A balance approach to analysis and reduction of nonlinear systems

Process controllers that are based explicitly on dynamic mathematical models have become increasingly popular in the chemical process industry. However, computational requirements grow with the complexity of the models. Many rigorous dynamic models require too much computation time to be useful for real-time model-based controllers. On the other hand, real-time control is important because the action of the controller must be computed in a time span that is less than the sampling period required for effective control of the plant. This presents a need for model reduction techniques. The method under investigation presents a novel approach to nonlinear model reduction that includes system analysis, nonlinearity quantification, and balanced model reduction. In a first step covariance matrices, which are an extension of linear gramians to any type of system, are computed. Nonlinearity measures are investigated that are based upon the difference between the covariance matrices of the linearized system and the covariance matrices, when data for the computation of these matrices is collected in a region of operation. The measures indicate how much the system dynamics deviates from the behavior of the linearized system. If a nonlinear model is required for the process description, then information contained in the covariance matrices can be used for nonlinear model reduction. A balancing transformation can be found from the covariance matrices and applied to the system. After the system is balanced, states corresponding to entries of zero in the diagonal of the balanced covariance matrix can be truncated and the system will still have the identical input-output behavior as the original system. Nonlinear models with states that give only minor contributions to the input-output behavior of the system can be truncated after they have been identified by examining the magnitude of entries on the diagonal of the balanced covariance matrices. This covariance matrix-based method has been applied in simulation to a series of reactors, a distillation column, and a polymerization reactor. This procedure reduced the number of differential equations in the model as well as the computation time required for the solution of these models.