Laplace transforms

Laplace transforms Laplace Transforms – Motivation convenience – differential eqns become algebraic eqns. – easy to handle time delays – frequency response analysis to determine how the system responds to oscillating inputs Block Diagram Algebra – doing math with pictures – arithmetic for manipulating dynamic components using boxes and arrows Laplace Transform – Review Given a function f(t) Notes f(t) defined for t from 0 to infinity f(t) suitably â€Å"well-behaved† – piecewise continuous, integrable Linearity of Laplace Transforms the Laplace transform is a linear operation we will use Laplace transforms to analyze linear dynamic systems if our models arent linear, then we will linearize Useful Laplace Transforms for Process Control We need a small library of Laplace transforms for – differentiation – step input – pulse/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 example: with V and F as constants Library of Useful Transforms differentiation – initial conditions disappear if we use deviation variables that are zero at an in initial steady state unit step function (Heaviside fn.) Library of Transforms exponential – exponentials appear in solutions of differential equations  » a provides information about the speed of the response 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 find 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 partial fraction expansion to express the Laplace expressions in a form that can be easily inverted. CSTR Example 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. Tank Example Solution 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 transforms CSTR Example Solution inverse Laplace transform – Can be determined using a complex integral easiest approach is â€Å"table lookup† Use Table 4-1, entry 5 Maple is good at inverting Laplace transforms too The Impulse Function limit of the pulse function (with unit area) as the width goes to zero and height becomes infinite transform CSTR 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 see 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 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 first ed.) Reactor Example with Time Delay Suppose 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 directly in time domain  » easy to solve with Laplace transforms Tank Example with Time Delay – Solutionresponse to step input in cA0 time response Final 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 zero provided dy/dt isnt infinite between t=0 and t ®Ã‚ ¥ (i.e y(t) is STABLE) This will be true if Y(s) is continuous for s ³0 Using the Final Value Theorem Step Response Reactor example final value after a step input What 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 dynamics relates input to output p(s), q(s) polynomials in s – q(s) will also contain exponentials if time delay is present Once 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: