Laplace transforms

Laplace transformsLaplace Transforms Motivationconvenience differential eqns become algebraic eqns. belatedly to handle time delays frequency response analysis to determine how the dust responds to oscillating inputs Block Diagram Algebra doing math with pictures arithmetic for manipulating energetic components utilize boxes and arrowsLaplace Transform ReviewGiven a matter f(t)Notes f(t) defined for t from 0 to infinityf(t) suitably well-be attaind piecewise continuous, integrableLinearity of Laplace Transformsthe Laplace transform is a linear operationwe go forth use Laplace transforms to analyze linear dynamic systemsif our models atomic number 18nt linear, then we will linearize practice sessionful Laplace Transforms for Process ControlWe contain a small library of Laplace transforms for speciality step input throb/impulse employments exponentials oscillating functions because these are common functions that we will encounter in our compares Lets think about a simple linear differential equation example with V and F as constantsLibrary of Useful Transformsdifferentiation initial conditions disappear if we use deviation variables that are secret code at an in initial steady stateunit step function (Heaviside fn.)Library of Transformsexponential exponentials appear in responses of differential equations a provides education about the speed of the response when the input changes. If a is a broad negative number, the exponential decays to zero quickly What happens if a is positive? later on we have done some algebra to look a declaration 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 partial fraction expansion to express the Laplace expressions in a form that can be tardily inverted. CSTR mannikin Transform Model (in deviation variables) apply our library of transforms, the Laplace transform of the model is For a step change in feed tautne ss at time zero starting from steady state. Tank compositors case Solution Solve for CA(s) If we like, we can rearrange to the form This is the solution in the Laplace domain. To witness 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, entry 5Maple is good at inverting Laplace transforms tooThe itch Function limit of the pulse function (with unit area) as the largeness goes to zero and height becomes infinite transformCSTR inclination Response physically dump some pure A into reactor, all at erst input function Transform time response Interpretation of Impulse Response dump a bag of reactant into the reactor in a very very short time we see an instantaneous spring to a new ducking due to the impulse input concentration then decays back to the original steady-state concentration Time-Shifted Functions Re displayation of Delays Laplace transform for function with time delay 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 root ed.)Reactor Example with Time DelaySuppose we add a long length of pipe to feed assume plug electric current It will take a time period, q minutes, before the change in concentration reaches the tank, and begins to influence cA delay differential equation demanding to lap directly in time domain easy to solve with Laplace transforms Tank Example with Time Delay Solutionresponse to step input in cA0 time response closing 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 true? 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 conti nuous for s0Using the Final Value Theorem Step Response Reactor example final prise 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 measure of the output variable, for a particular input change, even without inverting the Laplace transform.Laplace transforms are in the main used by control engineers who want to determine and analyze bump off functions.compact way of expressing process dynamicsrelates input to outputp(s), q(s) polynomials in s q(s) will also 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 dif ferent types of input changes