Solution of ODEs using Laplace Transforms - Queen's U

1 1 Solution of Solution of ODEs ODEs using using Laplace Laplace TransformsTransformsProcess Dynamics and ControlProcess Dynamics and Control2 Linear Linear ODEsODEs For linear ODEs, we can solve without integrating byusing Laplace Transforms Integrate out time and transform to Laplace domainMultiplicationMultiplicationIntegr ationIntegration3 Common TransformsCommon TransformsUseful Useful Laplace Laplace TransformsTransforms1. Exponential1. Exponential2. Cosine2. Cosine4 Common TransformsCommon TransformsUseful Useful Laplace Laplace TransformsTransforms3. Sine3. Sine5 Common TransformsCommon TransformsOperators1. Derivative of a function, ,2. Integral of a function6 Common TransformsCommon TransformsOperatorsOperators3. Delayed function7 Common TransformsCommon TransformsInput Signals1. Constant1. Constant2. Step2. Step3. Ramp function3.

2 Ramp function8 Common TransformsCommon TransformsInput SignalsInput Signals4. Rectangular Pulse4. Rectangular Pulse5. 5. Unit impulse9 Laplace Laplace TransformsTransformsFinal Value TheoremLimitations:Limitations:Initial Value Theorem10 Solution of Solution of ODEsODEsWe can continue taking Laplace Transforms and generate acatalogue of Laplace domain final aim is the Solution of ordinary Laplace Transform, solveResult11 Solution of Solution of ODEsODEsCruise Control Example Taking the Taking the Laplace Laplace transform of the ODE transform of the ODE yields (recalling the yields (recalling the LaplaceLaplacetransform is a linear operator)transform is a linear operator)Force ofEngine (u)FrictionSpeed (v)12 Solution of Solution of ODEsODEs IsolateIsolateand solveand solve If the input is kept constantIf the input is kept constantits its Laplace Laplace transformtransform Leading toLeading to13 Solution of Solution of ODEsODEs Solve by inverse Solve by inverse Laplace Laplace transform:transform.

3 (tables)(tables) Solution is obtained by aSolution is obtained by a getting the inverse getting the inverse Laplace Laplace transformtransformfrom a tablefrom a table AlternativelyAlternatively we can use partial fraction expansion to compute we can use partial fraction expansion to compute the Solution usingthe Solution using simple inverse transformssimple inverse transforms14 Solution of Linear Solution of Linear ODEsODEs DC MotorDC Motor System dynamics describes (negligible inductance)System dynamics describes (negligible inductance)15 Laplace Laplace TransformTransform Expressing in terms of angular velocityExpressing in terms of angular velocity Taking Taking Laplace Laplace TransformsTransforms SolvingSolving Note that this function can be written asNote that this function can be written as16 Laplace Laplace TransformTransformAssume then the transfer function gives directlyCannot invert explicitly, but if we can find such thatwe can invert using Partial Fraction Expansion to deal withsuch functions17 Linear Linear ODEsODEsWe deal with rational functions of the form where degree of> degree ofis called the characteristic polynomial of the functionThe roots of are the poles of the functionTheorem.

4 Every polynomial with real coefficients can be factored into theproduct of only two types of factors powers of linear terms and/or powers of irreducible quadratic terms,18 Partial fraction ExpansionsPartial fraction Expansions1. has real and distinct factorsexpand as2. has real but repeated factorexpanded19 Partial Fraction ExpansionPartial Fraction ExpansionHeaviside expansionFor a rational function of the formConstants are given by20 Partial Fraction ExpansionPartial Fraction ExpansionExampleExampleThe polynomialThe polynomialhas rootshas rootsIt can be factored asIt can be factored asBy partial fraction expansionBy partial fraction expansion21 Partial Fraction ExpansionPartial Fraction ExpansionBy By HeavisideHeaviside becomesbecomes By inverse By inverse laplacelaplace22 Partial Fraction ExpansionPartial Fraction ExpansionHeaviside expansionFor a rational function of the formConstants are given by23 Partial Fraction ExpansionPartial Fraction ExpansionExampleExampleThe polynomialThe polynomialhas rootshas rootsIt can be factored asIt can be factored asBy partial fraction

5 ExpansionBy partial fraction expansion24 Partial Fraction ExpansionPartial Fraction ExpansionBy By HeavisideHeaviside becomesbecomes By inverse By inverse laplacelaplace25 Partial Fraction ExpansionPartial Fraction Expansion3. 3. has an irreducible quadratic factorhas an irreducible quadratic factor Gives a pair of complex conjugates ifGives a pair of complex conjugates if Can be factored in two waysCan be factored in two waysa)a) is factored asis factored asb)b)oror asas26 Partial Fraction ExpansionPartial Fraction ExpansionHeaviside expansionFor a rational function of the formConstants are given by27 PartialPartial Fraction ExpansionFraction ExpansionExampleExampleThe polynomialThe polynomialhas rootshas rootsIt can be factored asIt can be factored asBy partial fraction expansionBy partial fraction expansion28 Partial Fraction ExpansionPartial Fraction ExpansionBy By HeavisideHeaviside,,which yieldswhich yieldsTaking the inverseTaking the inverse laplacelaplace29 PartialPartial Fraction ExpansionFraction ExpansionThe inverse The inverse laplacelaplaceCan be re-arranged toCan be re-arranged to30 PartialPartial Fraction ExpansionFraction ExpansionExampleExampleThe polynomialThe polynomialhas rootshas rootsIt can be factored as (It can be factored as ( ))

6 Solving for A and B,Solving for A and B,31 Partial Fraction ExpansionPartial Fraction ExpansionEquating similar powers of s in,Equating similar powers of s in,yieldsyieldshencehenceGivingGivingTak ing the inverseTaking the inverse laplacelaplace32 Partial Fraction ExpansionsPartial Fraction ExpansionsAlgorithm for Solution of ODEs Take Laplace Transform of both sides of ODE Solve for Factor the characteristic polynomial Find the roots (roots or poles function in Matlab) Identify factors and multiplicities Perform partial fraction expansion Inverse Laplace using Tables of Laplace Transforms33 Partial Fraction ExpansionPartial Fraction Expansion For a givenFor a given functionfunction The polynomialThe polynomial has three distinct types ofhas three distinct types of rootsroots Real rootsReal roots yields exponential termsyields exponential terms yieldsyields constant termsconstant terms complex rootsComplex roots yields exponentially weighted sinusoidalyields exponentially weighted sinusoidal signalssignals yieldsyields pure sinusoidalpure sinusoidal signalsignal A lot of information is obtained from the roots ofA lot of information is obtained from the roots of