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## 13. Digital Control

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Models for discrete time systems have been described in Chapter 1. There we saw that digital and continuous systems were actually quite close. Hence it is often true that digital responses approach the corresponding continuous response as the sampling period goes to zero. For this reason, in the remainder of the book we will present continuous and discrete ideas in parallel. The purpose of the current chapter is to provide a smooth transition to this latter work by highlighting the special issues associated with digital control. In particular, the chapter covers:

• why one cannot simply treat digital control as if it were exactly the same as continuous control, and
• how to carry out designs for digital control systems so that the at-sample response is exactly treated.

### Summary

• There are a number of ways of designing digital control systems:
• design in continuous time and discretize the controller prior to implementation;
• model the process by a digital model and carry out the design in discrete time;

• Continuous time design which is discretized for implementation;
• Continuous time signals and models are utilized for the design.
• Prior to implementation, the controller is replaced by an equivalent discrete time version.
• Equivalent means that simply maps s to (where is the delta operator).
• Caution must be exercised since the analysis was carried out in continuous time and the expected results are therefore based on the assumption that the sampling rate is high enough to mask sampling effects.
• If the sampling period is chosen carefully, in particular with respect to the open and closed loop dynamics, then the results should be acceptable.

• Discrete design based on a discretized process model:
• First the model of the continuous process is discretized.
• Then, based on the discrete process, a discrete controller is designed and implemented.
• Caution must be exercised with so called intersample behavior: the analysis is based entirely on the behavior as observed at discrete points in time, but the process has a continuous behavior also between sampling instances.
• Problems can be avoided by refraining from designing solutions which appear feasible in a discrete time analysis, but are known to be unachievable in a continuous time analysis (such as removing non-minimum phase zeros from the closed loop!).

• The following rules of thumb will help avoid intersample problems if a purely digital design is carried out:
• Sample 10 times the desired closed loop bandwidth.
• Use simple anti-aliasing filters to avoid excessive phase shift.
• Never try to cancel or otherwise compensate for discrete sampling zeros.
• Always check the intersample response.