Page - (000107) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

104 14 Summary Many specificmethods refine the deployment of feedback. For example, filters reduce the resonant peaks in system response. Controllers modulate dynamics to improvestabilitymargin. A large stabilitymarginmeans that the systemcanmaintain stability even if the true process dynamics depart significantly from the simple linear model used to describe thedynamics. 14.3 DesignTradeoffsandOptimization Awell-performingsystemmovesrapidlytowardthedesiredsetpoint.However,rapid response can reduce stability. For example, a strong response to error can cause a systemtoovershootitssetpoint.Ifeachovershootincreasestheerror,thenthesystem diverges fromthe target. The fast response of a high-performing systemmay destabilize the system or make itmore sensitive todisturbances.A tradeoffoccursbetweenperformanceand robustness. Manyothertradeoffsoccur.Forexample,controlsignalsmodulatesystemdynam- ics.Theenergy required toproduce control signalsmaybeexpensive.Thecosts of control signals tradeoff against thebenefitsofmodulating the systemresponse. Thesensitivityofasystemtoperturbationsvarieswiththefrequencyatwhichthe signaldisturbs thesystem.Often, reducedsensitivity toonesetof frequencies raises sensitivity toanother setof frequencies. Optimization provides a rigorous design approach to tradeoffs.Onemay assign costsandbenefitstovariousaspectsofperformanceandrobustnessortotheresponse at different frequencies.Onecan thenconsider howchanges in systemdesignalter the total balance of the various costs and benefits. Ideally, one finds the optimal balance. 14.4 FutureDirections Control theory remains a very active subject (Baillieul andSamad2015).Methods suchas robustH∞ analysis andmodelpredictivecontrol are recentdevelopments. Computational neural networks have been discussed for several decades as a method for the control of systems (Antsaklis 1990). Computational networks are looselymodeled after biological neural networks.Aset of nodes takes inputs from the environment. Each input node connects to another set of nodes. Each of those intermediate nodes combines its inputs to produce an output that connects to yet another set of nodes, and so on. The final nodes classify the environmental state, possibly takingactionbasedon that classification (Nielsen2015;Goodfellowet al. 2016).