The transfer function

Before an efficient control system can be designed it is necessary to consider how all sections of the control loop will behave under the influence of variations in load (i.e. what are the load rejection properties of the system ) and/or its set point (i.e. what are the set point following characteristics ). This requires experimental investigation, or a time-dependent mathematical analysis (i.e. in the unsteady state), or both. Although each section of the loop will necessitate a separate analysis, there 

Laplace transform of output Laplace transform of input 

Thus the transfer function is basically an input-output mathematical relationship. This is a most appropriate concept to use in conjunction with block diagrams (Section 7.2.1) which are themselves basically input-output schematic diagrams so that each block may be represented by the transfer function describing its behaviour. 

The advantage of defining a transfer function in terms of Laplace transforms of input and output is that the differential equations developed to describe the unsteady-state behaviour of the system are reduced to simple algebraic relationships (e.g. cf. equations 7.17 and 7.19). Such relationships are much easier to deal with, and normal algebraic laws can be used to relate the various transfer functions of each component in the control loop (see Section 7.9). Furthermore, the output (or response) of the system to a variety of inputs may be obtained without classical integration. 

The transfer function approach will be used where appropriate throughout the remainder of this chapter. Transfer functions of continuous systems will be expressed as functions of s, e.g. as G(j) or H(s). In the case of discrete time systems, the transfer function will be written in terms of the z-transform, e.g. as G(z) or H(z) (Section 7.17). An elementary knowledge of the Laplace transformation on the part of the reader is assumed and a table of the more useful Laplace transforms and their z-transform equivalents appears in Appendix 7.1. 

A powerful relationship used for analyzing digital filters is the tranter function, which is found by solving for the ratio of output (7) to input (X) in the Z-transformed filter expression. The transfer function for Equation 3.4 can be solved by using simple algebra  

The transfer fiinction is notated as H. H is the Z transform of the time-domain impulse response function / ( ). Transformation of x and 7 into the Z domain gives us a tool for talking about a fimction H (the Z transform of h) that takesXas input and yields Y. 

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Most thermometry using the KTTS direcdy requites a thermodynamic instmment for interpolation. The vapor pressure of an ideal gas is a thermodynamic function, and a common device for reali2ing the KTTS is the helium gas thermometer. The transfer function of this thermometer may be chosen as the change in pressure with change in temperature at constant volume, or the change in volume with change in temperature at constant pressure. It is easier to measure pressure accurately than volume thus, constant volume gas thermometry is the usual choice (see Pressure measurement). 

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 ... 

Capacity Element Now consider the case where the valve in Fig. 8-7 is replaced with a pump. In this case, it is reasonable to assume that the exit flow from the tank is independent of the level in the tank. For such a case, Eq. (8-22) still holds, except that/i no longer depends on hi. For changes in fi, the transfer function relating changes in to changes in is shown in Fig. 8-10. This is an example of a pure capacity process, also called an integrating system. The cross sectional area of the tank is the chemical process equivalent of an electrical capacitor. If the inlet flow is step forced while the outlet is held... 

Since /i is the inlet flow to tank 2, the transfer function relating changes in ho to changes in/i has the same form as that given in Fig. 8-4 ... 

Distance-Velocity Lag (Dead-Time Element) The dead-time element, commonly called a distance-velocity lag, is often encountered in process systems. For example, if a temperature-measuring element is located downstream from a heat exchanger, a time delay occurs before the heated fluid leaving the exchanger arrives at the temperature measurement point. If some element of a system produces a dead-time of 0 time units, then an input to that unit,/(t), will be reproduced at the output a.s f t — 0). The transfer function for a pure dead-time element is shown in Fig. 8-17, and the transient response of the element is shown in Fig. 8-18. 

The response produced by Eq. (8-26), c t), can be found by inverting the transfer function, and it is also shown in Fig. 8-21 for a set of model parameters, K, T, and 0, fitted to the data. These parameters are calculated using optimization to minimize the squarea difference between the model predictions and the data, i.e., a least squares approach. Let each measured data point be represented by Cj (measured response), tj (time of measured response),j = 1 to n. Then the least squares problem can be formulated as ... 

As an illustrative example, consider the simplified block diagram for a representative decoupling control system shown in Fig. 8-41. The two controlled variables Ci and Co and two manipulated variables Mi and Mo are related by four process transfer functions, Gpn, Gpi9, and pie, Gpii denotes the transfer function between Mi... 

The terms on the right are the transfer functions. With the two units in series. 

Combined Models, Transfer Functions The transfer function relation is... 

Although a transfer function relation may not be always invertible analytically, it has value in that the moments of the RTD may be derived from it, and it is thus able to represent an RTD curve. For instance, if Gq and Gq are the limits of the first and second derivatives of the transfer function G(.s) as. s 0, the variance is... 

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

The elements of a closed-loop control system are represented in block diagram form using the transfer function approach. The general form of such a system is shown in Figure 4.1. 

The transfer function relating R s) and C(.v) is termed the closed-loop transfer function. 

Equation (4.56) may be re-arranged to give the transfer function relating Xo(s) and... 

Flence, for a sinusoidal input, the steady-state system response may be calculated by substituting. v = )lu into the transfer function and using the laws of complex algebra to calculate the modulus and phase angle. 

Comparing equations (6.14) and (6.18), providing there are no zeros in the transfer function, it is generally true to say... 

The active network shown in Figure 6.30 has the transfer function given in equation (6.99)... 

Assuming that a sample and hold device is in cascade with the transfer function G. s), determine G z) for the following... 

The response-factor approach is based on a method in which the response factors represent the transfer functions of the wall due to unit impulse excitations. The real excitation is approximated by a superposition of such impulses (mostly of triangular shape), and the real response is determined by the superposition of the impulse responses (see Figs. 11.33 and 11.34). ... 

A sigmoid (s-shaped) is a continuous function that has a derivative at all points and is a monotonically increasing function. Here 5,p is the transformed output asymptotic to 0 < 5/,p I and w,.p is the summed total of the inputs (- 00 < Ui p < -I- 00) for pattern p. Hence, when the neural network is presented with a set of input data, each neuron sums up all the inputs modified by the corresponding connection weights and applies the transfer function to the summed total. This process is repeated until the network outputs are obtained. 

Normalize both the input and the target output data to fit the transfer function range. This implies that the data have to be scaled to fit between the minimum and maximum values of the selected transfer function. 

S. Ducruix, D. Durox, and S. Candel. Theoretical and experimental determination of the transfer function of a laminar premixed flame. Proceedings of the Combustion Institute, 28 765-773, 2000. 

In the catenary model of Fig. 39.14a we have a reservoir, absorption and plasma compartments and an elimination pool. The time-dependent contents in these compartments are labelled X, X, and X, respectively. Such a model can be transformed in the 5-domain in the form of a diagram in which each node represents a compartment, and where each connecting block contains the transfer function of the passage from one node to another. As shown in Fig. 39.14b, the... 

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 ... 

Fig. 44.2. An artificial neuron x,. ..Xp are the incoming signals wi... Wp are the corresponding weight factors and F is the transfer function.

The net input, NET, is then passed to the transfer function that transforms it into the output signal of the unit. Different transfer functions may be used, the most common non-linear one being the sigmoidal function (Fig. 44.5b). 

The output units receive the weighted output signals of the h hidden units. The weighted sums are calculated and passed through the transfer function to yield the final output of the network. 

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