Cranking up the gain Cranking up the gain Cranking up the gain: poles and step responseRoot locus for nonunity feedback systemsRoot locus terminologyRoot-locus sketching rulesRoot-locus sketching ruleslecture17_Matlab.pdftocDesigning CompensatorsSISO Design ToolComponents of the SISO ToolDesign Options in the SISO ToolLecture 17 – Monday, Oct. 172.004 Fall ’07 Cranking up the gain ☺Type 0 system (no disturbance)Type 1 system with disturbanceK0.3162s(s+2)Vref(s)X(s)Steady—state error due to step input:eR(∞)=0Steady—state error due to step disturbance:eD(∞)=−1KeD(∞) → 0asK →∞Steady—state error due to step input:eR(∞)=22+0.3162KeR(∞) → 0asK →∞K0.3162s+2Vref(s)V (s)K0.3162s+2K0.3162s(s+2)V (s)Vref(s)Vref(s)X(s)Lecture 17 – Monday, Oct. 172.004 Fall ’07 Cranking up the gain Type 1 system (no disturbance)K0.3162s(s+2)Vref(s)X(s)Closed—loop transfer functionX(s)Vref(s)=0.3162Ks2+2s +0.3162KPole locationsp1= −1+√1 − 0.3162Kp2= −1 −√1 − 0.3162KSystem becomes underdampe d⇒⇒ step response overshoots if1 − 0.3162K<0 ⇔ K>3.1626X(s)Vref(s)Lecture 17 – Monday, Oct. 172.004 Fall ’07 Cranking up the gain: poles and step responseK increasingK<3.1626K increasingK>3.1626RootLocusLecture 17 – Monday, Oct. 172.004 Fall ’07 Root locus for nonunity feedback systemsClosed loop TF:“Open loop” TF:G(s)H(s)Nise Figure 8.1Caveat: K>0Closed-loop pole locations1+KG(s)H(s)=0⇒½K =1/ |G(s)H(s)| ;6KG(s)H(s)=(2n +1)180◦.Forward transfer functionKG(s)Ea(s) C(s)R(s)+-InputOutputActuatingsignal (error) H(s)Feedback transferfunctionKG(s)1 + KG(s)H(s)C(s)R(s)Figures by MIT OpenCourseWare.Lecture 17 – Monday, Oct. 172.004 Fall ’07 Root locus terminology-j3-j2-j1j1-j1j10 1-1-1-j10 1 2 3 4-1-2-3-4j2j32 + j42 - j4-j2-j3-j4-j50-j4-j2-j6j2j4j6-2-5-10-2-3-4-2-3-4AsymptoteBranchesAsymptoteAsymptotereal-axis interceptReal-axissegmentBreakaway pointBreakaway pointBreak-in pointAsymptoteRL imaginary axisinterceptAsymptoteangles-planes-planes-planes-planeDeparture/Arrival anglesjωjωσσjωjωσ σ2j2j3j4j5ζ = 0.45j1X X X XXX XXX XFigure by MIT OpenCourseWare.Lecture 17 – Monday, Oct. 172.004 Fall ’07 Root-locus sketching rules• Rule 1: # branches = # poles• Rule 2: symmetrical about the real axis• Rule 3: real-axis segments are to the left of an odd number of real-axis finite poles/zerosRecall angle condi tion for closed—loop pole:6KG(s)H(s)=(2n + 1)180◦.Complex—p o le/zero contributions: cancelbecause of symmetryReal—pole/zero contributions: each is0◦from the left, 180◦from the right;total contributions from right must beodd number of 190◦’s to satisfy angle condition.Let G(s)=NG(s)DG(s),H(s)=NH(s)DH(s).⇒6G(s)H(s)=X6(poles) −X6(zeros).Image removed due to copyright restrictions. Please see: Fig. 8.8 in Nise, Norman S. Control Systems Engineering. 4th ed. Hoboken, NJ: John Wiley, 2004.Lecture 17 – Monday, Oct. 172.004 Fall ’07 Root-locus sketching rules• Rule 4: RL begins at poles, ends at zerosExample-j1j1-1-2-3-4s-planeσX XNise Figure 8.10If K → 0+(small gain limit)Closed—loo p TF(s) ≈KNG(s)DH(s)DG(s)DH(s)+²⇒closed—lo op denominator is denom inator of G(s)H(s )⇒closed—lo op poles are the poles of G(s)H(s).If K → +∞ (large gain limit)Closed—loop TF(s) ≈KNG(s)DH(s)² + KNG(s)NH(s)⇒closed—lo op denominator is numerator of G(s)H(s)⇒closed—loop poles are the zeros of G(s)H(s).Let G(s)=NG(s)DG(s),H(s)=NH(s)DH(s).⇒ Closed—loop TF(s)=KNG(s)DH(s)DG(s)DH(s)+KNG(s)NH(s).jωFigure by MIT OpenCourseWare.Please see the following selections from Mathworks, Inc. "Control System Toolbox Getting Started Guide." http://www.mathworks.com/access/helpdesk/help/pdf_doc/control/get_start.pdf Ch. 4, pp.
View Full Document
Unlocking...