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Root Locus Analysis & Design Root Locus Analysis & Design • A designer would like: – To know if the system is absolutely stable and the degree of stability. – To predict a system’s performance by an analysis that does NOT require the actual solution of the differential equations. – The analysis to indicate readily the manner or method by which this system must be adjusted or compensated to produce the desired performance characteristics. Mechatronics K. Craig Root Locus Analysis and Design 1 • Two Methods are available: – Root-Locus Approach – Frequency-Response Approach • Root Locus Approach – Basic characteristic of the transient response of a closed-loop system is closely related to the location of the closed-loop poles. – If the system has a variable loop gain, then the location of the closed-loop poles depends on the value of the loop gain chosen. Mechatronics K. Craig Root Locus Analysis and Design 2 – It is important to know how the closed-loop poles move in the s plane as the loop gain is varied. – From a Design Viewpoint: • Simple gain adjustment may move the closed-loop poles to desired locations. The design problem then becomes the selection of an appropriate gain value. • If gain adjustment alone does not yield a desired result, addition of a compensator to the system is necessary. – The closed-loop poles are the roots of the closed-loop system characteristic equation. Mechatronics K. Craig Root Locus Analysis and Design 3 – The Root Locus Plot is a plot of the roots of the characteristic equation of the closed-loop system for all values of a system parameter, usually the gain; however, any other variable of the open- loop transfer function may be used. – By using this method, the designer can predict the effects on the location of the closed-loop poles of varying the gain value OR adding open-loop poles and/or open-loop zeros. – A designer MUST know how to generate the root loci of the closed-loop system BOTH by hand and with a computer (e.g., MatLab). Mechatronics K. Craig Root Locus Analysis and Design 4 – Experience in sketching the root loci by hand is invaluable for interpreting computer-generated root loci, as well as for getting a rough idea of the root loci very quickly. Mechatronics K. Craig Root Locus Analysis and Design 5 • Underlying Principle: – Poles of the closed-loop transfer function are related to the zeros and poles of the open-loop transfer function and also to the gain. – The values of s that make the open-loop transfer function equal to –1 must satisfy the characteristic equation of the closed-loop system. – The root-locus plot clearly shows the contributions of each open-loop pole and zero to the locations of the closed-loop poles. Mechatronics K. Craig Root Locus Analysis and Design 6 – The root-locus plot also shows the manner in which the open-loop poles and zeros should be modified so that the response meets system performance specifications. Mechatronics K. Craig Root Locus Analysis and Design 7 • Example: Motor Position Control System A q (s) K G(s) =o ==J E(s) æöB s(sa+ ) Assume a = 2. ssç÷+ èøJ 2 C(s)KK wn ===222 R(s)s(s+2Ks) ++2s+Ks+2szwnn+w 1 w=K z= n K Mechatronics K. Craig Root Locus Analysis and Design 8 Problem: Determine the roots of the characteristic equation for all values of K and plot these roots in the s plane. Roots of the characteristic equation are given by: Root-Locus s=-1±-1K 1,2 Plot Note: s = -2 and s = 0 are the open-loop poles Mechatronics K. Craig Root Locus Analysis and Design 9 – For 0 < K < 1, the roots are real and lie on the real axis. For K > 1, the roots are complex. – Once the root-locus plot has been obtained, it is possible to determine the variation in system performance with respect to a variation in K. 2 C(s) wn = 22 R(s)s+2sVwnn+w 2 s1,2=-Vwnn±i1w-V si1,2d=-s±w Mechatronics K. Craig Root Locus Analysis and Design 10 – As K is increased from K = 1, we observe: • A decrease in the damping ratio z. This increases the overshoot of the time response. • An increase in the undamped natural frequency wn. • An increase in the damped natural frequency wd. • No effect on the rate of decay s. • No matter how much the gain is increased in this simple linear second-order system, the system can never become unstable. Mechatronics K. Craig Root Locus Analysis and Design 11 Second-Order System Unit Step Response 1.8 tr » rise time wn 4.6 ts » settling time Vwn -pV -st æös 2 y(t) =1-eç÷coswddt+wsint 1-V Mp =e (0£z<1) overshoot èøwd æöz =ç÷1- (0£z£0.6) èø0.6 Mechatronics K. Craig Root Locus Analysis and Design 12 Constant Parameter Curves on the S Plane Mechatronics K. Craig Root Locus Analysis and Design 13 1.8 4.6 w³n s³ t r ts z³0.6(1-Mp ) 0£z£0.6 Time-Response Specifications vs. Pole-Location Specifications Mechatronics K. Craig Root Locus Analysis and Design 14 • General effects of the addition of poles: – pull root locus to the right – lower system’s relative stability – slow down the settling of the response • Compare root locus plots of: 111 ÞÞ (s+4)(s+4)(s+2)(s+4s2s1)( ++)( ) Mechatronics K. Craig Root Locus Analysis and Design 15 • General effects of the addition of zeros: – pull root locus to the left – makes system more stable – speed up the settling of the response • Compare root locus plots of: 1(s+ 6) Þ (s++4)(s2)(s+1)(s+++4)(s2)(s1) (s++3)(s1.5) ÞÞ (s++4)(s2)(s+1)(s+++4)(s2)(s1) Mechatronics K. Craig Root Locus Analysis and Design 16 Step Response From: U(1) 2.5 2 zs1+ G(s) = s2 ++s1 1.5 To: Y(1) Amplitude 1 Mp • 0.5 Increasing z tr ¯ tp ¯ 0 0 2 4 6 8 10 12 Time (sec.) z = 0, 0.26, 0.61, 1.43, 3.33 Mechatronics K. Craig Root Locus Analysis and Design 17 Bode Diagrams From: U(1) 20 0 -20 -40 BW • -60 M • Increasing z -80 r 50 0 -50 Phase (deg); Magnitude (dB) -100 To: Y(1) Increasing z -150 -200 10-2 10-1 100 101 102 zs1+ z = 0, 0.26, 0.61, 1.43, 3.33 Frequency (rad/sec) G(s) = s2 ++s1 Mechatronics K. Craig Root Locus Analysis and Design 18 Step Response From: U(1) 1.4 1.2 1 Mp ¯ 0.8 Increasing p tr • To: Y(1) Amplitude 0.6 tp • 0.4 1 G(s) = 2 0.2 (ps+1)(s++s1) 0 0 5 10 15 20 p = 0, 0.26, 0.61, 1.43, 3.33 Time (sec.) Mechatronics K. Craig Root Locus Analysis and Design 19 1 Bode Diagrams G(s) = 2 (ps+1)(s++s1) From: U(1) 50 0 -50 BW ¯ Increasing p -100 Mr ¯ -150 0 -100 Phase (deg); Magnitude (dB) To: Y(1) -200 Increasing p -300 10-2 10-1 100 101 102 p = 0, 0.26, 0.61, 1.43, 3.33 Frequency (rad/sec) Mechatronics K. Craig Root Locus Analysis and Design 20 • Root-Locus Plots: Angle & Magnitude Conditions D(s) R(s) E(s) + C(s) + S G (s) S G (s) + c - B(s) H(s) C(s) G(sGs) ( ) = c R(s) 1G+ c (sGsHs) ( ) ( ) Mechatronics K. Craig Root Locus Analysis and Design 21 – The characteristic equation of the closed-loop system is: 1G+=c (sGsH) ( ) (s0) Gc (sGsH) ( ) (s1) =- – Here we assume that Gc(s)G(s)H(s) is a ratio of polynomials in s. – Gc(s)G(s)H(s) is a complex quantity: • Angle Condition o ÐGc (sGsH) ( ) (s) =±+180(2k1) (k = 0, 1, 2, ...) • Magnitude Condition Gc (sGsH) ( ) (s1) = Mechatronics K. Craig Root Locus Analysis and Design 22 – The values of s that fulfill both the angle and magnitude conditions are the roots of the characteristic equation, or the closed-loop poles. – A plot of the points of the complex plane satisfying the angle condition alone is the root locus. – The roots of the characteristic equation (the closed-loop poles) corresponding to a given value of the gain can be determined from the magnitude condition. Mechatronics K. Craig Root Locus Analysis and Design 23 – Gc(s)G(s)H(s) often involves a gain parameter K and the characteristic equation may be written as: K(s+z)(s++z) (sz) 10+=12mL (s+p1)(s++p2n)L(sp) – The root loci for the system are the loci of the closed-loop poles as the gain K is varied from zero to infinity. – To begin sketching the root loci we must know the location of the poles and zeros of Gc(s)G(s)H(s). Mechatronics K. Craig Root Locus Analysis and Design 24 – Consider: K(sz+ 1 ) Gs( )Gc (sHs) ( ) = (s++++p1234)(sp)(sp)(sp) ÐGc(sGsHs) ( ) ( ) =f11234-q-q-q-q Ksz+ 1 KB1 Gc (sGsHs) ( ) ( ) == s++++p123spspsp4AAAA1234 Mechatronics K. Craig Root Locus Analysis and Design 25 – Since the open-loop complex-conjugate poles and complex-conjugate zeros, if any, are always located symmetrically about the real axis, the root loci are always symmetrical with respect to the real axis. Mechatronics K. Craig Root Locus Analysis and Design 26 D(s) Generalized R(s) E(s) + C(s) + Block S G (s) S G (s) + c Diagram - B(s) H(s) C(s) G(s)G(s) = c Assume: D(s) = 0 R(s)1+ Gc (s)G(s)H(s) Nc (s) N(s)Nh (s) Gch(s)= G(s)== G(s) Dch(s)D(s)D(s) C(s) NND = ch Closed-Loop Zeros consist R(s)DcDDh+ NchNN of the zeros of Gc and G and the poles of H Mechatronics K.
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