tailieunhanh - Essentials of Process Control phần 5

Tham khảo tài liệu 'essentials of process control phần 5', ngoại ngữ, ngữ pháp tiếng anh phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Ị CIIAITBR 7 Laplace-Domain Dynamics 237 Therefore A. lim B __L_ T Ơ 1 TO 2 J 1 to Inverting term by term yields r kJi - Le- T - e-t T T TRANSFER FUNCTIONS Our primary use of Laplace transformations in process control involves representing the dynamics of the process in terms of transfer functions. These are output-input relationships and are obtained by Laplace-transforming algebraic and equations. In the following discussion the output variable of the process is The input variable or the forcing function is Multiplication by a Constant Consider the algebraic equation J i Laplace-transforming both sides of the equation gives X r X y t e st dt K U j e st dt Io Jo KU s where and are the transforms of and Note that is an arbitrary function of time. We have not specified at this point the exact form of the input. Comparing Eqs. and shows that the input and output variables are related in the domain in exactly the same way as they are related in the time domain. Thus the English and Russian words describing this situation are the same. Equation can be put into transfer function form by finding the output input ratio - K u x 238 PART TWO Laplace-Domain Dynamics and Control FIGURE Gain transfer function. For any input ơ s the output Y s is found by simply multiplying U s by the constant K. Thus the transfer function relating h V and U s is a constant or a gain. We can represent this in block-diagram form as shown in Fig. . Differentiation with Respect to Time Consider what happens when the time derivative of a function formed. is trans- ịỵ - f co dt Jo 47 e- dt dt Integrating by parts gives u dv d4d dt du -se st dt Therefore Jo dl v l x h 0 sye st dt Jo f X y t e sldt 0 0 - j i 0 s The integral is by definition just the transformation of which we call ịl dt 5y J - F 0 The result is the most useful of all the transformations. It says that the operation of differentiation in the time domain is .