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243 Control Systems Root_LocusMurphy

# 243 Control Systems Root_LocusMurphy - Root Locus...

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Root Locus Techniques William J. Murphy Washington University in St. Louis Reference: Nise (2004): Chapter 8

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10/12/04 ESE 441 2 TOPICS The Root Locus Method Closed-loop poles Plotting the root locus of a transfer function Choosing a value of K from root locus Closed-loop response Key MATLAB commands used: feedback, rlocfind, rlocus, sgrid, step, pzmap, zpk , rltool
10/12/04 ESE 441 3 Root Locus Method Closed-loop response depends on the location of closed-loop poles. If system has a variable design parameter (e.g., a simple gain adjustment or the location of compensation zero), then the closed-loop pole locations depend on the value of the design parameter. The root locus of a system is the plot of the paths (loci) of all possible closed loop poles as the design parameter takes on a range of possible values.

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10/12/04 ESE 441 4 Root Locus Method (cont.) The poles that provide the desired closed-loop response are selected and the proper value of the design parameter is thereby established. The closed-loop poles are the roots of the system's characteristic equation. Since finding the roots of polynomials of degree higher than 3 is laborious, graphical aids were devised in the late 1940s to help construct the root loci. Recently, computer-aided design tools such as Matlab provide a convenient computer solution. The older, graphical aids are still relevant since the ability to quickly sketch root loci by hand is invaluable in making fundamental decisions early in the design process and in checking Matlab results.
10/12/04 ESE 441 5 Closed-Loop Poles Closed-Loop Transfer Function Characteristic Equation C s R s G s G s H s ( ) ( ) ( ) ( ) ( ) = + 1 where G(s) is the forward-path transfer function and H(s) is the feedback-path transfer function The poles of the closed loop system are values of s such that 1 + G(s) H(s) = 0 or G(s) H(s) = -1 for G(s)H(s) =k num(s)/den(s) , then this equation has the form: den s knum s den s k num s ( ) ( ) ( ) ( ) + = + = 0 0

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10/12/04 ESE 441 6 Geometric Properties Therefore: |G(s) H(s)| = 1 Magnitude Condition angle(G(s) H(s)) = odd multiple of 180 deg Angle Condition ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 < + = + - - + + + = + + = = + = + = = 0 K for 180 * * 2 0 K 180 * ) 2 1 ( 1 1 & | 1 | | 1 | | | | | 1 1 ) ( ) ( N N 1 1 1 1 q for q s s N s s s s K s s s K s H s G j j k k m k k n N j j N N n N j j N m k k N τ τ τ τ τ τ
10/12/04 ESE 441 7 Root Loci - Their Number, Starting Points and Ending Points As k ranges from 0 to infinity the closed-loop poles migrate from the open-loop poles to the open-loop zeros . The path of a closed-loop pole on the s-plane is called a branch of the root locus. No matter what we pick k to be, the closed-loop system must always have n poles , where n is the number of poles of GH(s). The root locus must have n branches , each branch starts at a pole of GH(s) and goes to a zero of GH(s).

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