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Chapter9 StateFeedback A transfer function corresponds to a time-invariant, linear systemof ordinary dif- ferential equations. In an earlier chapter, I showed the general form of a transfer function inEq.2.5and theunderlyingdifferential equations inEq.2.6. For example, the transfer function P(s)=1/(s+a)with input u and output y corresponds to the differential equation x˙ =−ax+u, with output y= x. Here, x is the internal stateof theprocess.Models thatworkdirectlywith internal statesare called state-spacemodels. Transfer functionsprovide significant conceptual andanalytical advantages.For example, themultiplication of transfer functions and the simple rules for creating feedback loops allow easy creation of complex process cascades.With regard to systemresponse, aBodeplot summarizesmanyaspects ina simple,visualway. However, itoftenmakessensetoanalyzetheunderlyingstatesdirectly.Consider, for example, the regulation of an organism’s body temperature.We could model performance and cost in terms of body temperature. Alternatively, the underlying statesmayinclude theburningofstoredenergy, theriseandfallofvarioussignaling molecules, thedilationofbloodvessels, andsoon. Directanalysisof those internal statesprovidesadvantages.The individualstates mayhave associated costs,whichwecould studydirectly in our cost function.We couldconsider theregulatorycontrolof the individualstates rather thantemperature because temperature isanaggregateoutcomeof theunderlyingstates.Forexample, eachstatecouldberegulated throughfeedback, inwhich thefeedbackintoonestate maydependon thevaluesofallof thestates, allowingmore refinedcontrolofcosts andperformance. Whenwe use a state-space analysis, we do not have to give up all of the tools of frequencyanalysis thatwedevelopedfor transfer functions.Forexample,wecan consider the responseofa systemtodifferent input frequencies. State-space models can also describe time-varying, nonlinear dynamics. The responseofanonlinearsystemwillchangewithitsunderlyingstate,whereastransfer functionsystemshaveaconstant frequency response. ©TheAuthor(s)2018 S.A.Frank,ControlTheoryTutorial, SpringerBriefs inAppliedSciences andTechnology,https://doi.org/10.1007/978-3-319-91707-8_9 69