Sampled-Data Control Systems

Sampled-Data (Digital) Control Systems

Here we have a controller that requires Digital/Analog conversions.

• $D[z]$ discrete-time controller
• $r,d,u,y$ are all in continuous time
• A/D is the sampler (analog to digital converter)
  • samples at discrete intervals $kT,k\in \mathbb{Z}{\ge }_{}0$
• T is the sampling time of the system
• D/A is the zero order hold (Digital to Analog converter)
  • holds $u[k]$ for $t\in [kT,(k+1)T]\forall k\in \mathbb{Z}{\ge }_0$

This system in NOT LTI (Linear Time Invariant)! Thats becuase the signal can change if you shift the time off a bit (will sample and hold completely different signals)

As a result, what do we do?

• Attempt to choose a sampling time of the control architecture in such a way to make the system as LTI as possible
• Utilize LTI tools anyways
  • Approximating continuous time controllers with discrete controllers
  • or by directly designing a discrete controller
• Analyze and simulate the results on the simulated true system (accounting for the samplers and holders)

Example

1. Our sampler gets worse and worse the higher the frequency of the input signal.
2. We can cause instability if our sampling time is too long