Lecture03过程控制讲义3.ppt

Laplace Transforms Laplace Transforms of Common Functions Table 3.1. Laplace Transforms See page 54 of the text. * Chapter 3 Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important process control concepts and techniques. - Examples: Transfer functions Frequency response Control system design Stability analysis Definition The Laplace transform of a function, f(t), is defined as where F(s) is the symbol for the Laplace transform, L is the Laplace transform operator, and f(t) is some function of time, t. Note: The L operator transforms a time domain function f(t) into an s domain function, F(s). s is a complex variable: s = a + bj, Inverse Laplace Transform, L-1: By definition, the inverse Laplace transform operator, L-1, converts an s-domain function back to the corresponding time domain function: Important Properties: Both L and L-1 are linear operators. Thus, where: - x(t) and y(t) are arbitrary functions - a and b are constants - Similarly, Constant Function Let f(t) = a (a constant). Then from the definition of the Laplace transform in (3-1), Step Function The unit step function is widely used in the analysis of process control problems. It is defined as: Because the step function is a special case of a “constant”, it follows from (3-4) that Derivatives This is a very important transform because derivatives appear in the ODEs we wish to solve. In the text (p.53), it is shown that initial condition at t = 0 Similarly, for higher order derivatives: where: - n is an arbitrary positive integer - Special Case: All Initial Conditions are Zero Suppose Then In process control problems, we usually assume zero initial conditions. Reason: This corresponds to the nominal steady state when “deviation variables” are used, as shown in Ch. 4. Exponential Functions Consider where b 0. Then, Recta