Linearization and Perturbation Variables

As mentioned earlier, we must convert the rigorous nonlinear difTerential equations describing the system into linear differential equations if we are to be able to use the powerful linear mathematical techniques. 

The first question to be answered is just what is a linear differential equation. Basically it is one that contains variables only to the first power in any one term of the equatioiL If square roots, squares, exponentials, products of variables, etc., appear in the equation, it is nonlinear  

Mathematically, a linear differential equation is one for which the following two properties hold  

is a solution, then CX( is also a solution, where c is a constant. 

Linearization is quite straightforward. All we do is take the nonlinear functions, expand them in Taylor series expansions around the steadystate operating level, and neglect all terms after the first partial derivatives. 

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Since we will be using perturbation variables most of the time, we will often not bother to use the superscript p. It will be understood that whenever we write the linearized equations for the system all variables will be perturbation variables. Thus Eqs. (6.39) and (6.41) can be written... 

L2. Linearize the ODE describing the conical Utnk modeled in Prob. 3.1 and convert to perturbation variables. 

The variables can be either total or perturbation variables since the equations are linear (all Jc s and r s arc constant). Let us use perturbation variables, and therefore the initial conditions for all variables are zero. 

The Oi/s are all constants made up of the steadystate holdups, flow rates and compositions. Table 12.2 gives their values. The variables in Eqs. (12.72) to (12.79) are all perturbation variables. is the hydraulic constant, the linearized relationship between a perturbation in liquid holdup on a tray,, and the perturbation in the liquid flow rate L leaving the tray. 

K is the linearized relationship between the perturbations in vapor composition y and liquid composition x,. Note that this is not the same K value" used in VLE calculations which relates total x and y variables. The K s in Table 12.2 are the slopes of the equilibrium line and relate perturbation variables. 

These equations are linear in the variables Aa and A0 but still involve partial derivative forms. Comparing (10.28) and (10.29) with (10.26) and (10.27), we see that the perturbations are given by... 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

Dynamic matrix control (DMC) is also an MVC technique, but it uses a set of linear differential equations to describe the process. The DMC method obtains its data from process step responses and calculates the required manipulations utilizing an inverse model. Coefficients for the process dynamics are determined by process testing. During these tests, manipulated and load variables are perturbed, and the dynamic responses of all... 

Remark A.l. Condition (i) of Theorem A.l essentially means that the corresponding DAE system in Equation (2.45) has an index of two, which directly fixes the dimensions of the fast and slow variables to p and n—p, respectively. Condition (ii) of the theorem ensures that the (n — p)-dimensional slow C,-subsystem can be made independent of the singular term /e, thereby yielding the system in Equation (A.13) in the standard singularly perturbed form. While condition (n) is trivially satisfied for all linear systems and for nonlinear systems with pi = 1, it is not satisfied in general for nonlinear systems with p> > 1. 

The front is inherently unstable, however, and this is often studied by a linear stability analysis. Infinitesimal perturbations are applied to all of the variables to simulate reservoir heterogeneities, density fluctuations, and other effects. Just as in the Buckley-Leverett solution, the perturbed variables are governed by force and mass balance equations, and they can be solved for a perturbation of any given wave number. These solutions show whether the perturbation dies out or if it grows with time. Any parameter for which the perturbation grows indicates an instability. For water flooding, the rate of growth, B, obeys the proportionality... 

According to the theory of linear stability analysis, infinitesimally small perturbations are superimposed on the variables in the steady state and their transient behavior is studied. At this stage the difference between turbulent fluctuations and perturbations may be noted. Turbulence is the characteristic feature of the multiphase flow under consideration the mean and fluctuating quantities were given by Eq. (2). The fluctuating components result in eddy diffusivity of momentum, mass, and Reynolds stresses. The turbulent fluctuations do not alter the mean value. In contrast, the perturbations are superimposed on steady-state average values and another steady... 

On substituting in the Hamilton-Jacobi equation (7), 41, equation (9) would again result but in averaging subsequently over the unperturbed motion, H1(w°, J) would remain dependent on wp°. We cannot therefore apply the method without further consideration. The deeper physical reason for this is that the variables vfi, J°, with which the angle and action variables w, J of the perturbed motion are correlated, are not determined by the unperturbed motion on account of its degenerate character, other degenerate action variables, connected with the Jp° s by linear non-integral relations, could be introduced in place of the Jp° s, by a suitable choice of co-ordinates. 

That the time and frequency variables are conjugate in the same sense that the position and momentum variables are, was in principle clear in both classical and quantum mechanics. It is even possible to take the usual notion of the phase space, whose dimension equals twice the number of mechanical degrees of freedom and to add two more, those of time and energy, and this is possible also in quantum mechanics (29,30). The advantage of doing so in practice was first realized in the so called, linear response regime (31,32), where the change in the state of the system is linear in the perturbation. Spectroscopy with weak (i.e., ordinary) fields is a clear example and for reasons which we intend to discuss, the early applications were to simple molecules (typically diatomic)... 

The last term in this equation is zero. Therefore, Eqs. (2.54) and (2.46) are identical, except one is in terms of total variables and the other is in terms of perturbations. Whenever the original ODE is already linear, either total or perturbation variables can be used. Initial conditions will, of course, differ bv ihe steady-state values of all variables. ... 

Solutions of linear ODEs can also be found using the software tool MATLAB. To demonstrate this, let us consider the three-heated-tank process studied in Chapter 1. The process is described by three linear ODEs [Eqs. (1.10), (1.1 1), and ( 1.12)]. If flow rate F, volume V (assuming equal volumes in the three tanks), and physical properties p and Cp are all constants, these three equations are linear and can be converted into perturbation variables by inspection. 

Openloop process transfer function. These three ODEs are linear, so we do not have to linearize. Converting to perturbation variables, Laplace transforming, and solving for the transfer function between the controlled variable T, and the manipulated variable Q give... 

Step 4 The linearized equations are formulated in terms of perturbation variables that express the deviation from the stationary point (or steady state) x = x — x, y = y u = u — M , and d = d — d,. Substituting the perturbation variables into Eqs. (21.4) and ignoring higher-order terms ... 

We see that the new set of equations in linear and is identical to that in Section 14.3 with the exception that the Henry constant in Section 14.3 is now replaced with the slope of the adsorption isotherm at C. Thus if the moment method is applied on the perturbed variable AC at the exit of the column, we will obtain the first normalised moment and the second central moment as given in eqs. (14.3-3) and (14.3-4), respectively with K replaced by... 

Hamiltonian of orhit-lattice interaction, linear in lattice variables. The response of a paramagnetic crystal to different external perturbations (electric or magnetic field, hydrostatic pressure, uniaxial pressure), the dependence of spectra on temperature, and the spin-phonon interaction, are all determined by diffo-ent combinations of these parameters. 

Most work on the development of dynamic process models has been empirical this work is usually referred to as process identification. As mentioned earlier, two classes of empirical identification techniques are available one uses deterministic (step, pulse, etc.) functions, the other stochastic (random) identification functions. With either technique, the process is perturbed and the resulting variations of the response are measured. The relationship between the perturbing variable and the response is expressed as a transfer function. This function is the process model. Empirical identification of process models by the deterministic method has been reported by various workers [55-58]. A drawback of this method is the difficulty in obtaining a measurable response while restricting the process to a linear response (small perturbation). If the perturbation is large, the process response will be nonlinear and the representations of the process with a linear process model will be inaccurate. 

We introduce the change of variables into Equations 11.6a-c and neglect nonlinear terms in the perturbation variables 0, 4, and n. (The nonlinearities here are quadratic, but they will not be quadratic for the energy equation or for a viscoelastic constitutive equation like the PTT model.) We thus obtain the following linear equations ... 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

The isotopic difference obviously depends exclusively on the even derivatives of the perturbing potential, which implies that a linear potential, corresponding to a constant force, gives rise to no isotope effect. This may be easily understood in the following way if two functions of a variable x, one parabolic (fi(x)) and the other linear (fz x)), i.e.,... 

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