Page - (000058) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

50 6 Regulation 0.1 1 10 -80 -60 -40 -20 0 20 0.1 1 10 -80 -60 -40 -20 0 20 0.1 1 10 -80 -60 -40 -20 0 20 0 2 4 6 8 10 -1.0 -0.5 0.0 0.5 1.0 0 2 4 6 8 10 -1.0 -0.5 0.0 0.5 1.0 0 2 4 6 8 10 -1.0 -0.5 0.0 0.5 1.0 (a) (b) (c) (d) (e) (f) Fig. 6.3 Response of the process in Eq. 6.8 in blue and the transfer functions in Eq. 6.1, with Gud =Gηn in red,Gηd ingold, andGun ingreen.Top rowshowsBodemagnitudeplots.Bottom row shows impulse responses. The input signal weights in Eq. 6.4 areμ=1 and, for the three columns from left to right, ρ=0.25,1,4. Swapping values ofμ and ρ gives identical results, becauseof the symmetries inEq.6.1and6.4 Thebottomrowshowstheimpulseresponses.AswiththeBodeplots,anincrease inρ favors reduced responseofGηd, ingold, causingasmaller impulse response in the right plotwithhighρ relative to the left plotwith lowρ. Similarly, an increase in ρweakens the pressure on theGun transfer function in green, causing a larger impulse responsewith increasingρ. 6.4 FrequencyWeighting TheH2 norm sums a system’s gain over all input frequencies, as in Eq. 5.4. That sumweights all input frequenciesequally. Often,wewish toprotect against perturbations that occur primarily in a limited bandof frequencies. For example, disturbance loads,d, typically occur at low fre- quency,reflectinglong-termfluctuationsormisspecificationsinthesystem’sintrinsic processes. In that case, our optimizationmethod should emphasize reducing a sys- tem’s gain at low frequencywith respect to disturbance load inputs and accepting a tradeoff that allows agreater gain at high frequency.By reducing thegain at low frequency,weprotect against thecommonfrequencies for loaddisturbances. Tradeoffs between low-andhigh-frequencybands are common. Ifwe startwith aprocess transfer function G(s)= 10(s+1) s+10 , thenatzerofrequency,s= jω=0, thegainisone.Asfrequencyincreases, thegain approaches ten.