Exponentially asymptotically stable

A system represented by a set of equations describing a physical system is said to be stable around a steady state if the transients of the system are bounded when the system is subjected to small perturbations from the steady state. The system is said to be asymptotically stable if it is stable and it eventually returns to the steady state. If in addition, the steady state is approached exponentially, it is called exponentially asymptotically stable. If any magnitude of perturbation is allowed, it is said to be globally stable. The stability problem as related to reactor startup and control is concerned with the following questions (Luss 1977) ... 

Here, the second subscript denotes the steady state value. The roots of the quadratic characteristic equation (eigenvalues) of the matrix A determine the stability of the equations the system will converge exponentially to the steady state if all roots have a negative real part and, therefore, is asymptotically stable. It will show a limit cycle if the roots are imaginary with zero real parts. It is unstable if any of the roots has a positive real part. Since the perturbations will decay asymptotically if and only if all the eigenvalues of the matrix A have a negative real part, it follows that the necessary and sufficient conditions for local stability are ... 

On the other hand, the main characteristic of interval observers is the use of the aforementioned cooperativity property, which must hold on the estimation error dynamics. Now, let hypotheses Hlf-g be verified. That means that, on the one hand, some bounds are now available on the initial conditions and, on the other hand, b t) is considered in the following as unmeasured, but some lower and upper bounds — possibly time varying — are known. In such a situation, notice that model (16) may not be observable. Consequently, it may not possible to design an asymptotic observer such as (19). Nevertheless, its basic exponentially stable structure and its property of being independent of the nonlinearities may be used. The main idea developed in the following... 

The asymptotic behavior of [C] was analyzed by Agmon and Gopich in the most general case of a different u and u [135,172]. It was shown that at long times P,(t) experiences the transition from the power law for stable dissociation products ( the A regime ) to an exponential decay for highly unstable products... 

Finally when a >0 (Figure 5.1.5e), x becomes unstable, due to the exponential growth in the x-direction. Most trajectories veer away from x and head out to infinity. An exception occurs if the trajectory starts on the y-axis then it walks a tightrope to the origin. In forward time, the trajectories are asymptotic to the x-axis in backward time, to the y-axis. Here x =0 is called a saddle point. The y-axis is called the stable manifold of the saddle point x, defined as the set of initial conditions Xg such that x(z) x as t -> o . Likewise, the unstable manifold of X is the set of initial conditions such that x(z) x as z. Here the unstable manifold is the x-axis. Note that a typical trajectory asymptotically approaches the unstable manifold as z —> o , and approaches the stable manifold as z —> -oo. This sounds backwards, but it s right ... 

It is interesting, however, that the instability of a system without activator may take place even if neither of its subsystems is a damped oscillator all the steady states are stable nodes. We argue that the mechanism of the instability is still of a resonant type. The heuristic arguments are as follows. When a system of at least three variables is split by a differential flow, one of the subsystems involves at least two variables. We assume that the steady state of this subsystem is a stable node. The response of such a system to a perturbation is a linear combination of at least two exponential functions [i.e. exp(—Aif) + 0 exp(—A2f) (-1-...) Aj > 0]. Although this response asymptotically decays, for certain a it may initially grow. Then its Fourier spectrum will have a maximum at a finite frequency. Therefore, the subsystem is most sensitive to a perturbation at this frequency. This can be interpreted as a resonance (although with a small quality factor). The instability is thus caused by the resonance which is induced by the differential flow. For this reason we call the instability of a system without an activator differential flow induced resonance instability . It becomes clear now why only modes with wavenumbers within a finite range [Equation (41)] may be unstable all other modes are out of resonance . 

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