Linearization and Laplace Transformation

Since we are only interested in changes of Fout as a result of changes to changes in the inputs Tsteam and Ti will not be considered. Linearization of the energy balance, Eqa (15.6) results then in  

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Consider the coupled hydraulic tanks depicted in Fig. 4.7. The nonlinear characteristic of the valves is given by Bernoulli s well-known square root law. It is assumed that the constitutive equations of the valves have been linearised around an operating point so that the model equations are linear and Laplace transform can be applied. 

As we did in the membrane, again, we linearize and Laplace transform equation (8.61) to get the electrode impedance. 

After linearization and Laplace transformation, the following transfer function for the vapor flow as a function of the load can be found ... 

The overall transfer function between cc and c, can easily be obtained through linearization and Laplace transformation of Eqn. (12.27) ... 

The measured output variable is the feedstream temperature [7]. Using classical methods (i.e., deviation variables, linearization, and Laplace transforms) theoverail closed-loop transfer function for the control system is given by... 

These equations are perturbed, linearized, and Laplace-transformed from the time domain to the frequency domain to evaluate the transfer functions between various thermal-hydraulic parameters. The Laplace-transformed equations are solved simultaneously by means of a matrix equation. 

The linearized and Laplace-transformed equations of the models described above are used to evaluate the various system transfer functions as functions of the Laplace variables s = cr + jco, where a is the real part and co is the imaginary part of the complex variable s. a refers to the damping constant (or damped exponential frequency) and co refers to the resonant oscillation frequency of the system. 

Linearization and Laplace-transformation of (5.93) and (5.95) give the next equations. 

Fourier and Laplace transforms are linear transforms and are very often used for analyzing problems in various branches of science and engineering. Since receptivity is studied with respect to onset of instability, it is quite natural that these transform techniques will be the tool of choice for such studies. Fourier transform provides an approach wherein the differential equation of a time dependent system is solved in the transformed plane as. 

First order series/parallel chemical reactions and process control models are usually represented by a linear system of coupled ordinary differential equations (ODEs). Single first order equations can be integrated by classical methods (Rice and Do, 1995). However, solving more than two coupled ODEs by hand is difficult and often involves tedious algebra. In this chapter, we describe how one can arrive at the analytical solution for linear first order ODEs using Maple, the matrix exponential, and Laplace transformations. 

The problem posed by (3-119) with the boundary and initial conditions, (3-120b), is very simple to solve by either Fourier or Laplace transform methods.14 Further, because it is linear, an exact solution is possible, and nondimensionalization need not play a significant role in the solution process. Nevertheless, we pursue the solution by use of a so-called similarity transformation, whose existence is suggested by an attempt to nondimensionalize the equation and boundary conditions. Although it may seem redundant to introduce a new solution technique when standard transform methods could be used, the use of similarity transformations is not limited to linear problems (as are the Fourier and Laplace transform methods), and we shall find the method to be extremely useful in the solution of certain nonlinear DEs later in this book. 

Exercise 4.6 linear differential equations and Laplace transforms... 

It is readily shown that several simple mathematical operations such as differentiation, integration, and linear transformations (scaling and translation), as well as more complex operations such as convolution, deconvolution, and Laplace transformations (and inverse Laplace transformation) have the above linear operator property. 

Even though this equation is linear, it can still be written in terms of deviation variables and Laplace transformed, and since SFout 0, it can be written as ... 

For decades, the subject of control theory has been taught using transfer functions, frequency-domain analysis, and Laplace transform mathematics. For linear systems (like those from the electromechanical areas from which these classical control techniques emerged) this approach is well suited. As an approach to the control of chemical processes, which are often characterized by nonlinearity and large doses of dead time, classical control techniques have some limitations. 

Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater Adv. Catal., 1.3, 203 [1962]). Reactions of petroleum fractions are examples of this type. 

Differential equations and their solutions will be stated for the elementary models with the main lands of inputs. Since the ODEs are linear, solutions by Laplace transforms are feasible. 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

Application of the definition shows that the Laplace transform is a linear oper-ator " this property is represented in Eqs. (3-67) and (3-68). 

The preceding two equations are examples of linear differential equations with constant coefficients and their solutions are often found most simply by the use of Laplace transforms [1]. 

Equation (15.34) is the system model. It is a linear PDE with constant coefficients and can be converted to an ODE by Laplace transformation. Define... 

While the simple linear models in the previous sections have been solved by straightforward integration, the present model (and more complicated ones) are more conveniently solved by means of the Laplace transform. 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

Complex systems can often be represented by linear time-dependent differential equations. These can conveniently be converted to algebraic form using Laplace transformation and have found use in the analysis of dynamic systems (e.g., Coughanowr and Koppel, 1965, Stephanopolous, 1984 and Luyben, 1990). 

Classical process control builds on linear ordinary differential equations and the technique of Laplace transform. This is a topic that we no doubt have come across in an introductory course on differential equations—like two years ago Yes, we easily have forgotten the details. We will try to refresh the material necessary to solve control problems. Other details and steps will be skipped. We can always refer back to our old textbook if we want to answer long forgotten but not urgent questions. 

Let us first state a few important points about the application of Laplace transform in solving differential equations (Fig. 2.1). After we have formulated a model in terms of a linear or linearized differential equation, dy/dt = f(y), we can solve for y(t). Alternatively, we can transform the equation into an algebraic problem as represented by the function G(s) in the Laplace domain and solve for Y(s). The time domain solution y(t) can be obtained with an inverse transform, but we rarely do so in control analysis. 

An important property of the Laplace transform is that it is a linear operator, and contribution of individual terms can simply be added together (superimposed) ... 

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

Hence, we can view the transfer function as how the Laplace transform of the state transition matrix O mediates the input B and the output C matrices. We may wonder how this output equation is tied to the matrix A. With linear algebra, we can rewrite the definition of O in Eq. (4-5) as... 

The linear viscoelastic behavior of liquid and solid materials in general is often defined by the relaxation time spectrum 11(1) [10], which will be abbreviated as spectrum in the following. The transient part of the relaxation modulus as used above is the Laplace transform of the relaxation time spectrum H(l)... 

This equation is linear first-order and may be solved in a variety of fashions. One may use an integrating factor approach, Laplace transforms, or rearrange the equation and obtain the sum of the homogeneous and particular solutions. The solution is... 

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