Transfer Functions and Block Diagrams

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 is solved. 

By denoting X(s) as the Laplace transform of % and Xt s) as the transform of the final transfer function can be written as 

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Transfer Functions and Block Diagrams A very convenient and compact method of representing the process dynamics of linear systems involves the use or transfer functions and block diagrams. A transfer func tion can be obtained by starting with a physical model as... 

Now that transfer functions and block diagrams in Figs. 11.2 to 11.6 have been developed for the individual components of the feedback control system, we can combine this information to obtain the composite block diagram of the controlled system shown in Fig. 11.7. 

The use of block diagrams to illustrate cause and effect relationship is prevalent in control. We use operational blocks to represent transfer functions and lines for unidirectional information transmission. It is a nice way to visualize the interrelationships of various components. Later, they are crucial to help us identify manipulated and controlled variables, and input(s) and output(s) of a system. 

Consider a simple feedback loop (Fig. 7.3a) in which the feedback path consists of elements which approximate to a steady-state gain K (Fig. 7.37). In this instance, the equivalent unity feedback loop is determined by placing 1 IK in the set point input to the main loop and compensating for this by adding an additional factor K in the forward part of the loop prior to the entry of the load disturbance, as in Fig. 7.38. It is easy to confirm that the standard closed loop transfer functions and are the same for the block diagrams in Figs 7.37 and 7.38. 

Develop the block diagram of a generalized feedback control system with one disturbance, incorporating in each block the appropriate transfer function and on each stream the appropriate variable. 

Figure 5.56 Block diagrams and transfer functions, and rearrangement gives... 

The sensor transfer function is a gain and the controller, which has no derivative contribution, is a PI. The block diagram looks like the following (Figure 4.34) where we can see the transfer functions and their values. We Run the model and the results can be seen in Figure 4.35. We see that we can control the perturbation. 

In the previous two sections, we considered two design methods for feedforward control. The design method of Section 15.3 was based on a nonlinear steady-state process model, while the design method of Section 15.4 was based on a transfer function model and block diagram analysis. Next, we show how the two design methods are related. 

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... 

Figure 8-41 includes two conventional feedback controllers G i controls Cl by manipulating Mi, and G o controls C9 by manipidating Mo. The output sign s from the feedback controllers serve as input signals to the two decouplers D o and D91. The block diagram is in a simplified form because the load variables and transfer functions for the final control elements and sensors have been omitted. 

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... 

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). 

Example 4.6. Derive the transfer function Y/U and the corresponding state space model of the block diagram in Fig. E4.6. 

To derive the state space representation, one visual approach is to identify locations in the block diagram where we can assign state variables and write out the individual transfer functions. In this example, we have chosen to use (Fig. E4.6)... 

We first establish the closed-loop transfer functions of a fairly general SISO system. After that, we ll walk through the diagram block by block to gather the thoughts that we must have in synthesizing and designing a control system. An important detail is the units of the physical properties. 

For the rest of the control loop, Gc is obviously the controller transfer function. The measuring device (or transducer) function is Gm. While it is not shown in the block diagram, the steady state gain of Gm is Km. The key is that the summing point can only compare quantities with the same units. Hence we need to introduce Km on the reference signal, which should have the same units as C. The use of Km, in a way, performs unit conversion between what we dial in and what the controller actually uses in comparative tests. 2... 

This is a problem that we have to revisit many times in later chapters. For now, draw the block diagram of the dye control system and provide the necessaiy transfer functions. Identify units in the diagram and any possible disturbances to the system. In addition, we do not know where to put the photodetector at this point. Let s just presume that the photodetector is placed 290 cm downstream. 

Here, we use L to denote the major load variable and its corresponding transfer function is GL. We measure the load variable with a sensor, Gnu., which transmits its signal to the feedforward controller GFF. The feedforward controller then sends its decision to manipulate the actuating element, or valve, Gv. In the block diagram, the actuator transfer function is denoted by G v. The idea is that cascade control may be implemented with the actuator, Gv, as we have derived in Eq. (10-1). We simply use G v to reduce clutter in the diagram. 

The key points will be illustrated with a blending process. Here, we mix two streams with mass flow rates m, and m2, and both the total flow rate F and the composition x of a solute are to be controlled (Fig. 10.10). With simple intuition, we know changes in both m, and m2 will affect F and x. We can describe the relations with the block diagram in Fig. 10.11, where interactions are represented by the two, yet to be derived, transfer functions G12 and G21. 

For any input the output is found by simply multiplying by the constant K, Thus the transfer function relating and is a constant or a gain. We can represent this in block-diagram form as shown in Fig. 9.2. 

The above demonstrates one very important and useful property of transfer functions. The total effect of a number of transfer functions connected in series U just the product of all the individual transfer functions. Figure 9.7 shows this in block-diagram form. The overall transfer function is a third-order tag with three poles. 

Figure 10.2h gives a sketch of the feedback control system and a block diagram for the two-heated-tank process with a controller. Let us use an analog electronic system with 4 to 20 mA control signals. The temperature sensor has a range of 100°F, so the Gj transfer function (neglecting any dynamics in the temperature measurement) is... 

Draw a block diagram of a process that has two manipulated variable inputs (Mi and M]) that each affect the output (2T). A feedback controller Si is used to control X by manipulating Mi since the transfer function between Mj and X has a small time constant and smaU deadtime. 

A control system using PI control is represented by the block diagram shown in Fig. 7.48. The transfer functions describing the various blocks are as shown with = 10, t, = 1 min, = 0.8 and K, = O.S. By determination of the gain and phase margins, show the effect on the stability of the control system of introducing derivative action with r0 1 min. 

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