Singularly perturbed ODEs

Equation (2.29) has one distinct root, x2=0, and hence the above singularly perturbed ODE system is in standard form. Proceeding with our analysis, we obtain a reduced-order, uniform approximation of the slow component of the dynamics as... 

Viewing DAEs as the limiting case of singularly perturbed ODEs of the form... 

Remark 2.1. For a standard singularly perturbed model, the DAE system (2.10) has an index v=l, i.e., the variables x2 can be solved for directly from the algebraic equations (2.9) and the reduced-order (equivalent ODE) representation (2.13) is obtained directly. For systems that are in the nonstandard singularly perturbed form, the DAE system (2.10) obtained in the limit as —> 0 has an index v > 1 and an equivalent ODE representation for the slow dynamics is not always readily available. 

This chapter has reviewed existing results in addressing the analysis and control of multiple-time-scale systems, modeled by singularly perturbed systems of ODEs. Several important concepts were introduced, amongst which the classification of perturbations to ODE systems into regular and singular, with the latter subdivided into standard and nonstandard forms. In each case, we discussed the derivation of reduced-order representations for the fast dynamics (in a newly defined stretched time scale, or boundary layer) and the corresponding equilibrium manifold, and for the slow dynamics. Illustrative examples were provided in each case. 

The process response is presented in Figure 4.6. Observe that all the state variables exhibit a fast transient, followed by a slow approach to steady state, which is indicative of the two-time-scale behavior of the system, and is consistent with our observation that processes with impurities and purge are modeled by systems of ODEs that are in a nonstandard singularly perturbed form. 

Such nested applications of single-parameter singular perturbation theory (i.e., the extension of the analysis of two-time-scale systems presented in Chapter 2 to multiple-time-scale systems) have been used for stability analysis of linear (Ladde and Siljak 1983) and nonlinear (Desoer and Shahruz 1986) systems in the standard form. However, as emphasized above (Section 2.3), the ODE models of chemical processes are most often in the nonstandard singularly perturbed form, with the general multiple-perturbation representation... 

The reduction techniques which take advantage of this separation in scale are described below. They include the quasi-steady-state approximation (QSSA), the computational singular perturbation method (CSP), the slow manifold approach (intrinsic low-dimensional manifold, ILDM), repro-modelling and lumping in systems with time-scale separation. They are different in their approach but are all based on the assumption that there are certain modes in the equations which work on a much faster scale than others and, therefore, may be decoupled. We first describe the methods used to identify the range of time-scales present in a system of odes. 

We have shown an example of the application of the singular perturbation technique to PDE. Obviously, even for this simple example, the application still requires some knowledge of the solution behavior. Its application is not as straightforward as that for ODEs, but we present it here for the serious readers who may have practical problems in setting up the procedure. More often than not, when an analytical means is not available to solve the problem, we suggest that numerical methods should be tried first to get an overall picture of the... 

We end this chapter by noting that the application of the singular perturbation method to partial differential equations requires more ingenuity than its application to ODEs. However, as in the case of ODEs, the singular perturbation solutions can be used as a tool to explore parametric dependencies and, as well, as a valuable check on numerical solutions. The book by Cole (1968) provides a complete and formal treatment of partial differential equations by the singular perturbation method. 

Let, as in Section 2, (y z) denote the solution of the ODE system (2.4) and (yo,zo) the solution of the DAE system (2.5). Then the QSSA error of interest after one integration step is a = 11(2/) )( ) (2/Oj o)(t), where r is the timestep chosen by the applied numerical integrator. In the special situation, we can apply standard results from singular perturbation theory, in particular a quite well-known result of Vasilyeva [21] - see, for instance, the textbook [19]. If we assume the right-hand side F to be at least twice differentiable, the following asymptotic expansion is known to hold ... 

In general, if all (n = l,. .., A7e) are distinct, then A will be full rank, and thus a = A 1 /3 as shown in (B.32). However, if any two (or more) (< />) are the same, then two (or more) columns of Ai, A2, and A3 will be linearly dependent. In this case, the rank of A and the rank of W will usually not be the same and the linear system has no consistent solutions. This case occurs most often due to initial conditions (e.g., binary mixing with initially only two non-zero probability peaks in composition space). The example given above, (B.31), illustrates what can happen for Ne = 2. When ((f)) = ()2, the right-hand sides of the ODEs in (B.33) will be singular nevertheless, the ODEs yield well defined solutions, (B.34). This example also points to a simple method to overcome the problem of the singularity of A due to repeated (< />) it suffices simply to add small perturbations to the non-distinct perturbed values need only be used in the definition of A, and that the perturbations should leave the scalar mean (4>) unchanged. 

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