MATLAB/Simulink Based Study of Different Approaches Using Mathematical Model of Differential Equations

Transcription

1 I.J. Intelligent Systems and Appliations, 04, 05, -4 Published Online April 04 in MECS ( DOI: 0.585/ijisa MATLAB/Simulink Based Study of Different Approahes Using Mathematial Model of Differential Equations Vijay Nehra Department of Eletronis and Communiation Engineering, Bhagat Phool Singh Mahila Vishwavidyalaya, Khanpur Kalan, Sonipat, Haryana, India Abstrat A large number of diverse engineering appliations are frequently modeled using different approahes, viz., a differential equation or by a transfer funtion and state spae. All these desriptions provide a great deal of information about the system, suh as stability of the system, its step or impulse response, and its frequeny response. The present paper addresses different approahes used to derive mathematial models of first and seond order system, developing MATLAB sript implementation and building a orresponding Simulink model. The dynami analysis of eletri iruit and system using MATLAB/Simulink has been investigated using different approahes for hosen system parameters. Index Terms Mathematial Model, System Dynamis, MATLAB, ODE, Eletri Ciruit, Simulink, Symboli Computation, Transfer Funtion I. Introdution Differential equations play a pivotal role in diverse fields like mathematis, physis, eletrial, mehanial & ivil engineering and biologial population modeling et. They form a powerful means of modeling several engineering and sientifi problems. The diversity of engineering appliations employing these equations results into differential equations of different orders. A differential equation is a mathematial equation for an unknown funtion of one or several variables that relates the value of the funtion itself and its derivatives of various orders []. It is well evident that ordinary differential equations (ODE) play a vital role in iruit and system analysis, whih in turn play a fundamental role in the modern tehnologial system. Eletri iruits are systems that an be desribed in different ways using differential equations of first, seond and higher order. Often, eletri systems are modeled using different approahes suh as a differential equation, or by a transfer funtion and state spae. Any of these desriptions an provide information about the system dynamis [-4]. In this study, MATLAB/Simulink pakage has been seleted beause of its general purpose nature and its extensive use in modeling, simulation and analysis of dynamis systems. MATLAB is a general purpose ommerial simulation pakage used to perform numerial and symboli omputation. Generally, it is used to solve differential equations quikly and easily in an effetive manner. It also leads to a visual plot of the results [5-0]. Engineering simulation using graphial programming tool Simulink plays a vital role in understanding and assessing the operation of a system. Simulink, built upon MATLAB, is a powerful interative tool for modeling, simulating and analyzing dynamial system; thus, forming an ideal tool for qualitative and quantitative analysis of eletrial and eletroni network study. It has been identified as an ideal tool for laboratory projets, and has hene been adopted for teahing a variety of ourses in eletrial and eletronis engineering. The benefits of MATLAB and Simulink have been well doumented by several workers [-8]. The present work is an implementation of MATLAB text based desription m.file, graphial programming Simulink model; and data driven modeling using MATLAB and Simulink together. The MATLAB sript implementation is a text based m.file desription of system under referene that an be written in any text editor. The graphial approah uses Simulink model in terms of blok diagram realization for visualizing system dynamis. A Simulink model for a given problem an also be onstruted using building bloks from the Simulink library. The model onsists of various bloks from Simulink libraries arranged in a desired fashion, offering solution to various problems without having to write any odes. The work presented here an be implemented to investigate the response of other similar engineering appliations. One an implement the MATLAB sript and Simulink model presented here as a omputational projet based model in iruit and system ourse, that an ontribute to an improvement in learning and ultimately to eduational suess in engineering program. In this study, the same has been demonstrated by onsidering an example of st order eletri system. The behavior of seond order system has also been

2 MATLAB/Simulink Based Study of investigated for step exitation for various values of damping ratio. The organization of the present paper involves a brief introdution, followed by Setion whih fouses on mathematial modeling of first order system. Setion 3 and Setion 4 desribe seletion of system parameters, and response analysis of first order eletri iruit using MATLAB with different approahes. Setion 5 deals with various approahes for analysis of first order system using Simulink. Setion 6 summarizes data driven modeling followed by result and disussion in Setion 7. Setion 8 desribes the sope of work in diverse engineering appliations; and finally, Setion 9 emphasizes on implementation of work presented in this study using Silab/Xos as an alternative to MATLAB/Simulink; followed by a onlusion in Setion 0. II. Mathematial Modeling of RC System One of the first engineering appliations that students always pratie is first order series RC eletri iruit. The differential equation governing the series RC iruit of Fig. is given as: v i t) i( t) R i t dt C ( ) ( () where v i (t) is the foring funtion; v (t) is the output voltage aross the apaitor; C is the apaitane of the apaitor in Farad; R is resistane of the resistor in Ohm; and i (t) is urrent flowing in the iruit in Ampere The iruit urrent i (t) is given as: dv i C () dt The equation () an be rewritten as: dv RC v vi (3) dt v i(t) Fig. : Eletrial RC Ciruit R C v (t) Taking Laplae transform of (3) and rearranging, the system funtion is given as: V ( s) RC V ( ) ( ) i s src ( s ) ( s ) RC where RC is the time onstant of the system. Looking at RC iruit, urrent as being the output and voltage aross the apaitor as state variables, the state variable representation of the RC iruit is given as:. ( dv t) dt v vi RC RC The urrent aross the iruit is i v vi R R (4) (5) (6) The matrix A B C D for RC series iruit is given as: A RC C R B RC D R The mathematial model presented in this setion has been implemented using MATLAB/Simulink using different approahes for hosen system parameters. III. System Paramaters In this investigation, the hosen parameters of eletri iruit inlude: the resistane value: R =.5 KΩ; the iruit apaitane value: C =0.00 F. The system response has been investigated for step exitation of amplitude 0, ramp funtion of slope 0 and impulse exitation of magnitude 0. After setting the system parameters, the governing (3) for series RC iruit beomes: dv 5 v vi dt (7) Thus, the series RC iruit generates a mathematial model in terms of first order differential equation. Using the hosen design parameters, various approahes have been inorporated for deriving the solution of differential equation using MATLAB and Simulink. At first, the mathematial model presented in this setion has been implemented through different approahes using MATLAB sript implementation m.file.

3 MATLAB/Simulink Based Study of 3 In the nd part of the study, the Simulink models have been drawn diretly using different approahes governing the behavior of series RC iruit. The mathematial desription of the system in Simulink has been translated into blok diagram representation using the various elements of Simulink blok library, and finally, the solution of model has been obtained using built-in MATLAB solver. IV. System Response Analysis USING MATLAB In this investigation, response of first order series RC eletri iruit is presented using different approahes. The effet of varying system parameters is also disussed. 4. Symboli Simulation Tehnique, Approah I (Laplae Transform Method) In this setion, MATLAB symboli simulation tehnique using Laplae transform method has been employed to solve the model. The Laplae transform method (LTM) is useful for solving initial value problems involving linear onstant oeffiient equations. In this tehnique, by using Laplae transform governing, differential equations are onverted into algebrai equations that an be solved by simple algebrai tehnique. The inverse Laplae transform is then applied to give orresponding funtion whih forms the solution to the differential equation. The symboli built-in ommands suh as 'laplae', 'ilaplae', and 'heaviside' funtions have been used [5-6, 9-0]. The MATLAB sript for the mathe matial model of differential equation (7) using symboli implementation is shown below in Fig.. MATLAB Solution % MATLAB sript: RC.m %Symboli simulation of (7) of series RC iruit l; % lear the ommand window lear all; % lear all the variables lose all; %lose all figure windows syms t s v %define symboli variables v0 = 0; %define IC for v(0) vd0 = 0; %define IC for v'(0) f=0*(heaviside(t)); %define step signal F = laplae(f, s); %Laplae of step input v = s*v - v0; %implement Dv soltn = 5*v+v-F; %differential equation tf=solve(soltn,v); %solution of equations v_output=ilaplae(tf,t); %find v(t) by using IL pretty(v_output); subplot(3,,); ezplot(v_output,[0,40]); % plot of symboli objet ylim([0 ]) title('voltage versus Time for step using LTM'); ylabel('voltage' ); F = laplae(int(0*heaviside(t)),s); %LT of ramp solution = 5*v+v-F; %implement DE tf=solve(solution,v); %find sol of DE v_output=ilaplae(tf,t); %find v(t)by using IL pretty(v_output); subplot(3,,); ezplot(v_output,[0,40]); title('voltage versus Time for ramp using LTM'); F = laplae(diff(0*heaviside(t)),s); %LT impulse solution = 5*v+v-F; %differential equation tf=solve(solution, v); v_output=ilaplae(tf,t) ; %find v(t) by using ILT pretty (v_output); subplot (3,,3); ezplot(v_output,[0,40]); axis([ ]) title('voltage versus Time for impulse using LTM'); Fig. : MATLAB sript for analysis of RC iruit (RC.m) 4.. Analysis using 'dsolve' ommands One an also find the solution of differential equation using 'dsolve' funtions of symboli omputation. The MATLAB symboli funtion 'dsolve' is used to symbolially solve the ordinary differential equations speified by ODE as first argument and the boundary or initial onditions speified as seond argument. In fat, only one statement 'dsolve' is required to define the series RC iruit. The other ommands are required for plotting purpose and annotation of graphial output. The MATLAB implementation to solve the (7) using 'dsolve' funtion is shown in Figure 3. MATLAB Solution %MATLAB sript: RC.m %Symboli solution of (7) using dsolve funtion l; %lear ommand window lear all; % lear workspae lose all %lose all figure windows syms t s v % reating symboli objet out=dsolve('5*dv+v=0*heaviside(t)','v(0)=0'); pretty(out); subplot(3,,); ezplot(out,[0,40]); % plot of symboli objets title('voltage versus Time for step using dsolve'); ylim([0 ]) output=int(out); %finding ramp response pretty(output); subplot (3,,);

4 4 MATLAB/Simulink Based Study of ezplot(output,[0,40]); title('voltage versus Time for ramp using dsolve'); output=diff(out); %finding impulse response pretty(output); subplot(3,,3); ezplot(output,[0,40]); title('voltage versus Time for impulse using dsolve'); ylim([0,3]); Fig. 3: MATLAB sript using dsolve funtion (RC.m) 4.. Effet of varying the Ciruit Parameters The effet of varying the iruit parameters of first order series RC iruit has also been studied by implementing MATLAB ode as shown in Figure 4. Different values of R (, 5, 0 K Ohms) will be onsidered in the sript presented below. MATLAB Solution % MATLAB sript: RC3.m % Effet of varying the iruit parameters l; %lear ommand window lear all; %lear workspae lose all; %lose all figure windows % define iruit parameters R=, 5 and 0 K; C=e-3 syms t s v % reating symboli objet v_out=dsolve('4*dv+v=0*heaviside(t)','v(0)=0'); v_out=dsolve('0*dv+v=0*heaviside(t)','v(0)=0'); v_out=dsolve('40*dv+v=0*heaviside(t)','v(0)=0'); figure() ezplot(v_out,[0 40]); hold on ezplot(v_out,[0 40]); ezplot(v_out,[ 0 40]); axis([ ]) ylabel('voltage') legend('step, R=K', 'Step, R=5K', 'Step, R=0K') title ('Voltage aross apaitor for R=, 5 and 0K are') %voltage aross resistor figure () vr=0-v_out; vr=0-v_out; vr3=0-v_out; ezplot(vr,[0 40]); hold on ezplot(vr,[0 40]); ezplot(vr3,[0 40]) ; ylim([ 0 0]) title('voltage aross resistor for R=, 5 and 0K are') legend('voltage, R=K', 'Voltage, R=5K', 'Voltage, R=0K') figure(3) %urrent aross resistor ir=(0-v_out)/e3; ir=(0-v_out)/5e3; ir3=(0-v_out)/0e3; ezplot(ir,[0 40]); hold on; ezplot(ir,[0 40]); ezplot(ir3,[0 40]); ylim([ 0 6e-3]) title ('Current i(t)for R=, 5 and 0K are'); legend('current, R=K', 'Current, R=5K', 'Current, R=0K') ylabel('current'); Fig. 4: Sript for varying resistane (R) (RC3.m) 4. Numerial Solution of ODEs This setion demonstrates the usage and implementation of ommonly used ODE solver for solving differential equations. 4.. Simulation Using ODE MATLAB Funtions, Approahes II In general, mathematial model of diverse engineering appliations like eletri iruit are designed and solved to produe the behavior of the iruit and system under different onditions. MATLAB has a library of several built-in ODE funtions that an be used for partiular ases. The response of eletri iruit for different applied onditions has been presented here using ode45 solver. The ode for a first order ODE is very straightforward. One of the features that ode45 solver requires is that the system of equations must be organized in first order differential equations. The transformation of higher ODE of the system of DE to the first ODE is mandatory. ode45 is one of the most popular ode used to solve differential equations []. The differential equation of series RC iruit is already modeled in terms of first order differential equations as given by (7) for hosen iruit parameters. The syntax of MATLAB ODE solver is: [tout, yout]=solver_name (odefun, timespan, IC) where odefun is the given DE as string ontained in a m funtion file; time span is the range t t t 0 final over whih the solution is required (tspan=[t0: tfinal]) and IC represents the initial onditions

5 MATLAB/Simulink Based Study of 5 The differential equation an be handled in a few simple steps. Step: First of all, redue the system governing n th order differential as a set of n first order ordinary linear or nonlinear DEs. If it is already first order, no need to onvert it. Step: Write all the first order equation in a standard form speifying the interval of independent variable and initial value as given below. dy f ( t, y) for t t t dt f t. t 0 y y 0 with 0 Step3: Create a user defined funtion file or use an anonymous funtion to solve all the first order equations for a given values of t and y. It implies that DE must be first defined in a funtion file. Step4: Selet a method of solution i.e. hoose the built-in funtion of MATLAB ODE solver type. Step5: Solve the ODE and get the result from output. The funtion ode3 and ode45 are very similar. The only differene is that ode45 is fast, aurate and uses larger step sizes, but is still muh slower than ode3. The response obtained using ode45 is not as smooth as using ode3. The MATLAB sript for the mathematial model of series RC iruit using MATLAB ODE funtion is presented in Fig. 5 and 6 respetively. Fig. 5 depits MATLAB sript alling program file (RC4.m) for funtion file as shown in Fig.6. MATLAB Solution %MATLAB alling sript file: RC4.m %ODE solver approah l; %lear ommand window lear all; %lear workspae lose all; %lose all figure windows tspan=0: 40; %define time interval to solve ODE initial =0; %define initial ondition % solve the ODE diretly with ode45 [t,v]=ode45('ode3', [tspan],initial); %plot the step response plot(t,v,'mp'); ylim([0 ]); xlabel('time') ylabel('voltage') title('voltage versus Time for step input using ode45'); legend('step response') Fig. 5: Calling program file (RC4.m) % funtion file that defines the DE funtion dydt=ode3(t,v) % filename: ode3.m % define model parameters at R=.5e3; C=0.00; u=0 dydt=(u-v)/(r*c) end Fig. 6: MATLAB ODE funtion sript files for analysis of RC iruit 4.. Using Anonymous Funtion The solution of (7) an also be developed using anonymous funtion. The anonymous funtion an be defined in ommand window or be within sript. The solution applied in this study using anonymous funtion is oded in MATLAB program form as presented in MATLAB sript m.file (RC5.m) as depited in Fig. 7. MATLAB Solution %MATLAB sript: RC5.m % This program solves a system of ODE l; %lear ommand window lear all; %lear workspae lose all; %lose all Figure windows % define eletri iruit parameters R=input('Enter the resistane R:'); C=input('Enter the apaitane C:'); u=input('enter the input signal U:'); tspan=0: 40; %define time interval to solve ODE initial =0; %define initial ondition % solve the ode diretly with ODE45 ode=@(t,v)(u-v)/(r*c); [t,v]=ode45(ode, tspan,initial); plot(t,v,'r.'); ylim([0 ]) xlabel('time') ylabel('voltage') title('step response') legend ('Capaitor voltage') Fig. 7: MATLAB ODE sript for analysis of RC iruit using anonymous funtion (RC5.m) 4.3 Simulation Using Built-in MATLAB Funtions, Approahes III In this setion, transfer funtion and state spae methods have been used to solve the model The Transfer Funtion Method One an also study the response of eletri iruit using the various built-in funtions of MATLAB. The response of the RC system is investigated by subjeting the model to various hosen input funtions. The system an be entered in state spae form or as transfer funtion by means of numerator and denominator oeffiient or by means of zeros, poles and the gains. Indeed, there is no diret ommand to obtain the ramp response of the system, therefore for obtaining ramp response the

6 6 MATLAB/Simulink Based Study of transfer funtion an be expressed as: G ramp ( s) Gstep ( s) *. Finally, the ramp response s an be obtained by using MATLAB built-in funtion 'step'. The MATLAB sript implemented to solve the RC iruit is depited in Fig. 8. MATLAB Solution %MATLAB sript: RC6.m %Response of RC system using transfer funtion l; %lear ommand window lear all; %lear workspae lose all; %lose all figure windows %define eletri iruit parameters R=input('Enter the resistane R:'); C=input('Enter the apaitane C:'); U=input('Enter the amplitude of step input signal U:'); num=[u]; %define the numerator of the TF den=[r*c ];%denominator of the TF disp('the transfer funtion representation is:'); G=tf(num,den);% reate transfer funtion subplot(3,,) step(num,den);% plot the step response axis([0, 40, 0, ]); title('step response of series RC system'); subplot(3,,); num=[u]; den=[ R*C 0]; step(num,den)% plot the ramp response axis([0, 40, 0, 400]); title('ramp response of series RC system'); subplot(3,,3); impulse(num,den);% plot the impulse response axis([0, 40, 0, 3]); title('impulse response of series RC system'); Fig. 8: Sript for analysis of RC iruit (RC6.m) 4.3. The State Spae Method MATLAB also failitates step response of RC system using state spae approah by diretly assigning system matries/vetors as input argument [4]. The solution of RC iruit an also be developed using state spae representation of (5) and (6) as depited in Fig 9. MATLAB Solution %MATLAB sript: RC7.m %omputes the response of RC system using SS l; %lear ommand window lear all; %lear workspae lose all; %lose all figure windows R=input('Enter the resistane R:'); C=input('Enter the apaitane C:'); U=input('Enter the amplitude of step signal U:'); A= [-/(R*C)]; B= [/(R*C)]; C= [-/R]; D= [/R]; disp('the state spae representation:'); G=U*ss(A,B,C,D); step(g); % plot step response xlabel('time') ylabel('current') title('current response of RC system') legend ('Current response') axis([ e-3]) Fig. 9: Sript for analysis of RC iruit (RC7.m) 4.4 Analytial Computation of Time Domain Response In this setion, at first the time domain step, ramp and impulse response of eletri RC iruit has been omputed analytially from transfer funtion; and then the same has been symbolially solved using 'dsolve' ommand for solution of DEs. Finally MATLAB sript has been developed for time domain step, ramp and impulse analytial expression Computation of Time Domain Response Step Response Using (4), the system funtion of the iruit is given as: V ( s) G( s) V ( s) s Laplae transform of unit step signal is i L( u( t)) (8) s Substituting the (8) in (4), the Laplae transform of output signal is V ( s) s( s ) Applying inverse Laplae transform, the step response of the system is t / v ( e ) for t 0 In other words, for step input with amplitude A i.e v i Au( t) the step response is given by t / v A( e ) for Ramp Response t (0) Laplae transform of ramp signal is (9)

7 MATLAB/Simulink Based Study of 7 L( r( t)) () s Substituting the () in (4), the Laplae transform of output signal is V ( s) s ( s ) () Applying inverse Laplae transform, the ramp response of the system is given as: v t ( t) t ( e ) for t 0 In other words, if we have ramp input with slope i.e v i A r( t) the ramp response is given by v t) A ( t ( e )) for 0 ( Impulse Response t Laplae transform of impulse signal is A t (3) L( ) (4) Substituting the (4) in (4), the Laplae transform of output signal is ( s) ( s ) ( s ) V (5) Applying inverse Laplae transform, the impulse response is v t / e for t 0 In other words, for an impulse voltage input with area or strength A i.e A v i The Impulse response is give by v A RC t / e for 0 t (6) 4.4. Analytial Computation of Time Domain Response Using Symboli Solution In this part, the time domain analytial expression of the apaitor voltage v (t) for step, ramp and impulse exitation has been obtained by the solution of (3) using 'dsolve' funtion. Using (4) and ross multiplying, the equation that desribes the eletri iruit is sv ( s) V ( s) V ( s) (7) i Converting (7) bak to a differential equation, we get ' v v v (8) i For o mputing step response v i for t 0 Substituting the same in (8), the governing differential equation for step signal is: ' v v (9) For omputing ramp response v i t fort 0 The governing differential equation for ramp signal is: ' v v t (0) For omputing impulse response v i dira( t) fort 0 Substituting the same in (8), the governing differential equation for impulse signal is: ' v v dira( t) () The MATLAB sript developed to ompute time domain step, ramp and impulse response given by (9), (0) and () is developed below in Fig.0. MATLAB Solution MATLAB sript: RC8.m %Computation of time domain response of RC iruit %Symboli sol.of DE (9) and (0) using dsolve funtion syms tau y t % define symboli variables % Solution of DE using dsolve funtion % Time domain step solution of DE step=dsolve('tau*dy+y=','y(0)=0') % Time domain ramp solution of DE ramp=dsolve('tau*dy+y=t','y(0)=0') Fig. 0: Differential equation solution using dsolve funtion (RC8.m) 4.5 Plotting of Analytial Computation of Time Domain Response, Approahes IV The time domain analytial expression of the apaitor voltage v (t) for step, ramp and impulse exitation is obtained by the solution of (3) and is given by (0), (3) and (6) respetively. The system response of eletri iruit under referene has also been plotted by implementing a MATLAB sript of time domain representation of (0), (3) and (6) for hosen system paramaters. A typial MATLAB solution m.file developed for time domain response plotting is depited in Fig..

8 8 MATLAB/Simulink Based Study of MATLAB Solution %MATLAB sript: RC9.m %plotting of analytial time domain response l; %lear ommand window lear all; %lear workspae lose all; %lose all figure windows %define parameters of eletri iruit R=input('Enter the resistane R:'); C=input('Enter the apaitane C:'); A=input('Enter the step input amplitude A:'); A=input('Enter the ramp input slope A:'); A=input('Enter the impulse input strength A:'); T=R*C; %ompute RC time onstant of the iruit t=0: T/00:8*T; %reate time index vetor subplot(3,,) v=a*(-exp(-t/t)); %time domain step expression plot (t, v,'r'); % plot the variables axis([ ]); title('voltage versus Time for step exitation'); subplot(3,,) v=a*(t-t*(-exp(-t/t))); %time domain ramp response plot (t, v,'m'); title('voltage versus Time for ramp exitation'); subplot(3,,3) v3=(a/t)*exp(-t/t); % time domain impulse relation plot (t, v3); title ('Voltage versus time for impulse exitation'); Fig. : MATLAB sript for plotting of analytial time domain response of RC iruit (RC9.m) 4.6 Comparison of Symboli ds olve Obtained Solution with a Diret Solution ODE S olver, Approahes V This setion gives a omparison of symboli solution with analytial solution. The MATLAB sript for omparing symboli solution with diret ode45 solver is shown in Fig.. MATLAB Solution %MATLAB sript: RC0.m %solve a system of ODE using anonymous funtion %Solution using MATLAB ODE45 funtion %Symboli solution of (7) using dsolve funtion l; %lear ommand window lear all; %lear workspae lose all; %lose all figure windows syms t s v %reating symboli objet % define parameters of eletri iruit R=input('Enter the resistane R:'); C=input('Enter the apaitane C:'); u=input('enter the input signal u:'); out=dsolve('5*dv+v=0*heaviside(t)','v(0)=0'); pretty(out); ezplot(out,[0,40]); %symboli plot of step response hold on %solve the ode diretly with ODE45 ode=@(t,v)(u-v)/(r*c); [t,v]=ode45(ode,0:40,0); plot(t,v,'r+','linewidth',); %plot the response axis([ ]); title('voltage versus Time for step exitation'); legend('symboli solution', 'ode solver solution') Fig. : MATLAB sript for omparison of symboli dsolve obtained solution with a diret solution ODE solver (RC0.m) The forthoming setion presents the simulation and analysis of first order eletri iruit using Simulink. The mathematial model of RC iruit desribed in setion () has been utilized in Simulink model development, and finally, some results of simulation have been presented. V. System Response Analysis Using SIMULINK Simulink is a powerful interative simulation tool built upon MATLAB. In Simulink, a typial model onsists of the soure, the system being modeled and sinks. The soure blok provides an input signal to a system in order to extrat its behavior. A system is an objet or a olletion of interonneted objets or blok diagram representation of proess being modeled whose behavior an be investigated using Simulink methods. The Simulink model is reated using various bloks of library available in Simulink. The sink bloks are display devies used to visualize the output. Engineering systems are often modeled using different approahes suh as differential equation, transfer funtion and state spae. The Simulink model an be drawn diretly using different approahes governing the behavior of iruit or system under referene. Simulink onverts graphial representation into a state spae representation onsisting of the set of simultaneous first order differential/differene equations. These equations are then solved by using MATLAB integrating funtions [-3]. The steps for Simulink solution is as follows: Step: Firstly, desribe the mathematial model of system under referene. Step: Open a new Simulink model window. Step3: Selet the required number of desired bloks from Simulink library and opy into model window.

9 MATLAB/Simulink Based Study of 9 Step4: Connet the appropriate blok and ustomize eah blok as desired. Save the model. Step5: Make the required hanges in the simulation parameters. Step6: Run the simulation model and observe the result. In the forthoming subsetion, the various approahes for solving the first order RC iruit is presented. 5. Simulation Model Using Differential Equations, Approahes I On solving the (3), the highest derivates term is given as: dv vi v () dt RC In this methodology, the differential equation governing the RC eletri iruit is arranged as () for highest order derivates in the left side [-7]. In order to develop the Simulink model using (), firstly, the number of terms inside the differential equation to be solved is identified. Eah term requires a series of operation. The Simulink blok needed for solving differential equation () are dragged from respetive blok libraries as per requirement and interonneted to reate the model. The various bloks suh as signal generator, step and onstant blok in soure library; produt and subtration blok in the Simulink math library; the integrator blok in Simulink ontinuous library and the sope, To workspae, To file, XY graph are dragged from sink blok library. The integrator is the basi building bloks for solving differential equations. The Clok blok generates the time and the same is opied to workspae. The Simulink mode l solution for first order differential equation () is depited in Fig. 3. In this Simulink model, one an hoose the input exitation from signal generator, ramp or impulse using multiport swith. Depending on numerial onstant speified in the ontrol port, one an selet the orresponding signal. There is a provision to selet multiple inputs from signal generator suh as sine and square. Constant Simulink model solution for solving DE of series RC iruit voltage Signal Generator Ramp r(t) Multiport Swith Clok u Time To Workspae RC.mat To File Step Step Add u(t) Impulse Sope3 Manual Swith u Add (u-v) Produt /s Integrator Sope Measured output Step v RC.mat To File Constant v R R C Produt Divide du/dt Derivative Produt Measured urrent C Fig.3: Simulink Model (RCS.mdl) for RC Ciruit of () After onstrution of the model, all system parameters for model are set by double liking on eah blok as hosen before running simulation. After ustomization of Simulink model, simulation is performed for speified time period to analyze its behavior. One an visualize the input and output plot on the same sope using mux blok as indiated in Sope of model. The 'To Wokspae' and 'To File' blok is used to save the data in the Workspae and to MAT files for later proessing. 5. Simulation Model Using Transfer Funtion, Approah II Simulink enables to simulate ontinuous time system either using state spae or transfer funtion of eletri iruit forming the system using individual bloks from the Simulink ontinuous blok library. The transfer funtion of series RC iruit as given by (4) is: V ( s) RC V ( ) ( ) i s src ( s ) ( s ) RC

10 0 MATLAB/Simulink Based Study of Alternatively, one an develop the Simulink model using (4) and study the behavior of RC series iruit using the various bloks of ontinuous bloks library suh as step, transfer funtion and sope. The Simulink model solution for (4) using transfer funtion blok is shown in Fig. 4. In a nut shell, the step is a soure blok whih generates a step input signal. This signal is transferred through line in the diretion indiated by arrows to the transfer funtion blok. The transfer funtion blok modifies its input signal and outputs a new signal on another line to the sope using Go to and From blok. The sope is a sink blok used to display a signal of the system output muh like osollosope. The hosen system parameters are fed into these bloks depending upon the modeling type of the system either in terms of transfer funtion or state spae or zero pole gain representation. Fig. 4: Simulink flow diagram using transfer funtion of (4) (RCS.mdl) The Simulink model to investigate the pulsed response of series RC iruit onsisting of pulse generator blok, transfer funtion blok and sope is depited in Fig.5. The input and output waveform are both visualized on the sope. Fig. 5: Simulink Model (RCS3.mdl) for RC Ciruit of (4) An alternative Simulink model reated for solving differential equations () along with transfer blok is shown in Fig.6. In this model, there is a provision to selet step or ramp exitation. Simulink blok diagram for series RC iruit Clok Time voltage To Workspae RC3.mat Ramp u u To File Step u Manual Swith u Add (u-v) Produt /s Integrator Sope Measured output Constant R du/dt Clok XY Graph R C C Produt Divide Input signal v R*C.s+ Transfer Fn Derivative Produt Measured input & output Measured urrent Measured input Measured output RC.mat To File Fig. 6: Simulink flow diagram (RCS4.mdl) of RC Ciruit using DE () and TF (4) 5.3 Simulation Model Using State Spae, Approah III Complex eletri iruit an be alternatively solved using the system state equation and output equation. Mathematially, the state of the system is desribed by a set of first order differential equation in terms of state variables [8]. dx Ax Bu (3) dt

11 MATLAB/Simulink Based Study of y Cx Du (4) The equation (3) represents the state equation and (4) represents the output equation. The variable x represents the state of the system, u is the system input, and y is the output, A is system dynamis matrix representing the oeffiient of state variable, B is the matrix of input representing the oeffiient of input in state equation,c is the matrix of output representing the oeffiient of state variable in the output equation, D is the diret exposure matrix input to output. The matrix A, B, C, D for RC series iruit an be obtained diretly and is given as: A RC C R B RC D R The Simulink model solution for (5) and (6) is depited in Fig. 7. The model onsists of step, ramp, manual swith, state spae blok and sope. In this model, one an selet the input from ramp or impulse blok using manual swith. There is a provision to selet step input or ramp and impulse signal using manual swith. Ramp Step3 Add Manual Swith Impulse Sope3 u Manual Swith x' = Ax+Bu y = Cx+Du State-Spae Sope simout Step To Workspae u Step Fig. 7: Simulink flow diagram (RCS5.mdl) setup using SS 5.4 Modified Simulink Model using Transfer Funtion and State Spae Together The modified form of Simulink model using transfer funtion and state spae approah is shown in Figure 8. In this model transfer funtion and state spae blok both are used together. There is a provision to visualize system dynamis using step, ramp or impulse signal. Using multiport swith one an selet the desired input. The seleted input signal is applied simultaneously through state spae and transfer funtion blok using manual swith. Constant Clok Time Signal Generator Ramp Step3 Step Step Add Impulse u Multiport Swith Sope3 u Manual Swith x' = Ax+Bu y = Cx+Du State-Spae R*C.s+ Transfer Fn Sope Sope4 Fig. 8: Simulink flow diagram based on state spae representation (RCS6.mdl) simout To Workspae Sope

12 MATLAB/Simulink Based Study of 5.5 Simulink Model Implementation Using User Defined Funtions Library 5.5. Alternative Model Development Using Fn Bloks, Approahes IV Alternatively, one an develop Simulink model of the series RC iruit using the Fn Blok of User-Defined Funtions Library of Simulink pakages. The Simulink flow diagram set up using the fn blok is shown in Fig. 9. Simulink blok diagram for series RC iruit Clok Time voltage To Workspae RC.mat Ramp r(t) To File u Manual Swith f(u) /s v v Sope Step u(t) u(3)(u()-u()) Integrator Measured Output Constant v Clok XY Graph R R C Produt Divide du/dt Derivative Produt Measured Ciruit Current C Fig. 9: Simulink model of () using Fn blok (RCS7.mdl) 5.5. Using S-funtion in Models In this setion, the Simulink model of (7) has been implemented using S funtion bloks of User Defined Funtions Library. The S-funtion blok is generally used to reate ustomized or to add new general purpose blok to Simulink or to build general purpose blok that an be used several times in a model. Firstl of all, funtion m.file is developed and saved using name RC_kt.m as depited in Fig. 0. MATLAB Solution funtion dx = RC_kt (t,x,vs) %MATLAB funtion sript: RC_kt.m % define model parameters R =.5e3; C = 0.00; dx = -/(R*C)*x+/(R*C)*Vs end funtion[sys, x0, str, ts]=rc_kt_sfn(t, x, u, flag, xinit) % RC_kt_sfn is the name of S-funtion % This is the m-file S-funtion RC_kt_sfn.m % %M file S funtion implements a series RC iruit xinit = 0; swith flag ase 0 % initialization str = []; ts = [0 0]; x0 = xinit; % alternatively, the three lines above an be ombined into a single line as % [sys, x0, str, ts]=mdlinitializesizes(t,x,u) sizes = simsizes; sizes.numcontstates = ; sizes.numdisstates = 0; sizes.numoutputs = ; sizes.numinputs = ; sizes.dirfeedthrough = 0; sizes.numsampletimes = ; sys = simsizes(sizes); ase % Continuous state derivatives Vs = u; sys = RC_kt(t,x,Vs); ase 3 % output sys = x; ase { 4 9} % : Disrete state updates % 3: altimeshit % 9: termination sys = []; otherwise error (['unhandled flag=', numstr (flag)]); end Fig. 0: MATLAB funtion sript for analysis of RC iruit

13 MATLAB/Simulink Based Study of 3 The Simulink model for S-funtion file is depited in Figure. To inorporate an S-funtion into a Simulink model, the S funtion blok from the User-Defined Funtions Library is dragged into the model window. The name of the S-funtion is speified in the S-funtion name field of the S-funtions bloks in dialog box. waveform both are visualized on the sope. In the model, the impulse response is obtained by differentiating the step response, and ramp response is obtained by integrating the step signal Time Domain Model Development, Approahes V The Simulink may be easily adopted in representing time domain response using various elements of Simulink blok library. The orrosponding step, impulse and ramp response for the hosen design paramater is given as: 0.t v 0( e ) (3) v 0 e 5 t 5 (4) Fig.: Simulink model using S-funtion blok (RCS8.mdl) The S-funtion name RC_ kt_sfn is defined in the blok parameter dialog box. The model onsists of step blok, S funtion blok and sope. The input and output v t 5 0( t 5( e )) (5) The system response of hosen iruit is also investigated by onstruting a Simulink model using analytial time domain representation of (3), (4) and (5), and using TF approah as depited in Fig.. Response using frequeny domain and time domain representation Step 5s+ Transfer Fn Step Response Constant Clok -K- Gain /RC e u Math Funtion Add Gain5 5 0 Gain Sope Gain Impulse Response Add -K- Gain4 Ramp Response Fig. : Simulink model (RCS9.mdl) using time domain mathematial model of (3), (4) and (5) and transfer funtion blok given by (4) The Simulink model onsists of lok bloks whih generate time index vetor, gain blok, math funtion blok, add blok, integrator and derivative blok. These bloks are dragged from the respetive libraries, interonneted, and ustomized to set the model paramaters. The upper part of the model onsists of step, transfer funtion and sope. The step response is obtained here using transfer funtion blok and is depited on the sope named Step Response. The lower part of the model onsist of time domain implementation of step, impulse and ramp response and is displayed on sope,3,4 respetively. On blok named sope, along with step response the ramp and impulse response is obtained using integration and derivatives bloks as

14 4 MATLAB/Simulink Based Study of depited in the model. The impulse response is obtained by taking derivatives of step response, and the ramp response is obtained by integration of step response. VI. Data Driven Modeling One an use Simulink together with MATLAB in order to speify data and parameters to the Simulink model. In data driven modeling, one an speify the system parameters ommands in the MATLAB ommand window or as ommands in an m.file in MATLAB sript. The two ases are presented here: 6. Diretly From MATLAB Command Window In this approah, firstly the hosen system parameter variables are set in the funtional parameters dialog box window for desired blok of the model. The variable referred in funtional parameters dialog window of the model are speified in the ommand window and simulation model is run from Simulink. An alternative Simulink model having the system parameters R, C speified in Simulink model transfer fn blok and U speified in step blok funtional parameter is depited in Fig. 3. Fig. 3: Simulink model simulated from the ommand window (RCS0.mdl) The system variables defined in the ommand window are as follows: >>R=.5e3 >>C=0.00 >>U=0 6. Using m. file Besides speifying the system variables in the MATLAB ommand window, it is a good pratie to reate Simulink model of governing equation in Simulink, and then onfigure & run the simulation using m.file. 'Simset' and 'sim' is the most useful ommand here. The 'simset' ommand is used to onfigure the simulation parameters and the 'sim' ommand is used to run the simulation. The variable referred in the m.file is set in hosen value field in the parameters window for eah blok. 6.. MATLAB Sript to Run the Simulink Model The MATLAB sript used to all Simulink model of RCS0.mdl is shown in Fig.4. The MATLAB sript RC_modelall.m file defines system parameters, exeutes the simulation for given parameters, and plots the result. One an quikly hange parameters of the model to simulate multiple ases through sript file. MATLAB Solution % MATLAB sript: RC_modelall.m % Run the simulation model RCS0.mdl from sript % Call simulink model RCS0.mdl l; %lear ommand window lear all; %lear workspae lose all; %lose all figure windows R=.5e3; %Resistane of the iruit in Ohms C=0.00; %Capaitane of the iruit in Farads U=0; % Step input signal sim('rcs0') %run the simulink model 'modelall.m' % ********plot the simulation results**************

15 MATLAB/Simulink Based Study of 5 subplot (,,) % plot simulation time index vetor versus step input plot(sopedata(:,), SopeData(:,),'LineWidth',) hold on % plot simulation time index vetor versus step response plot(sopedata(:,), SopeData(:,3),'m.','LineWidth',) title ('Voltage versus time for step exitation'); legend('step Input','Step Response') axis ([ ]) subplot(,,) % plot time index vetor versus urrent response plot(sopedata(:,), SopeData(:,),'LineWidth',) ylabel('current'); title ('Current Response of RC iruit for step exitation'); % put the axis subplot (,,3) % plot simulation time index vetor versus ramp input plot(sopedata(:,), SopeData(:,),'LineWidth',) hold on % plot simulation time index vetor versus ramp response plot(sopedata(:,), SopeData(:,3),'m.','LineWidth',) title ('Voltage versus time for ramp exitation'); legend('ramp Input','ramp Response') axis ([ ]) subplot(,,4) % plot time index vetor versus ramp urrent response plot(sopedata3(:,), SopeData3(:,),'LineWidth',) ylabel('current'); title ('Current Response of RC iruit for ramp exitation'); % put the axis Fig. 4: MATLAB sript (RC_modelall.m) used to all Simulink Model RCS0.mdl Results of system response to different inputs of various approahes using MATLAB simulation tehnique are presented in Fig. 5 to 35. The results of MATLAB sript (RC.m) using Laplae transform method symboli simulation with step, ramp and impulse signals are presented in Fig.5. Fig. 6 shows the simulation results of MATLAB symboli simulation sript RC.m and plot of voltage versus time for step, ramp and impulse exitations using dsolve funtion. The symboli simulation results are also presented using sript RC3.m by hanging system parameters. The results of simulations on exeuting the MATLAB sript (RC3.m) are presented in Fig. 7, 8 and 9. Fig. 5: Simulation result using symboli simulation of (7) using LTM with step, ramp and impulse signal (RC.m) VII. Results and Disussion Fig.6: Simulation result using 'dsolve' funtion for analysis of RC iruit (RC.m) Fig. 7: Voltage aross apaitor for different R using (RC3.m)

16 6 MATLAB/Simulink Based Study of Fig. 7 shows the voltage aross apaitor for R=, 5 and 0 K Ohm respetively using step exitation. The variation in R implies the variation in time onstant of the iruit. It is observed from Figure 7 that the larger is the time onstant, slower is the response. The voltage aross resistor for R=, 5 and 0 K Ohm, respetively is depited in Figure 8. Fig. 9 shows the urrent aross resistor for different R=, 5 and 0K Ohm respetively. Fig. 8: Voltage aross resistor for different R using (RC3.m) Fig. 9: Current aross resistor for different R using (RC3.m) It is observed that symboli solution obtained using Laplae transform method of MATLAB sript 'RC.m' fully agrees with the plot generated using dsolve funtion of sript 'RC.m'. Upon exeuting the MATLAB alling sript (RC4.m), the result of MATLAB ODE sript for analysis of RC iruit is presented in Fig. 30. The running of sript file (RC5.m) by typing the file name RC5 without having.m suffix in the ommand prompt, result into a solution that is graphially depited in Fig. 3. Fig. 3: Simulation result using MATLAB ODE funtion handle using (RC5.m) Fig. 30: Simulation result of MATLAB ODE sript for analysis of RC iruit (RC4.m)

17 MATLAB/Simulink Based Study of 7 The exeution of MATLAB ode 'RC6.m' to evaluate the solution of RC iruit using transfer funtion for step, ramp and impulse exitation is shown in Fig.3. On exeuting the MATLAB sript RC0.m, the results of MATLAB sript are presented in Fig.35. Fig. 3: Simulation result of RC system using TF (RC6.m) Fig. 35: Simulation result omparison of symboli dsolve solution with a diret solution ODE solver (RC0.m) The exeution of MATLAB sript RC7.m results into the urrent response of RC system as presented in Fig. 33. Fig. 33: Simulation result using sript (RC7.m) The exeution of MATLAB sript 'RC9.m' results into a solution in the form of a plot as presented in Fig. 34. The plot depits apaitor voltage v (t) versus time for hosen parameters. Fig. 34: Simulation result using time domain expression of (0), (3) and (6) respetively (RC9.m) The system responses to different inputs using Simulink for first order RC iruit are presented in Figure 37-46, respetively. Fig presents the response of Simulink model (RCS.mdl) for step and ramp exitations using differential equations for hosen system and simulation parameters. After running the model, one an visualize the output of Simulink model on the sope blok by double liking, or plot the result of simulation output by entering the plotting ommands in ommand window or by writing sript.m file. In this model, besides temporarily storing the simulation output in workspae, the result of simulation of model RCS.mdl is also written to a MAT file named RC.mat, and RC.mat using the 'To File' blok. Fig. 37 depits the voltage and urrent aross apaitor for step exitation. Fig.38 shows the voltage and iruit urrent for ramp driving signal. After the simulation is omplete, one an also load the MAT file using sript as shown in Fig.36. % MATLAB sript: RCMP.m % plot the output of Simulink model RCS.mdl l; %lear ommand window lose all; %lose all figure windows load RC %load RC mat file %SopeData ontains the time, input and output vetor subplot (,,) % plot simulation time index vetor versus step input plot(sopedata(:,), SopeData(:,),'LineWidth',) hold on % plot simulation time index vetor versus step response plot(sopedata(:,), SopeData(:,3),'m.','LineWidth',) title ('Voltage versus time for step exitation'); legend('step Input','Step Response') axis ([ ])

18 8 MATLAB/Simulink Based Study of subplot(,,) % plot time index vetor versus urrent response plot(voltage3(,:),voltage3(,:),'r.','linewidth',) ylabel('current'); title ('Current Response of RC iruit for step exitation'); axis ([ e-3]) Fig. 36: MATLAB sript for response plotting using MAT file of Simulink model (RCS.mdl) Fig. 39: Measured output of Transfer Funtion blok using (RCS.mdl) On running the model RCS3.mdl using pulse exitation, for hosen system and simulation parameters results the response as depited in Fig. 40. Fig. 37: Voltage and urrent aross apaitor for step exitations using Simulink model (RCS.mdl) of DE () Fig. 40: Pulse response using ustomized transfer funtion blok (RCS3.mdl) Fig. 38: Voltage and urrent aross the apaitor for ramp exitations using Simulink model (RCS.mdl) of DE () The result of simulation of Simulink model (RCS.mdl) reated using ontinuous blok library is presented in Fig. 39. The output of Simulink model (RCS.mdl) reated using transfer funtion and DE (RCS.mdl) approahes are idential. Fig. 4: Variation of iruit urrent using state spae model (RCS5.mdl) The input and output voltage aross the apaitor using Transfer Funtion approah in Simulink model

19 MATLAB/Simulink Based Study of 9 RCS4.mdl is obtained on sope named 'Measured input &output' and onfirm with mathematial model as reated using ordinary differential equations. It means, both ways of reating Simulink model are orret. The dynamis of Simulink model RCS5.mdl is being solved using the state spae approahes and the value of matrix A, B, C, D are fed diretly to the state spae blok. The result of simulation leading to urrent response using RCS5.mdl is shown in Fig. 4. On running the model RCS6.mdl, the results of simulation agree with earlier obtained results of Simulink model as reated in setion 5. to 5.3. The input, output voltage and urrent response for ramp exitation leading to result of simulation using Simulink model RCS7.mdl reated using bloks of user defined funtion library is depited in Fig. 4. The step response of the model fully agrees with the result obtained earlier. Fig. 4: Simulation result of RC iruit model reated using bloks of user defined funtion library using RCS7.mdl On running the model RCS8.mdl using step exitation, for hosen system and simulation parameters results, the voltage step, ramp, impulse response and urrent response are as depited in Fig. 43 and 44, respetively. Fig. 43: Step, ramp and impulse response using ustomized S-funtion blok (RCS8.mdl) Fig. 44: Ciruit urrent using ustomized S-funtion blok (RCS8.mdl)

20 0 MATLAB/Simulink Based Study of The results of simulation of Simulink model RCS9.mdl reated from time domain step, ramp and impulse response (0), (3) and (6) are depited in Fig. 45. Fig. 45: Simulation result using RCS9.mdl developed using (0), (3) and (6) The Simulink model RCS9.mdl and the model reated in setion 5., 5., 5.4 and 5.5 is being simulated using data driven modeling by defining the system paramaters as variables in the orrosponding bloks of Simulink model. The system paramater variable set in the paramater dialog box of Simulink model are either defined in the MATLAB ommand window or by alling the model using m.file with the system and simulation paramaters defined in the MATLAB sript. The simulation results onfirm with the result of Simulink model RCS.mdl to RCS9.mdl. The Simulink model RCS0.mdl is also simulated by running MATLAB sript RC_modelall.m as depited in Fig. 4. The results of simulation sript are shown in the Fig. 46. Fig. 46: Simulation results from MATLAB sript file It is evident that same results are obtained by both speifying the parameters in ommand window and by running the model from MATLAB sript. It is observed that running the model with different apporhes results into same orrosponding voltage/time, urrent/time plot for given input signal. It is evident

21 MATLAB/Simulink Based Study of upon inspetion that all these methods produe the same result. Simulation and analysis of diverse engineering appliations (first order RC iruit) based on differential equation, transfer funtion, state spae representation, user defined funtion, time domain response using Simulink model is depited in various suggested models as shown in setion 5. The Simulink flow diagram of first order RC system shown in setion 5 merely rereates the same simulation as using MATLAB sript implementation depited in setion 4. It is evident from these plots that the system response to various exitation onditions shows a similar output. In ase of step signal, it is observed that smaller the time onstant of the system, the faster is its response to a step input; or the larger is the time onstant, slower is the response. At the output of the system equals to t t step amplitude. In ase of ramp response, as the output of the system also beomes infinity and error signal beomes equals to time onstant. The greater the time onstant of the system, the greater is the error. In ase of impulse signal, as t the output of the system deays to zero. The larger the time onstant of the system, the greater will be the time required to bring the system output to zero. The response of RC iruit using various approahes under different input onditions seems to be very similar and results are ideal under different exitations. The omponent value an easily be hanged and the iruit an be re-simulated. In a similar manner, these approahes may be useful in deriving a solution for other diverse engineering appliations by desribing the behavior of various eletrial, mehanial, biologial, hemial or any other systems by the governing equations. VIII. Sope of the work The MATLAB sript implementation and Simulink model presented in this study an be used in numerous diverse engineering appliations. These equations our in numerous settings ranging from mathematis itself to its appliations to omputing, eletri iruit analysis, dynamis systems, biologial fields et. An approah similar to the one presented in this work an be used to model various engineering appliations. Some of the appliations for study may inlude: Biologial proess modeling Bateria growth in a jar Tumour growth in body Inset population modeling Fish growth modeling Water disharge modeling from hole in a tank Radioative deay of material Transient analysis of first order system Mass spring damper system Response analysis of series and parallel RLC iruit Automobile Suspension System Seond Order System Investigation In general, many ommon systems are represented by a linear differential equation of nd order as follows: d y dy( t) y( t) u( t) n n n d t dt (6) where the variab le y (t) and (t) u represents the output and input of the system, respetively. n represents the natural frequeny and ζ is the damping oeffiient. The transfer funtion of a general seond order system is as follows: H( s) Y( s) X ( s) s n (7) where Y(s) and X (s) represent the Laplae transform of output and input and H(s) represents the transfer funtion of the system. In the seond order system investigation, three distint ases are enountered for various values of damping ratio ζ. The different values of ζ deide the behaviour of the system. Any linear system an be studied under different values of damping ratio. This fator an have following three ases:. ζ >, the roots of the harateristi equation are real and the system orresponds to over damped system.. ζ<, the roots of the harateristi equation are omplex onjugate and the system orresponds to under damped system. The time response is damped sinusoid. 3. ζ=, the roots of the harateristi equation are real and repeated and the system represents ritially damped system. 4. ζ= 0, the roots of the harateristi equation are imaginary and the system provides osillating output. The various MATLAB and Simulink simulation approahes presented in setion 4 and 5 an be applied to find the solution of various seond order systems suh as (6) and (7). With referene to seond order system Simulink model using TF for three ases, 0 and 0., 0.4, 0.6 and, is depited in Fig. 47 and the n result of simulation is shown in Fig. 48 and 49. In the n n

22 MATLAB/Simulink Based Study of same way, we an examine the effet of varying for various seond order systems, and system behavior an be re-simulated for all the three ases. appliations as stated above. But, in a nutshell, the response of the first and seond order systems an be ahieved using the methodology presented here through MATLAB and Simulink. In fat, MATLAB presents several tools for modeling and simulation in iruit and system. These tools an be used to solve DEs arising in suh models and to visualize the input and output relation. The work presented here an therefore be used in MATLAB Simulink based studies for a variety of appliations. Fig. 47: Simulink model of seond order system using transfer funtion blok given by (7) IX Migrating from MATLAB to Open Soure Pakages The work presented here an also be implemented using free and open soure pakage Silab and Xos. Moreover, with some modifiation and moderate additional efforts, the solution of governing differential equation model developed in Simulink an be replaed by an equivalent free open soure environment Silab and Xos [9]. Fig. 48: Simulation result of seond order system for, 0 and using (7) n Fig. 49: Simulation result of seond order system for 0., 0.4 & 0.6 and n using (7) Due to the limitations of the length of the paper, it is not possible to over other diverse engineering X Conlusion In the present study, first and seond order system equations have been modeled, simulated and analyzed using different approahes involving MATLAB and Simulink. Moreover, through this ommuniation, an attempt has also been made to demonstrate the usage of majority of bloks of Simulink sinks and ontinuous blok library, and to present a brief idea about data driven modeling. The methodology presented here using MATLAB and Simulink may serve as an inspiration for solving similar st and nd order ODE systems governing the behavior of diverse engineering systems suh as those arising in many ontexts inluding mathematis, physis, geometry, mehanis, astronomy, mehanial, ivil, thermal, biologial, population modeling and many others too. In fat, any ontinuous time system that an be modeled using differential equation, by transfer funtion, or the state variable an be simulated using MATLAB and Simulink. The proess used here ould very profitably be employed in the analysis of mehanial engineering ourse systems suh as automobile, ship or air plane systems; and in hemial engineering ourses in temperature ontrol system, fluid level systems or in modelling of a hemial reator. Thus, in onlusion, it an be said that with model based design, it is feasible to bridge the gap between the theoretial foundation and pratial appliations, thereby ultivating innovation talents and promoting undergraduate level researh. The work presented here prepares the students to aelerate innovation through simulation based engineering and sienes.

23 MATLAB/Simulink Based Study of 3 Referenes [] Zafar A, Differential equations and their appliations, PHI, 00. [] Agarwal A and Lang J H, Foundation of Analog and Digital Eletroni Ciruits, Elsevier, 0. [3] Choudhury D R, Networks and System, New Age International Publisher, nd Edition, 00. [4] Karris S T, Signal and System with MATLAB Computing and Simulink Modeling, Orhard Publiations, 007. [5] Kalehman M, Pratial MATLAB appliations for engineers, CRC Press, 009. [6] Palamids A. and Veloni A., Signals and systems laboratory with MATLAB, CRC Press, 00. [7] Blaho M, Foltin M, Fodrek P and Poliaik P, Preparing advaned MATLAB users, WSEAS Transation on Advanes in Engineering Eduation, 00, 7(7): [8] Jain S, Modeling and simulat ion using MATLA B- Simulink, Wiley India, 0. [9] Blaho M, Foltin M, Fodrek P and Murgas J, Eduation of Future Advaned MATLAB Users, MATLAB-A Fundamental Tool for Sientifi Computing and Engineering Appliations -Volume 3, INTECH, 0. [0] Bober W and Stevens A, Numerial and Analytial Methods with MATLAB for Eletrial Engineers, CRC Press, 03. [] Dabney J B and Harman T L, Mastering Simulink, Pearson Eduation, 004. [] Karris S T, Introdution to Simulink with engineering appliations, Orhard Publiations, 006. [3] Harold K and Randal A, Simulation of dynamial system with MATLAB and Simulink, nd edition, CRC Press, 0. [4] Ibrahim D, Engineering Simulation with MATLAB: improving teahing and learning effetiveness, Proedia Computer Siene, 0, 3: [5] Feng P, Mingxiu L, Dingyu X, Dali C, Jianjiang C, Appliation of MATLAB in teahing reform and ultivation of innovation talents in universities, IEEE proeedings of nd International Workshop on Eduation Tehnology and Computer Siene,00, [6] Tahir H H, Pareja T F, MATLAB pakage and siene subjets for undergraduate studies, International Journal for Cross-Disiplinary Subjets in Eduation, 00, (): [7] Osowski S, Simulink as an advaned tool for analysis of dynamial eletrial systems, Computational Problems of Eletrial Engineering, 0, ():5-60. [8] J Ryan A Tarini, Preparing students to aelerate innovation through simulation based engineering and sienes, IEEE nd International Workshop proeedings on Eduation Tehnology and Computer Siene, 00, : [9] Abihandani P, Primerano R and Kam M, Symboli sientifi software skills for engineering students, In the proeedings of IEEE nd International Workshop on Eduation Tehnology and Computer Siene, 00. [0] Kazimovih Z M and Guverin S, Appliations of symboli omputation in MATLAB, International Journal of Computer Appliations, 0, 4(8):-5. [] Ahmed W K, Advantages and disadvantages of using MATLAB/ode45 for solving differential equations in engineering appliations, International Journal of Engineering, 03, 7():5-3. [] Klegka J S, Using Simulink as a design tool, In Proeedings of Amerian Soiety for Engineering Eduation Annual Conferene and Exposition, 00. [3] Frank W. Pietryga, P.E., Solving differential equations using MATLAB/Simulink, In Proeeding of Amerian Soiety for Engineering Eduation Annual Conferene and Exposition, 005. [4] Maddalli R K, Modeling ordinary differential equations in MATLAB Simulink, Indian Journal of Computer Siene and Engineering, 0, 3(3): [5] Nehra V, Engineering Simulation Using Graphial Programming Tool Simulink: Putting Theory into Pratie, pp , In Proeeding of TEQIP Sponsored National Conferene on Contemporary Tehniques and Tehnologies in Eletronis Engineering, held at DCRUS&T, Murthal, Marh 3-4, 03. [6] Niulesu T, Study of Indutive-Capaitive Series Ciruits Using the Simulink Software Pakage, Tehnology and Engineering Appliation & Simulink, InTeh, 0. [7] Kisabo A B Osheku C A, Adetoro M.A Lanre and Aliyu Funmilayo A., Ordinary Differential Equations: MATLAB/Simulink Solutions, International Journal of Sientifi and Engineering Researh, 0, 3(8):-7. [8] Ossman K A K, Teahing state variable feedbak to tehnology students using MATLAB and Simulink, In Proeedings of Amerian Soiety for Engineering Eduation Annual Conferene and Exposition, 00. [9] Leros A. and Andreatos A., Using Xos as a teahing tool in simulation ourse, In Proeedings

24 4 MATLAB/Simulink Based Study of of the 6th International Conferene on Communiations and Information Tehnology (CIT '), Marh 7-9, 0,-6, Reent Researhes in Communiations, Information Siene and Eduation, World Sientifi and Engineering Aademy and Soiety (WSEAS) Stevens Point, Wisonsin, USA. Author s Profiles respetively. Dr Vijay Nehra obtained B.Teh in Eletronis and Communiation Engineering from JMIT, Radaur Kurukshtra University and M.E in Eletronis from Punjab Engineering College, Chandigarh in 000 and 00 He started his areer as Leturer at Tehnologial Institute of Textile and Sienes, Bhiwani in 00. Meanwhile he enrolled for Ph.D at Maharshi Dayanand University, Rohtak and earned his Ph.D in Eletronis and Communiation Engineering in 008. He has been with the Chaudhary Devilal Memorial Engineering College, Panniwala Mota and Guru Jambheshwar University of Siene and Tehnology, Hisar from 006 to 008. During his areer he had taught various UG and PG ourses in Eletronis Devie and Ciruits, Linear Integrated Ciruit, Network Theory, Signal and System, Control System, Antenna and Wave Propagation, Digital Signal Proessing, Problem Solving Using MATLAB, Modeling and Simulation of Dynami System and many more along with their labs. He has also served as Dean and Head Shool of Engineering and Sienes, Bhagat Phool Singh Mahila Vishwavidyalaya, Khanpur Kalan, Sonipat, Haryana. He has professional experiene of years in teahing, researh, urriulum planning, laboratory development, eduational administration, planning, management and exeution. His urrent area of interest inlude engineering eduation teahing strategies that promote professionalism and areer development, learning style and innovative laboratories that rossut the urriulum. His researh papers have been published in international and national journals of repute. He is a life member of various professional soieties suh as ISTE, CSI, IETE, Institution of Engineers, Plasma Siene Soiety of India. How to ite this paper: Vijay Nehra,"MATLAB/Simulink Based Study of Different Approahes Using Mathematial Model of Differential Equations", International Journal of Intelligent Systems and Appliations(IJISA), vol.6, no.5, pp.- 4, 04. DOI: 0.585/ijisa