Difference equation characteristic root

The oniy difference between the roots given in equation (5.9) and those in equation (5.i0) is the sign of the reai part. If the real part cr is negative then the system is stabie, but if it is positive, the system wiii be unstabie. This iioids true for systems of any order, so in generai it can be stated If any of the roots of the characteristic equation have positive reai parts, then the system wiii be unstabie . 

Suppose that the two feedback controllers Gci and Gc2 are tuned separately (i.e., keeping the loop under tuning closed, and the other open). Then we cannot guarantee stability for the overall control system, where both loops are closed. The reason is simple Tuning each loop separately, we force the roots of the characteristic equations (24.12) for the individual loops to acquire negative real parts. But the roots of these equations are different from the roots of the characteristic equation (24.13), which determines the stability of the overall system with both loops closed. 

In solving ODE, we assumed the existence of solutions of the form y = A exp(/%) where r is the characteristic root, obtainable from the characteristic equation. In a similar manner, we assume that linear, homogeneous finite difference equations have solutions of the form, for example, in the previous extraction problem... 

Table 5.1 Roots of second-order characteristic equation for different values of K...

The usual steadystate performance specification is zero steadystate error. We will show below that this steadystate performance depends on both the system (process and controller) and the type of disturbance. This is different from the question of stability of the system which, as we have previously shown, is only a function of the system (roots of the characteristic equation) and does not depend on the input. 

At this point it might be useful to pull together some of the concepts that you have waded through in the last several chapters. We now know how to look at and think about dynamics in three languages time (English), Laplace (Russian) and frequency (Chinese). For example, a third-order, underdamped system would have the time-domain step responses sketched in Fig. 14.10 for two different values of the real TOOt. In the Laplace domain, the system is represented by a transfer function or by plotting the poles of the transfer function (the roots of the system s characteristic equation) in the s plane, as shown in Fig. 14.10. In the frequency domain, the system could be represented by a Bode plot of... 

Rhee et al. developed a theory of displacement chromatography based on the mathematical theory of systems of quasi-linear partial differential equations and on the use of the characteristic method to solve these equations [10]. The h- transform is basically an eqmvalent theory, developed from a different point of view and more by definitions [9]. It is derived for the stoichiometric exchemge of ad-sorbable species e.g., ion exchange), but as we have discussed, it can be applied as well to multicomponent systems with competitive Langmuir isotherms by introducing a fictitious species. Since the theory of Rhee et al. [10] is based on the use of the characteristics and the shock theories, its results are comprehensive e.g., the characteristics of the components that are missing locally are supplied directly by this theory, while in the /i-transform they are obtained as trivial roots, given by rules and definitions. 

Typical DHa6T OTFT I-V characteristics are reported in figure 3. The curves are relevant to OTFT of different channel length, namely 1 and 0.2 mm. The square root of I vs. Vg curves are reported as inset of the figures as well (figures 3a and 3b insets). The devices were tested as p-channel materials and the field effect mobilities in saturation regimes were extracted using the well-know equation [10] ... 

The Newton-Raphson method for the solution of n equations in n unknowns takes the form given by Eq. (15-3). In the application of this method, it is recommended that the convergence characteristics be checked by solving a wide variety for examples. The use of different initial sets of values for the variables should also be investigated. Also, if only positive roots of the functions are desired, provisions should be made for an alternative selection of variables for the next trial when one or more negative values are computed by an intermediate trial. 

The Nyquist stability criterion developed in Chapter 11 can be directly applied to multivariable processes. As you should recall, the procedure is based on a complex variable theorem that says that the difference between the number of zeros and poles of a function inside a closed contour can be found by plotting the function and looking at the number of times it encircles the origin. We can use this theorem to find out if the closedloop characteristic equation has any roots or zeros in the right half of the s plane. The s variable follows a closed contour that completely surrounds the entire right half of the 5 plane. Since the closedloop characteristic equation is given in Eq. (I 2.36), the function of interest is... 

Let us consider different particular forms of relation (5.272). If the surfactant in the solution exists mainly in form of micelles, and using the approximation (5.263) for the roots of characteristic equation (5.262), we obtain... 

A characteristic square-root dependence is frequently observed between the spatial extent of a transient diffusion process and the time elapsed, for example, 8 /Dt, where 5 is a measure of the spatial extent of the diffusion process. This equation can be used as a helpful way to roughly estimate the extent of progress of many different types of transient diffusions processes in materials as a function of time. 

Elimination of [R ] by means of equation 54 leads to an expression for the kinetic chain length v that shows the dependence of the different kinetic parameters. One important characteristic of the free radical polymerization is hereby well illustrated The sizes of the macromolecules produced are inversely proportional to the square root of initiator concentration. Increasing the initiator concentration leads to smaller size polymer molecules. 

The theoretical prediction is supported by the experiments. Patterns that spontaneously form from the uniform state have multiple domains with different characteristic angles. The root-mean-square width A rms of the observed angular distribution function changes with the bifurcation parameter in qualitative accord with theory compare Figure 8b with Figure 8a [13]. A quantitative comparison of experiment and theory would require an evaluation of the coefficients in the Landau-Ginzburg equation from the chemical kinetics and diffusion coefficients of the reactants [47]. 

In certain problems it may be necessary to locate all the roots of the equation, including the complex roots. This is the case in finding the zeros and poles of transfer functions in process control applications and in formulating the analytical solution of linear nth-order differential equations. On the other hand, different problems may require the location of only one of the roots. For example, in the solution of the equation of state, the positive real root is the one of interest. In any case, the physical constraints of the problem may dictate the feasible region of search where only a subset of the total number of roots may be indicated. In addition, the physical characteristics of ihe problem may provide an approximate value of the desired root. 

If geometrically similar cyclones or swirl tubes of different sizes are operated at the same inlet velocity, Vrcs and vecs will also be similar. The equation therefore shows that the cut size is roughly proportional to the square root of the vortex finder diameter. Thus, in geometrically similar cyclones, the cut size will be proportional to the square root of the characteristic cyclone dimension, say D. Incidentally, since vecs and Vrcs are proportional to the inlet and outlet velocities, it can be also observed from inspection of Elquation (5.2.1) that the cut size for geometrically similar cyclones is inversely proportional to the square root of any characteristic velocity such as the gas superficial inlet or outlet velocity. 

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