Block diagram algebra

In Fig. 7.11 Ga(j), Gb(s) and Gc( ) are transfer functions describing the input-output relationships for each block respectively, where  

The transfer function of the whole system is given by Gx(j) = yl3c,. This can also be obtained by multiplying together the three blocks in series, viz.  

The transfer function of the whole system is Gr(.s) = y/x, and by the principle of superposition y is obtained from the additive effects of yt,y2 and y2.Therefore  

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Using block diagram algebra and Laplace transform variables, the controlled variable C(.s) is given by... 

The time delay effect is canceled out, and this equation at the summing point is equivalent to a system without dead time (where the forward path is C = GCGE). With simple block diagram algebra, we can also show that the closed-loop characteristic polynomial with the Smith predictor... 

Block Diagram Algebra for Sampled Data Systems... 

A generalized model for a servo-control system can be depicted as shown in Fig. 3 (Skogestad and Postlethwaite 2005). By applying block diagram algebra (Franklin et al. 2005 Ogata 1997), the true output y(s) can be expressed in terms of the reference (r(s)), disturbance (d(s)), and sensor noise (v(i)) as ... 

This is called the transfer-function model, where the output is related to the inputs m(5) and d(s) through a relation involving the transfer functions gu(s) and This is the basis of block diagram algebra, which will be discussed in a later section. 

In this subsection, we present the typical control loop (as shown in Fig. 5.53) with its transfer functions, inputs-outputs, and their relations. It is also a simple introduction to block diagram algebra. 

Note that Ygp and D are the independent input signals for the controlled process because they are not affected by operation of the control loop. By contrast, U and D are the independent inputs for the uncontrolled process. To evaluate the performance of the control system, we need to know how the controlled process responds to changes in D and Ygp. In the next section, we derive expressions for the closed-loop transfer functions, Y(s)/Ysp(s) and Y(s)/D(s). But first, we review some block diagram algebra. 

As a starting point for the stability analysis, consider the block diagram in Fig. 11.8. Using block diagram algebra that was developed earlier in this chapter, we obtain... 

The following closed-loop relation for IMC can be derived from Fig. 12.66 using the block diagram algebra of Chapter 11 ... 

Cascade control can improve the response to a set-point change by using an intermediate measurement point and two feedback controllers. However, its performance in the presence of disturbances is usually the principal concern (Shinskey, 1996). In Fig. 16.4, disturbances in D2 are compensated by feedback in the inner loop the corresponding closed-loop transfer function (assuming = Di = 0) is obtained by block diagram algebra ... 

However, if the second feedback controller is in the automatic mode with Y2sp = 0, then, using block diagram algebra. 

To evaluate the effects of control loop interactions further, again consider the block diagram for the 1-1/2-2 control scheme in Fig. 18.3a. Using block diagram algebra (see Chapter 11), we can derive the following expressions relating controlled variables and set points ... 

These equations can be derived easily using the block diagram algebra of Chapter 11. They illustrate how the three external inputs (D, N, and Ygp) affect three output variables the actual output Y, the error E, and the controller output U, The nine transfer functions in (J-1) to (J-3) completely characterize the closed-loop performance of... 

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