《轮机仿真及控制技术》课程教学课件（讲稿）02 Applied control engineering mathematics

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

CalculationofInverseLaplaceTransformProcedureoutlinedforrational functionsofs.G(s) -b*+b-bs+h _ ()s"+a-s"--+.as+aoP(s)P0=iG-A)UNon-repeated linear factors:之,k,=m[(s-~),G()] =...nG(s)=)Bs-AIn effect:=Lfe'),e C, 1≥0= g(0)=2k-exp(an),12011Calculationof InverseLaplaceTransformProcedureoutlinedforrational functionsof s.G(s)=b+b+bs+h_2()s"+a.-s"+.as+aoP(s)In the case that the characteristic polynomial obtains a pair of complex conjugateroots, Ae,, the factorized form includes the quadratic factor(s-)-(s-)=2 -2 Re()-$+)Ineffect:kc+ko→ke.=(ke)-2.Re()s+--126

6 11 Calculation of Inverse Laplace Transform ( ) 1 1 1 0 1 1 1 0 ( ) ( ) m m m m n n n b s b s b s b Q s G s s a s a s a P s − − − − + + + = = + + + Non-repeated linear factors: [ ] 1 1 ( ) ( ) ( ) , lim ( ) ( ) , 1,., v n v v n v v v s v v P s s k G s k s G s v n s λ Π λ λ λ = → = = − ⇓ = = − ⋅ = − ∑ In effect: { } 1 1 , , 0 ( ) exp( ), 0 n t v v v e t g t k t t s λ λ λ λ = = ∈ ≥ ⇒ = ⋅ ≥ − L ∑ Procedure outlined for rational functions of s. 12 Calculation of Inverse Laplace Transform ( ) 1 1 1 0 1 1 1 0 ( ) ( ) m m m m n n n b s b s b s b Q s G s s a s a s a P s − − − − + + + = = + + + In the case that the characteristic polynomial obtains a pair of complex conjugate roots, , λ λ C C ∗ , the factorized form includes the quadratic factor 2 2 ( ) ( ) 2 Re( ) C C C C s s s s λ λ λ λ ∗ − ⋅ − = − ⋅ ⋅ + In effect: ( ) 2 2 1 2 Re( ) C C C C C C C C k k k k s s s s λ λ λ λ ∗ ∗ = + ∗ ⇒ ∗ = − ⋅ ⋅ + − − Procedure outlined for rational functions of s.

Calculation of Inverse Laplace TransformProcedure outlined for rational functions of s.G(s)=-b+b-bs+h_2()s"+a.-s"-l +..a,s+aoP(s)Repeated linear factors:P(0)=(s-2) (s-2)= G(s)=2+2_k0台s-台(s-）di-μ(-)[s-) G] = koμ=lim-1).4In effect:(u-1)13Solutionof LinearODEswithconstantcoefficientsNon-homogeneous equation with constant coefficientsL[y](t)= f(t)↑a,y("(t)+an--y(n-'(t)+...+ay(t)+aoy(t)=f(t),a,+0Initial conditionsy(to) = %, y (to) = Yi..,y(-l(t) = Y-!The Laplace Transform can be used for the both the general and theparticular solution.147

7 13 Calculation of Inverse Laplace Transform ( ) 1 1 1 0 1 1 1 0 ( ) ( ) m m m m n n n b s b s b s b Q s G s s a s a s a P s − − − − + + + = = + + + Repeated linear factors: ( ) ( ) ( ) ( ) ( ) 0 0 0 1 1 1 0 0 0 ( ) ( ) ( ) , 1 lim ( ) ( ) , 1,., ! n n l l v v v v v l l l s k k P s s s G s s s d k s G s l l ds µ µ µ µ µ µ λ λ Π λ λ λ λ µ µ = = = − → − = − − ⇒ = + − −   = ⋅ − ⋅ =       −   ∑ ∑ In effect: ( ) ( ) ( ) 0 1 1 0 1 1 1 ! t t e s µ λ µ λ µ − −       =   − −   L Procedure outlined for rational functions of s. 14 Solution of Linear ODEs with constant coefficients [ ] ( ) ( 1) (1) 1 1 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ), 0 n n n n n L y t f t a y t a y t a y t a y t f t a − − = + + + + = ≠  Non-homogeneous equation with constant coefficients The Laplace Transform can be used for the both the general and the particular solution. ( 1) 0 0 0 1 0 1 ( ) , '( ) , , ( ) n n y t y t y t γ γ γ − = = = . − Initial conditions

SolutionofLinearODEswithconstantcoefficientsConversiontothecomplexfrequencydomain:a.y("(t)+ar--y(a-)(t)+..+a,y("(t)+aoy(t)= f(t),a,0(t)= . (t)= ... (a-()= .-(a.s"-(0)-2a-n---(=F(s)al15SolutionofLinearODEswithconstantcoefficientsSolution in the complexfrequency domain:-Y(s)-22a-rμs"+=F(s)==0 1=P()-Y(s)=0(s)+F(s),P(s)=as"@(s)=2a.-s=m=0M=0 k=IY(s)=2()+F(2)P(s)y()=L'(r(s)168

8 15 Solution of Linear ODEs with constant coefficients Conversion to the complex frequency domain: ( ) ( ) ( ) ( ) ( 1) (1) 1 1 0 1 0 0 0 1 0 1 1 0 1 ( ) ( ) ( ) ( ) ( ), 0 ( ) , '( ) , , ( ) n n n n n n n n m m m k m m k m k a y t a y t a y t a y t f t a y t y t y t a s Y s a s F s γ γ γ γ − − − − − − = = + + + + = ≠   → = = =    → − =     ∑ ∑ . L L 16 Solution of Linear ODEs with constant coefficients Solution in the complex frequency domain: { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } 1 1 0 0 1 1 0 0 1 1 , , n n m m m k m m k m m k n n m m m k m m k m m k a s Y s a s F s P s Y s Q s F s P s a s Q s a s Q s F s Y s P s y t Y s γ γ − − − = = = − − = = = − ⋅ − = ⇒ ⋅ = + = = ⇒ + = → ∑ ∑∑ ∑ ∑∑ L = L

Systems of Linear OrdinaryDifferentialEquationsNik.Xiros17Definitionof Systems of LinearDifferential Equations1'-order system (expanded notation)α.(t)x+...+a.(t)x,+B..(t)x,+...+β.(o)x=f(t)α.(t)j+..+αm(t)x.+βa(t)x++β..(t)x,=f.(t)1'-order system (compact notation)α(0)+β()x=(0),1≤1≤nA()()+B()(0)=f(0)k=lwhere:α (t)][B.(t)B (t)7[α (t).Ba中Ae.α,(t)αa (t)][B. (t)βm (t)X=[ (t)..x (t)]',f=[f (t).. f.(t)])189

9 Nik. Xiros 17 Systems of Linear Ordinary Differential Equations 18 Definition of Systems of Linear Differential Equations 1 st-order system (expanded notation) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 1 1 11 1 1 1 1 1 1 1 n n n n n nn n n nn n n t x t x t x t x f t t x t x t x t x f t α α β β α α β β + + + + + = + + + + + =      1 st-order system (compact notation) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 ,1 n n lk k lk k l k k α β t x t x f t l n t t t t t ∅ ∅ = = ∑ ∑+ = ≤ ≤ ⇔ ⋅ + ⋅ = A x B x f     where: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) 11 1 11 1 1 1 1 1 , , n n n nn n nn T T n n t t t t t t t t x t x t f t f t α α β β α α β β ∅ ∅         = =     = = A B x f   . .

GenericSystemsofFirst-orderDifferential EquationsStandard (state-space or phase-plane)formn: ×(1)=会=F(x():(0)Dynamic equation:dtAlgebraic equation:y(t)=H(x(t);u(t))XExample:y=sin(x)(-3x,+u)) Straightforward numerical solution method (for linear ones analytical solution)→ Most 'practical' problems involving ODEs can be reduced to 1"-order systems19NumericalSolutionofFirst-orderODESystemsStandard (state-spaceorphase-plane)formDynamic equation:dx*~ x(t+ 4r)- x(t)dt4tx(t)=F(x(t);u(t)=x(t+4t)=x(t)+F(x(t);u(t))-4x(t =0)=XoAlgebraic equation:y(t)=H(x(t):u(t))Example: x(4t) = x, +F(x;u(t= 0)- 4r2010

10 19 Generic Systems of First-order Differential Equations Standard (state-space or phase-plane) form Dynamic equation: ( ) ( ) ( ) ( ) ; d t t t dt = x x F x u 
Algebraic equation: y H x u (t t t ) = ( ( ); ( )) Example: ( ) ( ) 1 2 1 2 2 1 , sin 3 x x y x x x u x     = =       − +    
Straightforward numerical solution method (for linear ones analytical solution)
Most ‘practical’ problems involving ODEs can be reduced to 1st-order systems 20 Numerical Solution of First-order ODE Systems Standard (state-space or phase-plane) form Dynamic equation: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 ; ; 0 t d t t t dt t t t t t t t t t t t ∆ ∆ ∆ ∆ ∆ → + −  ≈   =  ⇒ + = + ⋅  = =   x x x x F x u x x F x u x x  Algebraic equation: y H x u (t t t ) = ( ( ); ( )) Example: x x F x u (∆ ∆ t t t ) = + = ⋅ 0 0 ( ; 0 ( ))