Loop transfer function

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

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

A control system may have several feedback control loops. For example, with a ship autopilot, the rudder-angle control loop is termed the minor loop, whereas the heading control loop is referred to as the major loop. When analysing multiple loop systems, the minor loops are considered first, until the system is reduced to a single overall closed-loop transfer function. 

Find the closed-loop transfer function for the system shown in Figure 4.2. 

The complete, or overall closed-loop transfer function can now be evaluated... 

Note that in Figure 4.12 there is a positive feedback loop. Flence the closed-loop transfer function relating and C (.v) is... 

The system element dynamic equations can now be combined in the block diagram shown in Figure 4.31. Using equation (4.4), the inner-loop transfer function is... 

Comparing the closed-loop transfer function given in equation (4.113) with the standard form given in (3.42)... 

Assuming that the temperature of the surroundings O it) remains constant, the closed-loop transfer function (using equation (4.130)) for the temperature control system, is... 

The response to a step change in the desired temperature of 0-20 °C for the closed-loop transfer function given by equation (4.134) is shown in Figure 4.35. 

The closed-loop transfer function for any feedback control system may be written in the factored form given in equation (5.41)... 

The position of the closed-loop poles in the. v-plane determine the nature of the transient behaviour of the system as can be seen in Figure 5.5. Also, the open-loop transfer function may be expressed as... 

The open-loop transfer function for a control system is... 

A control system has the open-loop transfer function given in Example 5.8, i.e. 

Closed-loop control systems are classified according to the number of pure integrations in the open-loop transfer function. If... 

The open-loop transfer function is third-order type 2, and is unstable for all values of open-loop gain K, as can be seen from the Nichols chart in Figure 6.33. From Figure 6.33 it can be seen that the zero modulus crossover occurs at a frequency of 1.9 rad/s, with a phase margin of —21°. A lead compensator should therefore have its maximum phase advance 0m at this frequency. Flowever, inserting the lead compensator in the loop will change (increase) the modulus crossover frequency. 

A process plant has an open-loop transfer function... 

The laser guided missile shown in Figure 5.26 has an open-loop transfer function (combining the fin dynamics and missile dynamics) of... 

From the partitioned matrix in equation (9.161), the closed-loop transfer function matrix relating yi and uj is... 

Since H =, the system has unity feedback, and the closed-loop transfer function and step response is given by... 

The closed-loop transfer function and step response is given by... 

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. 

The command cloop is used to find the closed-loop transfer function. The command max is used to find the maximum value of 20 logio (mag), i.e. Mp and the frequency at which it occurs i.e. tUp = uj k). A while loop is used to find the —3 dB point and hence bandwidth = ca (n). Thus, in addition to plotting the closed-loop frequency response gain diagrams,/ gd29.7 will print in the command window ... 

Figure 2.12. (a) Simple negative feedback loop, and (b) its reduced single closed-loop transfer function form. 

The RHS of (2-63) is what we will refer to as the closed-loop transfer function in later chapters. 

Example 2.14. Derive the closed-loop transfer function C/R and C/L for the system as shown in Fig. E2.14. 

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). 

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