Laplace transforms

Laplace interpretsLaplace Transforms Motivationconvenience differential eqns become algebraic eqns. easy to handle measure delays frequency reply analysis to determine how the system responds to oscillating introduces Block Diagram Algebra doing math with pictures arithmetic for manipulating dynamic components using boxes and arrowsLaplace Transform ReviewGiven a tend f(t)Notes f(t) defined for t from 0 to infinityf(t) suitably well-behaved piecewise continuous, integrableLinearity of Laplace Transformsthe Laplace transform is a additive operationwe go out use Laplace transforms to analyze linear dynamic systemsif our models bent linear, then we will linearize Useful Laplace Transforms for Process ControlWe need a small library of Laplace transforms for differentiation step input meter/impulse functions exponentials oscillating functions because these are common functions that we will encounter in our equations Lets think about a simple linear differential equation r ole model with V and F as constantsLibrary of Useful Transformsdifferentiation sign conditions disappear if we use deviation variables that are zero at an in initial steady stateunit step function (Heaviside fn.)Library of Transformsexponential exponentials appear in solutions of differential equations a provides information about the speed of the repartee when the input changes. If a is a large negative number, the exponential decays to zero quickly What happens if a is positive? After we have done some algebra to strike a solution to our ODEs in the Laplace domain, we must invert the Laplace transform if we want to get a solution in the time domain. We sometimes use uncomplete fraction expansion to express the Laplace expressions in a form that can be easily inverted. CSTR archetype Transform Model (in deviation variables) using our library of transforms, the Laplace transform of the model is For a step change in feed concentration at time zero starting from steady state. T ank Example beginning Solve for CA(s) If we like, we can rearrange to the form This is the solution in the Laplace domain. To find the solution in the time domain, we must invert the Laplace transformsCSTR Example Solutioninverse Laplace transform Can be determined using a complex integraleasiest approach is table lookupUse Table 4-1, entrance 5Maple is good at inverting Laplace transforms tooThe Impulse Function limit of the pulse function (with unit area) as the width goes to zero and height becomes blank transformCSTR Impulse Response physically dump some pure A into reactor, all at once input function Transform time response Interpretation of Impulse Response dump a bag of reactant into the reactor in a very very short time we distinguish an instantaneous jump to a new concentration due to the impulse input concentration then decays back to the original steady-state concentration Time-Shifted Functions Representation of Delays Laplace transform for function with time d elay Just pre-multiply by an exponential. How could we prove this? change of variables in integration in expression for Laplace Transform (see p. 103 of Marlin, p. 115 in prototypical ed.)Reactor Example with Time DelaySuppose we add a long length of pipe to feed assume plug flow It will take a time period, q minutes, before the change in concentration reaches the tank, and begins to influence cA delay differential equation difficult to solve instantaneously in time domain easy to solve with Laplace transforms Tank Example with Time Delay Solutionresponse to step input in cA0 time response final exam exam Value Theorem An easy way to find out what happens to the output variable if we wait a long time. We dont have to invert the Laplace transform Why is it authoritative? Consider the Laplace transform of a time derivative now let s approach zeroprovided dy/dt isnt infinite between t=0 and t (i.e y(t) is STABLE) This will be true if Y(s) is continuous for s0Using the Final Value Theorem Step Response Reactor example final value after a step inputWhat can we do with Laplace Transforms so far.Take Laplace transforms of linear ODEs (in deviation variables).Substitute Laplace transform expressions for different kinds of inputs we are interested in Steps, pulses, impulses (even with dead time)Solve for the output variable in terms of s.Invert the Laplace transform using Table 4.1 to get the solution in the time domain. Find the final steady state value of the output variable, for a particular input change, even without inverting the Laplace transform.Laplace transforms are mostly used by control engineers who want to determine and analyze transfer functions.compact way of expressing process dynamicsrelates input to outputp(s), q(s) polynomials in s q(s) will as well contain exponentials if time delay is presentOnce we know the transfer function of the process, we can use it to find out how the process responds to different types of input changes