Lyapunov stability

Model reference adaptive control is based on a Lyapunov stability approach, while the hyperstability method uses Popov stability analysis. All of the above methods have been tested on experimental systems, both SISO and MIMO (53), (54), (55). The selftuning regulator is now available as a commercial software package, although this method is not satisfactory for variable time delays, an important industrial problem. 

Definition of Lyapunov stability Consider a reference solution with given initial data at some time to] define a second solution, which is obtained by slightly varying the initial data. The reference solution is called stable, if for any t > to the distance between the two solutions can be made smaller than e by an appropriate choice of the variations of the initial conditions. 

There are various definitions of stability which describe how a dynamic system being at an equilibrium reacts on a small disturbance. In machining science the Lyapunov stability and the asymptotic stability are of high interest. Consider the continuous dynamic system ... 

For the analysis of local Lyapunov stability of rest points, a traditional and very reliable ritual exists, which in fact is an extension of the approach demonstrated in Section 3.6.2. 

Leaving aside individual reactions, mention will be made here only of one of the methods commonly used at present. This method involves elimination of time from the kinetic equations and attempts at finding stable solutions in terms of the Lyapunov stability theory. In the simple case of two variables X and Y (e.g. of two active centers, or of one active center and temperature), from the kinetic equations dx/dt = y) dy/dt = y(x, y) (x and y are either... 

Box 18.1 Kinetic Equations and Lyapunov Stability Theory An Example... 

To prove Lyapunov stability let us surround the point O by a sphere 5 of radius e. Let > 0 be the minimum of the function V (x) on the surface of the sphere (it is strictly positive because all points of the sphere lie at a finite distance from the origin). Since V is continuous and V (O) = 0, it follows that for any point xq chosen sufficiently close to O the value of the function V (x) is strictly less than V. ... 

In his paper On Governors , Maxwell (1868) developed the differential equations for a governor, linearized about an equilibrium point, and demonstrated that stability of the system depended upon the roots of a eharaeteristie equation having negative real parts. The problem of identifying stability eriteria for linear systems was studied by Hurwitz (1875) and Routh (1905). This was extended to eonsider the stability of nonlinear systems by a Russian mathematieian Lyapunov (1893). The essential mathematieal framework for theoretieal analysis was developed by Laplaee (1749-1827) and Fourier (1758-1830). 

Proof. By analyzing the steady-state behavior of AD model (11), four possible equilibrium points are obtained (see Table 1). However, it is observed that only the point P4 has physical meaning under NOC. This means that under NOC, AD model (11) has a single equilibrium point P4, which depends on the process kinetics and the influent composition. Now, in order to evaluate the stability of the internal d mamics of AD model (11), the following candidate Lyapunov function (CLP) is proposed... 

Until further notice we use stable for what is technically called asymptotically stable in the sense of Lyapunov see, e.g., J. La Salle and S. Lefshetz, Stability by Liapunov s Direct Method (Academic Press, New York 1961). 

Show that F(0) is a Lyapunov function ), which can be used to prove the global stability. 

The stability matrix carries the necessary information related to the vicinity of the trajectory and provides an efficient numerical procedure for computing the response function. It plays an important role in the field of classical chaos the sign of its eigenvalues (related to the Lyapunov exponents) controls the chaotic nature of the system. Interference effects in classical response functions have a different origin than their quantum counterparts. For each initial phase-space point we need to launch two trajectories with very close initial conditions. [For 5(n) we need n trajectories.] The nonlinear response is obtained by adding the contributions of these trajectories and letting them interfere. 

For periodic orbits that undergo a bifurcation, some Lyapunov exponents may vanish so that the orbit becomes of neutral linear stability in the critical directions [32]. In such cases, the dynamics transverse to the periodic orbit... 

An analysis of chemical reactor stability and control-VIII The direct method of Lyapunov. Introduction and applications to simple reactions in stirred vessels (with R.B. Warden and N.R. Amundson). Chem. Eng. ScL 19, 149-172 (1963). 

J.M. (1996) A new method for assessing the thermal stability of semi-batch processes based on Lyapunov exponents. Chemical Engineering Science, 51 (11), 3089-96. 

Strozzi, F., Alos, M.A. and Zaldivar, J.M. (1994) A method for assessing thermal stability of batch reactors by sensitivity calculation based on Lyapunov exponents experimental verification. Chemical Engineering Science, 49, 5549-61. 

Since rest points are particular cases of the phase trajectories < ( , k, < ) = c0, the above definitions of stability according to Lyapunov are also valid for them. A rest point is stable according to Lyapunov if, for any e > 0 there exists values of 3 > 0 such that after a deviation from this point within 3, the system remains close to it (within the value) for a long period of time. A rest point is asymptotically stable if it is stable according to Lyapunov and there exists values of S > 0 such that after the deviation from this point within 3 the system tends to approach it at t - oo. 

Many other definitions for stability are known and they do not always look alike. Each of them characterizes a required property of the solution under study. Though the Lyapunov definition of stability seems to be the most natural, in many cases it cannot be used. No stability definition can fit every real case. Some other versions of this concept are given in ref. 20 in which further references can be found. 

So far we have defined the local stability ("there exists such <3 as. . . ). Now let us define the global stability for rest points. The rest point c0 is called globally asymptotically stable (as a whole) within the phase space D if it is stable according to Lyapunov, and for any initial conditions d0e D the solution c(t, k, cLa) tends to approach c 0 at t - oo. 

In principle, to study the local stability of a stationary point from a linear approximation is not difficult. Some difficulties are met only in those cases where the real parts of characteristic roots are equal to zero. More complicated is the study of its global stability (in the large) either in a particular preset region or throughout the whole phase space. In most cases the global stability can be proved by using the properly selected Lyapunov function (a so-called second Lyapunov method). Let us consider the function V(c) having first-order partial derivatives dY/dCf. The expression... 

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. 

The term to the right of the equal sign in Eq. (12.32) is the excess entropy production. Equations (12.31) and (12.32) describe the stability of equilibrium and nonequilibrium stationary states. The term 82S is a Lyapunov functional for a stationary state. 

Since a definite function 82S 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 have... 

A general criterion for stability of a state is given by the Lyapunov function. A physical system x may be defined by an m dimensional vector with elements of x, (/ = 1, 2,. .., m) and parameters a, and we have... 

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. 

Equations (12.27) and (12.64) show the stability of the nonequilibrium stationary states in light of the fluctuations Sev The linear regime requires P > 0 and dP/dt < 0, which are Lyapunov conditions, as the matrix (dAJdej) is negative definite at near equilibrium. 

The Lyapunov function resembles the thermodynamic entropy production function and the asymptotic stability principle. If the eigenvalues of the coefficient matrix of the quadratic form of the entropy production are very large, then the convergence to equilibrium state will be rapid. 

Linear stability analysis does not provide information on how a system will evolve when a state becomes unstable. It does not distinguish between metastable and stable states when multiple local states are possible for given boundary conditions. Boundary conditions affect the value of the Lyapunov functional, and cause changes between stable and metastable states, hence altering the relative stability. An unstable state corresponds to the saddle points of the functional and defines a barrier between the attractors. Approximate solutions of nonlinear evolution equations may help us to understand how the system will behave in time and space. 

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... 

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