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14. Hybrid Control

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Chapter 13 gives a traditional treatment of digital control based on analyzing the at-sample response. Generally we found that this was a simple and problem free approach to digital control design. However, at several points we warned the reader that the resultant continuous response could contain nasty surprises if certain digital controllers were implemented on continuous systems. The purpose of this chapter is to analyze this situation and to explain:

  • why the continuous response can appear very different from that predicted by the at-sample response
  • how to avoid these difficulties in digital control
The general name for this kind of analysis where we mix digital control and continuous responses is "hybrid control".

Summary
  • Hybrid analysis allows one to mix continuous and discrete time systems properly.
     
  • Hybrid analysis should always be utilized when design specifications are particularly stringent and one is trying to push the limits of the fundamentally achievable.
     
  • The ratio of the magnitude of the continuous time frequency content at frequency $\omega$ to frequency content of the staircase form of the sampled output is

\begin{displaymath}\nonumber \Theta (s)=\frac{G_o(s)}{\left[G_oG_{h0}\right] _q(e^{s\Delta})} \end{displaymath}

  • The above formula allows one to explain apparent differences between the sampled and continuous response of a digital control system.
     
  • Sampling zeros typically cause $\left[G_oG_{h0}\right] _q(e^{j\omega\Delta})$ to fall in the vicinity of $\omega=\frac{\pi}{\Delta}$, i.e. $\vert\Theta (j\omega)\vert$ increases at these frequencies.
     
  • It is therefore usually necessary to ensure that the discrete complementary sensitivity has been reduced significantly below 1 by the time the folding frequency, $\frac{\pi}{\Delta}$, is reached.
     
  • This is often interpreted by saying that the closed loop bandwidth should be 20%, or less, of the folding frequency.
     
  • In particular, it is never a good idea to carry out a discrete design which either implicitly or explicitly cancels sampling zeros since this will inevitably lead to significant intersample ripple.