LECTURETWO-2Applied control engineeringmathematicsLEARNINGOBJECTIVESTobeabletoemploytheLaplaceTransformtothesolutionofordinary differential equationsTo understand howto convert a set of coupled differentialequationstostate-spaceformTo perform linearization of nonlinear dynamic equationsToapply theLaplaceTransformandOrdinaryDifferentialEquationsTheorytotheanalysisofmarineshaftingsystemsToderivetheempiricaltransferfunctiondepictingthefundamentaldynamicsofmarinepropulsionengines
1 LECTURE TWO – 2 Applied control engineering mathematics 1 LEARNING OBJECTIVES • To be able to employ the Laplace Transform to the solution of ordinary differential equations • To understand how to convert a set of coupled differential equations to state-space form • To perform linearization of nonlinear dynamic equations • To apply the Laplace Transform and Ordinary Differential Equations Theory to the analysis of marine shafting systems • To derive the empirical transfer function depicting the fundamental dynamics of marine propulsion engines 2
TheLaplaceTransformforLinearOrdinaryDifferentialEquationsNik.Xiros3Definition of the LaplaceTransformLetf(t) bea function defined on the interval [0,oo).The Laplace transform of fis the function Fdefined by the integralF(s)f()exp(-st)dtThe domain of F(s) is all the values of se C for which the definition integral aboveexists.Alternatively, the Laplace transform of f(t) is denoted as (f(t)2
2 Nik. Xiros 3 The Laplace Transform for Linear Ordinary Differential Equations 4 Definition of the Laplace Transform Let f(t) be a function defined on the interval [0, ) ∞ . The Laplace transform of f is the function F defined by the integral ( ) 0 F s f t st dt ( ) ( )exp +∞ − ∫ The domain of F(s) is all the values of s ∈ for which the definition integral above exists. Alternatively, the Laplace transform of f(t) is denoted as L{ f t( )}
Linearityof theLaplaceTransformL(f(t)+f,(t))=L(f())+L(f(t)fcf(t)=c.L(f()In effect, the principle of superposition holds5Calculationof theLaplace TransformConvenient relationship:exp(β),βe C(ex(β)=_βProof:Lfexp(Br)= exp(βt)exp(-s)dt= [ exp(-(s-β)dt=[exp(-(s-β)t)d(s-β)t=s-βexp(-(s-RS-BS-βRegion of convergence ??3
3 5 Linearity of the Laplace Transform { } { } { } { } { } 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) f t f t f t f t cf t c f t + = + = ⋅ L L L L L In effect, the principle of superposition holds 6 Calculation of the Laplace Transform Convenient relationship: ( ) ( ) 1 { } 1 exp , exp t t s β β β β − ∈ = ← → − L L L Proof: { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 exp exp exp exp 1 1 exp exp 1 t t st dt s t dt s t d s t s t s s s β β β β β β β β β +∞ +∞ +∞ +∞ = − = − − = = − − − = − − − = − − = − ∫ ∫ ∫ L Region of convergence ??
Some useful LaplaceTransform pairsexp(Bi),βe Cc(exp(β)=s-βcos(2t)=[exp(j2t)+exp(-j2n)]=Cosine function:1c(cos(21)=(2(s-jQs+jQs+2sin(2t)=[exp(j2t)-exp(-j2n)]=2Sine function:12AC[sin(21) =2s-j252+22s+jd7LaplaceTransformpropertiesAssumption:L{f(t))=F(s)cfe f(t)}=F(s-β)Translation in s:L(f(t)=sF(s)-f(t=0)Laplacetransform of derivative:L[f(()=s"F(s)-2+ (-)(t=0)Higher-order‘time'derivatives:lal((0)=(-1F(s)Laplace transformderivation:ds*804
4 7 Some useful Laplace Transform pairs ( ) ( ) 1 { } 1 exp , exp t t s β β β β − ∈ = ← → − L L L Heaviside step function: ( ) { } 0, 0 1 ( ) exp 0 ( ) 1, 0 step step t u t t u t t s < = = ⇒ = ≥ L Cosine function: [ ] { } 2 2 1 cos( ) exp( ) exp( ) 2 1 1 1 cos( ) 2 t j t j t s t s j s j s Ω Ω Ω Ω Ω Ω Ω = + − ⇒ = + = − + + L Sine function: [ ] { } 2 2 1 sin( ) exp( ) exp( ) 2 1 1 1 sin( ) 2 t j t j t j t j s j s j s Ω Ω Ω Ω Ω Ω Ω Ω = − − ⇒ = − = − + + L 8 Laplace Transform properties Assumption: L{ f t F s ( )} = ( ) Translation in s: { ( )} ( ) t e f t F s β L = − β Laplace transform of derivative: L{ f t sF s f t ′( ) ( 0) } = − = ( ) Higher-order ‘time’ derivatives: { } ( ) ( ) ( 1) 1 ( ) ( 0) n n n n k k k f t s F s s f t − − = L = − = ∑ Laplace transform derivation: { } ( ) 1 ( ) ( ) n n n n d t f t F s ds L = −
SomemoreLaplaceTransformpairsm!Generalization:"-exp(βt),βe Cc(r"exp(βt) =(s-β)"*Proof:By use of mathematical induction.The induction step follows.(r exp(βr)= m-exp(βr)+ Pr" exp(β)UsL[t" exp(βt) =mL(rm-l exp(βt)+βL(t" exp(βt)Uc[r exp(βr)=-"r [(r- exp(Br)s-βPolynomial function: "c{r]-"SDefinitionoftheInverseLaplaceTransformGiven a function of F(s), if there is a function f(t), continuous on the interval [0,0),that satisfiesL[f(t)= F(s)then we say that f(t) is the Inverse Laplace Transform of F(s) and employ thenotation f(t)='{F(s)).The Inverse Laplace Transform is linear.{f(t)+f(t)='{f,(t)+{f(t)cf(t)=c.r'(f(t)105
5 9 Some more Laplace Transform pairs Generalization: ( ) ( ) { } ( ) 1 1 ! exp , exp m m m m t t t t s β β β β − + ⋅ ∈ = ← → − L L L Proof: By use of mathematical induction. The induction step follows. ( ) ( ) ( ) ( ) { } ( ) { } ( ) { } ( ) { } ( ) { } ( ) 1 1 1 exp exp exp exp exp exp exp exp m m m m m m m m d t t mt t t t dt s t t m t t t t m t t t t s β β β β β β β β β β β − − − ⋅ = + ⇓ = + ⇓ = ⋅ − L L L L L Polynomial function: 1 { } 1 m m ! m m t t s − + ← → = L L L 10 Definition of the Inverse Laplace Transform Given a function of F(s), if there is a function f(t), continuous on the interval [0, ) ∞ , that satisfies L{ f t F s ( ) ( ) } = then we say that f(t) is the Inverse Laplace Transform of F(s) and employ the notation { } 1 f t F s ( ) ( ) − = L . The Inverse Laplace Transform is linear. { } { } { } { } { } 1 1 1 1 2 1 2 1 1 ( ) ( ) ( ) ( ) ( ) ( ) f t f t f t f t cf t c f t − − − − − + = + = ⋅ L L L L L