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Review of First- and Second-Order System Response 1 First ...

MASSACHUSETTS INSTITUTE OF TECHNOLOGYDEPARTMENT OF MECHANICAL Advanced System Dynamics and ControlReview of First - and Second-Order System Response11 First -Order Linear System Transient ResponseThe dynamics of many systems of interest to engineers may be represented by a simple modelcontaining one independent energy storage element. For example, the braking of an automobile,the discharge of an electronic camera flash, the flow of fluid from a tank, and the cooling of a cupof coffee may all be approximated by a First -order differential equation, which may be written in astandard form as dydt+y(t) =f(t)(1)where the System is defined by the single parameter , the System time constant, andf(t) is aforcing function. For example, if the System is described by a linear First -order state equation andan associated output equation: x=ax+bu(2)y=cx+du.(3)and the selected output variable is the state-variable, that isy(t) =x(t), Eq.

Solution: The tank is represented as a °uid capacitance Cf with a value: Cf = A ‰g (i) where A is the area, g is the gravitational acceleration, and ‰ is the density of water. In this case Cf = 2=(1000£9:81) = 2:04£10¡4 m5/n and Rf = 1=10¡6 = 106 N-s/m5. The linear graph generates a state equation in terms of the pressure across the °uid

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Transcription of Review of First- and Second-Order System Response 1 First ...

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGYDEPARTMENT OF MECHANICAL Advanced System Dynamics and ControlReview of First - and Second-Order System Response11 First -Order Linear System Transient ResponseThe dynamics of many systems of interest to engineers may be represented by a simple modelcontaining one independent energy storage element. For example, the braking of an automobile,the discharge of an electronic camera flash, the flow of fluid from a tank, and the cooling of a cupof coffee may all be approximated by a First -order differential equation, which may be written in astandard form as dydt+y(t) =f(t)(1)where the System is defined by the single parameter , the System time constant, andf(t) is aforcing function. For example, if the System is described by a linear First -order state equation andan associated output equation: x=ax+bu(2)y=cx+du.(3)and the selected output variable is the state-variable, that isy(t) =x(t), Eq.

2 (3) may be rearrangeddydt ay=bu,(4)and rewritten in the standard form (in terms of a time constant = 1/a), by dividing throughby a: 1adydt+y(t) = bau(t)(5)where the forcing function isf(t) = ( b/a)u(t).If the chosen output variabley(t) is not the state variable, Eqs. (2) and (3) may be combinedto form an input/output differential equation in the variabley(t):dydt ay=ddudt+ (bc ad)u.(6)To obtain the standard form we again divide through by a: 1adydt+y(t) = dadudt+ad bcau(t).(7)Comparison with Eq. (1) shows the time constant is again = 1/a, but in this case the forcingfunction is a combination of the input and its derivativef(t) = dadudt+ad bcau(t).(8)In both Eqs. (5) and (7) the left-hand side is a function of the time constant = 1/aonly, andis independent of the particular output variable Rowell 10/22/041 Example 1A sample of fluid, modeled as a thermal capacitanceCt, is contained within an insulatingvacuum flask. Find a pair of differential equations that describe 1) the temperature ofthe fluid, and 2) the heat flow through the walls of the flask as a function of the externalambient temperature.

3 Identify the System time 1: A First -order thermal model representing the heat exchange between a laboratory vacuumflask and the :The walls of the flask may be modeled as a single lumped thermal resistanceRtand a linear graph for the System drawn as in Fig. 1. The environment is assumedto act as a temperature sourceTamb(t). The state equation for the System , in terms ofthe temperatureTCof the fluid, isdTCdt= 1 RtCtTC+1 RtCtTamb(t).(i)The output equation for the flowqRthrough the walls of the flask isqR=1 RtTR= 1 RtTC+1 RtTamb(t).(ii)The differential equation describing the dynamics of the fluid temperatureTCis founddirectly by rearranging Eq. (i):RtCtdTCdt+TC=Tamb(t).(iii)from which the System time constant may be seen to be = differential equation relating the heat flow through the flask isdqRdt+1 RtCtqR=1 RtdTambdt.(iv)This equation may be written in the standard form by dividing both sides by 1/RtCt,RtCtdqRdt+qR=CtdTambdt,(v)and by comparison with Eq.

4 (7) it can be seen that the System time constant =RtCtand the forcing function isf(t) =CtdTamb/dt. Notice that the time constant isindependent of the output variable The Homogeneous Response and the First -Order Time ConstantThe standard form of the homogeneous First -order equation, found by settingf(t) 0 in Eq. (1),is the same for all System variables: dydt+y= 0(9)and generates the characteristic equation: + 1 = 0(10)which has a single root, = 1/ . The System Response to an initial conditiony(0) isyh(t) =y(0)e t=y(0)e t/ .(11)024681001234 Time (secs)tSystem responsey(t)t = -3t = -5t = -10t = 10t = 5t = < 0stablet > 0t infiniteFigure 2: Response of a First -order homogeneous equation y+y(t) = 0. The effect of the systemtime constant is shown for stable systems ( >0) and unstable systems ( <0).A physical interpretation of the time constant may be found from the initial condition responseof any output variabley(t).

5 If >0, the Response of any System variable is an exponential decayfrom the initial valuey(0) toward zero, and the System isstable. If <0 the Response growsexponentially for any finite value ofy0, as shown in Fig. , and the System isunstable. Althoughenergetic systems containing only sources and passive linear elements are usually stable, it is possibleto create instability when an active control System is connected to a System . Some sociological andeconomic models exhibit inherent instability. The time-constant , which has units of time, is thesystem parameter that establishes the time scale of System responses in a First -order System . Forexample a resistor-capacitor circuit in an electronic amplifier might have a time constant of a fewmicroseconds, while the cooling of a building after sunset may be described by a time constant ofmany is common to use a normalized time scale,t/ , to describe First -order System responses.

6 Thehomogeneous Response of a stable System is plotted in normalized form in Fig. 3, using both thenormalized time and also a normalized Response magnitudey(t)/y(0):y(t)/y(0) =e (t/ ).(12) time t/ty(t)/y(0) 3: Normalized unforced Response of a stable First -order t/ y(t)/y(0) =e t/ ys(t) = 1 e t/ 1: Exponential components of First -order System responses in terms of normalized timet/ .The third column of Table 1 summarizes the homogeneous Response after periodst= ,2 , ..Aftera period of one time constant (t/ = 1) the output has decayed toy( ) =e 1y(0) or of itsinitial value, after two time constants the Response isy(2 ) = (0).Several First -order mechanical and electrical systems and their time constants are shown in For the mechanical mass-damper System shown in Fig. 4a, the velocity of the mass decays fromany initial value in a time determined by the time constant =m/B, while the unforced deflectionof the spring shown in Fig.

7 4b decays with a time constant =B/K. In a similar manner thevoltage on the capacitor in Fig. 4c will decay with a time constant =RC, and the current inthe inductor in Fig. 4d decays with a time constant equal to the ratio of the inductance to theresistance =L/R. In all cases, if SI units are used for the element values, the units of the timeconstant will be (t)mBmBF(t)BKV(t)BKV(t)v = 0refv = 0refRC+-V(t)V(t)V = 0refRCLRI(t)I(t)V = 0refLRFigure 4: Time constants of some typical First -order 2A water tank with vertical sides and a cross-sectional area of 2 m2, shown in Fig. 5, isfed from a constant displacement pump, which may be modeled as a flow sourceQin(t).A valve, represented by a linear fluid resistanceRf, at the base of the tank is alwaysopen and allows water to flow out. In normal operation the tank is filled to a depth ofvalveRftankCfcp (t)Q (t)outQ (t)p = prefRCffinQ (t)inatmFigure 5: Fluid tank m.

8 At timet= 0 the power to the pump is removed and the flow into the tank the flow through the valve is 10 6m3/s when the pressure across it is 1 N/m2,determine the pressure at the bottom of the tank as it empties. Estimate how long ittakes for the tank to :The tank is represented as a fluid capacitanceCfwith a value:Cf=A g(i)where A is the area,gis the gravitational acceleration, and is the density of this caseCf= 2/(1000 ) = 10 4m5/n andRf= 1/10 6= 106N- linear graph generates a state equation in terms of the pressure across the fluidcapacitancePC(t):dPCdt= 1 RfCfPC+1 CfQin(t)(ii)which may be written in the standard First -order formRfCfdPCdt+PC=RfQin(t).(iii)The time constant is =RfCf. When the pump fails the input flowQinis set to zero,and the System is described by the homogeneous equationRfCfdPCdt+PC= 0.(iv)The homogeneous pressure Response is (from Eq. (11)):PC(t) =PC(0)e t/RfCf.(v)With the given parameters the time constant is =RfCf= 204 seconds, and theinitial depth of the waterh(0) is 1 m; the initial pressure is thereforePC(0) = gh(0) =1000 1 N/m2.

9 With these values the pressure at the base of the tank as itempties isPC(t) = 9810e t/204N/m2(vi)which is the standard First -order form shown in Fig. time for the tank to drain cannot be simply stated because the pressure asymptot-ically approaches zero. It is necessary to define a criterion for the complete decay of theresponse; commonly a period oft= 4 is used sincey(t)/y(0) =e 4< as shownin Table 1. In this case after a period of 4 = 816 seconds the tank contains less than2% of its original volume and may be approximated as The Characteristic Response of First -Order SystemsIn standard form the input/output differential equation for any variable in a linear First -ordersystem is given by Eq. (1): dydt+y=f(t).(13)The only System parameter in this differential equation is the time constant . The solution withthe givenf(t) and the initial conditiony(0) = 0 is defined to be thecharacteristic First -orderhomogeneous solutionis of the form of an exponential functionyh(t) =e twhere = 1/.

10 The total responsey(t) is the sum of two componentsy(t) =yh(t) +yp(t)=Ce t/ +yp(t)(14)whereCis a constant to be found from the initial conditiony(0) = 0, andyp(t) is aparticularsolutionfor the given forcing functionf(t). In the following sections we examine the form ofy(t)for the ramp, step, and impulse singularity forcing The Characteristic Unit Step ResponseThe unit stepus(t) is commonly used to characterize a System s Response to sudden changes in itsinput. It is discontinuous at timet= 0:f(t) =us(t) ={0t <0,1t characteristic step responseys(t) is found by determining a particular solution for the stepinput using the method of undetermined coefficients. From Table , with a constant input fort >0, the form of the particular solution isyp(t) =K, and substitution into Eq. (13) givesK= complete solutionys(t) isys(t) =Ce t/ + 1.(15)The characteristic Response is defined when the System is initiallyat rest, requiring that att= 0,ys(0) = 0.}


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