8 IntroductionSome root loci examples (K from zero to infinity)
9 Sketching a Root Locus Definition 1 Definition 2 The root locus is the values of s for which 1+KL(s)=0 is satisfied as K varies from 0 to infinity (pos.).Definition 2The root locus is the points in the s-plane where the phase of L(s) is 180°.The angle to a point on the root locus from zero number i is yi.The angle to a point on the root locus from pole number i is fi.Therefore,
10 Sketching a Root Locus s0 f1 p y1 f2 p* A root locus can be plotted using Matlab: rlocus(sysL)
11 Root Locus characteristics Rule 1 (of 6)The n branches of the locus start at the poles L(s) and m of these branches end on the zeros of L(s).
12 Root Locus characteristics Rule 2 (of 6)The loci on the real axis (real-axis part) are to the left of an odd number of poles plus zeros.Notice, if we take a test point s0 on the real axis :The angle of complex poles cancel each other.Angles from real poles or zeros are 0° if s0 are to the right.Angles from real poles or zeros are 180° if s0 are to the left.Total angle = 180° + 360° l
13 Root Locus characteristics Rule 3 (of 6)For large s and K, n-m branches of the loci are asymptotic to lines at angles fl radiating out from a point s = a on the real axis.flaFor example
14 Root Locus characteristics n-m=1n-m=2n-m=3n-m=4For ex.,n-m=3
15 Root Locus3 poles:q = 3Rule 4: The angles of departure (arrival) of a branch of the locus from a pole (zero) of multiplicity q.f1,dep = 60 deg.f2,dep = 180 deg.f3,dep = 300 deg.
16 Root Locus characteristics Rule 5 (of 6)The locus crosses the jw axis at points where the Routh criterion shows a transition from roots in the left half-plane to roots in the right half-plane.Routh: A system is stable if and only if all the elements in the first column of the Routh array are positive.
17 Root Locus characteristics Rule 6 (of 6)The locus will have multiplicative roots of q at points on the locus where (1) applies.The branches will approach a point of q roots at angles separated by (2) and will depart at angles with the same separation.Characteristic equation
18 Root locus for double integrator with P-control ExampleRoot locus for double integrator with P-controlRule 1: The locus has two branches that starts in s=0. There are no zeros. Thus, the branches do not end at zeros.Rule 3: Two branches have asymptotes for s going to infinity. We get
19 Rule 2: No branches that the real axis. Rule 4: Same argument and conclusion as rule 3.Rule 5: The loci remain on the imaginary axis. Thus, no crossings of the jw-axis.Rule 6: Easy to see, no further multiple poles. Verification:
20 ExampleRoot locus for satellite attitude control with PD-control
23 Selecting the Parameter Value The (positive) root locusA plot of all possible locations of roots to the equation 1+KL(s)=0 for some real positive value of K.The purpose of design is to select a particular value of K that will meet the specifications for static and dynamic characteristics.For a given root locus we have (1). Thus, for some desired pole locations it is possible to find K.
24 Selecting the Parameter Value ExampleSelection of K using Matlab: [K,p]=rlocfind(sysL)
25 Dynamic Compensation Some facts Controller design using root locus We are able to determine the roots of the characteristic equation (closed loop poles) for a varying parameter K.The location of the roots determine the dynamic characteristics (performance) of the closed loop system.It might not be possible to achieve the desired performance with D(s) = K.Controller design using root locusLead compensation (similar to PD control)overshoot, rise time requirementsLag compensation (similar to PI control)steady state requirements
30 Lead Compensation Selecting p and z Usually, trial and error The placement of the zero is determined by dynamic requirements (rise time, overshoot).The exact placement of the pole is determined by conflicting interests.suppression of output noiseeffectiveness of the zeroIn general,the zero z is placed around desired closed loop wnthe pole p is placed between 5 and 20 times of z
31 Lead Compensation Design approach (A) Initial design z=wn, p=5z Noise: additional requirement, max(p)=20IterationsFirst design: z=wn, p=5z(neglecting the add. req.)Second design: z=wn, p=20Third design: new z, p=20
32 Lead Compensation Possible pole location Matlab – design 1 sysG = tf(,[1 1 0])sysD = tf([1 7],[1 5*7])rlocus(sysD*sysG)
33 Lead Compensation No possible pole location ! Matlab – design 2 sysG = tf(,[1 1 0])sysD = tf([1 7],[1 20])rlocus(sysD*sysG)
34 Lead Compensation Possible location New zero location Matlab – design 3sysG = tf(,[1 1 0])sysD = tf([1 4],[1 20])rlocus(sysD*sysG)
35 Lead Compensation Design approach (B) Pole placement from the requirementsNoise: Additional requirement, max(p)=20. So, p=20 to minimize the effect of the pole.
37 Lead Compensation Pole location Matlab – design B sysG = tf(,[1 1 0])sysD = tf([1 5.4],[1 20])rlocus(sysD*sysG)
38 Lead Compensation Exercise Design a lead compensation D(s) to the plant G(s) so that the dominant poles are located at s = -2 ± 2jYou can use, trianglea/sin(A) = b/sin(B)r(t)e(t)controllerD(s)u(t)plantG(s)y(t)+-
39 Lead Compensation Answer we have three poles and one zero f1 y f2 Notice, trianglea/sin(A) = b/sin(B)
41 Lag Compensation Lag compensation Steady state characteristics (requirements)Position error constant KpVelocity error constant KvLet us continue using the example systemand the designed controller (lead compensation)
42 Lag Compensation Calculation of error constants Suppose we want Kv ≥ 100We need additional gain at low frequencies !
43 Lag Compensation Selecting p and z To minimize the effect on the dominant dynamics z and p must chosen as low as possible (i.e. at low frequencies)To minimize the settling time z and p must chosen as high as possible (i.e. at high frequencies)Thus, the lag pole-zero location must be chosen at as high a frequency as possible without causing any major shifts in the dominant pole locations
44 Lag Compensation Design approach Increase the error constant by increasing the low frequency gainChoose z and p somewhat below wnExample system:K = 1 (the proportional part has already been chosen)z/p = 3, with z = 0.03, and p = 0.01.
45 Lag Compensation Lead-lag Controller Notice, a slightly different pole locationLead-lagController
49 Notch compensation Example system We have successfully designed a lead-lag controllerSuppose the real system has a rather undampen oscillation about 50 rad/sec.Include this oscillation in the modelCan we use the original controller ?
50 Notch compensationStep response using the lead-lag controller
51 Notch compensation Aim: Remove or dampen the oscillations PossibilitiesGain stabilizationReduce the gain at high frequenciesThus, insert poles above the bandwidth but below the oscillation frequency – might not be feasiblePhase stabilization (notch compensation)A zero near the oscillation frequencyA zero increases the phase, and z ≈ PM/100Possible transfer functionIf zn < 1, complex zerosand double pole at w0
53 Notch compensation Cancellation Root Locus The lead-lag controller with notch compensationw0 = 50z = 0.01
54 Notch compensation Too much overshoot !! Also, exact cancellation might cause problems due to modelling errorsThus, we need different parameters
55 Notch compensation Modified parameters Notch compensation Dynamic characteristicsIncrease zn to obtain less overshootMake sure that the roots move into the LHPIn general, obtain a satisfactory dynamic behaviorAfter some trial and error
56 Notch compensationThe overshoot has been lowered to an acceptable level !The oscillations have almost been removed !So, we have reached a final design !
57 Notch compensation Root Locus The lead-lag controller with notch compensationw0 = 60z = 0.3
58 Time Delay Notice Root locus analysis Solutions Time delay always reduces the stability of a system !Important to be able to analyze its effectIn the s-domain a time delay is given by e-lsMost applications contain delays (sampled systems)Root locus analysisThe original method does only handle polynomialsSolutionsApproximation (Padé) of e-lsModifying the root locus method (direct application)