CH02-2 Laplace transform.ppt

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1、Ch.2MathematicalModelingofDynamicSystems2-1微分方程2-2传递函数2-3动态结构图及等效变换2-4信号流图及梅逊公式2-5控制系统的传递函数拉普拉斯变换Hugedifficultyinsolvingcomplicateddifferentialequations(求解困难)Difficultyinstudyingdynamicperformance(难以分析研究系统动态性能)Disadvantagesindifferentialequations:线性微分方程(时域t)代数方程(复数域s)代数方程的解(复

2、数域s)求解拉氏变换拉氏反变换线性微分方程的解(时域t)?Laplacetransform(拉普拉斯变换)LaplaceoperatorvariableOne-onemappingDefinitionofLaplaceTransformLaplaceoff(t)LaplacetransformIff(t)=0,whent<0Laplacetransform——典型时间函数的拉氏变换1.PulseFunction脉冲函数2.StepFunction阶跃函数3.RampFunction斜坡函数4.ExponentialFunction指数函数5.Si

3、nusoidalFunction正余弦函数LaplacetransformoftypicalfunctionsLaplacetransform——典型时间函数的拉氏变换1.PulseFunction脉冲函数f(t)0t0tA/t0RectangleFunction矩形窗函数Laplacetransform——典型时间函数的拉氏变换f(t)0tNote：ItiscalledunitpulsefunctionwhenA=1,denotedasLaplacetransform——典型时间函数的拉氏变换2.StepFunction阶跃函数f(t)A0tN

4、ote：whenA=1，itiscalledunitstepfunction,anddenotedas1(t).Stepfunctionisundefinedatt=0.Laplacetransform——典型时间函数的拉氏变换3.RampFunction斜坡函数f(t)t0A1Note：ItiscalledunitrampfunctionwhenA=1.Laplacetransform——典型时间函数的拉氏变换4.ExponentialFunction指数函数Laplacetransform——典型时间函数的拉氏变换EulertheoremLa

5、placetransform——典型时间函数的拉氏变换5.SinusoidalFunction正余弦函数Laplacetransform——典型时间函数的拉氏变换时间函数f(t)拉氏变换F(s)δ(t)1A(t)[1(t)]A/s[1/s]At[t]A/s2[1/s2]sin(ωt)cos(ωt)Laplacetransform——拉氏变换定理TheoremofLaplacetransform1.Linearity(线性定理)Laplacetransform——拉氏变换定理2.TimeShifting(时移)0atLaplaceoff(t)Shi

6、ftintimeTimeShifting:Multipliedbyexponentialincomplexfrequencydomain.Laplacetransform——拉氏变换定理Shiftingincomplexfrequency----Multipliedbyexponentialintime.multipliedby3.FrequencyShifting(频移)ExampleLaplacetransform——拉氏变换定理4.Scalingintime(时间尺度变换)ProofLaplacetransform——拉氏变换定理5.The

7、Time-Differentiation(微分定理)here,f(0)istheinitialvalueoff(t)att=0.Thus,operatorsinfrequencydomaincanbetreatedasdifferentialoperatorintimedomain:Similarly,forthen-thderivativeoff(t),wehave6.TheTime-IntegrationTheorem(积分定理)whereisthevalueofatt=0.Laplacetransform——拉氏变换定理Laplacetra

8、nsform——拉氏变换定理Proof：denoteFromthedifferentiationtheorem,wehaveLaplac