Control System Theory

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TIMEDOMAINGENERALNON- LINEARCASE ODEREPRESENTATION SSREPRESENTATION TF/IRREPRESENTATION dy {ict)= 5-(✗(t),ult))da = f(Ylt),ult)) > - - ✗(f)= ✗(t) - - s YCS)= GCS). Us) Onu, FORCED RESPONSE GCS)= GssIT(S- qi)+✗(a)= ✗o µIT(S- pi) RECALL (Hp: ✗(a)=-0)ult)Isa) ult)= Slt)e > UCS)= 1 -t ult)= STEPH)e s U(s)= %"""" 1 ' i Ynlt)= hlt)*sct) >t LIynlt))= HLS)>TF ult)= RAMPA)< > Ucs)= 11g, ""'^ j >t f-(ynlt))= Hcjw)-FR r RESPONSECHARACTERIZATION IMPULSERESPONSEUcs)- -1 STEPRESPONSEUCS)- _ 11s • Finalvaluetheorem: y(t→ a)= limSYCS)= limSGCSJUCS) y(t→a)= eimSGCS) ylt→a)= limGCS)- - G" s→o s→o Soso Soso Omen ✗(g)= limGCS)= {gssm.in• Initiovaluetheorem: YCÒ)= limSYCS)= limSGCSNCS) ycot)= limSGCS) Sasao Sasao Sasao Sasao ÌCÒ)= lims'YCS)- SY(Ot) flot)= limsls- a)Gcs) [(Ot)= limSGCS)-- {G" man s→ao s→• +ao m=m S→ co ✗(t)SPECIFICCASES Mt)è Gcs) s - ^ considerinotoendosethePLANTGCS)ina SIMPLECLOSEDLOOPwitha CONSTANTGAINK ti • GCS)isa1°ORDERTF(orgenericM- M-1) GAIN Gss, GSS G(g)= Gss^ ✓(5)= ①(s) ZEROESX (1+Ts) 1+KGCS)= 1+KGS> +Ts →> POLES1+146" -1 IMPULSERESPONSE @OPENLOOP = IL-0 So1+124%1 ✗(tra)= eimSG% OPENLOOPRESPONSEs→o(s+1Gss)= ° Ylt→a)= 0 CLOSEDCOOPRESPONSE Y(Ot)= c'%thatis §' ,inIabslis <ità)@CLOSED ggssy, %(0+1=-4%-2→ loop 1 , ylò)= lim>→00(s+Ayers)= G% le,iii. y.co/-)=limSGss/Tc'%(si- s'- s'È"" )= - gsspu.mg» ) ; ; sia (s-11Gss)- SCÈ= him (si1tk)S→00 1-2 2 22 . . . STEPRESPONSE @OPENLOOP Gssy, = Mioso1+114%1 OPENLOOPRESPONSE ✗(t→ao)= lim ,+mass CLOSEDCOOPRESPONSE s→o(s+1Gss)= €» y(+→a)= Gss- thatis >thouthe ✗(Ot)= 0 thatis @ÈOSED y.co.za" " ,, ✗(Ot)= limG» /T t → loop S→ao(s+1 Gss)= ① I Ì(Ot)= limsessista(s- 1 Gss)- 6= G" , • GCS)isa2°ORDERTF(orgenericM- M- -2) withCOMPLEXandCONJUGATEDPOLES GCS)= Gss lui s'+2Guns+Un' P«=±jÙ×#PARAMETERS *NaturalfrequencyWn → Un'= papa→ ifPs=…