Systems Analysis and Control

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1 Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 13: Root Locus Continued

2 Overview In this Lecture, you will learn: Review Definition of Root Locus Points on the Real Axis Symmetry Drawing the Real Axis What Happens at High Gain? The effect of Zeros Asymptotes M. Peet Lecture 13: Control Systems 2 / 31

3 Root Locus Review Root Locus Imaginary Axis Real Axis Definition 1. The Root Locus of G(s) is the set of all poles of as k ranges from 0 to kg(s) 1 + kg(s) M. Peet Lecture 13: Control Systems 3 / 31

4 Root Locus Review G(s) = (s z 1) (s z m ) (s p 1 ) (s p n ) < s-p 1 Im(s) < s-z Re(s) < s-p 2 For a point on the root locus: m n G(s) = (s z i ) (s p i ) = 180 i=1 i=1 M. Peet Lecture 13: Control Systems 4 / 31

5 Root Locus Demo 1 Wiley+ Root Locus Demo 1 M. Peet Lecture 13: Control Systems 5 / 31

6 Property of Symmetry Constructing the Root Locus Root Locus Imaginary Axis Real Axis Symmetry: Complex roots come in pairs: a ± bı. Points on the root locus are mirrored above/below the real axis We can divide points on root locus into Points on the real axis Symmetric Pairs off the real axis. M. Peet Lecture 13: Control Systems 6 / 31

7 Points on the Real Axis Phase Contribution from Zeros What is the PHASE of a point (s = a) on the Real Axis? We need: m n (a z i ) (a p i ) = 180 i=1 i=1 Phase Contribution from Zeros (from z i ): Im(s) < (a-z)=180 o Case 1: If z i is on the Real Axis If a > z i, then (a z i ) = 0 a is right of the zero. a z i Re(s) If a < z i, then (a z i ) = 180 a is left of the zero. Contribution is 0 or 180! M. Peet Lecture 13: Control Systems 7 / 31

8 Root Locus on the Real Axis Phase Contribution of Complex Zeros Case 2: If the z i is Complex: Complex Roots some in pairs. z 1,2 = b ± cı First Zero: Trigonometry (a z 1 = a b + cı) (a z 1 ) = tan 1 ( a b c Second Zero: ( (a z 2 ) = tan 1 a b ) c = (a z 1 ) ) < a-z 1 < a-z 1 = - < a-z 2 Since (a z 1 ) = (a z 2 ), the total contribution to phase is 0! (a z 1 ) + (a z 2 ) = 0 Im(s) a Re(s) M. Peet Lecture 13: Control Systems 8 / 31

9 Root Locus on the Real Axis Phase Contribution from Real Poles The Poles Similar to zeros m n (a z i ) (a p i ) = 180 i=1 i=1 Case 1: If p i is Real If a is right of p i, then (a p i ) = 0 If a is left of p i, then (a p i ) = 180 Im(s) < s-p=180 o Re(s) M. Peet Lecture 13: Control Systems 9 / 31

10 Root Locus on the Real Axis Phase contribution from Complex Poles Case 2: If the p i are Complex: Complex Roots some in pairs. z 1,2 = b ± cı First Pole: (a p 1 ) = tan 1 ( a b c Second Pole: ) < a-p 1 Im(s) (a p 2 ) = (a p 1 ) Re(s) < a-p 1 = - < a-p 2 Again, (a p 1 ) = (a p 2 ) so the Contribution is 0! M. Peet Lecture 13: Control Systems 10 / 31

11 Root Locus on the Real Axis ON-OFF Summary: A point on the real axis: s = a Complex poles and zeros don t matter Real poles and zeros contribute 0 or if the pole/zero is to the left of a 180 if the pole/zero is to the right of a The PHASE of G(a) is G(a) = 180 (# of poles and zeros to the right of a) A Simple Rule: If the # of poles and zeros to the right of a is EVEN. We are OFF the root locus. If the # of poles and zeros to the right of a is ODD. We are ON the root locus. M. Peet Lecture 13: Control Systems 11 / 31

12 Root Locus on the Real Axis Examples M. Peet Lecture 13: Control Systems 12 / 31

13 Root Locus on the Real Axis Examples 4 Root Locus 3 2 Imaginary Axis Real Axis M. Peet Lecture 13: Control Systems 13 / 31

14 Root Locus on the Real Axis Examples G(s) = 1 s Root Locus Imaginary Axis Real Axis M. Peet Lecture 13: Control Systems 14 / 31

15 Root Locus on the Real Axis Examples The inverted pendulum with some derivative feedback: ˆK(s) = k(1 + s) 0.8 Root Locus Imaginary Axis Real Axis M. Peet Lecture 13: Control Systems 15 / 31

16 The Root Locus at High Gain k Now lets look at what happens when gain increases. 8 Root Locus 6 4 Imaginary Axis Real Axis Figure : Suspension Problem Conclusion: Some stable poles oscillate more. M. Peet Lecture 13: Control Systems 16 / 31

17 The Root Locus at High Gain G(s) = (s + 3)(s + 4) (s + 1)(s + 2) Conclusion: Nothing Much Happens. M. Peet Lecture 13: Control Systems 17 / 31

18 The Root Locus at High Gain Again, Inverted Pendulum with derivative feedback. 0.8 Root Locus Imaginary Axis Real Axis Conclusion: Poles get More Stable. M. Peet Lecture 13: Control Systems 18 / 31

19 The Root Locus at High Gain G(s) = s + 3 s(s + 1)(s + 2)(s + 4) Conclusion: Poles Destabilize. Notice the Asymptotes. M. Peet Lecture 13: Control Systems 19 / 31

20 The Root Locus at High Gain So what happens when k is large? Logically, there are TWO CASES: Poles can remain small. Poles can get big. Imaginary Axis Root Locus Consider the Small Poles ( s < ). G(s) = n(s) d(s) OL zeros are roots of n(s) = 0. OL poles are roots of d(s) = 0. Now, closed loop: kg(s) 1 + kg(s) = kn(s) d(s) + kn(s) CL poles are roots of d(s) + kn(s) = 0 Imaginary Axis Real Axis Root Locus Real Axis M. Peet Lecture 13: Control Systems 20 / 31

21 The Root Locus at High Gain At high gain, small CL poles are roots of d(s) + kn(s) = 0 Im(s) If s is small, then d(s) is small Hence as k, d(s) + kn(s) = kn(s) As k, small poles satisfy Re(s) d(s) + kn(s) = kn(s) = 0 which means n(s) = 0!!! n(s) = 0 means s is an OL zero! At high gain, small CL poles are attracted by OL Zeros. M. Peet Lecture 13: Control Systems 21 / 31

22 The Root Locus at High Gain Asymptotics Now consider the other possibility: s also gets Very Large d(s) + kn(s) = Root Locus In this case, d(s) is not small. Very Large solutions of 1 + kg(s) are called asymptotics. asymptotics increase forever with k limk s = Questions: Do asymptotes exist? Where do they go? Imaginary Axis Real Axis M. Peet Lecture 13: Control Systems 22 / 31

23 The Root Locus at High Gain Asymptotics Consider a point s which is Very Large Recall that a point is on the root locus if G(s) = 180 Which means: m (s z i ) i=1 n (s p i ) = 180 i=1 < s-p 1 Im(s) However, when s, All Angles are the Same!!! < s-z = < s-p 1 = < s-p 2 = < s-p 3 Re(s) (s z 1 ) = (s z 2 ) =... = (s z m ) = (s p 1 ) = (s p 2 ) =... = (s p n ) = M. Peet Lecture 13: Control Systems 23 / 31

24 Constructing the Root Locus Asymptotics This makes life easier. just solve for one angle,. (m n) = 180 where m is the number of OL zeros n is the number of OL poles < s-p 1 Im(s) < s-z = < s-p 1 = < s-p 2 = < s-p 3 Re(s) So asymptotics occur at for integers l = 0, 1, 2,. = 1 m n ( l) M. Peet Lecture 13: Control Systems 24 / 31

25 Asymptotes Case 1: n m = 0 G(s) = (s + 3)(s + 4) (s + 1)(s + 2) Count: 2 zeros, 2 poles. m n = 0 = = No Asymptotes. M. Peet Lecture 13: Control Systems 25 / 31

26 Asymptotes Case 2: n m = 1 G(s) = 1 s K(s) = k(1 + s) Root Locus Imaginary Axis Count: 1 zeros, 2 poles m n = 1 Real Axis = asymptote at 180. M. Peet Lecture 13: Control Systems 26 / 31

27 Asymptotes Case 3: n m = 2 Root Locus The suspension system. Count: 2 zeros, 4 poles. m n = 2 Imaginary Axis Real Axis = 1 2 ( l) = l = 90, 270 Poles MAY destabilize at large gain. 2 vertical asymptotes at 90 and 270. M. Peet Lecture 13: Control Systems 27 / 31

28 Asymptotes Case 4: n m = 3 Root Locus The suspension system with integral Feedback. Count: 2 zeros, 5 poles. m n = 3 Imaginary Axis Real Axis = 1 3 ( l) = l = 60, 180, 300 Poles WILL destabilize at large gain. 3 asymptotes at 60, 180 and 300. M. Peet Lecture 13: Control Systems 28 / 31

29 Asymptotes Case 5: n m = 4 n(s) is degree 2, d(s) is degree 6. G(s) = s 2 + s + 1 s 6 + 2s 5 + 5s 4 s 3 + 2s Root Locus Count: 2 zeros, 6 poles. m n = 4 Imaginary Axis Real Axis = 1 4 ( l) = l = 45, 135, 225, 315 Poles WILL destabilize at large gain. 4 asymptotes at 45, 135, 225 and 315. M. Peet Lecture 13: Control Systems 29 / 31

30 Asymptotics Summary Asymptotes depend only on relative number of poles and zeros. Location of poles/zeros doesn t matter At least not for the angle One pole goes to each zero. When there are more poles than zeros: Cases: n m = 0 - No Asymptotes n m = 1 - Asymptote at 180 n m = 2 - Asymptotes at ±90 n m = 3 - Asymptotes at 180, ±60 n m = 4 - Asymptotes at ±45 and ±135 Im(s) Re(s) M. Peet Lecture 13: Control Systems 30 / 31

31 Summary What have we learned today? Review Definition of Root Locus Points on the Real Axis Symmetry Drawing the Real Axis What Happens at High Gain? The effect of Zeros Asymptotes Next Lecture: Centers of Asymptotes, Break Points and Departure Angles M. Peet Lecture 13: Control Systems 31 / 31

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