Digital Divide

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6 IEEE CONTROL SYSTEMS MAGAZINE » AUGUST 2006

Years ago, while I was teachinganalog-to-digital (A/D) conver-sion to a lab class, I stressed the

difficulty of obtaining true 18-bit data.The students seemed to follow thediscussion about noise without anyproblem, so a question at the endcaught me by surprise: “Isn’t the SonyPlayStation 32 bit?” Somehow my lec-ture failed to explain why the dataacquisition system in the lab was atleast 14 bits inferior to a householdvideo game.

In control applications, we continu-ally cross the fine line between the ana-log and digital worlds. Both worldshave their advantages and disadvan-tages—analog signals have theoreticallyinfinite resolution but are more suscep-tible to noise, whereas digital technolo-gy packs the analog continuum intoneat bins but introduces quantizationand discretization errors. There’s a lotto worry about on both sides.

Given our reliance on digital tech-nology, I sometimes wonder whethercontrol theory has somehow slighteddiscrete-time systems. Among theseveral hundred books on systemsand control sitting on my shelves,only a handful of books take a discrete-time approach. Why?

When I raise this question in casualconversation, I find that some of mycolleagues take a view that can besummarized as “digital effects are adetail.” Digital systems are continuallygetting faster, they argue, while suffi-ciently fast sampling makes discrete-time systems indistinguishable fromcontinuous-systems. I guess the keyword is “sufficient” since I’m thinkingabout those applications in whichsampling speed never seems to catchup with requirements.

I believe, however, that there is amore basic reason why many theoristsand practitioners favor the continuous-time setting, namely, that we have amuch better feel for continuous-timedynamics than for their discretizations.Since we spend a lot of time thinkingabout the physics of the plant before weset up the A/D and D/A hardware, we

become resistant to burying continuous-time insights in a blizzard of ks andk+1s. Unfortunately, dynamics aremessier and less intuitive in discretetime. I’ve even seen engineers approxi-mate digital control systems with s-domain block diagrams simply to giveother engineers a better understandingof how the systems work.

Tyrone Vincent and Dennis Bernstein overlooking Nelson Bay, near Newcastle, Australia.Photo taken courtesy of a friendly Australian.

Dennis Bernstein on a hike inside Barrington Tops, a World Heritage area with a rain forestclimate north of Newcastle, Australia.

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AUGUST 2006 « IEEE CONTROL SYSTEMS MAGAZINE 7

Discrete-time systems are alsomessier because of chaos. It’s easy tofind chaos in one-dimensional dynam-ics since trajectories can jump overeach other, but one has to examinethree-dimensional dynamics in contin-uous time to find it, which is exactlywhat Poincare and Lorenz did. Chaoswould have undoubtedly been discov-ered much earlier had there been moreinterest in discrete-time dynamics.

Kalman’s breakthrough state esti-mation paper was written for dis-crete-time systems. The analoganalogue entails much more sophisti-cated and challenging mathematics,thereby providing the opportunityfor researchers to publish impressivetheory papers. Interesting math isperhaps one motivation for cherish-ing continuous-time models.

If our goal were to validate continu-ous-time models, we could use sampleddata—the only real kind of data—toestimate physical parameters. But formodel-based digital controller tuning, itmakes sense to use the sampled data toconstruct discrete-time models. A true-blue digital control engineer would takediscrete-time data, identify a discrete-time model, tune a discrete-time con-troller, and never look back.

The interface between the continu-ous-time and discrete-time worldsalso provides opportunities forenhancing stability and performance.Instead of seeing aliasing as a scourgeto be prevented, some researchersexploit the aliased frequency responseto modify zeros, especially nonmini-mum-phase zeros, which drasticallyimpede performance in analog linear

time-invariant control. Likewise,replacing the traditional zero-orderhold with a nonconstant sampled-datahold function provides the means tomove nonminimum-phase zeros inadvantageous ways. These effects aremathematically subtle but have realpractical ramifications.

As a community of researchersand practitioners, we see these issuesdifferently because we work withdifferent problems with differentfeatures. Some of us work on oneside of the dividing line, some workon the other, and some straddle theboundary looking for opportunities.This is what we do.

Dennis S. BernsteinEditor-in-Chief

IEEE Control Systems Magazine