Page - (000024) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

14 2 ControlTheoryDynamics We can combine processes by simply multiplying the transfer functions. For example,supposewehaveanintrinsicexponentialdecayprocess,P(s), thatisdriven byoscillating inputs,U(s).That combinationproducesanoutput Y(s)=U(s)P(s)= ω0 (s+a)(s2+ω20) , (2.10) which describes a third-order differential equation, because the polynomial of s in thedenominatorhasahighestpowerof three. We could have easily obtained that third-order process by combining the two systemsofdifferentialequationsgivenabove.However,whensystemsincludemany processes incascades, includingfeedback loops, itbecomesdifficult tocombine the differentialequationsintoveryhigh-ordersystems.Multiplyingthetransferfunctions through the system cascade remains easy. That advantagewas nicely summarized byBode (1964), oneof the foundersof classical control theory Thetypical regulatorsystemcanfrequentlybedescribed, inessentials,bydifferentialequa- tionsofnomore thanperhaps the second, thirdor fourthorder.…Incontrast, theorder of the set of differential equations describing the typical negative feedback amplifier used in telephony is likely tobeverymuchgreater.Asamatter of idle curiosity, I oncecounted to findoutwhat theorderof thesetofequations inanamplifier Ihadjustdesignedwouldhave been, if I hadworkedwith thedifferential equationsdirectly. It turnedout tobe55. 2.4 Frequency,Gain,andPhase Howdosystemsperformwhenparametersvaryorwhen thereareexternal environ- mentalperturbations?Wecananalyzerobustnessbyusing thedifferentialequations to calculate the dynamics formany combinations of parameters and perturbations. However, suchcalculationsare tediousanddifficult toevaluate formore thanacou- pleofparameters.Usingtransfer functions,wecanstudyawiderangeofconditions byevaluatinga function’soutput response tovarious inputs. Thischapteruses theBodeplotmethod.Thatmethodprovidesaneasyandrapid way inwhich toanalyzea systemovervarious inputs.Wecanapply thismethod to individual transferfunctionsor tocascadesof transferfunctions thatcompriseentire systems. This section illustrates the method with an example. The following section describes thegeneral concepts andbenefits. Consider the transfer function G(s)= a s+a, (2.11) whichmatches the function for exponential decay inEq.2.9.Here, Imultiplied the functionbya so that thevaluewouldbeonewhen s=0.