Laplace transform transfer functions

THE LAPLACE TRANSFORM, TRANSFER FUNCTIONS AND SYSTEM MOMENTS... 

Exponential / 7.2.5 Exponential Multiplied by Time / 7.2.6 Impulse (Dirac Delta Function 8 d) Inversion of Laplace Transforms Transfer Functions... 

An equivalent representation of a state-space system ((5.3), (5.4)) is the Laplace transform transfer function description. 

To illustrate how Laplace transforms work, consider the problem of solving Eq. (8-2), subjec t to the initial condition that = 0 at t = 0, and Cj is constant. If were not initially zero, one would define a deviation variable between and its initial value (c — Cq). Then the transfer function would be developed using this deviation variable. Taking the Laplace transform of both sides of Eq. (8-2) gives ... 

The term in parentheses in Eq. (8-17) is zero at steady state and thus it can be dropped. Next the Laplace transform is taken, and the resulting algebraic equation solved. Denoting X s) as the Laplace transform of and X,(.s) as the transform of 4, the final transfer Function can be written as ... 

The combination of reac tor elements is facihtated by the concept of transfer functions. By this means the Laplace transform can be found for the overall model, and the residence time distribution can be found after inversion. Finally, the chemical conversion in the model can be developed with the segregation and maximum mixed models. 

A transfer function is the Laplace transform of a differential equation with zero initial conditions. It is a very easy way to transform from the time to the. v domain, and a powerful tool for the control engineer. 

G(.v) is the transfer function, i.e. the Laplace transform of the differential equation for zero initial conditions. 

Fig. 4.1 Block diagram of a closed-loop control system. R s) = Laplace transform of reference input r(t) C(s) = Laplace transform of controlled output c(t) B s) = Primary feedback signal, of value H(s)C(s) E s) = Actuating or error signal, of value R s) - B s), G s) = Product of all transfer functions along the forward path H s) = Product of all transfer functions along the feedback path G s)H s) = Open-loop transfer function = summing point symbol, used to denote algebraic summation = Signal take-off point Direction of information flow.

Fig. 39.14. (a) Catenary compartmental model representing a reservoir (r), absorption (a) and plasma (p) compartments and the elimination (e) pool. The contents X, Xa, Xp and X,. are functions of time t. (b) The same catenary model is represented in the form of a flow diagram using the Laplace transforms Xr, Xa and Xp in the j-domain. The nodes of the flow diagram represent the compartments, the boxes contain the transfer functions between compartments [1 ]. (c) Flow diagram of the lumped system consisting of the reservoir (r), and the absorption (a) and plasma (p) compartments. The lumped transfer function is the product of all the transfer functions in the individual links. 

If X (0 and Xjit) are the input and output functions in the time domain (for example, the contents in the reservoir and in the plasma compartment), then XJj) is the convolution of Xj(r) with G(t), the inverse Laplace transform of the transfer function between input and output ... 

Since we are doing inverse transform using a look-up table, we need to break down any given transfer functions into smaller parts which match what the table has—what is called partial fractions. The time-domain function is the sum of the inverse transform of the individual terms, making use of the fact that Laplace transform is a linear operator. 

After Laplace transform, a differential equation of deviation variables can be thought of as an input-output model with transfer functions. The causal relationship of changes can be represented by block diagrams. 

We can extend these results to find the Laplace transform of higher order derivatives. The key is that if we use deviation variables in the problem formulation, all the initial value terms will drop out in Eqs. (2-13) and (2-14). This is how we can get these clean-looking transfer functions later. 

The final step should also has zero initial condition C (0) = 0, and we can take the Laplace transform to obtain the transfer functions if they are requested. As a habit, we can define x = V/Qin s and the transfer functions will be in the time constant form. 

Example 2.16. Derive the closed-loop transfer function X,/U for the block diagram in Fig. E2.16a. We will see this one again in Chapter 4 on state space models. With the integrator 1/s, X2 is the Laplace transform of the time derivative of x,(t), and X3 is the second order derivative of x,(t). 

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

Hence, we can view the transfer function as how the Laplace transform of the state transition matrix O mediates the input B and the output C matrices. We may wonder how this output equation is tied to the matrix A. With linear algebra, we can rewrite the definition of O in Eq. (4-5) as... 

Consider the stirred-tank heater again, this time in a closed-loop (Fig. 5.4). The tank temperature can be affected by variables such as the inlet and jacket temperatures and inlet flow rate. Back in Chapter 2, we derived the transfer functions for the inlet and jacket temperatures. In Laplace transform, the change in temperature is given in Eq. (2-49b) on page 2-25 as... 

The next order of business is to derive the closed-loop transfer functions. For better readability, we ll write the Laplace transforms without the 5 dependence explicitly. Around the summing point, we observe that... 

Instead of spacing out in the Laplace-domain, we can (as we are taught) guess how the process behaves from the pole positions of the transfer function. But wouldn t it be nice if we could actually trace the time profile without having to do the reverse Laplace transform ourselves Especially the response with respect to step and impulse inputs Plots of time domain dynamic calculations are extremely instructive and a useful learning tool.1... 

A particular vessel behavior sometimes can be modelled as a series or parallel arrangement of simpler elements, for example, some combination of a PFR and a CSTR. Such elements can be combined mathematically through their transfer functions which relate the Laplace transforms of input and output signals. In the simplest case the transfer function is obtained by transforming the linear differential equation of the process. The transfer function relation is... 

For a model with a known transfer function the several moments can be obtained directly without need for inversion of the transform. This a consequence of a property of the derivative of the Laplace transform, namely certain limits as s= 0 ... 

Second, the fluctuation is delayed by a time 5t which is a function of the residence time t, of the element in the reservoir. For an infinite residence time the argument of the tangent tends towards n/2 and the delay 5f towards T/4, while for a short residence time, the delay tends towards zero. As expected, reactive elements respond more rapidly than inert elements. The phase shift and the damping factor relating input to output concentrations represent the angular phase and argument of a complex function known as the transfer function of the reservoir. Such a function, however, is most conveniently introduced via Laplace and Fourier transforms. Applications of these geochemical concepts to the dynamics of volcanic sequences can be found in Albarede (1993). 

Most of you have probably been exposed to Laplace transforms in a mathematics course, but we will lead off this chapter with a brief review of some of the most important relationships. Then we will derive the Laplace transformations of commonly encountered functions. Next we will develop the idea of transfer functions by observing what happens to the differential equations describing a process when they are Laplace-transformed. Finally, we will apply these techniques to some chemical engineering systems. 

Our primary use of Laplace transformations in process control involves representing the dynamics of the process in terms of "transfer functions." These are output-input relationships and are obtained by Laplace-transfonning algebraic... 

The 1/s is an operator or a transfer function showing what operation is performed on the input signal. This is a completely different idea than the simple Laplace transformation of a function. Remember, the Laplace transform of the unit step function was also equal to l/s. But this is the Laplace transformation of a function. The 1/s operator discussed above is a transfer function, not a function. 

Returning now to Eq. (9.94), let us Laplace-transform and solve for the ratio of output to input the system transfer function... 

J0i Use Laplace transforms to prove mathematically that a P controller produces steadystate ofiMt and that a PI controller does not. The disturbance is a step change in the load variable. The process openloop transfer functions, Gm and G[, are both liist-order lags with dUTerent gains but identical time constants. 

Laplace transformation converts time functions into Laplace transforms and converts ordinary differential equations into transfer functions G j. 

Inversion of the Laplace transformation gives the time function. If we have a transfer function the unit step function is C [G(,y s] and the impulse response is C" [G(,)]. 

In Chap. 18 we will define mathematically the sampling process, derive the z transforms of common functions (learn our German vocabulary) and develop transfer functions in the z domain. These fundamentals are then applied to basic controller design in Chap. 19 and to advanced controllers in Chap. 20. We will find that practically all the stability-analysis and controller-design techniques that we used in the Laplace and frequency domains can be directly applied in the z domain for sampled-data systems. 

We know how to find the z transformations of functions. Let us now turn to the problem of expressing input-output transfer-function relationships in the z domain. Figure 18.9a shows a system with samplers on the input and on the output of the process. Time, Laplace, and z-domain representations are shown. G(2, is called a pulse transfer function. It will be defined below. 

Defining in this way permits us to use transfer functions in the z domain [Eq. (18.57)] just as we use transfer functions in the Laplace domain. G,, is the z transform of the impulse-sampled response of the process to a unit impulse function <5( . In z-transforming functions, we used the notation =... 

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