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Nonlinear Model Reduction of Differential Algebraic Equation (DAE) Systems

Most process models resulting from first principles consist of not only nonlinear differential equations but also contain nonlinear algebraic equations, resulting in nonlinear DAE systems. Since large-scale nonlinear DAE systems are too complex to be used for real-time optimization or control, model reduction of these types of models is a strategy that needs to be applied for online applications. However, in the past, model reduction techniques mainly focused on differential equations, and no general model reduction methods specifically geared towards reducing DAE systems have been proposed. Since in most cases, the number of algebraic equations by far exceeds the number of differential equations, it is insufficient to simply reduce the differential equations by conventional methods (POD, balancing, etc.) and leave the rest of the model intact. This paper presents a novel technique for reducing nonlinear DAE systems. This method can reduce both the order of the model by eliminating some of the differential equations as well as the number and complexity of the algebraic equations. This is achieved by a 3-step approach: 1) performing order reduction of the differential equations and algebraic equations; 2) identifying correlation in the variables that connect the retaining differential equations to the algebraic ones; 3) reduction of the “input-output” behavior of the algebraic equations via system identification techniques. This procedure has the advantages over other methods in that it addresses both reduction of the algebraic and the differential equations and that it results in a system where the algebraic equations can be represented by a feedforward neural network. This last property is important insofar as the reduced model does not require a DAE solver for its solution but can instead be computed by regular ODE solvers. A more detailed description of the model reduction procedure is provided next: in a first step, the controllability and observability covariance matrices for the differential variables are computed. While a controllability covariance matrix can be computed for the algebraic as well as the differential variables, it is important to point out that the information contained in this matrix is only meaningful for the description of the input-to-state behavior of the differential equations. At the same time the covariance matrix can be used to determine the relationship between the algebraic variables. Since algebraic variables have no dynamics, the perturbation on the algebraic variables cannot introduce output responses which are not already reflected in the perturbation of the differential states. Therefore describing a state-to-output behavior for the algebraic variables is not meaningful. In a second step, balancing is applied to compute the state transformation matrix for the differential variables; and singular value decomposition is applied to determine the degree of correlation between the algebraic variables. The two transformation matrices obtained from these computations can then be applied to the system. The resulting model still has the same order and identical input-output behavior to the original system. However, it has the advantage that it can easily be determined how much a model can be reduced without loosing the important parts for the input-output behavior of the system. The model reduction itself is performed by balanced truncation or residualization for the differential equations and by replacing the algebraic equations with an explicit expression obtained from system identification. Feedforward neural networks are used in this work for the reduction of the algebraic equations. This technique is illustrated with a case study. The behavior of reduced-order models of a distillation column with 32 differential equations and 32 algebraic equations is compared.