Finite-gain stability

Definition 9—Finite-gain/initial conditions stability. A system S n x = Su is finite-gain stable over the set U for initial conditions s[0] in the set S if the following inequality holds ... 

To gain a better understanding of the stability criterion, consider the explicit finite difference formulation for an interior node of a plane wall (Eq. 5 47) for the case of no heat generation,... 

This starts to increase around To and is saturated to the maximum value at the low temperature limit. With increasing the system size N, Q abruptly increases around To. Stability of the = 1 states is attributed to the low-lying excited states around the = 1 states. The high density of the low-lying fluctuations around = 1 states contributes to the entropy gain and stabilizes the = 1 states at finite temperature. 

It is obvious that we obtain a stability condition that is not much different from the stability condition of the initial value equation. If At is larger than 2 y/MfK, (the cosine is smaller than —1), the solution grows exponentially and is numerically unstable. Hence, in the straightforward boundary value formulation of classical mechanics, we gain very little in terms of stability and step size compared to the solution of the initial value differential equation. The difficulty is not in the philosophical view (global or local) but in the estimate of the time derivative, which is approximated by a local finite difference expression. 

The closed-loop stability of the batch motion can be established with the application of the standard singular perturbation [25] or small gain theorems [8, 10] available in the nonlinear dynamical systems literature, in eonjunction with the definition (7) of finite-time motion stability. In a chemical process context this closed-loop stability assessments can be seen in the cascade control of a continuous reactor [22], the cascade control of a continuous distillation [21, 24], and in the calorimetric estimation [15] of a batch polymer reactor. The closed-loop motion stability is ensured if the observer gain ( o) is tuned slower than the characteristic frequency ( j) of the fastest unmodeled dynamics, and the observer ( ), secondary ( ,.), and primary ( p) gains are sufficiently separated. This is. 

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