A sufficient condition for stability

Linearized or asymptotic stability analysis examines the stability of a steady state to small perturbations from that state. For example, when heat generation is greater than heat removal (as at points A— and B+ in Fig. 19-4), the temperature will rise until the next stable steady-state temperature is reached (for A— it is A, for B+ it is C). In contrast, when heat generation is less than heat removal (as at points A+ and B— in Fig. 19-4), the temperature will fall to the next-lower stable steady-state temperature (for A+ and B— it is A). A similar analysis can be done around steady-state C, and the result indicates that A and C are stable steady states since small perturbations from the vicinity of these return the system to the corresponding stable points. Point B is an unstable steady state, since a small perturbation moves the system away to either A or C, depending on the direction of the perturbation. Similarly, at conditions where a unique steady state exists, this steady state is always stable for the adiabatic CSTR. Hence, for the adiabatic CSTR considered in Fig. 19-4, the slope condition dQH/dT > dQG/dT is a necessary and sufficient condition for asymptotic stability of a steady state. In general (e.g., for an externally cooled CSTR), however, the slope condition is a necessary but not a sufficient condition for stability i.e., violation of this condition leads to asymptotic instability, but its satisfaction does not ensure asymptotic stability. For example, in select reactor systems even... 

Because instability occurs when a > 0, it follows that a sufficient condition for stability is... 

We wish to investigate stability with / = 0. If fQ has a positive time integral for every f(t), then stability is assured. Further, if instability exists, then the system will become neutrally stable when one relation is satisfied among the parameters of the system (jS,Q, ). Neutral stability implies the existence of a sinusoidal motion with / = 0. If, then, it can be shown that (time average of fQ) is positive for all frequencies of variation w, and all values of and jS, then stability follows. Note that this is only a sufficient condition for stability. The necessary condition is the usual one on the location of the poles of the Fourier transform of (24). 

It is of interest that (28) remains a sufficient condition for stability even when the amplitude of the motion becomes too large to approximate by 1 + Q. The following proof, which is physical, but clumsy and unrigorous, is taken from [1]. 

For systems with multiple (o or Wg, the Bode stability criterion has been modified by Hahn et al. (2001) to provide a sufficient condition for stability. 

If (a) above is satisfied, then the necessary and sufficient condition for stability is either... 

Equation (8.29) provides no guarantee of stability. It is a necessary condition for stability that is imposed by the discretization scheme. Practical experience indicates that it is usually a sufficient condition as well, but exceptions exist when reaction rates (or heat-generation rates) become very high, as in regions near thermal runaway. There is a second, physical stability criterion that prevents excessively large changes in concentration or temperature. For example. An, the calculated change in the concentration of a component that is consumed by the reaction, must be smaller than a itself Thus, there are two stability conditions imposed on Az numerical stability and physical stability. Violations of either stability criterion are usually easy to detect. The calculation blows up. Example 8.8 shows what happens when the numerical stability limit is violated. 

The book includes many good examples illustrating the practical use of general stability theory with regard to particular schemes to assist the users in subsequent implementations. Stability is probably the most pressing problem in any algorithm, since it is a necessary rather than a sufficient condition for accuracy. 

If the characteristic polynomial passes the coefficient test, we then construct the Routh array to find the necessary and sufficient conditions for stability. This is one of the few classical techniques that we do not emphasize and the general formula is omitted. The array construction up to a fourth order polynomial is used to illustrate the concept. 

Huckel s 4n+2 //-electron rule is a necessary but not a sufficient condition for aromaticity. Coplanarity and electronegativity restrictions of constituent atoms represent the most important restrictions. Phos-phole is a marginally aromatic five-membered heterocycle76 (see further examples and discussion). Mesoionic compounds, mesomeric betaines, and 2H-and 4i+-pyrone have all been considered to be weakly aromatic or non-aromatic, though their conjugated acids are aromatic. Spectroscopic data evidenced the aromaticity of dioxolium and oxathiolium cations 59 (Scheme 28) and mesoionic oxathioles not in the classical sense but by their ring currents and chemical stability.77... 

This is a necessary but not sufficient condition for stability, as we will see in more detail in Chap. 10. 

The method is a necessary but not sufficient condition for stability of a closedloop system with integral action If the index is negative, the system will be unstable for any controller settings (this is called integral instability ). If the... 

Based on Eqs. (22.18), (22.21), and (22.25), a sufficient condition for the existence of a robust controller which stabilizes all perturbed plants with desired performance, subject to some uncertainty bound, can be obtained as follows ... 

From these experiments it would appear that the presence of a free aldehyde group is a necessary but not a sufficient condition for the formation of diamides in the above reactions. Once formed, the aldehyde group must be stabilized long enough to allow it to react with one or more amide molecules, the latter arising from the ammonolysis of the ester groups present in the acylated sugar. 

Knowledge of the expressions for the chemical potentials of each of the components allows theoretical prediction of the critical concentration boundaries of the phase diagram for ternary solutions of biopolymeri + biopolymer2 + solvent. According to Prigogine and Defay (1954), a sufficient condition for material stability of this multicomponent system in relation to phase separation at constant temperature and pressure is the following set of inequalities for all the components of the system ... 

Equation (3.20) implies that the system will be thermodynamically stable if the addition of an infinitely small amount of any component leads to a decrease in chemical potentials of all the other constituent components. The fulfilment of the second inequality in equation (3.20) is a sufficient condition for the stability of the multicomponent system with respect to mutual diffusion. 

All of the photochemical cycloaddition reactions of the stilbenes are presumed to occur via excited state ir-ir type complexes (excimers, exciplexes, or excited charge-transfer complexes). Both the ground state and excited state complexes of t-1 are more stable than expected on the basis of redox potentials and singlet energy. Exciplex formation helps overcome the entropic problems associated with a bimolecular cycloaddition process and predetermines the adduct stereochemistry. Formation of an excited state complex is a necessary, but not a sufficient condition for cycloaddition. In fact, increased exciplex stability can result in decreased quantum yields for cycloaddition, due to an increased barrier for covalent bond formation (Fig. 2). The cycloaddition reactions of t-1 proceed with complete retention of stilbene and alkene photochemistry, indicative of either a concerted or short-lived singlet biradical mechanism. The observation of acyclic adduct formation in the reactions of It with nonconjugated dienes supports the biradical mechanism. 

Condition 1 The necessary and sufficient condition for stability of a single steady state is B < B. ... 

For the low-temperature steady state y in Figure 4 (A-2) a similar analysis shows that this steady state is stable as well. However, for the intermediate steady-state temperature yi and Sy > 0 the heat generation is larger than the heat removal and therefore the system will heat up and move away from y2. On the other hand, if 5y < 0 then the heat removal exceeds the heat generation and thus the system will cool down away from 2/2 -We conclude that yi is an unstable steady state. For 2/2, computing the eigenvalues of the linearized dynamic model is not necessary since any violation of a necessary condition for stability is sufficient for instability. 

For to prove instability, we only have to show some circumstance, no matter how special, under which there will be a tendency to fly away from the steady state. To put it in another way, we can say that Eq. (7.5.1) is a sufficient condition for instability, while its converse, dUrldT < dUJdT is a necessary, but not sufficient, condition for stability. 

The complete set of necessary and sufficient conditions for stability, as first given by Amundson in 1955, is derived in a rather different way. The basic idea is to focus attention on small perturbations away from a given steady state. If they are sufficiently small, they can be described by linear equations and we shall be able to see just how they grow or die away. It can be proved that this establishes local stability, in the sense that sufficiently small perturbations will certainly die out. It does not say anything... 

A quasi-stationary state is stable if a small excursion from it is self-correcting, but is unstable if the excursion escalates. Specifically, in a system as described here, stability is ensured if the net rates rx and rY decrease if the concentrations of X and Y increase. If this is true for one of the intermediates, but not for the other, the stabilizing and destabilizing tendencies counteract one another, and the (necessary and sufficient) condition for stability becomes... 

---