控制技术概论3

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3.2 the transfer function
The transfer function of a block diagram is defined as the output divided by its input when represented in the Laplace domain with zero initial conditions. The transfer function G(s) of the block diagram shown in Fig. 3-1 is
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3.2 the transfer function
In summary, we observe that for cascaded elements the overall transfer function is equal to the product of the transfer function of each element, whereas the overall transfer function for parallel elements is equal to the sum of the individual transfer function. When a control system is represented in block diagram form we have not only forward paths but feedback paths as well. By giving due consideration to the direction of the signals we can derive a single transfer function to represent the entire system. This overall transfer function will be seen to be the ratio of the input and the output of the entire system.
Overview of control technology
Chapter 3 Control System representation
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Chapter 3
Control System Representation
3.1 introduction 3.2 the transfer function 3.3 closed loop systems
Y ( s ) G 3 ( s ) X 3 ( s ) G 3 ( s )[ G 2 ( s ) X
2
( s )] G 3 ( s ) G 2 ( s ) G 1 ( s ) X 1 ( s )
which can be written as
Y (s) X 1 (s)
G 3 ( s )G 2 ( s )G 1 ( s ) G ( s )
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3.2 the transfer function
X 1(s) G 1(s) X 2(s) G 2(s) X 3(s) Y(s ) G 3(s) X 1(s) Y(s ) G (s) (b) F ig . 3 - 2 C a s c a d e d e le m e n ts (a)
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For applications where it is required to generate a signal which is the sum of two signals we define a summer or summing junction as shown in Fig.3-3a. If the difference is required, then we define a subtractor as shown in Fig. 3-3b. Subtractors are often called error detecting devices since the output signal is the difference between two signals of which one is usually a reference signal. Examples of several components used for summing and subtracting signals were given in the previous chapter.
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3.1 introduction
As an alternate way of system representation in the last chapter we also obtained a set of first-order differential equations. Later in this chapter we shall use such a set to show how a control system can be compactly represented in matrix notation. Such a representation is also called state space representation and the variables used are known as state variables. We will show that such a representation is directly related to the governing firstorder differential equations as well as the system transfer function. The analysis of control systems using the state space representation constitutes what is referred to as the modern method.
Y ( s ) [ G 1 ( s ) G 2 ( s ) G 3 ( s )] X ( s )
The overall transfer function shown in Fig.3-4b is
Y (s) X (s) G (s)
where
G (s) G1 (s) G 2 (s) G 3 (s)
The overall transfer function then is simply the product of individual transfer functions as shown in Fig. 3-2b.
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3.2 the transfer function
X
2
(s)
X 1 (s) X X
3 2
G1 (s) G 2 (s) G 3 (s)
X
2
(s) G1 (s) X 1 (s)
(s) (s)
X 3 (s) G 2 (s) X
2
(s)
Y (s) X
3 (s)
Y (s) G 3 (s) X 3 (s)
and substitution yields
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3.2 the transfer function
The combination of block diagrams in parallel is shown in Fig. 3-4a. Form the definition of the transfer function we have
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3.2 the transfer function
We have seen how the transfer function of an individual component can be obtained in the previous chapter. Owing to the occurrence of many transfer functions in series (or cascade) and parallel connections it is necessary that we have rules to combine them systematically. Consider the block diagram of cascaded elements shown in Fig. 3-2a. From the definition of a transfer function we have
X (s) Y (s) G (s)
Here the path of the signals X(s) and Y(s) is a forward path.
Y (s) G (s) X (s)
F ig .3 -1 T r a n s f e r f u n c tio n o f a b lo c k d ia g r a m
X 1 (s ) + + X 1 (s )+ X 2 (s ) X 1 (s ) + X 1 (s )-X 2 (s )
X 2 (s ) (a ) Sum m er
X 2 (s ) (b ) S u b tra c to r (E rro r d e te c tin g d e v ic e )
F ig .3 -3 A d d itio n o r s u b tra c tio n o f s ig n a ls
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