Stability of dynamic systems

From equations (5.1)-(5.4), it ean be seen that the stability of a dynamie system depends upon the sign of the exponential index in the time response funetion, whieh is in faet a real root of the eharaeteristie equation as explained in seetion 5.1.1. 

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Before we can demonstrate the connection between process control and Eq. (A.20), we need to introduce the concept of Lyapunov functions (Schultz and Melsa. 1967). Lyapunov functions wnre originally designed to study the stability of dynamic systems. A Lyapunov function is a positive scalar that depends upon the system s state. In addition, a Lyapunov function has a negative time derivative indicative of the system s drive toward its stable operating point where the Lyapunov function becomes zero. Mathematically we can describe these conditions as... 

A. M. Lyapunov (1857 1918) was an outstanding Russian mathematician who created the theory of stability of dynamic systems. 

M. Shub. Global stability of dynamical systems. With the collab. of Albert Fathi and Remi Langevin. Transl. from the FYench by Joseph Christy. New York etc. Springer-Verlag, 150 p., 1987. 

Calculations of mutual locations of poles and zeros for these TF models allow to trace dynamics of moving of the parameters (poles and zeros) under increasing loads. Their location regarding to the unit circle could be used for prediction of stability of the system (material behavior) or the process stationary state (absence of AE burst ) [7]. 

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

The important point is that the dynamics and stability of the system are governed by the closed-loop characteristic polynomial ... 

The closed-loop system is stable if all the roots of the characteristic polynomial have negative real parts. Or we can say that all the poles of the closed-loop transfer function he in the left-hand plane (LHP). When we make this statement, the stability of the system is defined entirely on the inherent dynamics of the system, and not on the input functions, fn other words, the results apply to both servo and regulating problems. 

Oligomerization of nucleobases can be advantageous to reinforce the H-bonding supramolecular motifs when supramacromolecular polymers are desired. Moreover the different interconverting outputs that may form by oligomerization define a dynamic polyfunctional diversity which may be extracted selectively under the intrinsic stability of the system or by interaction with external factors by polymerization in the solid state. 

The locations of the zeros of the transfer function have no effect on the stability of the system They certainly affect the dynamic response, but they do not affect stability. 

Such was the state of the art when Amundson and Bilous s paper was published in the first volume of the newly founded A.I.CH.E. Journal (Bilous and Amundson, 1955). This for the first time treated the reactor as a dynamical system and, using Lyapounov s method of linearization, gave a pair of algebraic conditions for local stability. One of these corresponded to the slope condition of previous analyses, and there was a brief flurry of attempts to invest the other with a similarly physical explanation. For the global picture they introduced the phase plane (another feature of the theory of dynamical systems) and, with consummate skill, Bilous conjured the now classic figures from a Reeves electronic analogue computer. Even in this early paper, they had touched upon the consecutive reaction scheme A - B - C and had shown that up to five steady states might be expected under some conditions. 

The Relationship between Entropy and Dynamic Stability of a System 301... 

THE RELATIONSHIP BETWEEN ENTROPY AND DYNAMIC STABILITY OF A SYSTEM... 

The concept of stability is central in the study of dynamical systems. Loosely speaking, stability is a dynamical system s property related to good long-run behavior of that system. Although stability by itself may not necessarily guarantee satisfactory performance of a dynamical system, it is not conceivable that a dynamical system may perform well without being stable. 

In the preceding sections, the possible rest points for the gradostat equations were determined and their stability analyzed. The problem that remains is to determine the global behavior of trajectories. In this regard, the theory of dynamical systems plays an important role. First of all, some information can be obtained from the general theorem on inequalities discussed in Appendix B. We illustrate this with an application to the gradostat equations. 

In the previous chapter the gradostat was introduced as a model of competition along a nutrient gradient. The case of two competitors and two vessels with Michaelis-Menten uptake functions was explored in considerable detail. In this chapter the restriction to two vessels and to Michaelis-Menten uptake will be removed, and a much more general version of the gradostat will be introduced. The results in the previous chapter were obtained by a mixture of dynamical systems techniques and specific computations that established the uniqueness and stability of the coexistence rest point. When the number of vessels is increased and the restriction to Michaelis-Menten uptake functions is relaxed, these computations are inconclusive. It turns out that unstable positive rest points are possible and that non-uniqueness of the coexistence rest point cannot be excluded. The main result of this chapter is that coexistence of two microbial populations in a gradostat is possible in the sense that the concentration of each population in each vessel approaches a positive equilibrium value. The main difference with the previous chapter is that we cannot exclude the possibility of more than one coexistence rest point. 

The procedure used for the evaluation of the performances is composed of two steps. First, the seven initial days of dry weather proposed in BSMl are repeated until stabilization of the system is achieved. The system is considered as stabilized when the relative differences of all states between the beginning and the end of the week are inside a given tolerance. Then, the seven days of rain weather proposed in BSMl are used for the evaluation of the performance index. In fact, the first step of this procedure performs the stabilization of the plant with a dynamic influent, while the second one performs the evaluation of the performances when facing an important disturbance. 

In addition to computer simulations, what drives the research in this direction is elaborated perturbation theories developed almost simultaneously. In particular, the Kolmogorov-Arnold-Moser (KAM) theorem, which has shown the existence of invariant tori under a small perturbation to completely inte-grable systems, and the Nekhoroshev theorem, which has proved exponentially long-time stability of trajectories close to completely integrable ones, are landmarks in this field. Although a lot of works have been done, there still remain unsolved important questions, and the Hamiltonian system is being studied as one of important branches in the theory of dynamical systems [3-5]. 

For such supramolecular systems, diffusion NMR may be extremely important for the determination of the structure and dynamics of the system. Diffusion is also extremely important in determining the association constant in systems, which on their formation induce a small change in the NMR spectra, and in systems in which chemical shift changes occur for reasons other than complexation, for example when protonation takes place. Diffusion NMR can also easily be used to probe the kinetic stability of multicomponent systems just by monitoring the effect of a small excess of one of the components on the diffusion coefficient of the supramolecular system. 

Another contemporary and noteworthy review article by Koper follows yet another concept. Koper first stresses the importance of the electric circuit by evaluating, in a rigorous way, the stability of electrochemical systems by frequency response methods. He then thoroughly discusses the dynamics of selected examples, including some semiconductor systems, which are not included in this chapter, with special emphasis on how they relate to the frequency response theory. 

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