Course Outline. Problem Identification. Engineering as Design. Amme 3500 : System Dynamics and Control. System Response. Dr. Stefan B.

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1 Course Outlie Amme 35 : System Dyamics a Cotrol System Respose Week Date Cotet Assigmet Notes Mar Itrouctio 8 Mar Frequecy Domai Moellig 3 5 Mar Trasiet Performace a the s-plae 4 Mar Block Diagrams Assig Due 5 9 Mar Feeback System Characteristics 6 5 Apr Root Locus Assig Due 7 Apr Root Locus 8 9 Apr Boe Plots No Tutorials 6 Apr BREAK 9 3 May Boe Plots Assig 3 Due May State Space Moelig 7 May State Space Desig Techiques 4 May Avace Cotrol Topics 3 3 May Review Assig 4 Due 4 Spare Amme 35 : Itrouctio Slie Egieerig as Desig Problem Ietificatio Egieerig Desig is the systematic, itelliget geeratio a evaluatio of specificatios for proucts whose form a fuctio achieve state objectives a satisfy specifie costraits The purpose of esig is to erive from a set of specificatios a escriptio of a prouct sufficiet for its realizatio Slie 3 Desig costraits must be ietifie Restrictios or limitatios o a behaviour or a value or some other aspect of a esige object s performace Requiremet aalysis allows us to verify that we have reache a solutio a escriptio of the requiremets that characterise a solutio the criteria to be use to verify that the requiremets have bee met prouces a uerstaig of the ature a scope of the remaiig activities Slie 4

2 Respose Vs. Pole Locatio Respose Vs. Pole Locatio As we saw previously, a qualitative uerstaig of the effect of poles a zeros o the system respose ca help us to quickly estimate performace stable Im{s} ustable Re{s} We woul ow like to gai a eeper isight ito the system performace as a fuctio of pole a zero locatios This will help us to escribe the system performace quatitatively We will also see how this allows us to esig a system to meet certai esig costraits Slie 5 Slie 6 Geeral First Orer System Geeral First Orer System A first orer system without zeros ca be escribe by: The resultig impulse respose is Whe >, pole is locate at <, expoetial ecays stable <, pole is at >, expoetial grows ustable Hs () ( s + ) ht () e t Step Respose yt () e t ( ) We efie the time costat of the system as τ This is the time take for the system to ecay to / e or 37% of its iitial value or rise to 63% of step respose Slie 7 Slie 8

3 Geeral First Orer System The Rise Time T r is efie as the time for the waveform to go from. to.9 of its fial value yt () e.9 e. e T t t R t t.9 t Slie 9 Geeral First Orer System The Settlig Time T s is efie as the time for the waveform to reach a stay withi a certai percetage of its fial value Values of %, % a 5% are ofte use.98 e T T T S S S 4 T s (%) 4.6 (%) 3 (5%) Slie A system has a trasfer fuctio 5 Gs () s + 5 The time costat is: /. Settlig time (%) is: 4/.8 Rise Time is:./.44 Example Amplitue Step Respose Time (sec) Slie Geeral Seco Orer System May systems of iterest are of higher orer For a first orer system we have a sigle, real pole Seco orer systems are quite commo a are geerally writte i the followig staar form Gs () C s + ς s + Slie 3

4 Natural Frequecy a Dampig Ratio The Natural Frequecy,, of a secoorer system is the frequecy of oscillatio of the system without ampig The Dampig Ratio,!, of a seco-orer is the ratio of the expoetial ecay frequecy a the atural frequecy Geeral Seco Orer System For a seco orer system, we ca have four istict combiatios of real a imagiary poles Two real poles -, - Two ietical real - Two complex poles - ± j Two imagiary ± j Slie 3 Slie 4 Natural Frequecy a Dampig Ratio How oes this relate to the pole locatio i terms of their real a imagiary parts? s ± j Complex poles are always i complex cojugate pairs so s () ( s+ j )( s+ + j ) ( s + ) + Natural Frequecy a Dampig Ratio By multiplyig out a comparig the eomiator of the two forms we fi that ( s+ ) + s + ξ s+ ζ ζ Slie 5 Slie 6 4

5 Natural Frequecy a Dampig Ratio Natural Respose of Seco Orer System We ca ow relate the atural frequecy a ampig ratio to the s-plae ζ ζ θ si ( ζ) We like to fi the atural respose of a geeral orer system y +!" y + " y s Y(s) # sy # y +!" sy(s) # y ( Y(s) s +!" )y + y s +!" s + " For the case of zero iitial velocity we have Cs s () y + ς s s ( ) + " Y(s) + ς + Slie 7 Slie 8 Natural Respose of Seco Orer System We like to fi the atural respose of a geeral orer system After completig squares Take Iverse Laplace a simplify Cs s () y + ς s s Cs () y + ς + ζ ( s + ζ) + ζ ζ ( ) ( ) s + ζ + ζ ( s + ) + y ( s + ) + yt ( ) ye t cost+ sit Slie 9 Natural Respose of Seco Orer System ( ) t cos + si We ca use this to estimate the system parameters By isplacig the system a measurig its respose we ca ietify the peaks a estimate atural frequecy a ampig ratio This is kow as the log ecremet yt ye t t π π ζ y ye π ζ y π l ζ y ζ y l ζ π y y ζ ; y e y l π y Slie 5

6 A Familiar Mechaical Example M f(t) y(t) M!! y (t ) + K!y (t ) + Ky (t ) f (t ) Earlier, we cosiere this mechaical system Now we ca cosier the atural a force respose M ( s Y (s )! sy ()!!y ()) + K ( sy (s )! y ()) + KY (s ) F (s ) Example: Natural Respose The atural respose of the system ca be fou by assumig o iput force ( Ms + K s + K )Y(s) ( Ms + K )y() + M y () ( Y(s) Ms + K )y() + M y () Ms + K s + K ( ) Takig iverse Laplace trasform will give us the system respose Slie Slie Example: Natural Respose Suppose we wish to fi the atural respose of the system to a iitial isplacemet of.5m Assume the mass of the system is kg a the systems costat k is Nsec/m a k is N/m Example: Natural Respose The atural respose of the system ca be fou by assumig o iput force K s + M s + ς Y( s) y() y K K s s s + s+ M M K K ζ, M M + ς + Slie 3 Slie 4 6

7 Example: Natural Respose Substitutig values, we fi!",", ",!,," 9 % y(t) y e #$t cos" t + $ ( ' si" " t* & ).5 System: h Time (sec): Amplitue: ( * -. ) -..5e #t % ' cos 9t + si 9t & 9 Amplitue Impulse Respose System: h Time (sec):.88 Amplitue:.79 Example: Natural Respose We ca work backwars from the respose to estimate the parameters Impulse Respose cycles ;.88s ra.69hz 4.36 s y l ζ ; π y.5 l 4π System: h Time (sec): Amplitue: Amplitue System: h Time (sec):.88 Amplitue: Time (sec) Time (sec) Slie 5 Slie 6 Step Respose of Seco Orer System We like to fi the step respose of the staar form After some partial fractio expasio a completig squares Take Iverse Laplace a simplify Cs () s s ( + ςs+ ) ζ ( s + ζ) + ζ ζ Cs () s ( s+ ζ) + ( ζ ) ζ ( s + ) + ζ s ( s+ ) + ct ( ) e t cost+ sit Step Respose vs. Dampig Ratio Dampig ratio etermies the characteristics of the system respose Slie 7 Slie 8 7

8 Step Respose vs. Dampig Ratio ct ( ) Acos( t φ) ( φ) t c( t) Ae cos t c() t K e + K te t t ct () Ke + Ke t t Step Respose vs. Dampig Ratio The ampig ratio escribes the egree of ampig i the system For a uerampe system, risig ampig will lower the overshoot of the system Slie 9 Slie 3 Time Domai Specificatios Specificatios for a cotrol system esig ofte ivolve requiremets associate with the time respose of the system Peak Time, t p, is the time take to reach maximum overshoot Overshoot, M p, is the maximum amout the system overshoots its fial value Settlig time, t s, is the time for system trasiets to ecay Rise time, t r, is the time it takes for the system to reach the viciity of its ew set poit Time Domai Specificatios Rise time, settlig time a peak time yiel iformatio about the spee of respose of the trasiet respose This ca help a esiger etermie if the spee a ature of the respose is appropriate Slie 3 Slie 3 8

9 Peak time is fou by ifferetiatig a fiig the first zero crossig after t After some maipulatio we fi t p occurs whe si t Peak Time L[! c(t )] sc (s ) t p! s +!" +! π π ζ Slie 33 Percet Overshoot Substitutig the peak time back ito the step respose yiels Notice this is oly a fuctio of the ampig ratio. We ca ivert to fi the require ampig ratio for a particular overshoot ct ( p) e cosπ + siπ M p + e π πζ ζ π e, ζ < ζ l( M ) π + l ( M ) p p Slie 34 Percet Overshoot Settlig Time We ca plot the relatioship betwee percet overshoot a ampig ratio Two frequetly use specificatios are M p 5% for ζ.7 M p 6% for ζ.5 6% Mp,% Overshoot vs. Dampig Ratio for seco orer system The settlig time ca be erive i a similar maer as for a first orer system The settlig time is essetially ecie by the trasiet expoetial e ζ s t so t s. 4 ζ 4 5% ζ Slie 35 Slie 36 9

10 Rise Time Time Domai Specificatios Rise time is ot as straightforwar to compute for this case A precise aalytical relatioship oes ot exist Istea we use a approximatio as suggeste i Frakli (Sectio 3.4.). Nise suggests a alterative approach base o a graph of ormalise rise time t r.8 Lies of costat peak time,t p, settlig time, T s, a percet overshoot, %OS a rise time Note: T s < T s ; T p < T p ; %OS < %OS Slie 37 Slie 38 Step respose a Pole Locatio Step resposes of seco-orer uerampe systems as poles move: a. with costat real part ; b. with costat imagiary part j; c. with costat ampig ratio ζ Slie 39 Example : Trasformig Specificatios Fi allowable regio i the s-plae for the poles of a trasfer fuctio to meet the requiremets t r.6 sec M p % t s 3 sec.8 3. ra / sec t r ζ Im(s) si - Slie 4 Re(s)

11 Example : Trasiet Respose through Compoet Desig Give the system show here, fi J a D to yiel % overshoot a a settlig time of secos for a step iput torque T (t) Example : Trasiet Respose through Compoet Desig First the ifferetial equatio J!!! + D! + K! T Fi the trasfer fuctio Gs () s / J D K + s+ J J The seco orer properties are K D a ζ J J Slie 4 Slie 4 Example : Trasiet Respose through Compoet Desig From problem statemet 4 Ts ζ Hece D 4 J 4a ζ J K For a % overshoot, ζ.456 so J.5 K From problem statemet K 5 so J.6kgm a D.4 Nms/ra Aitioal Poles a Zeros The preceig evelopig hols for a seco orer system with o zeros What happes to the system respose if we a aitioal poles or zeros? This epes o their locatio i the s- plae Slie 43 Slie 44

12 Aitioal Poles Aitioal Poles Cosier a aitioal pole at By partial fractio expasio a completig squares we have the step respose H() s ( s+ α)( s + ζ + ) A Bs ( + ζ) + C T() s + + s ct () Aut () ( s + ζ ) + ( s+ α ) ( cos si ) D ζ t α e B t t C t De Slie 45 Slie 46 Aitioal Poles Aitioal Poles As the pole locatio is move to the left i the LHP, the effect o the respose becomes less a the trasiet ies out more quickly How much further from the omiat poles oes the thir pole have to be for its effect o the seco-orer respose to be egligible? This epes o the accuracy require by the esig We ofte assume the expoetial ecay is egligible after five time costats The accuracy of this assumptio shoul always be verifie Slie 47 Slie 48

13 Step Respose Time (sec) Cosier a zero at β For large β, the zero acts as a scalig factor For smaller β, the erivative icreases the spee of respose a overshoot Zeros H () s z ( s + ) β ( s + ζ + ) s + s + ζ + s + ζ + sh () s + β H () s Amplitue Zero at - Zero at -5 Zero at -3 No zero β If we compare the ormalize step respose, we fi that the erivative term icreases the resposiveess of the system Zeros Slie 49 Slie 5 We have see that a pole i the RHP makes a system ustable A zero i the RHP causes the respose to be epresse If the erivative term is larger tha the scale respose, this ca lea to o-miimum phase behaviour Zeros Effects of Pole-Zero Patters For a seco orer system with o fiite zeros, the trasiet respose parameters are approximate by Rise time : Overshoot : Settlig Time (%) :.8 tr 5%, ζ.7 6%, ζ.5 %, ζ.45 4 t s M p Slie 5 Slie 5 3

14 Effects of Pole-Zero Patters A aitioal pole i the left half-plae (LHP) will icrease the rise time sigificatly if the extra pole is withi a factor of 5 of the real part of the complex poles A zero i the LHP will icrease the overshoot if the zero is withi a factor of 4 of the real part of the complex poles A zero i the RHP will epress the overshoot (a may cause the step respose to be omiimum phase) Coclusios We have looke at the characteristics of first a seco orer systems We have erive specificatios that escribe the respose of the system to a step iput We have also looke at the effect of extra poles a zeros o the system respose Slie 53 Slie 54 Further Reaig Nise Sectios Frakli & Powell Sectio Slie 55 4