(15a) Nonlinear Model Order Reduction Using Block Structured Models for Large Processes

Typically mathematical models of physical and chemical processes are large and complex because of complications and nonlinearities of physical processes. The importance of mathematical models is evident for optimization and control purposes. Large rigorous mathematical models are cause of high computational loads and computational times, restricting the use of such models for online applications. This provides the opportunity to use reduced models, which are match of NL processes within defined 'operational domain' which can achieve low computational loads and faster simulation times.

Block structured models (in particular Hammerstein structure) have been used for identification purposes efficiently, but have not been used for model reduction purposes. Block structured models have advantage in approximation and identification; the approximated or identified block structures give insight to the complex and complicated process, hence providing handle for reduction. In this study, input-state (IS) Hammerstein block structure is used for the approximation and model reduction of NL process. Initially input-output Hammerstein block structure has been used for approximation (identification) of NL process. The methodology has been extended to IS Hammerstein structure. In this work, it is shown that I/S Hammerstein structure can be derived by
Taylor series expansion (under trivial assumptions). Mathematically, I/S Hammerstein structure is given (as in equation below) and is shown in Fig. (1);

                        MACROBUTTON MTPlaceRef \* MERGEFORMAT  SEQ MTEqn \h \* MERGEFORMAT (1)

Where, J = Jacobian; C = output state matrix; y = output; xss = g(u) is steady-state scheduling (from lookup table); x_ = xss; u^* = u.

Fig.(1): Input-State Hammerstein Structure

The accuracy of the IS Hammerstein approximation structure is improved by estimating Jacobian online using state x and input u information. The accuracy of approximation is improved further by including higher order terms; estimation of Jacobians not only using (current) state information but also using steady-state information. The second order approximation block (I/S Hammerstein structure) has been derived from
Taylor series (shown in Fig.(2)).

 Fig. (2): Input-State Hammerstein Structure with online Jacobians update

Fig. (3): Approximation results for the high purity distillation column (by IS Hammerstein)

The approximation results are shown for a high purity distillation benchmark (pp splitter), which are acceptable for the kind of application. Order reduction of 70% is possible using the methodology with satisfactory results (high accuracy). The computational time reduction by factor 3.44 has been observed (shown in Fig.(3)).

The methodology has been implemented on fluid cracking unit as well. The simulation time reduction by factor 1.45 has been noticed (since the system is already small). The maximum estimation error of 0.5% has been calculated for this specific case over the operating domain; which means that IS-Hammerstein structure identifies the original system precisely.

Moreover it is proved (and shown) that Input-State Hammerstein structure can be derived from a
Taylor expansion. The approximation model's accuracy can be improved by including higher order terms.

Work on the following tasks is done presently or is to be considered in future:

i)                     The computational load reduction for the benchmark example (high purity distillation column) is to be investigated. The computational load and simulation time reduction has to be compared with original NL model.

ii)                  It is planned to extend the methodology to industrial case. The industrial models make use of dynamic link library (dll) files (as foreign process) to compute different task (such as thermodynamic properties). Such foreign processes buildup overhead costs, resulting in increased computational load. It is anticipated, that transformation of large DAE model to ODE structure will reduce the computational effort (and simulation time), since the algebraic computations are vanished in ODE structure, replaced by NL mapping.