Ro LOcus BAsICS BASIC F∈∈06 CK SYSTEM GA PLANT TRANSFER FUNCTION CONTROLLER T.F. KNC 2.D, MoNIC. CONT ROL COMM ANOS PoLYNoMIALS工川s SYSTEM OUT PUT 「; REFERENCE INPUT(EN·) RESPONSE ERRoR WE WILL 0ISCVSS THE PER FORMANCE GOALS N№R0 ETAIL LATER,f0RNu可s7 CONCENTRATE ON LOCAT IDN oF SySTEM POLES THIS IS THE WITy FEEOBACK FORM WE COULD PUT THE CONTRoL G IN THE FeEOBACK ATH WITHOUT CHANGING THE POLE LOCATIONS DIST URBANCES AD0ED LATER
BASIC QUESTIONS: ANALYSIS GIVEN Nc 0c WHERE Do THE CLOSED-LooP POLES GO A5 A FUACTIO of THE GAIN K. I.E. WHAT K To AclE- YNTHESIS GⅣENNP,D HOW SHOULO WE PICk K,Ne,Dc To GeT THE CLDSED POLES WHERE WE WANT THEM 小DE:BoPE5M∈ THAT WE KNOW WHERE THE CLOSED-LooP POLES SHOULD BE LOCATED, THIS THEME WILL RUN TH ROUGHOUT THE ENTIRE COURSE uElL5PE川AL7 F TIME LooK|NGAT工T。 BLOCK DIAGRAM ANALYSIS EASy To SHow THAT y P L 十G CLOSED-LooP CL K Np Dp k Nc Np TRANSFER FUNCTIO THE DENOMINATOR IS CHLLED THE (4) CHARACTERISTIc EQvATIDN +THE RD0TS OF ARE CALLED THE CLOSED-LooP POLES. POLES ARE CLEARLY A FUNCTIDN of K (FOR GIVEN Ne, Np, De, Dp)+ A"LOCUS OF AooTp
OBSE RVATIONS LOCUS IS CONSISTENT WITH FIXING E COMPENSATOR DYN AM 0THE小 CHANGING THE GAIA K ROOT LOCUS ENABLES US TO DETERMINE KEy FEATU RES OF THE CLOSE0-L0oP /STEM TRANSIENT) RESPONSE (FROM THE POLE LOCATLONS GIVEN THE OPEN-LooP N FORMATION 0c.k WILL SEE THAT IT IS HARD To INFER SoME PERFoRMANCE PRo PERTIES FRoM THIS LOCUS sE800ETEc川uT00 PPROACH SUGGESTE D BY W.R. EVANS Evans, W.R. Graphical analysis of control systems, AIEE Transactions, vol 67,, 1948, pp 547-551; and Control system synthesis by root locus method, AIEE Transactions, vol. 69, 1950, pp. 88-89
2 ROOT LOCUS ANALYSIS IN GENERAL THE FULL ROOT LDCUS IS VERy COMPLEX AND REQUIRES MATLAB TOOLS LIKE LOCus(AM,E心) EVANS ORIGINAL PAfER USES A SoURCES +SINks RNAL⊙GY FULL PLOTTING RULES A FPE AAGE 260 → NEED T0 EVELof SoMEβsc0RWNG 5KLsD心 E CAN Do“ BACK oF THG ENVEL0作 0ESG小s BAsIc POINTS: ASSUME Nc, Dc KNOWN LET T帖EN8c-A+KLd K REAL, PosITIVE CAN USE THIS TO ANSWER BASIC QUESTo N 5s。0A1 HE ROoT LoCs2 As:0NL工F凵(s)。\30。360°·L L6,1,2, k PosITive 1飞0°LDcs NEGATIVE LOCUs
BASIC QUESTIONS O WHERE DoEs THE LOCUS START POLES ib WHERE DOEs THE LoCUs END? ZERo s ASYMPTOTES ③WNEN/wERE工 s THE LOCUS ON THE REAL LINE LOCUs POINTS ON THE REAL LIE ARE千0 E LEFT0fN ODD NUMBER OF REAL-AXIs PoL∈SAA0zER0s ④G叫小MAT5。工5 ON THE LOCUS HAT GAIN1sNEE0ED16T训S CLDSED-LDoP PoLE kls: k De