Laplace transform, identifiability

G( ) is the transfer function relating 0O and 0X. It can be seen from equation 7.18 that the use of deviation variables is not only physically relevant but also eliminates the necessity of considering initial conditions. Equation 7.19 is typical of transfer functions of first order systems in that the numerator consists of a constant and the denominator a first order polynomial in the Laplace transform parameter s. The numerator represents the steady-state relationship between the input 0O and the output 0 of the system and is termed the system steady-state gain. In this case the steady-state gain is unity as, in the steady state, the input and output are the same both physically and dimensionally (equation 7.16h). Note that the constant term in the denominator of G( ) must be written as unity in order to identify the coefficient of s as the system time constant and the numerator as the system... 

To connect the equations in (B.l) through the Fourier-Laplace transform, we need to define suitable complex contours to make the transforms convergent. Specifically we identify the contours C by the lines in upper and lower complex planes defined by CU ( id — oo — id + oo), where d > 0 may be arbitrary. Using the Heaviside function, 0(f), and the Dirac delta function, 5(f), we can characterize positive and negative times (with respect to f = 0) as linked with appropriate contours C as... 

This has exactly the same form as the Laplace transform of the GLE in Eq. (11.71), and the different terms may be identified according to... 

Though the preceding equations represent a convenient way to express the relationship between stress and strain, it is necessary that they be consistent with the integral formulation of Section 16.2 (6). Consequently, taking the Laplace transform of both formulations and identifying them, we obtain... 

T (t) given by eq. (8.4) is the solution to our initial differential equation (5.3). Indeed, taking the Laplace of eq. (8.4), it yields eq. (8.3). The procedure by which we find the time function when its Laplace transform is known is called the inverse Laplace transformation and is the most critical step while solving linear differential equations using Laplace transforms. To summarize the solution procedure described in the example above, we can identify the following steps ... 

Part III (Chapters 6 through 12) is devoted to the analysis of static and dynamic behavior of processing systems. The emphasis here is on identifying those process characteristics which shape the dynamic response for a variety of processing units. The results of such analysis are used later to design effective controllers. Input-output models have been employed through the use of Laplace transforms. 

This method is simple in theory and is the most widely used, although it becomes quite cumbersome with large models. First we note that if a linear model is identifiable with some input in an experiment, it is identifiable from impulsive inputs into the same compartments. That allows one to use impulsive inputs in checking identifiability even if the actual input in the experiment is not an impulse. Take Laplace transforms of the system differential equations and solve the resulting algebraic equations for the transforms of the state variables. Then write the Laplace transform for the observation function (response function). That will be of the form... 

Here we need the convolution theorem which identifies the inverse Laplace transform of a product as being a convolution ... 

More complex equations can be used to identify higher order models. For example Equation (2.23) was developed by applying a z-transform to the Laplace form of a second order process (with lags tj and X2, and lead x ) as shown in Equation (2.68). Lead is required if there is PV overshoot (xj, > 0) or inverse response (T3 < 0). 

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