A New Accurate Analytical Expression for Rise Time Intended for Mechatronics Systems Performance Evaluation and Validation

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1 Iteratioal Joural of Automatio, Cotrol ad Itelliget Systems Vol., No., 05, pp A New Accurate Aalytical Expressio for Rise Time Iteded for Mechatroics Systems Performace Evaluatio ad Validatio Ayma A. Aly,, *, Farha A. Salem, 3 Mechatroics Egieerig Program, College of Egieerig, Taif Uiversity, Taif, Saudi Arabia Mechatroics Egieerig Sectio, Dept. of Mechaical Egieerig, Assuit Uiversity, Assiut, Egypt 3 Alpha ceter for Egieerig Studies ad Techology Researches, Amma, Jorda Abstract Most Mechatroics systems are desiged with syergy ad itegratio to operate with exceptioal high levels of accuracy ad speed despite adverse effects of system oliearities, ucertaities ad disturbaces. Both rise time ad peak time are used as a measure of swiftess of respose, meawhile, the closeess of the respose to the desired respose, is measured by the overshoot ad settlig time. Most used formulae ad expressios for determiig such performace specificatios i texts lack accuracy, sice it is more difficult to determie the exact aalytical expressios of most used specificatios. This paper proposes derivatio of more accurate aalytical expressios for rise time that ca be applied to reflect the actual Mechatroics system rise time ad used i accurate performace evaluatio ad verificatio. Keywords Mechatroics Systems, Performace, Rise Time, Aalytical Expressio Received: Jue 8, 05 / Accepted: Jue 7, 05 / Published olie: July 3, 05 The Authors. Published by America Istitute of Sciece. This Ope Access article is uder the CC BY-NC licese. Itroductio Most used formulae ad expressios for performace specificatios i texts e. g. [-8] lack accuracy. The determied performace specificatios usig these expressios, rarely accurate compared with actual results ad measuremets sice it is more difficult to determie the exact aalytical expressios of most used specificatios ad most itroduced expressios are rough approximatio of actual values. Mechatroics systems are supposed to operate with high accuracy ad speed despite adverse effects of system oliearities ad ucertaities, therefore, accuracy i Mechatroics systems performace is of cocer, ad the eed for precise aalytical expressios for mechatroics systems performace specificatios calculatio, is highly desired. This paper exteds writer's previous work [9], ad proposes derivatio of accurate aalytical expressio for a importat performace measure 'rise time' iteded for research purposes i accurate verificatio/validatio of Mechatroics systems performace evaluatio as well as for the applicatio i educatioal process.. Case Classificatio The step respose is the measured reactio of the cotrol system to a step chage i the iput, step respose has uiversal acceptace ad popularity, because of simplicity of its form facilitates mathematical aalysis, modelig, ad experimetal verificatios, as well as it is easy to geerate ad has several measuremet techiques for recordig the time domai respose. A typical step respose ad its associated performace specificatios of secod order systems are show i Figure. The most used performace specificatios are; Time costat T, Rise time T R, Settlig * Correspodig author address: draymaelaggar@yahoo.com (A. A. Aly)

2 5 Ayma A. Aly ad Farha A. Salem: A New Accurate Aalytical Expressio for Rise Time Iteded for Mechatroics Systems Performace Evaluatio ad Validatio time T s, Peak time, T P, Maximum overshoot M P, Maximum udershoot M u, Percet overshoot OS%, Delay time T d, The decay ratio D R, Dampig period T O ad frequecy of ay oscillatios i the respose, thee swiftess of the respose ad the steady state error e ss. Figure. Secod-order uderdamped respose specificatios [9]... Rise Time T R I cotrol theory applicatios, rise time is defied as "the time required for the respose to rise from x% to y% of its fial value", with 0%-00% rise time commo for uderdamped secod order systems, 5%-95% for critically damped ad 0%-90% for overdamped [0]. The trasiet respose of the system may be describedd i terms of two factors; a) The swiftess of respose, as represeted by the rise time ad the peak time. b) The closeess of the respose to the desired respose, as represeted by the overshoot ad settlig time. Therefore, the rise time yields iformatio about the speed of the trasiet respose. This iformatio ca help a desiger determie if the speed ad the ature of the respose do or do ot degrade the performace of the system []. It is, for example determies speed of rise of flow or pressure (e.g. volume or pressure cotrol modes).. Rise Time for First Order Systems First order systems without zeros ad systems that ca be approximated as first order systems are described by first order differetial equatio ad trasfer fuctio, derived as give by Eq. (). As show i Figure, the respose is characterized by time costat T, rise time T R, settlig time T s ad steady state error e ss, where the oly parameter required to characterize respose is time costat T, ad whe first order system is subjected to a uity step iput, R(s) = /s, as by Eq. () the respose for these systems is either atural decay or growth geerated by the system pole: d a x( t) + b x( t) = u( t) dt a sx( s) + b X( s ) = U( s) a X( s) s + b = U ( s ) b X( s) [ Ts + ] = U( s) b X( s) / b KDC G(s) = = = U( t) Tss + Ts + C( s) = R( s)* G( s) = sts + Takig the iverse trasform, the (solutio) step respose is give by Eq. (3). Rise time is foud by solvig Eq. (3) for the differece i time, e.g. rise time for 0% to 90% criterio is foud by solvig Eq. (3) for the differece i time at c(t) = 0.9 ad c(t) = 0., as give by Eq.(4) T T R R = ( e K DC c( s) = e αt (3) ) ( e ) αα t90 αt = = a a a () () (4)

3 Iteratioal Joural of Automatio, Cotrol ad Itelliget Systems Vol., No., 05, pp The rise time ca be measured i terms of the time costat, ad give by Eq. (5): t90 t0 T T T = ( e ) ( e ) =.306T 0.054T =.97T (5) R Figure (a). First order system respose to step, ad performace specificatios. 5 Step Respose 0 5 Amplitude 0 5 Time costat,t Rise time, Tr Settlig time, Ts 5T Time (sec) Figure (b). Performace specificatios of first order PMDC motor step respose.

4 54 Ayma A. Aly ad Farha A. Salem: A New Accurate Aalytical Expressio for Rise Time Iteded for Mechatroics Systems Performace Evaluatio ad Validatio.3. Rise Time for Secod Order System For secod order systems, ad systems that ca be approximated as secod order systems, whe subjected to step iput, R(s) = A/s, the respose depeds o pole locatio o complex pla give by Eq. (6), that i turs, depeds o dampig ratio ζ, ad udamped atural frequecy, where dampig ratio determies how much the system oscillates as the respose decays toward steady state ad udamped atural frequecy, determies how fast the system oscillates durig ay trasiet respose, based o this, there are four cases of stable respose to cosider; udamped, uderdamped, critically damped ad overdamped respose. P = ± j (6) For uderdamped case; 0<ζ< ad two complex cojugate poles give by Eq.(6), allow us to rewrite geeral form of secod order system to have the form give by Eq.(7). To obtai iverse Laplace trasform we eed to expad by partial fractios ad solve, this all gives: C( s) = = R( s) s s ( s j )( s j ) C( s) = R( s) s ( s ) + s + s + s + = + = + s + s + ( ) s s + s + d s s + s + = + = + s + s + ( ) s s + s + d s s + = s ( s + ) + ( + ) + d d d s d s + = s ( s + ) + ( + ) + d d s d Takig iverse Laplace trasform, gives: ct ( ) = e t (cosdt + si dt) This ca be rewritte to have the followig forms: c( t) = e t cos( dt φ) c t e t ( ) t = si( d + cos ) (7) (8) (9) (0) Eq.(8),(9)ad(0) show that the damped atural frequecy d, give by = ζ,is the frequecy at which the system d will oscillate if the dampig is decreased to zero. Eq.(8) shows that performace of secod order system depeds o dampig ratio ζ ad udamped atural frequecy. Plots of the step respose as fuctios of the ormalized time t for various dampig ratio values of 0 ζ.5 are illustrated i Figure 3 (a), the curves show that he respose becomes more oscillatory as ζ decreases i value, up to ζ=, whe ζ, the step respose does ot exhibit ay overshoot or oscillatory behavior, also whe ζ betwee 0.5 ad 0.8 the system reaches fial value more rapidly. Plots of the step respose for various are illustrated i Figure 3 (b), the resposes show that has a direct effect o the rise time, delay time, ad settlig time but does ot affect the overshoot [9]. It is difficult to determie precise aalytical expressios for rise time T R [] for secod order systems. Differet approximate formulae for the rise time appear i differet texts[-8]. Oe reaso for that is because of differet defiitios of the rise time [7], as well as the required accuracy. A alterative measure to represet the rise time is as the reciprocal of the slope of the step respose at the istat that the respose is equal to 50% of its fial value [3][9], that is at delay time T D. The exact values of rise time for give rage of dampig ratio, ca be determied directly from the resposes of Figure, or rise time is foud by solvig Eq. (8) for the differece i time e.g. rise time for 0% to 90% is foud by solvig Eq. (8) for the differece i time at c(t) = 0.9 ad c(t) = 0.. For uderdamped case; 0% to 00% of its fial value, the rise time ca be obtaied by equatig Eq.(8) with uity ad solve for time t, that is rise time, this show i equatios below. Also MATLAB code ca be writte to retur actual values of rise time, ad plot it agai give rage of zetas, a example code, is writte ad applied i this paper. R c( t) = e T (cosd + si d) = Sice e 0, we have: cosd + si d = 0 ta d = ta T d R d = = ta d π φ π ta ( / ) = = 0 < < d () Where, referrig to Figure 4 (a), ϕ is defied by the followig Eqs.: φ = cos ( ) φ = si ( ) φ = ta ( )

5 Iteratioal Joural of Automatio, Cotrol ad Itelliget Systems Vol., No., 05, pp I the limit as 0, this equatio ca be approximated as: π π / π = = I the limit as, this equatio ca be approximated as: π 0 π = = () These equatios imply that rise time icreases as dampig approaches uity. A approximatio techiques ca be used to estimate approximate values; by plottig ormalized time T R versus rage of 0 ζ.5, ad the approximate the curve by a straight lie or over the rage of 0 < ζ <. We first desigate T R as the ormalized time variable ad select a value for ζ. Usig the computer, we solve for the values of T R that yield c(t) = 0.9 ad c(t) = 0.. Subtractig the two values of T R yields the ormalized rise time, T R, for that value of ζ [4], cotiuig i like fashio with other values of ζ, ad we obtai the results plotted i Figure 4(b). for =, this plot shows that icrease i dampig ratio leads to icrease i the rise time that is ot desirable..8 Closed-Loop Step for various zeta Amplitude zeta=0. zeta=0. zeta=0.3 zeta=0.4 zeta=0.5 zeta=0.6 zeta=0.7 zeta=0.8 zeta=0.9 zeta= zeta=. zeta=. zeta=.3 zeta=.4 zeta= ormalized time,omega *t (sec) Figure 3 (a). Plots of the step respose for various 0 ζ.5 with =0..6 Closed-Loop Step for various Omega.4. Amplitude Time (sec) omega = omega = omega =3 omega =4 omega =5 omega =6 omega =7 Figure 3 (b). Plots of the step respose for various with ζ =0..

6 56 Ayma A. Aly ad Farha A. Salem: A New Accurate Aalytical Expressio for Rise Time Iteded for Mechatroics Systems Performace Evaluatio ad Validatio 3. Derivig Aalytical Expressios for Rise Time T R 3.. Expressios for Rise Time T R of 0% to 00% Criterio Applyig curve fittig to curve show i Figure 4(b), to derive a approximate third order approximatio give by Eq.(3): < < 0.9 (3) Quadratic approximatio ca result i expressios give by Eq.(4) < < < < (4) Referrig to [3] the rise time T R, for secod order uderdamped system, ca be approximated as a straight lie give by give by Eq.(5): < < (5) Referrig to [4] the liear approximatio of rise time is give by Eq.(6): T R = (6) Referrig to [5] rise time is give by give by Eq.(7):. = (7) Referrig to [6] rise time is give by give by Eq.(8): = 4.7. = >. <. (8) All these equatios shows that rise time is proportioal to ζ ad iversely proportioal. based these equatios, it ca be stated that, the maximum overshoot ad the rise time coflict with each other, as overshoot icreases, the rise time decreases, this meas that we ca ote make both the maximum overshot ad the rise time smaller simultaeously, if oe is made smaller the other will become larger. Most of these derived expressio are rough approximatios ad mostly has huge deviatio at actual values, this is show i Figure 4(c) that shows the plots of differet approximatio for rise time agaist dampig ratio (PO). Aalyzig actual curve rise time agaist dampig ratio, show i Figure 4(b), show that the curve ca be fit as ramp i some regios ad of secod order i others. applyig curve fittig ad trial ad error approaches, a better ad more accurate expressios, ca be suggested for 0 ζ < 0.4, for 0.4 ζ <., ad for ζ >., suggested aalytical expressios are give by Eq.(9): = 0 < < = 0.4 < = >. (9) Plottig actual rise time ad a rise time obtaied usig suggested expressios, both versus ormalizes time, are show i Figure 4(d), aalysis of both plots show that the suggested expressios match the actual values with maximum upper error of 0.06 secods for 0.34 < ζ < 0.4, ad maximum upper error of 0.06 secods for 0.6 < ζ < 0.7. this ca lead us to coclude, that the suggested expressios ca be used to aalytically calculate rise time with error of ± 0.0 secods. 3.. Expressios for Rise Time T r of 0% to 00% Criterio Applyig the same procedure, expressios give by Eq.(0) are proposed. Plottig actual rise time ad a rise time obtaied usig proposed expressios, both versus ormalizes time, are show i Figure 5, = 0 < < = 0.4 < = 0.85 < = >. (0) 4. Testig Proposed Expressios Agaist MATLAB To test the proposed expressios, agaist actual values, a MATLAB code is writte to calculate rise time for a give I or II order systems. Rise time is to be calculated, applyig each of the followig: MATLAB cotrol toolbox, proposed expressios, rise time as the reciprocal of the step respose slope at delay time T D, Rise time for the differece i time for 0% to 90% by solvig Eq. (8)

7 Iteratioal Joural of Automatio, Cotrol ad Itelliget Systems Vol., No., 05, pp Comparig aalytical expressios for accuracy Actual Tr Suggested expressio 4.5 Normalised Tr*W Zeta Figure 4 (d). Rise time plot of actual ad usig suggested expressios. Figure 4 (a). Defiitio of agle ϕ. Comparig aalytical expressios for accuracy Comparig aalytical expressios for accuracy Actual Tr:, 0% to 90% criterio Calculated Tr Normalised time, Tr*W Zeta Figure 4 (b). Normalized rise time versus ζ, for secod-order system. Comparig aalytical expressios for accuracy 8 Actual Tr 7 6 Normalised Tr*W Normalised Tr*W Zeta Figure 5. Rise time, 0%-90%; comparig actual ormalized rise time agaist obtaied usig derived expressios. Testig for I or II order systems give by Eq.(-3), will result i rise time values actual ad calculated give i tables -3. Aalysis of plots of both actual rise time ad calculated usig suggested expressios, both versus ormalizes time show i Figures 4,5, as well as,calculated data i tables-3, show that the suggested expressios match the actual values with very small deviatio from actual value, this ca lead us to coclude, that the suggested may be used to reflect the actual value ad ca be applied i accurate calculatio, evaluatio ad verificatio of rise time with error of ± 0.0 secods G(s) = s s () Zeta Figure 4 (c). Plots of differet approximatio for rise time. G(s) = s 4 +.4s + 4 ()

8 58 Ayma A. Aly ad Farha A. Salem: A New Accurate Aalytical Expressio for Rise Time Iteded for Mechatroics Systems Performace Evaluatio ad Validatio G(s) = (3) s + Table. Testig for II order system give by Eq.(). Proposed ( ) Proposed (0.0-) MATLAB. Step properties : ( ) MATLAB (step properties:0-) at TD slope Calculated T R Actual T R Deviatio Table. Testig for II order system give by Eq.(). Proposed ( ) Proposed (0.0- ) MATLAB Step properties : ( ) MATLAB (step properties:0-) at TD slope Calculated T R Actual T R Deviatio Table 3. Testig for I order system give by Eq.(3). Proposed ( ) =.97*T MATLAB (step properties) Rise time at TD T R Coclusios More precise aalytical expressios for accurate calculatio of ''rise time'' with miimum deviatio from actual values are derived ad tested. The suggested expressios ca be applied to aalytically calculate rise time with deviatio to ± 0.0 secods at actual value. Suggested expressios are iteded to be used i systems dyamics performace aalysis, cotroller desig verificatio, ad related scieces, as well as for the applicatio i educatioal process. The Applied MATLAB Code clc, clear all, close all, um=iput(' Eter Numerator: '); de=iput(' Eter Deomiator: '); sys=tf(um, de); q=legth(de); if q== root=roots(de); poles=abs(root); if ((poles < 0) (real(poles)<0)) uu=' The system is stable'; else uu=' The system is ustable'; ed Time_costat = -/( poles); % rise time calculatios, Tr Calculatios rise_time 9=.97/poles; rise_time_0_=4.6/poles; Ess= /(+ dcgai(sys)); home, pritsys(um,de,'s') disp( ' '), sys; step(sys) disp('============================'); disp( ' Stability aalysis: ') disp(' '); fpritf( ' The system pole is: P = %.f, %s \',poles,uu) disp(' '); disp('==========================='); disp(' Rise time,tr Calculatios :') disp(' '); fpritf( 'Calculated Rise time,( criterio), Tr = %.5f Secods \',rise_time 9) fpritf( 'Calculated Rise time,( criterio), Tr = %.5f Secods \',rise_time_0_) disp('==========================='); disp( ' ') elseif q==3 disp( ' ') % disp(' Please wait: it takes 3-5 secods ') zeta=de()/(*sqrt(de(3)*de())); omega_=sqrt(de(3)/de()); omega_d=omega_*sqrt(-zeta^); system_poles=roots(de); if system_poles < 0 uu= ' The system is stable'; else uu= ' The system is ustable'; ed Time_costat=/(zeta*omega_) zeta=[0:0.0:.5]';

9 Iteratioal Joural of Automatio, Cotrol ad Itelliget Systems Vol., No., 05, pp sys={}; for i=:legth(zeta) sys{i}=tf(omega_*omega_, [, *zeta(i)*omega_, omega_*omega_]); % omega_= sys=tf(omega_*omega_, [, *zeta*omega_, omega_*omega_]); ed [y,t]=step([sys{:}]); % Calculatig Rise time,tr ; 0%:90% A=logical( y(:,:)>=0.); A9=logical( y(:,:)>=0.9); time_=[]; time_9=[]; Rise_time 9=[]; for i=:legth(zeta) tmp= t(a(:,i)); tmp9= t(a9(:,i)); time_(i)=tmp(i); time_9(i)=tmp9(i); ed Rise_time 9=time_9-time_; aswer3=[ zeta, Rise_time 9']; % format bak % P=zeta a3=fid(aswer3(:,)==zeta); actual_tr 9=aswer3(a3,); if ((0< zeta) && (0.4 > zeta)) Tr_calcul 9=(.5-0.*zeta+.55.*zeta^)/omega_ ; elseif 0.4 <= zeta && (zeta < 0.85) Tr_calcul 9=( *zeta+.58.*zeta^)/omega_ ; elseif 0.85 <= zeta && (zeta <=.) Tr_calcul 9=( *zeta+.58.*zeta^)/omega_ ; else Tr_calcul 9=(4.69*zeta-.9)/omega_; ed % Calculatig Rise time,tr ; 0%:00% A0=logical( y(:,:)>=0); A=logical( y(:,:)>=0.999); time_0=[]; time_=[]; Rise_time_0_=[]; for i=:legth(zeta) tmp0= t(a0(:,i)); tmp= t(a(:,i)); time_0(i)=tmp0(i); time_(i)=tmp(i); ed Rise_time_0_=time_-time_0; aswer4=[ zeta, Rise_time_0_']; a4=fid(aswer4(:,)== zeta); actual_tr_0_=aswer4(a4,); if ((0< zeta) && (0.4 > zeta)) Tr_calcul_0_ = (. -0.*zeta+3.*zeta^)/omega_ ; elseif 0.4 <= zeta && (zeta <.) Tr_calcul_0_=(.6-0.5*zeta+.58.*zeta^)/omega_ ; else Tr_calcul_0_=(4.67*zeta-.)/omega_; ed Tr_calcul_0_; y=step(sys,t); DC_gai=y(legth(t)); home, pritsys(um,de,'s') disp( ''), sys; step(sys) disp( '') disp(' Time costat,t Calculatios :') disp('====================='); disp( ' Stability aalysis: ') disp( [system_poles(,);system_poles(,)]) fpritf( ' %s \',uu), disp(' '); fpritf( ' The dampig ratio,zeta = %.5f Secods \',zeta) fpritf( ' The UNdamped atural frequecy, Omega_= = %.5f rad/s \',omega_ ) fpritf( ' The Damped atural frequecy, Omega_d= = %.5f rad/s \',omega_d ) disp(' '); fpritf( ' Time costat,t = %.5f Secods \',Time_costat) disp(' Rise time,tr Calculatios :') disp('====================='); fpritf( 'Actual Rise time,( criterio), Tr = %.5f Secods \',actual_tr 9) fpritf( 'Calculated Rise time,( criterio), Tr = %.5f Secods \',Tr_calcul 9) disp(' '); fpritf( 'Actual Rise time,( criterio), Tr = %.5f Secods \',actual_tr_0_) fpritf( 'Calculated Rise time,( criterio), Tr = %.5f Secods \',Tr_calcul_0_) disp('====================='); ed Refereces [] Ashish Tewari, Moder Cotrol Desig with MATLAB ad SIMULINK, Joh Wiley ad sos, February 00. [] LTD, 00 Eglad Katsuhiko Ogata, moder cotrol egieerig, third editio, Pretice hall, 997. [3] Farid Golaraghi Bejami C. Kuo, Automatic Cotrol Systems, Joh Wiley ad sos INC, 00. [4] Norma S. Nise cotrol system egieerig, Sixth Editio Joh Wiley & Sos, Ic, 0.

10 60 Ayma A. Aly ad Farha A. Salem: A New Accurate Aalytical Expressio for Rise Time Iteded for Mechatroics Systems Performace Evaluatio ad Validatio [5] Gee F. Frakli, J. David Powell, ad Abbas Emami-Naeii, Feedback Cotrol of Dyamic Systems, 4th ed., Pretice Hall, 00. [6] html. [7] Bill Goodwie, Egieerig Differetial Equatios Theory ad Applicatios, Spriger 0. [8] Dale E. Seborg, Thomas F. Edgar, Duca A. Mellichamp,Process dyamics ad cotrol, secod editio, Wiley 004. [9] Farha A. Salem, Precise Performace Measures for Mechatroics Systems, Verified ad Supported by New MATLAB Built-i Fuctio, Iteratioal Joural of Curret Egieerig ad Techology, Vol.3, No., Jue, 03. [0] Levie, William S. (996), The cotrol hadbook, Boca Rato, FL: CRC Press, p. 548, ISBN [] Norma S. Nise, Cotrol system egieerig, 6 editio, Joh Wiley & Sos, 0.