Page - (000036) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

26 3 BasicControlArchitecture open loopdirectly fromthe input to theoutput,L, dividedbyoneplus thepathway around the feedback loop,1+L. Ifwe assume in Fig.3.2a that there is no feedforward filter, so that F =1, and wedefine themainopen loopasL =CP, then theoutputη in response to the three inputs is η= L 1+Lr+ P 1+Ld− L 1+Ln, (3.7) in which each term is the open loop between the input signal and the output, η, dividedbyoneplus thepathwayaround the full loop,L. Ifwedefine S= 1 1+L T = L 1+L S+T =1, (3.8) with S as the sensitivity function andT as the complementary sensitivity function, then theerror is r−η= Sr−PSd+Tn. (3.9) This expression highlights the fundamental design tradeoffs in control that arise because S+T =1. Ifwe reduceT and the sensitivity tonoise,we increase S.An increase in S raises the error in relation to the reference signal, r, and the error in relation to the loaddisturbance,d. IfwereduceS,we increaseT and thesensitivity tonoise,n.These sensitivity tradeoffs suggest twoapproaches todesign. First, thesensitivitiesS(s)andT(s)dependon the input, s.Thus,wemayadjust the tradeoff at different inputs. For example, wemay consider inputs, s= jω, at variousfrequencies,ω.Sensornoise,n,oftenarisesasahighfrequencydisturbance, whereas the reference input, r, and the loaddisturbance,d, often followa low fre- quencysignal. Ifso, thenwecanadjust thesensitivity tradeoff tomatchthecommon input frequencies of the signals. In particular, at low frequency forwhich r and d dominate, wemay choose low S values whereas, at high frequency for which n dominates,wemaychoose lowT values. Second,wemayaddanadditionalcontrolprocess thatalters thesensitivity trade- off. For example, wemay use the feedforward filter, F, in Fig.3.2a, tomodulate the reference input signal.With thatfilter, the transfer function fromthe input,r, to theerroroutput, r−η, becomes1−FT . Ifweknowthe formofT with sufficient precision, we can choose FT ≈1, and thuswe can remove the sensitivity of the error to the reference input. Note that adjusting the tradeoff between S and T only requires an adjustment to the loop gain, L, which usually does not require precise knowledge about the system processes. By contrast, choosing F to cancel the reference input requires precise information about the form of T and the associated system processes. In otherwords, feedback is relativelyeasyand robust because it dependsprimarilyon adjusting gainmagnitude,whereas feedforward requires precise knowledge and is not robust tomisinformationorperturbation.