Page - (000034) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

24 3 BasicControlArchitecture C(s) y P(s) ur F(s) e d y n Controller Process (a) P d C u y n P y w C u z (b) (c) Fig. 3.2 Closed-loop feedback.aAnextended feedback loopwith inputs for disturbance,d, and noise,n.The functionF(s)maybeused tofilter the reference input, providingaseconddegreeof freedominadditiontothemaincontroller,C(s).Thesystemcanbedividedintointrinsicprocesses that cannot be adjusted directly and designed processes of control that can be adjusted.Note the inputs foreachblock:r and y for thecontroller, andu,d, andn for theprocess.b In thispanel, the blocks P andC represent themulticomponent process and control blocks from the upper panel. The reference signal is assumed tobe zero, allowingone to focus on the roles of disturbance and noise inrelation tosystemstability.cAnabstractionof thefeedbackprocess, inwhich thevector y includesall thesignals fromtheprocess to thecontroller,u includesall thecontrol input signals to the process,w includes all the extrinsic inputs, and z includes any additional signal outputs from theprocess. RedrawnfromÅströmandMurray (2008),©PrincetonUniversityPress Second,wecanaddintegralcontrolbyincludingthetermki/s tothecontroller.We canunderstandwhy this termisan integratorbyconsideringa fewstepsofanalysis thatextendearlierequations.MultiplyingEq.2.5by1/s increases thedenominator’s order of its polynomial in s. That increase in the exponents of s corresponds to an increaseintheorderofdifferentiationforeachtermontheleftsideofEq.2.4,which isequivalent to integratingeachtermontherightsideof thatequation.Forexample, ifwestartwith x˙ =u and then increase theorderofdifferentiationon the left side, x¨ =u, this newexpression corresponds to the original expressionwith integration of the input signal, x˙ = ∫ udt. Integratingtheinputsmoothsouthigh-frequencyfluctuations,actingasafilterthat passes low-frequency inputs andblockshigh-frequency inputs. Integrationcausesa slower, smoother, and oftenmore accurate adjustment to the input signal. A term such as a/(s+a) is an integrator for large s and a pass-through transfer function withvalueapproachingone for small s.