillates quite a bit. This is probably due to very light damping. You can check this by computing the open-loop poles. Type

• % Open-loop poles of discrete model
  damp(Hd) 
      Eigenvalue         Magnitude    Equiv. Damping    Equiv. Freq.
                                                                                 
  9.87e-01 + 1.57e-01i     9.99e-01         6.32e-03             3.16e+01        
  9.87e-01 - 1.57e-01i     9.99e-01         6.32e-03             3.16e+01        
  

The poles have very light equivalent damping and are near the unit circle. You need to design a compensator that increases the damping of these poles.

Deriving the Model

Adding a Compensator Gain