Exp1 DCMotorTimeResponse

 

Department of Mechanical Engineering Michigan State University East Lansing, MI 48824-122

ME451 Laboratory

Modeling and Experimental Va Validation of a First Order Plant Model: DC er!o Motor

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ME4"1 La#oratory Man$al %ages, Last &evise'( 1)14)2*1*

 

ME 451: Control Systems Laboratory Sen' comments to( Dr+ lar &a'cli.e, %rofessor

Mo'eling an' E/perimental 0ali'ation of a irst r'er %lant Mo'el( D Servo Motor

2

 

ME 451: Control Systems Laboratory

References: C.L. Phillips and R.D. Harbor Feedback Harbor  Feedback Control Control System Systemss Prenti!e Hall 4th Ed. Se Se!t !tio ion n 2." 2." pp. # #$% $%4# 4#:: El Ele! e!tr trom ome! e!ha hani ni!a !all Syst System emss Se!tion Se!t ion 4.1 pp. 11&%1 11&%12': 2': (i (ime me Respo Response nse o) *ir *irst st +rder System Systemss ,ppendi-  pp. %&5': Lapla!e ( (rans)orm rans)orm /Parti!0larly the *inal al0e (heorem3

1. Objective Linear time%inariant time%inariant dynami!al systems are !ate6ori7ed 0nder )irst%order systems se!ond%order systems and hi6her%order systems. (he trans)er )0n!tion o) all )irst%order systems has a standard )orm. (his enables 0s to inesti6ate the response o) )irst%order systems !olle!tiely !olle!tiely )or any spe!i)i! inp0t. (he response o) a )irst%order system depends on its DC 6ain  K   and time !onstant τ   . oth  K   and τ    are )0n!tion o) system parameters. (he ob8e!tie o) this e-periment is to model a )irst%order system and inesti6ate the e))e!t o) system parameters on its response to a step inp0t. 9e !hoose to e-periment ith an armat0re !ontrolled DC seromotor hi!h behaes as a )irst%order system hen the armat0re olta6e is the inp0t and the an60lar speed is the o0tp0t. 9e obtain the trans)er )0n!tion o) the motor and identi)y spe!i)i! parameters o) the system that a))e!t system response. Spe!i)i!ally Spe!i)i!ally e identi)y system  parameters that that indiid0ally a))e!t the DC 6ain and the time !onstant and ary these pa parameters rameters to e-perimentally eri)y the !han6e in system response.

2. Background 2.1. First-order systems (he standard )orm o) trans)er )0n!tion o) a )irst%order system is: Y / s 4  K  G / s 4 = = U / s 4  s + 14 /τ  

 /1

here Y(s) and U(s) are the Lapla!e trans)orms o) the o0tp0t and inp0t ariables respe!tiely respe!tiely  K  is the DC 6ain and

τ  

 is the time !onstant. *or a 0nit step inp0t U(s)

Y / s 4 =

Y / s 4 U / s 4

U / s 4 =

= 1 /s

 K 

1

 s + 14  s /τ  

 the response o) the system is:

=

 K 

/2

 s /τ    s + 14

(he inerse o) the res0ltin6 Lapla!e trans)orm !an be easily )o0nd /see the ,ppendi- in yo0r te-t. (ypi!ally (ypi!ally the inerse is aailable aailable in standard tables. ;n this !ase

    K   1    −1  −t < 4  y /t 4 = L−1    = K /1 − e  = K  L  τ   τ    s /  s 1 4  s /  s 1 4 + +          

τ  

/#

;t is !lear )rom /# that  y → K   as t  → ∞ . (he DC 6ain !an there)ore there)ore be interpreted as the ))inal inal al0e o) the o0tp0t )or a 0nit step inp0t. (he time !onstant is the time re=0ired )or  y(t)  to rea!h &#.2> o) its )inal al0e. ;ndeed at t ? τ     y(t) = '. K   )or a 0nit step step inp0t. *or a 0nit step step inp0t the !han6e !han6e in inp0t iiss one /1. ;n 6eneral )or a step inp0t o) ma6nit0de  A  at t ? τ     y(t)

=

'. KA .

(he response response o) the )irst% )irst%order order system

to a 0nit step inp0t is shon shon in *i6.1a )or to !ases. !ases. *or a system 6ain  K   < 1  the system@s o0tp0t !han6e is less than the inp0t !han6e applied. applied. *or a system 6ain 6ain  K   > 1 . the system@s o0tp0t !han6e is more than the inp0t !han6e applied. (he res0lts plotted are )or a system operat operatin6 in6 )or small positie inp0t and o0tp0t deiations )rom 7ero /the ori6in.

Mo'eling an' E/perimental 0ali'ation of a irst r'er %lant Mo'el( D Servo Motor

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ME 451: Control Systems Laboratory

Fig"re 1a: First#Order ystem tep $esponse

Fig"re 1b: Periodi% First#Order ystem tep $esponse (he step response o) the DC motor ill be eal0ated ith a s=0are ae inp0t !omposed o) a series o) positie and ne6atie steps. ,s shon in *i60re 1b these steps prod0!e repea repeated ted positie and ne6atie !han6es in a 1 st order system@ss o0tp0t. ,ss0min6 the positie system@s system@ system@s response rea!hes steady%state )or ea!h positie and ne6atie inp0t the 6ain and time !onstant parameters !an be separately eal0ated eal0ated )or both positie and ne6atie inp0t !han6es. (he spe!i)i! al0es )or the 6ain and time !onstant parameters parameters )or the aboe systems are !omp0ted belo. Aoti!e that the system 6ains are e=0al b0t that the positie and ne6atie !han6e time !onstants are are not. *or both parameters an aera6e is typi!ally 0sed as a representatie al0e. (he ariation )rom the aera6e aera6e indi!ates the repeatability o) the meas0rement. (able 1: (ypi!al Bain and (ime Constant Meas0rements and Comp0tations ;np0t Chan6e +0tp0t Chan6e (ime to &#.2> !han6e ∆;np0t ∆+0tp0t Risin6 '.%'.' ? '. '.&5 '.$%'.' ? '.$ *allin6 '.'%'.?%'. '.&5 &.2%5.' ? 1.2 ,era6e %%%%%%%%%%% %%%%%%%%%%%%%% %%%%%%%%%%%%%%

System Bain

(ime Constant

∆+0tp0t