Closed loop transfer function

Any system in which the output quantity is monitored and compared with the input, any difference being used to actuate the system until the output equals the input is called a closed-loop or feedback control system. 

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. 

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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... 

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)... 

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. 

Roots of denominator of 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. 

The final result is to close the big loop. The resulting closed-loop transfer function is ... 

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). 

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. 

Closed-loop transfer functions and characteristic polynomials... 

The next order of business is to derive the closed-loop transfer functions. For better readability, we ll write the Laplace transforms without the 5 dependence explicitly. Around the summing point, we observe that... 

We will not write out the entire closed-loop function C/R, or in this case, T/Tsp. The main reason is that our design and analysis will be based on only the characteristic equation. The closed-loop function is only handy to do time domain simulation, which can be computed easily using MATLAB. Saying that, we do need to analysis the closed-loop transfer function for several simple cases so we have abetter theoretical understanding. 

In this section, we will derive the closed-loop transfer functions for a few simple cases. The scope is limited by how much sense we can make out of the algebra. Nevertheless, the steps that we go through are necessary to learn how to set up problems properly. The analysis also helps us to better understand why a system may have a faster response, why a system may become underdamped, and when there is an offset. When the algebra is clean enough, we can also make observations as to how controller settings may affect the closed-loop system response. The results generally reaffirm the qualitative statements that we ve made concerning the characteristics of different controllers. 

With a given problem statement, draw the control loop and derive the closed-loop transfer functions. 

All analyses follow the same general outline. What we must accept is that there are no handy dandy formulas to plug and chug. We must be competent in deriving the closed-loop transfer function, steady state gain, and other relevant quantities for each specific problem. 

In our examples, we will take Gm = Ga = 1, and use a servo system with L = 0 to highlight the basic ideas. The algebra tends to be more tractable in this simplified unity feedback system with only Gc and Gp (Fig. 5.6), and the closed-loop transfer function is... 

Example 5.1 Derive the closed-loop transfer function of a system with proportional control and a first order process. What is the value of the controlled variable at steady state after a unit step change in set point ... 

Recall Eq. (5-11), the closed-loop characteristic equation is the denominator of the closed-loop transfer function, and the probable locations of the closed-loop pole are given by... 

Example 5.4 Derive the closed-loop transfer function of a system with proportional-derivative control and a first order process. 

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