Forward-path transfer function

The closed-loop transfer function is the forward-path transfer function divided by one plus the open-loop transfer function. 

Transfer function (4.7) now becomes the complete forward-path transfer function as shown in Figure 4.4. 

The block diagram for the control system is shown in Figure 4.27. From the block diagram, the forward-path transfer function G(.v) is... 

Process reaction curve This can be obtained from the forward-path transfer function... 

A closed-loop control system has a nominal forward-path transfer function equal to that given in Example 6.4, i.e. 

In this example, the inner loop is solved first using feedback. The controller and integrator are cascaded together (numpl, denpl) and then series is used to find the forward-path transfer function (numfp, denfp ). Feedback is then used again to obtain the closed-loop transfer function. 

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.

The important observation is that when we "close" a negative feedback loop, the numerator is consisted of the product of all the transfer functions along the forward path. The denominator is 1 plus the product of all the transfer functions in the entire feedback loop ( .e., both forward and feedback paths). The denominator is also the characteristic polynomial of the closed-loop system. If we have positive feedback, the sign in the denominator is minus. 

Step 1 Define transfer functions in the forward path. The values of all gains and time constants are arbitrarily selected. 

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. 

The numerator of an overall closed-loop transfer function is the product of the transfer functions on the forward path between the set point or the load and the controlled output. Thus ... 

Uf = product of the transfer functions in the forward path from Zj to Z... 

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