Given a sparse set of differential and algebraic equations (DAEs), it is always recommended to exploit the structure of the system’s sparsity (e.g., tridiagonal blocks matrix, band matrix, and staircase matrix, etc.), thus to use tailored numerical solvers in order to reduce the computation time. Very frequently, though, while highly structured, a couple of elements enter the description which make it difficult for the solvers to reach a solution. They are common in process control applications, where the states added to the plant description by the integral parts of the controllers introduce unstructured elements in the otherwise very structured Jacobian of the mathematical model. Such systems are characterized by a partially structured Jacobian, which inhibits the use of the solvers tailored to fit problems with fully structured matrices. In such cases, one can either use a solver with lower performance, resulting in larger computation times, or alternatively one seeks an approximation for the unstructured points. A solution to the handling of “dirty” Jacobians is presented, which is implemented in a DAE solver package available freely on the Internet. This novel DAE solver fully exploits the overall structure of the system’s sparsity, without compromising CPU computation time and precision of the results. A numerical comparison with different approaches is given by solving a DAE model representing an existing nonequilibrium distillation column.

Model predictive control (MPC) is an online application based on dynamic models. Its application faces two major obstacles: (i) computational constraints and (ii) the need to accurately simulate the process by a model that properly predicts how the plant will behave in the future. Implementation of MPC is not always possible in large-scale or industrial applications due to the computational complexity of MPC and to the dimensionality of the models. To facilitate MPC implementations, this paper proposes a self-adaptive approach based on simplified (or reduced-order) nonlinear models. The proposed methodology yields an MPC that adjusts the dimension of the model according to both the current process conditions and the control objectives. The self-adaptive approach is described and validated on an industrial case study, a C4-splitter.

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The balanced truncation method for reducing the size of a model was originally developed for linear systems. When extended to nonlinear systems, some considerations must be faced. First of all, the calculation of the balancing transformation matrix is not unique. This may results in nonphysical values for the reconstructed states, which may lead to failure, for example, in thermodynamic routines. To reduce this problem, it is recommended to include all the states in the balancing outputs. To further reduce the effect of nonlinearties in the original model, it is recommended to use a linearizing static transformation of the states, if available. In this paper, distillation column models are used as a case study, and, in this case, a logarithmic transformation of the compositions is beneficial.

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