Linear lumping

The simplest form of lumping is linear lumping, where the new variables are linear combinations of the original ones, 

For exact lumping Mf(c) must be a function of c so that the reduced system can be expressed in terms of the new variables. Therefore, we need to know the inverse of M since 

The inverse mapping from the c space to the c space is equally important as the forward mapping, not only because it provides a link between the lumped species and the original species, but because its existence is a necessary and sufficient condition for exact lumping. For a reduced system h n, so that the standard definition of an inverse will not apply. Therefore, we use the concept of a generalized inverse. The generalized inverse of an m X n matrix A satisfies the following criteria  

For non-linear systems, such as those found in combustion, J (c) will not be a constant matrix and, hence, converting it directly into a canonical form will be a difficult task. It is possible to express J (c) in the form 

The simplest technique for finding the invariant subspaces is to determine eigenvalues and eigenvectors of A. If the eigenvector matrix is 

We saw in the previous section that chemical lumping is often based on defining new species whose concentrations are linear combinations of those of the starting species within a mechanism. This approach can be generalised within a mathematical framework. The formal definition of lumping is the transformation of the original vector of variables Y to a new transformed variable vector Y using the transformation function h  

The dimension h of the new variable vector Y is smaller than that of the original concentration vector h N )- Due to the transformation above, a new kinetic system of ODEs is formed  

An important feature is the ability to recover the original vector of concentrations from the transformed variables Y using the inverse transformation function h  

Ftmction h is not unique, since several different functions h may belong to the same transformation function h. This inverse mapping is as important as the forward mapping not only because it provides the link between the lumped variables and the original species concentrations, but because its existence is a necessary condition of exact lumping. 

If the function h is linear, then in chemical kinetics, this approach would be termed linear species lumping and is essentially a formalisation of the chemical lumping approach described in the previous section. In the linear case the transformation is simply a matrix multiplication operation  

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Mechanical systems are usually considered to comprise of the linear lumped parameter elements of stiffness, damping and mass. 

By comparison with previous work on linear lumping, it is clear that finding the non-linear lumping function h depends on finding canonical forms for the operator A and/or on finding the invariant manifolds of the original system. This parallels the linearized case where we searched for canonical forms for the Jacobian J and its invariant subspaces. 

This example illustrates that chemical lumping is nothing other than a special case of linear lumping, and that this system is a good example of the application of exact linear lumping. In fact, we can express the relationship between the original and lumped species in the same formal way as we described in the previous section ... 

APPROXIMATE LUMPING IN SYSTEMS WITH TIME-SCALES SEPARATION 4.9.1 Linear lumping in systems with time-scale separation... 

Application of approximate non-linear lumping to the hydrogen oxidation example... 

G. Li, A.S. Tomlin, H. Rabitz and J. Toth, A General-Analysis of Approximate Non-Linear Lumping in Chemical-Kinetics, 1 Unconstrained Lumping, J. Chem. Phys. 101 (1994) 1172-1187. 

A successful equivalent circuit approach to Wagner s theory was worked out by Hoar and Price. They developed a simple voltage divider circuit which gives quantitative formulas for emf and scaling rate that are very similar to those derived more rigorously by Wagner. A linear lumped version of their proposed circuit is shown in Fig. 3. The subscripts 1, 2 and 3 refer to M cations, X anions and electrons respectively. 

Wei and Kuo (1969) have shown that the necessary and sufficient condition of exact linear lumping is the following equation ... 

We include here a short example of linear lumping taken from Li and Rabitz (1989) in order to illustrate the approach. Consider the first-order reaction system involving reversible reactions between three species as follows ... 

Linear Lumping in Systems with Timescale Separation... 

We now briefly provide a formalised framework for linear lumping in systems with timescale separation which is based on a similar approach to that presented in Sect. 6.3. We start with the initial value problem... 

The response in Figure 3.32 is based on a simple linear lumped parameter model thus, for large disturbances or set-point changes that exceed the limits of the linear assumptions and other operating point, quite different responses will be obtained. 

The previous simple analysis example follows a pre-computing classical approach where a simple linearized lumped parameter model of the system was developed. In the pre-computing or classical approach this simple model was solved by the application of analytical methods such as Laplace transforms and frequency-response analysis. For completeness, these methods will be briefly introduced here. The interested reader should refer to the texts that take this pre-computing classical approach, such as... 

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