Lyapunov’s function

The method of Lyapunov s function consists of constructing a function V(x,y) with the following properties ... 

Other types of Lyapunov s functions may also be constructed. 

For the general treatment in regard to Lyapunov s function, the reader may see Khalil (1996). 

In the framework of such an approach Lyapunov s function is used as a criterion, for the selection of steps by their significances ... 

This condition can be rewritten in terms of Ruelle s function defined as the generating function of the Lyapunov exponents and their statistical moments ... 

Conditions (66) and (67) ensure the existence of Lyapunov s convex function for eqns. (17) GGjdNi = fit. With a known type of the potentials /i, for which condition (1) is fulfilled, one can obtain Lyapunov s thermodynamic functions for various (including non-isothermal) conditions. Thus, for an ideal gas and the law of mass action [16]... 

In the case of nonlinear systems that cannot be reduced to a linearized system, stability is much more difficult to assess. Lyapunov s direct method [1] requires a suitable energy function to be found. Often, only numerical time integration gives an indication of the dynamic behaviour and stability that cannot be proven otherwise. 

Dynamic nonlinear analysis techniques (Isidori 1995) are not directly applicable to DAE models but they should be transformed into nonlinear input-affine state-space model form by possibly substimting the algebraic equations into the differential ones. There are two possible approaches for nonlinear stability analysis Lyapunov s direct method (using an appropriate Lyapunov-function candidate) or local asymptotic stability analysis using the linearized system model. 

It is convenient to formulate the problem so that its solution would not change with smooth changes of variables. For example, let us determine the limits for In s)lt and In 0 (t, e)/t at t - 00 (i.e the Lyapunov indices for these functions). For smooth rough two-dimensional systems, if e is sufficiently small we will obtain... 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

Figure 12.2. Stability at far from equilibrium. Due to the two terms, deSand d,S, the second law does not impose the sign of entropy variation dS= d,S+ deS. Therefore, there is no universal Lyapunov function, which creates ambiguity in the stability of states far...

The second law for isolated systems shows that the excess entropy, A.V S SKI < 0, increases monotonically in time, d(AS)/dt > 0. Therefore, it plays the role of a Lyapunov function, and defines a global stability. So, dfi/dt is a Lyapunov function that guarantees the global stability of stationary states that are close to global equilibrium. 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

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... 

Hence, the system s dynamics is fully relaxational, i.e. the evolution of the system is characterized by a monotonic decrease of the Lyapunov functional and the approach to a final stationary state, which excludes the possibility of oscillatory instabilities, spatio-temporal chaos etc. 

The term to the right of the equal sign in the equation above is the excess entropy production. The term 8 S is a Lyapunov functional for a stationary state. 

Since a definite function 6 S leads to the stability condition, it operates as a Lyapunov function and assures the stability of a stationary state. As the entropy production is the sum of the products of flows J and forces X, we... 

A function L satisfying the equation above is called a Lyapunov function. The second variation of entropy L = —d S may be used as a Lyapunov functional if the stationary state satisfies dKidf > 0. A functional is a set of functions that are mapped to a real or complex value. Hence, a nonequilibrium stationary state is stable if... 

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