Course Outline. Designing Control Systems. Proportional Controller. Amme 3500 : System Dynamics and Control. Root Locus. Dr. Stefan B.

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1 Amme 3500 : System Dyamics ad Cotrol Root Locus Course Outlie Week Date Cotet Assigmet Notes Mar Itroductio 8 Mar Frequecy Domai Modellig 3 5 Mar Trasiet Performace ad the s-plae 4 Mar Block Diagrams Assig Due 5 9 Mar Feedback System Characteristics 6 5 Apr Root Locus Assig Due 7 Apr Root Locus 8 9 Apr Bode Plots No Tutorials 6 Apr BREAK 9 3 May Bode Plots Assig 3 Due 0 0 May State Space Modelig 7 May State Space Desig Techiques 4 May Advaced Cotrol Topics 3 3 May Review Assig 4 Due 4 Spare Amme 3500 : Itroductio Slide Desigig Cotrol Systems We have had a quick look at a umber of methods for specifyig system performace We have examied some methods for desigig systems to meet these specificatios for first ad secod order systems We will ow look at a graphical approach, kow as the root locus method, for desigig cotrol systems As we have see, the root locatios are importat i determiig the ature of the system respose Proportioal Cotroller Whe the feedback cotrol sigal is made to be liearly proportioal to the system error, we call this proportioal feedback We have see how this form of feedback is able to miimize the effect of disturbaces R(s) E(s) C(s) + - K G(s) Slide 3 Slide 4

2 Proportioal Cotroller The closed look trasfer fuctio is give by Assumig we have two poles, G(s)/(s +a)(s+b) KG( s) + KG ( s ) K ( ) s + a + b s + ab + K Proportioal Cotroller We ca also look at the system parameters as a fuctio of the gai, K Give fixed values for the roots of the plat, we ca fid K to meet performace specificatios K s + ( a + b) s + ab + K C s ab + K a + b " ab + K a + b ab ess ab + K + " + Slide 5 Slide 6 Root Locatio The locatio of the roots, ad hece the ature of the system performace, are a fuctio of the system gai K I order to solve for this system performace, we must factor the deomiator for specific values of K We defie the root locus as the path of the closed-loop poles as the system parameter varies from 0 to Example: Secod order system A system to automatically track a subject i a visual image ca be modelled as follows We ca solve for the closed loop trasfer fuctio as a fuctio of the system parameter, K Slide 7 Slide 8

3 Example: Secod order system Example: Secod order system We ca also determie the closed loop poles as a fuctio of the gai for the system The idividual pole locatios The root locus Slide 9 Slide 0 Properties of the Root Locus Properties of the Root Locus We ca easily derive the root locus for a secod order system What about for a geeral, possibly higher order, cotrol system? Poles exist whe the characteristic equatio (deomiator) is zero KG( s) + KG ( s ) H ( s ) + KG( s) H ( s) 0 How do we fid values of s ad K that satisfy the characteristic equatio? This holds whe " " + KG( s) H ( s) 0 KG( s) H ( s) KG( s) H ( s) (k + )80 zero agles pole agles (k + )80 K pole legth G ( s ) H ( s ) zero legth Slide Slide 3

4 Properties of the Root Locus The precedig agle ad magitude criteria ca be used to verify which poits i the s-plae form part of the root locus It is ot practical to evaluate all poits i the s-plae to fid the root locus We ca formulate a umber of rules that allow us to sketch the root locus Basic Root Locus Rules Rule : Number of Braches the braches of the root locus start at the poles + KG( s) H ( s) 0 De( s) + KNum( s) 0 For K0, this suggests that the deomiator must be zero (equivalet to the poles of the OL TF) The umber of braches i the root locus therefore equals the umber of ope loop poles Slide 3 Slide 4 Basic Root Locus Rules Rule : Symmetry - The root locus is symmetrical about the real axis. This is a result of the fact that complex poles will always occur i cojugate pairs. Basic Root Locus Rules Rule 3 Real Axis Segmets Accordig to the agle criteria, poits o the root locus will yield a agle of (k +)80 o. O the real axis, agles from complex poles ad zeros are cacelled. Poles ad zeros to the left have a agle of 0 o. This implies that roots will lie to the left of a odd umber of real-axis, fiite ope-loop poles ad/or fiite ope-loop zeros. Slide 5 Slide 6 4

5 Basic Root Locus Rules Rule 4 Startig ad Edig Poits As we saw, the root locus will start at the ope loop poles + KG( s) H ( s) 0 De( s) + KNum( s) 0 The root locus will approach the ope loop zeros as K approaches Sice there are likely to be less zeros tha poles, some braches may approach Cosider the system at right The closed loop trasfer fuctio for this system is give by Difficult to evaluate the root locatio as a fuctio of K Example K( s + 3)( s + 4) ( + K) s + (3 + 7 K) s + ( + K) Slide 7 Slide 8 Example Example Ope loop poles ad zeros First plot the OL poles ad zeros i the s- plae This provides us with the likely startig (poles) ad edig (zeros) poits for the root locus Real axis segmets Alog the real axis, the root locus is to the left of a odd umber of poles ad zeros Slide 9 Slide 0 5

6 Startig ad ed poits The root locus will start from the OL poles ad approach the OL zeros as K approaches ifiity Eve with a rough sketch, we ca determie what the root locus will look like Example Basic Root Locus Rules Rule 5 Behaviour at ifiity For large s ad K, -m of the loci are asymptotic to straight lies i the s-plae The equatios of the asymptotes are give by the realaxis itercept, a, ad agle, a % fiite poles $ % fiite zeroes a $ m (k + ) " a $ m Where k 0, ±, ±, ad the agle is give i radias relative to the positive real axis Slide Slide Basic Root Locus Rules Why does this hold? We ca write the characteristic equatio as m s + b s + + bm + K 0 s + b s + + b m This ca be approximated by + K 0 ( s " ) " m For large s, this is the equatio for a system with -m poles clustered at sσ Here we have four OL poles ad oe OL zero We would therefore expect -m 3 distict asymptotes i the root locus plot Example Slide 3 Slide 4 6

7 We ca calculate the equatios of the asymptotes, yieldig % % Example fiite poles $ fiite zeroes a $ m ( $ $ $ 4) $ ( $ 3) 4 $ 3 3 (k + ) " a $ m " / 3 ( k 0) " ( k ) 5 " / 3 ( k ) Slide 5 Agles of Departure ad Arrival For poles o the real axis, the locus will depart at 0 o or 80 o For complex poles, the agle of departure ca be calculated by cosiderig the agle criteria Slide 6 Agles of Departure ad Arrival A similar approach ca be used to calculate the agle of arrival of the zeros Imagiary Axis Crossig We may also be iterested i the gai at which the locus crosses the imagiary axis This will determie the gai with which the system becomes ustable Slide 7 Slide 8 7

8 Usig Available Resources All of this probably seems somewhat complicated Fortuately, Matlab provides us with tools for plottig the root locus It is still importat to be able to sketch the root locus by had because This gives us a uderstadig to be applied to desigig cotrollers It will probably appear o the exam Root Locus as a Desig Tool As we saw previously, the specificatios for a secod order system are ofte used i desigig a system The resultig system performace must be evaluated i light of the true system performace The root locus provides us with a tool with which we ca desig for a trasiet respose of iterest Slide 9 Slide 30 Root Locus as a Desig Tool We would usually follow these steps Sketch the root locus Assume the system is secod order ad fid the gai to meet the trasiet respose specificatios Justify the secod-order assumptios by fidig the locatio of all higher-order poles If the assumptios are ot justified, system respose should be simulated to esure that it meets the specificatios Slide 3 Root Locus as a Desig Tool Recall that for a secod order system with o fiite zeros, the trasiet respose parameters are approximated by Rise time : Overshoot : Settlig Time (%) : M p.8 tr " " 5%, 0.7 $ % 6%, 0.5 & 0%, t s " Slide 3 8

9 Example: Secod order system Recall the system preseted earlier Determie a value of the gai K to yield a 5% percet overshoot For a secod order system, we could fid K explicitly Slide 33 Example: Secod order system K Examiig the trasfer s + 0s + K fuctio Solve for K give the C s + " s + desired dampig ratio specified by the desired K overshoot " 0 for 5% overshoot, " 0.7 $ 5 % therefore K & ' ( 0.7 ) 5 Slide 34 Example: Secod order system Alteratively, we ca examie the Root Locus + KGH 0 KGH K s( s + 0) K ( j)( j + 0) K 5.0 S5+5.j x x Im(s) x θsi - ζ x Slide 35 Re(s) Example: Secod order system We ca use Matlab to geerate the root locus s i x % defie the OL system g a m systf(,[ 0 0]) I % plot the root locus rlocus(sys) A Root Locus Real Axis Slide 36 9

10 Example: Secod order system Root Locus as a Desig Tool We also eed to verify the resultig step respose l % set up the closed loop TF p cl5*sys/(+5*sys) A % plot the step respose step(cl) e d u t i m System: cl Time (sec): 0.6 Amplitude:.05 Step Respose Time (sec) Cosider this system This is a third order system with a additioal pole Determie a value of the gai K to yield a 5% percet overshoot Slide 37 Slide 38 Root Locus as a Desig Tool Root Locus as a Desig Tool With the higher order poles, the d order assumptios are violated However, we ca use the RL to guide our desig ad iterate to fid a suitable solutio s i x A g 0 a m I Root Locus The gai foud based o the d order assumptio yields a higher overshoot We could the reduce the gai to reduce the overshoot e d u t i l p m A System: utitled Time (sec): Amplitude:. Step Respose Time (sec) Real Axis Slide 39 Slide 40 0

11 Geeralized Root Locus The precedig developmets have bee preseted for a system i which the desig parameter is the forward path gai I some istaces, we may eed to desig systems usig other system parameters I geeral, we ca covert to a form i which the parameter of iterest is i the required form Slide 4 Geeralized Root Locus Cosider a system of this form The ope loop trasfer fuctio is o loger of the familiar form KG(s) H(s) Rearrage to isolate p Now we ca sketch the root locus as a fuctio of p 0 + ( + ) s p s p ( + ) s s p s 0 s + s + 0 p ( s + ) + s + s + 0 Slide 4 Geeralized Root Locus This results i the followig root locus as a fuctio of the parameter p Coclusios We have looked at a graphical approach to represetig the root positios as a fuctio of variatios i system parameters We have preseted rules for sketchig the root locus give the ope loop trasfer fuctio We have begu lookig at methods for usig the root locus as a desig tool Slide 43 Slide 44

12 Nise Sectios Frakli & Powell Sectio Further Readig Slide 45